value of hedging
TRANSCRIPT
THE VALUE OF HEDGING
A DISSERTATION
SUBMITTED TO THE DEPARTMENT OF
MANAGEMENT SCIENCE & ENGINEERING
AND THE COMMITTEE ON GRADUATE STUDIES
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
Thomas C. Seyller
March 2008
UMI Number: 3302870
Copyright 2008 by
Seyller, Thomas C.
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I certify that I have read this dissertation and that, in my opinion, it is fully adequate in
scope and quality as a dissertation for the degree of Doctor of Philosophy.
(Ronald A. Howard) Principal Adviser
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in
scope and quality as a dissertation for the degree of Doctor of Philosophy.
(Ali E. Abbas)
I certify that I have read this dissertation and that, in my opinion, it is fully adequate in
scope and quality as a dissertation for the degree of Doctor of Philosophy.
j T ^ i \ y L A A x y t \j7 M A / ^
(Samuel S. Chiu)
Approved for the Stanford University Committee on Graduate Studies.
in
Abstract
In this dissertation I introduce a definition of hedging which is based on the comparison
of two indifferent buying prices. With that definition, it then becomes possible to ascribe
a monetary value to the hedging provided by a specific deal with respect to the decision
maker's existing portfolio. The value of hedging concept can also be used to identify
within a list of several deals the ones that best complement the portfolio.
Next, I present some fundamental properties of the value of hedging, including some
which only arise if the decision-maker's u-curve satisfies the delta property. I also shed
some light on the probabilistic phenomena which are at the source of hedging: in that part
of the thesis, I show that hedging can be thought of as moment reengineering, in other
words, as an opportunity to favorably reshape the moments of the decision-maker's
portfolio by adding other deals to it.
The last concept I introduce is that of the value of perfect hedging. While the value of
hedging captures how well a specific deal would fit within the existing portfolio, the
value of perfect hedging captures the decision-maker's willingness to pay for the best
hedges one can construct to complement his portfolio, based on a specific uncertainty.
Such an analysis provides three significant benefits to the decision-maker: first, it enables
him to decide how much of his resources he should devote to searching for hedges;
secondly, it allows him to identify the uncertainties on which it is most valuable to hedge,
and therefore to focus his search on the most promising classes of deals; and finally, the
value of perfect hedging can help him establish an upper bound on his personal
indifferent buying price for any uncertain deal which he might be considering acquiring.
Throughout the dissertation, I illustrate the approach through practical examples and
applications.
IV
Acknowledgements
I believe that few experiences in someone's life can be as instructive and transforming as
working on a doctoral thesis. My deepest gratitude goes to the wonderful people who
made this experience even more enjoyable than I imagined it would be; I especially wish
to thank my wife Aditi for her love, patience and constant encouragements, and my
parents, my sister and my brother for their support and for all they have taught me.
I am very much indebted to my dissertation advisor, Ronald Howard, for his teachings
and his friendship. From him I have learnt the importance of clarity in thought and
communication, and the efficiency and elegance of simple solutions. I would also like to
thank Ross Shachter, Ali Abbas, Daphne Koller, Samuel Chiu and Jim Matheson for their
guidance and for many thought-provoking classes and discussions. Their love of and
dedication to teaching has been an inspiring example for me.
I finally wish to thank my friends and colleagues Ibrahim Al Mojel, Somik Raha,
Debarun Bhattacharjya, Ram Duriseti and Christopher Han for their support and for
numerous suggestions on this research. My years at Stanford University have been
especially enriched by my experiences as a teaching assistant for decision analysis at
their side. Never let it be said that it is impossible to conciliate an efficient work ethic
with the desire to have fun.
v
Table of Contents
Abstract iv
Acknowledgements v
Table of Contents vi
Index of Tables & Figures ix
List of Notations Used xi
Chapter 1 - Introduction 1
1. Dissertation Overview 1
2. Motivation 2
3. Research Questions 5
4. Elements of Decision Analysis 7
a) The Six Elements of Decision Quality 8
b) The Five Rules of Actional Thought 9
c) U-curves, Indifferent Buying and Selling Prices, and Delta Property 11
d) The Decision Analysis Cycle 15
5. Hedging in the Financial Literature 32
a) Traditional Definitions of Hedging 33
b) Mean-Variances Approaches 35
c) Utility-Based Approaches 39
Chapter 2 -Definition & Valuation of Hedging 41
1. Definition of Hedging 42
2. Definition of the Value of Hedging 49
a) The Value of Hedging as a Difference of Two PIBPs 50
b) The Value of Hedging as the PIBP of a Translated Deal 52
3. Influence Diagram Representation 56
4. Basic Properties of the Value of Hedging 59
vi
a) Hedging in the Risk-neutral Case 59
b) Existence of Negative Hedging 61
c) Sign and Monotonicity of the Value of Hedging in the Risk-Averse Case.. 62
d) Value of Hedging for Two Deals Which Differ by a Constant 65
e) PIBPfor an Uncertain Deal and Value of Hedging 68
5. Extension of the Value of Hedging Concept to the Sale of Deals 69
Chapter 3 - Value of Hedging in the Delta Case 72
1. A Simpler Formula to Compute the Value of Hedging 73
2. Special Properties of the Value of Hedging in the Delta Case 76
a) Symmetry 76
b) Upper Bound on the Value of Hedging 77
c) Value of Hedging for Multiples of a Deal with Respect to Itself 83
d) Value of Hedging and Irrelevance 91
3. The Chain Rule for the Value of Hedging and its Implications 95
a) Chain Rule 95
b) Toward an Irrelevance-Based Value of Hedging A Igebra 101
Chapter 4 - Hedging as Moment Reengineering 106
1. General Principle 107
2. Variance Reengineering 109
3. Reengineering Moments of Higher Order 116
Chapter 5 - The Value of Perfect Hedging 120
1. Definition 121
2. Basic Properties 126
a) Risk Attitude and Sign of the Value of Perfect Hedging 126
b) PIBPfor an Uncertain Deal and Value of Perfect Hedging 127
c) Joint Value of Perfect Hedging 130
3. Impact of Relevance on the Value of Perfect Hedging 134
a) Complete Relevance 135
b) Irrelevance 139
c) Incomplete Relevance 142
d) Relevance-Based Dominance of one Hedge over Another 146
vn
e) Summary - Impact of Relevance on the Value of Perfect Hedging 150
f) Influence of the U-Curve on the Choice of a Perfect Hedge 151
4. General Upper Bound on the PIBP of an Uncertain Deal 153
5. Approximation of the Value of Perfect Hedging in the Delta Case 156
6. Similarities between Value of Perfect Hedging and Value of Clairvoyance 166
7. The Value of Perfect Hedging as a New Appraisal Tool 168
Chapter 6 - Conclusions & Future Work 172
1. Contributions 172
2. Limitations 174
3. Directions for Future Work 177
List of References 179
Probability, Bayesian Methods and Decision Analysis: 179
Finance and Hedging: 184
Index of Terms 188
Vlll
Index of Tables & Figures
Figure 1.1 - Our valuation of Yean be greatly impacted by X. 2
Figure 1.2 - The six elements of decision quality 8
Table 1.1 - Common u-curves and some of their characteristics 13
Figure 1.3 - The decision analysis cycle 15
Figure 1.4 - A decision hierarchy 16
Table 1.2 - Influence diagram semantics -possible nodes and their meanings 17
Table 1.3 — Influence diagram semantics - possible arrows and their meanings 18
Figure 1.5 - An influence diagram 19
Figure 1.6 - A tornado diagram 21
Figure 1.7 — Cumulative distribution functions for two alternatives 23
Figure 1.8 - Sensitivity to the risk-aversion coefficient 26
Figure 1.9 - Open loop and closed loop sensitivity analyses 28
Figure 1.10 - Minimum-variance hedging for foreign currency 36
Figure II. 1 - Hedging as the comparison of two PIBPs 43
Figure II. 2 - Hedging with complete relevance between X and Y. 44
Figure II. 3 — Hedging with partial relevance between X and Y 46
Figure II. 4 - An equivalent definition of the value of hedging 52
Figure II. 5 - Hedging with partial relevance between Xand Y 55
Figure II. 6 — Using influence diagrams to compute the value of hedging 56
Figure II. 7 -An example for which VoHfY | w, X) is not monotonic in y 63
Figure II. 8 - Sensitivity of VoH(Y \ w, X) to y 63
Figure 11.9 - Hedging provided by a deal which differs from Y by a constant 67
Figure III. 1 - Best hedge for X 79
Figure III.2 - Computation of an upper bound on the PIBP ofY 82
Figure III.3 - Sensitivity ofVoHflX \ X) to k 90
Figure III.4 - Value of hedging for two irrelevant deals 94
Figure III. 5 - Mnemonic for the chain rule for the value of hedging 98
IX
Figure 111.6 - A chain rule example 99
Figure 111. 7 - Applying the contraction property for the value of hedging 104
Figure IV.1 - Example of variance reengineering 109
Table IV. 1 - Decomposition of certain equivalents along the first two cumulants I l l
Table IV.2 — Decomposition of certain equivalents along the first three cumulants 112
Figure IV.2 — Accuracy of approximation depending on the number of cumulants 113
Figure IV. 3 - Hedging as reengineering of moments of order 3 and above 116
Table TV.3 - Decomposition of certain equivalents along the first five cumulants 117
Figure IV.4 - Accuracy of approximation depending on the number of cumulants 117
Figure IV.5 - Probability mass functions ofX and (X u Y) 118
Figure V.l - The value of perfect hedging: an example 123
Figure V.2- Upper bound on the PIBPfor Y based on the value of perfect hedging.... 128
Figure V. 3 - Joint perfect hedging on Oil Price and Volume 132
Figure V.4- Gold Price is only relevant to the value through Oil Price 144
Figure V.5 - Impact of the relevance between S and S' on VoPHfS"1 \ X, w) 150
Table V.l - Impact of the risk tolerance on the selection of a perfect hedge 152
Figure V.6 - Distribution over profit for the second oilfield. 155
Figure V. 7 -Accuracy of our value of perfect hedging approximation: an example .... 160
Figure V.8 - Influence of risk-aversion on the accuracy of the approximation 162
Figure V.9 — Influence of risk-aversion on the accuracy of the approximation (2) 163
Table V.2- Comparison of value of clairvoyance and value of perfect hedging 167
Figure V.10- The value of perfect hedging in the decision analysis cycle 168
Figure V.l I -Example for the use of perfect hedging as an appraisal tool 170
Figure VI. 1 - Hedging with negative side effects 175
Figure VI.2 - Equivalent probability of clairvoyance 177
x
List of Notations Used
{A | B, &} Probability assigned to A given B and the background state of
information & of the individual.
A J_ B | C, & A is irrelevant to (or some might say "probabilistically
independent of) B given C and the background state of
information & of the person. This means that in the opinion of the
person making the statement, {A | B = b;, C, &} = {A | B = bj, C,
&} for any possible outcomes bj and bj of B. Relevance statements,
just like probability assignments, are matters of information and
not matters of logic - two individuals might disagree as to whether
two uncertainties are relevant or irrelevant given a third.
X= {(pi, Xj)je[i,n]} The uncertain deal X comprises n prospects x;, with their
associated probabilities p;.
<Y | &> Mean of Y given &.
V(Y | &> Variance of Y given &.
~(Y | w, X) The decision-maker's PISP for deal Y given that he also owns
wealth w and a portfolio of deals denoted by X. In the delta case,
we will use the notation S~(X | 0) to refer to S~(X | w), since w does
not have any effect on the value of certain equivalents.
RPs(Y | w, X) The decision-maker's selling risk premium for deal Y given that he
also owns wealth w and a portfolio of deals denoted by X. It is
defined as <Y | &> - S~<Y | w, X>.
XI
~(Y | w, X) The decision-maker's PIBP for deal Y given that he already owns
wealth w and a portfolio of deals denoted by X. In the delta case,
we will use the notation S~(X | 0) to refer to B~(X | w), since w does
not have any impact on the value of certain equivalents and since
the PIBP for any deal is equal to the PISP for the same deal.
RPB(Y I w, X) The decision-maker's buying risk premium for deal Y given that
he already owns wealth w and a portfolio of deals denoted by X. It
is defined as <Y | &> - B~(Y | w, X).
xn
Chapter 1 - Introduction
"Humanity is constantly struggling with two contradictory processes.
One of these tends to promote unification, while the other aims at
maintaining or re-establishing diversification. "
Claude Levi-Strauss (b. 1908),
Race et Histoire
1. Dissertation Overview
In this dissertation we will study hedging through a decision analytic lens.
I will first introduce a definition of hedging which is entirely grounded on the principles
of personal indifferent buying and selling prices. This will also allow me to establish a
method through which we can assign a monetary value to the hedging that a deal
provides with respect to the decision-maker's existing portfolio. Many of the properties
of the so-defined value of hedging will be presented and illustrated through practical
examples.
In the later parts of the dissertation, I will present a concept which provides further
insights to the decision-maker: the value of perfect hedging. Given a specific portfolio,
the value of perfect hedging is the amount that the decision-maker should be willing to
pay for the most favorable deal that can be constructed by hedging on some particular
uncertainty or set of uncertainties. The value of perfect hedging thus helps us place an
upper bound on the resources that should be devoted to hedging the existing portfolio.
1
2. Motivation
In decision analysis, we often observe that our valuation of uncertain deals can be greatly
affected by the existence of other deals we own. It is a common oversight to
underestimate how pervasive the phenomenon actually is; many students in decision
analysis, including some who have received training in the discipline for several quarters,
often believe that it does not hold for decision-makers who follow the delta property. The
students' intuition is that since the certain equivalents of such decision-makers for
uncertain deals do not depend on their initial wealth, they should not depend on the
unresolved deals which they own either.
A simple example such as the one presented in the next figure is enough for those
students to become aware of their mistake. In that example, the decision-maker, who
follows the delta property, owns deal X and has a certain equivalent of $2,000 for it. The
decision-maker is then offered deal Y for a fee. At first it might be tempting to reason
that the decision-maker should be willing to pay up to $2,000 for Y, because it comprises
the same probabilities of the same prospects as X.
r ^ — $10,000 -$5,000 0.5
O
—— -$5,000 $10,000
Figure 1.1 — Our valuation of Y can be greatly impacted by X
However, such logic is flawed; it does not take into account the fact that deals X and Y
are probabilistically relevant. In fact, by combining X and Y into the same portfolio, the
decision-maker is assured of a profit of $5,000 once both deals are resolved, irrespective
2
of whether si or S2 is realized. The decision-maker should thus be willing to pay as much
as $3,000 for deal Y, since it can help him transition from a situation worth $2,000 to one
worth $5,000.
The difference between the two possible answers, the erroneous one ($2,000) and the
correct one ($3,000), is sizable. It should remind us that, just like information and control,
hedging can be regarded as an alternative which might significantly affect the value of
the decision-maker's portfolio.
Another element which should make that alternative worthy of our consideration is its
availability. Many financial instruments have been used extensively and for several
decades for hedging purposes, from insurance contracts to derivatives such as forwards or
swaps. Hedging instruments have even been gaining in accessibility in recent years, be it
for organizations or private investors; for instance, companies such as HedgeStreet have
started offering financial derivatives which are designed to give individuals more
flexibility as they try to manage the risk of their portfolio.
The primary ambition of this dissertation is to raise our sensitivity as decision analysts to
the value that hedging can bring to decision-makers. I will strive to present enough
evidence for the reader to be able to decide whether he wants to apply those views on
hedging in his professional practice; but it is not one of my intentions to dissect the
ideological differences between the approach to hedging presented here and that which
can be found in most of the traditional finance literature. It is true that philosophical
differences between the decision analytic framework and the financial framework abound,
the treatment of uncertainty being one of the most notable examples; but I share the
opinion of Edwin T. Jaynes [Jaynes, E.T., 1976], who argued that the true test of the
merits of two competing statistical theories should consist in applying them to the same
examples and deciding which one provides the most sensible results:
Let me make what, I fear, will seem to some a radical, shocking suggestion: the
merits of any statistical method are not determined by the ideology which led to
it. For, many different, violently opposed ideologies may all lead to the same
final 'working equations' for dealing with real problems. Apparently, this
phenomenon is something new in statistics; but it is so commonplace in physics
that we have long since learned how to live with it. Today, when a physicist says,
3
"Theory A is better than theory B", he does not have in mind any ideological
considerations; he means simply, 'There is at least one specific application
where theory A leads to a better result than theory B".
I suggest that we apply the same criterion in statistics: the merits of any
statistical method are determined by the results it gives when applied to specific
problems. The Court of Last Resort in statistics is simply our commonsense
judgment of those results.
I thus hope to provide a sufficiently ample number of practical examples throughout this
dissertation for the reader to get a solid grasp of the decision analytic approach to
hedging I am advocating, and of its possible benefits and drawbacks.
4
3. Research Questions
The fundamental research questions which I intend to address in this dissertation are
congruent with the motivation and objectives I exposed above:
[ 1 ] What is hedging? What is a clarity-test definition of hedging?
[2] How can I measure the value of hedging?
[2.a] Given the decision-maker's current portfolio, and given a specific
deal which he does not own but is considering buying, how can I
ascribe a monetary value to the hedging that the deal provides with
respect to the portfolio?
[2.b] How can I extend the value of hedging concept to the case of a
specific deal which he owns but is considering selling?
[2.c] What remarkable properties does the value of hedging have? Does
it have any special properties in the case in which the decision
maker's u-curve satisfies the delta property?
[3] Which probabilistic phenomena are at work in hedging? How can the
existence of hedging be explained?
[4] Given the decision-maker's portfolio, how can I help him identify the
kinds of deals which best complement it?
[4.a] How can I detect the uncertainties on which it is most valuable to
perform hedging?
[4.b] Is there a way to quantify the decision-maker's willingness to pay
for the ability to perform hedging on a given uncertainty or set of
uncertainties?
[5] How can analyses related to hedging be incorporated into the existing
decision analysis process? How can decision analysts best apply those
tools and methods to practical situations?
5
I will now give an overview of some of the existing research which will help us answer
those questions. I will start with a short introduction to decision analysis, the engineering
discipline whose principles and techniques will serve as a foundation for this dissertation.
After that, I will provide a succinct account of the works on hedging which can be found
in the finance literature.
6
4. Elements of Decision Analysis
The term "decision analysis" was coined by Ronald A. Howard [1966a] and refers to
"applied decision theory". The objective of a decision analysis is to help the decision
maker achieve clarity of action - in other words, to help him identify what is the best
course of action given the information he has, given his preferences, and given the
alternatives that are available to him.
Decision analysis is a normative discipline, and not a descriptive discipline; as such, its
sole intent is to provide axioms and norms for how people should make decisions, and
not to identify the intellectual and psychological processes which account for how people
actually make decisions.
Over the next few pages, I will provide an overview of some of the most fundamental
concepts of decision analysis, with an emphasis on the concepts which will be most
critical to understanding this dissertation. I recommend to the readers who are least
familiar with the field that they consult the works which are cited in the list of references,
in particular those of Ronald A. Howard [1964, 1965, 1966a, 1966b, 1968, 1992, 2004],
Peter C. Fishburn [1964], or Howard Raiffa [1968].
7
a) The Six Elements of Decision Quality
It is difficult to articulate what constitutes a good decision in no more than a sentence.
In decision analysis, we consider that there are six elements to decision quality
[Matheson, D., and Matheson, J. E., 1998]. They are enumerated on the figure below.
Any decision or decision process can be rated along those six dimensions, from 0% to
100%. 100% indicates the point at which trying to do better on that specific dimension
would yield an improvement which would not be worth the additional expense of
resources required to obtain it.
Three of the six elements of decision quality, namely the alternatives, information and
preferences of the decision-maker, are also collectively referred to as the "decision basis".
Meaningful, Reliable Information
Creative, Doable
Alternatives
Appropriate Frame
Clear Values & Trade-offs
Commitment To Action
Logically Correct
Reasoning
Figure 1.2 - The six elements of decision quality
b) The Five Rules of Actional Thought
Like any other normative system, decision analysis is entirely based on a few principles
which serve as axioms. There are five of them [Howard, R. A., 1992], and they are
commonly known as the "five rules of actional thought":
• The probability rule requires the decision-maker to be able to characterize every
alternative he faces by a possibility tree - that is, a tree showing the different
scenarios which might unfold following the selection of that particular alternative.
The decision-maker should also be able to assign probabilities to all branches of
the tree, and he may or may not choose to assign value measures to characterize
the quality of all of the corresponding possible futures. Once the probability rule
has been applied, the decision-maker obtains a list of prospects, in other words a
list of all of the possible futures he might face as a result of the decision situation.
• The order rule requires the decision-maker to order every prospect in the list from
best to worst, according to his own preference. Ties are allowed: the decision
maker might declare that he is indifferent between several prospects.
• The equivalence rule states that for any triplet of prospects which are at different
levels in the ordered list, A, B and C where A > B > C (">" denotes the
preference order), the decision-maker should be able to assign a number p,
comprised between 0 and 1, such that he would be just indifferent between
receiving B for sure and an uncertain deal in which he would receive A with
probability/) and C with probability 1 -p. Such a number/? is called a preference
probability.
• The substitution rule states that for any triplet of prospects A, B and C where A >
B > C, if the decision-maker faces an uncertain deal in which he assigns a
probability p to his receiving A versus 1 — p of his receiving C, where p is also
equal to his preference probability for B in terms of A versus C, then the decision
maker should be indifferent between keeping the uncertain deal and exchanging it
forB.
9
• Finally, let us suppose that there are two prospects A and B at different levels in
the ordered list of prospects, and that A >B; let us also suppose that the decision
maker needs to choose one of two uncertain deals: in the first, he will receive
either A with probability r or B with probability \ - r, while in the other deal he
will receive either A with a different probability s or B with a probability 1 - s.
Then, the choice rule requires that the decision-maker choose the deal which
offers the better chance of the prospect he likes better, namely A: he should select
the first deal if r > s, the second if r < s.
Decision analysis can only be of help to decision-makers who choose to subscribe to the
five rules of actional thought. By successively and carefully applying the rules, one after
the other, to a particular decision situation, a decision analyst can help a decision-maker
infer what the best alternative available to him is. Conversely, to the decision-maker who
does not subscribe to at least one of the five rules, counsel from a decision analyst is of as
little use as are the theorems of Euclidian geometry to someone who rejects one of its
postulates.
10
c) U-curves, Indifferent Buying and Selling Prices, and Delta Property
It would be a tedious process to repeatedly apply the equivalence rule to elicit from the
decision-maker a preference probability for each prospect of a large-scale decision
situation: if there are JV> 3 prospects {Pi}ie[i,AG, Pi being the best and PN the worst, one
would need at least N- 2 assessments, one for each prospect within the list {Pi}ie[2, jv-i] in
terms of Pi and PN- Fortunately, it is possible to greatly simplify the assessment process
by introducing a few more basic concepts.
Let us first observe that in any decision situation, the course of action which is
recommended by the five rules of actional thought does not change if all preference
probabilities are arbitrarily multiplied by the same positive constant, or if any real
number is added to all of them. We can thus lift the requirement that preference
probabilities should be contained within the interval [0, 1], All we need to do is introduce
a new term to refer to those unrestricted preference probabilities, as the analogy with a
probability is lost once we allow those numbers to dwell outside of [0, 1]; we will call
them u-values instead [Howard, R. A., 1998].
We can then simplify the assessment process for preference probabilities by assuming
that the u-values all follow a particular functional form, which we will call u-curve. If the
value measure which the decision-maker cares about is his wealth, w, we can denote the
u-curve by u(w). If in addition the decision-maker prefers more money to less, u(.) should
be an increasing function.
Once it is assessed from the decision-maker, the u-curve can be used to solve decision
situations without resorting to preference probabilities anymore: it can be shown that
subscribing to the five rules of actional thought is equivalent to making decisions by
selecting the alternative which yields the highest expected u-value.
With u-curves, it also becomes possible to compute the decision-maker's Personal
Indifferent Buying Price (PIBP) for something he does not own, or his Personal
Indifferent Selling Price (PISP) for something he owns [Howard, R. A., 1998]. The PIBP
for an item is defined as the sum of money such that the decision-maker is just indifferent
between not acquiring the item at all and acquiring the item at that price; more formally,
if we consider a deal X with probabilities pj and prospects x;, ie[l , n], which the
11
decision-maker is considering acquiring, and if we denote his present wealth by w and his
PIBP for X by b, the indifference statement can be translated into the following equation:
u(w) = X PiUCw + Xj-b) ie[l,n]
Similarly, the PISP for an item is defined as the sum of money such that the decision
maker is just indifferent between retaining the item and selling the item at that price. The
PISP for an uncertain deal is also called certain equivalent. For an uncertain deal with
probabilities q* and prospects y\, ie[l, m] which the decision-maker owns and is
considering selling, if we denote his PISP for Y by s, then:
u(w + s)= 2 9 , u ( w + yi) ie[l,m]
The decision-maker's risk attitude is directly reflected in the shape of his u-curve, and
more specifically in its concavity or convexity. We will call risk-neutral a decision-maker
who assigns to an uncertain deal a certain equivalent which is exactly equal to the
probability-weighted average of its monetary prospects (an amount usually referred to as
"the e-value of the monetary prospects" of the deal). A risk-averse decision-maker is
defined as one whose certain equivalent for an uncertain deal is inferior to the e-value of
the monetary prospects of the deal; for example, a decision-maker who has a certain
equivalent of $450 for an uncertain deal in which he assigns a 50% chance to receiving
$1,000 and a 50% chance to receiving nothing is risk-averse. Such a decision-maker has a
concave u-curve. In contrast, a risk-seeking decision-maker is one who assigns to an
uncertain deal a certain equivalent which is greater than the e-value of the monetary
prospects of the deal; his u-curve is convex.
It is important to understand that nothing in the five rules of actional thought prohibits u-
curves which are concave on some intervals but convex on others. A decision-maker
using such a u-curve would exhibit a risk-averse behavior for some deals and a risk-
seeking behavior for others. Piecewise concave and convex u-curves are seldom used in
practice, however; it is not surprising if we consider that very few decision-makers, once
they understand what the term risk-seeking truly means in decision analysis, would want
to adopt that sort of risk attitude in an actual decision situation.
12
The next table shows some of the u-curves which can be most commonly found in the
literature [Bickel, E. J., 1999]:
Table 1.1 — Common u-curves and some of their characteristics
Among the u-curves listed above, two of them, the linear and exponential form u-curves,
possess a fascinating characteristic called the delta property [Howard, R. A., 1998]:
consider a decision-maker who has a certain equivalent s for an uncertain deal X = {(p;, x0ie[i,n]}; what will be his certain equivalent s' for an uncertain deal X' = {(pi, Xj + §)ie[i,
n]}, in other words, for a deal which differs from X by a constant 8? It is a natural
desideratum for many decision-makers, especially if they regard the magnitude of the
monetary prospects involved as relatively small compared to their total wealth, to declare
that s' should be equal to s + 8. If that is the case, the decision-maker's u-curve is said to
satisfy the delta property.
13
What makes the delta property particularly remarkable is the fact that it entails many
additional properties for the decision-maker:
• The decision-maker's valuation of any uncertain deal remains the same whether
he includes his initial wealth in his characterization of the deal's monetary
prospects or not. Therefore, it is not necessary to take initial wealth into account
when computing his PIBP or PISP for a deal.
• For any uncertain deal, the decision-maker's PIBP and PISP for it are equal.
• The decision-maker's u-curve can only be of one of two forms: linear or
exponential. Because u-curves are unique up to a positive linear transformation,
as mentioned earlier, this also implies that the only number which needs to be
assessed to determine the decision-maker's u-curve in its entirety is y, which is
often called the decision-maker's risk-aversion coefficient. A y of zero
corresponds to a risk-neutral decision-maker, a positive y to a risk-averse
decision-maker, and a negative y to a risk-seeking decision-maker.
Therefore, one question is all decision analysts need to ask in order to assess y and
thereby the whole u-curve - at least in theory, since in practice it is still advisable
to proceed to several measurements of y in order to assess it more accurately.
• The value of any information gathering process can be computed as the difference
between the value of the deal with the help of the additional information and the
value of the deal without.
14
d) The Decision Analysis Cycle
The decision analysis process [Howard, R.A., 1966a] is iterative. It consists of four
phases, which are depicted below:
Formulation Evaluation Appraisal
0 . . • Deterministic ^ Probabilistic . . . • , ._ . . Structure . • . , • , , • Appraisal —b. n p r i « i n n Analysis Analysis uccisiuu
t Figure 1.3 - The decision analysis cycle
It is another common oversight to believe that the decision analysis process merely boils
down to the construction of a mathematical model and its analysis - in reality, the entire
process can be thought of as a continuous conversation between the decision-maker and
the decision analyst, intended to guide the decision-maker towards clarity of action, and a
mathematical model is no more than one of the recurrent and most visible constituents of
that conversation.
15
Formulation Phase
The main objective of the formulation phase is to frame the decision situation: decision
maker and decision analyst seek to distinguish the issues which should be under
consideration in the analysis from those which should not.
Framing is more of an art than a procedure with strictly established steps and guidelines.
However, an important element of virtually every framing exercise is to determine
exactly which decisions need to be made at this epoch in time. In many practical
examples, especially if many stakeholders are involved in the decision process, it is no
simple question to answer. A decision hierarchy is a tool which helps address the issue; it
prompts the decision-maker to separate his decisions into three distinct categories: those
that can be taken as givens; those which are important enough to be the object of the
current analysis; and, finally, those which are thought to have a small enough impact on
the value for them to see their examination safely deferred until later.
TAKE AS GIVEN
V ANALYZE NOW
y EXAMINE LATER
Figure 1.4 - A decision hierarchy
Once all decisions have been sorted into those three separate containers, we can focus our
attention on the middle category. The next important point to ponder is the list of the
uncertainties which should be included in the analysis, given the decisions which are to
16
be analyzed, and how those decisions and uncertainties all affect one another as well as
the decision-maker's value. Influence diagrams, which are also referred to as decision
diagrams in parts of the literature, are powerful tools which assist analyst and decision
maker as they communicate about such matters [Howard, R. A., and Matheson, J. E.
1981]. They offer a graphical representation of the structure of the decision situation, and
of the relationships which exist between all the issues involved.
The semantics of influence diagrams are outlined in the following tables; a total of four
kinds of nodes and four kinds of arrows, each with a specific meaning, can be
encountered in influence diagrams:
Table 1.2 - Influence diagram semantics -possible nodes and their meanings
17
Meaning
D,
D
D
D,
Relevance Arrow
Informational
Arrow
Functional
Arrow
Influence Arrow
A and B are probabilistically
relevant given their parent nodes
Decision Di and the outcome of
uncertainty C are both observed
before making decision D2
Deterministic node F is a function
of decision D and the outcome of E
The probability distribution over the
degrees of C varies depending on
the decision which is made at D
Table 1.3 - Influence diagram semantics -possible arrows and their meanings
An example of an influence diagram for an oil exploration and drilling decision is shown
in the next figure; it is based on one of the models which Ross Shachter presents in his
introduction to the subject [Shachter, R. D., 1997]:
18
Figure 1.5 — An influence diagram
It should also be noted that the role of influence diagrams can be extended beyond their
use as a representation of the structure of the decision situation: in fact, it is possible to
encode probabilities and value measures into the influence diagram, and to then solve it
in order to identify the best alternative and its associated certain equivalent [Shachter, R.
D., 1986 and 1988].
19
Evaluation Phase
During the evaluation phase, a mathematical model of the decision situation, as framed
during the formulation stage, is built and analyzed in order to identify the best alternative.
Deterministic Analysis
Naturally, the first step consists in building such a mathematical model of the decision
situation. It should include:
• all of the alternatives under consideration;
• all of the uncertainties listed on the influence diagram; for each of them, the
decision-maker is asked to provide three possible values, either by himself or with
the help of a designated expert: a base value, corresponding to the median of the
probability distribution over the uncertainty; a low value, corresponding to the
10th percentile of the distribution; and a high value, corresponding to the 90th
percentile;
• calculations showing the value which any possible scenario, that is to say any
possible selection of an alternative followed by any possible realization of the
uncertainties, would yield for the decision-maker. In many cases, it is the net
present value of the scenario which is evaluated.
Such models are often built using spreadsheet programs, as they allow for a rapid
recalculation of the value for various scenarios - a feature which will prove valuable on
many occasions throughout the analysis, as we will see later.
The next task for the decision analyst consists in assessing the relative importance of the
uncertainties by comparing their individual effects on the value. Tornado diagrams fill
that need [McNamee, P., and Celona, J., 1987]. For a specific alternative, the analyst first
computes what is called its base value - in other words, the value of the alternative when
all uncertainties are set to their base values. Then, the analyst perturbs each uncertainty,
one by one and one at a time, swinging it from base to low and from base to high. The
results are recorded and the uncertainties are sorted, from the one triggering the greatest
20
swing in value to the one triggering the smallest. A tornado diagram such as the one
shown below is built in order to display the results in a way which makes it easier to
discuss and interpret them. The whole procedure we just described is sometimes also
called deterministic sensitivity analysis, in recognition of the fact that the probability of
each possible scenario has not yet been taken into account.
-100.0
Market Size I
Revenue per jjnit
Efficacy of the Drug
Number of Compi rtitors
Side Effects of the Drug
Phase III Trial Duration
Phase III Trial Cost
I
Production Cost per Unit
New Plant Construction Cost
-50.0 Profit ($ million)
0.0 50.0 100.0 150.0
Figure 1.6-A tornado diagram
The analyst and the decision-maker can then read off the tornado diagram the names of
the uncertainties which have the greatest potential impact on value, and decide which
variables they want to model probabilistically throughout the remainder of the analysis
[Howard, R. A., 1966; Matheson, J. E., and Howard, R. A., 1968]. Uncertainties which
are not selected will be modeled as deterministic quantities instead. In addition, the
analyst and the decision-maker may choose to exclude an alternative from the analysis, if
they observe during the deterministic sensitivity phase that it is systematically dominated
by at least one of its rivals.
21
Probabilistic Analysis
A few assessments need to be made before the analyst can proceed with the evaluation of
the probabilistic model: the analyst should first elicit probability distributions for the
uncertainties which were selected at the term of the deterministic phase, with the help of
the decision-maker himself or of a designated expert.
Probability encoding is a time-consuming and complex procedure [Spetzler, C. S., and
Stael von Holstein, A. S., 1975]; like anyone else, decision-makers, experts or other
stakeholders in the decision-making process are often the victims of biases which cloud
their judgment, and it is crucial that the analyst help them discern their true belief from
the distortions that those biases impose. I encourage the reader to refer to the works of
Amos Tversky and Daniel Kahneman [1974] for an overview of such biases, and to those
of Carl S. Spetzler and Axel S. Stael von Holstein [1975] for an introduction to
techniques which decision analysts can use to combat their influence during probability
encoding.
Armed with those probability distributions, the analyst can then plot for each alternative a
cumulative distribution function over the value measure. Such a curve shows, for each
possible realization of the value measure, the probability assigned to obtaining a result
which would be inferior to that value. An example of two cumulative distribution
functions corresponding to two different alternatives is shown on the next figure.
Next, the analyst can assess the decision-maker's u-curve over the range of prospects
involved in the decision situation; as mentioned earlier, if the decision-maker is
comfortable with following the delta property over that range, the task is considerably
easier, since only one parameter needs to be assessed in that case. Once the u-curve has
been elicited, the analyst can compute the certain equivalents of all the alternatives and
identify the best course of action.
22
roba
bili
ty
0-4>
"3 g s U
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
It is at times possible to infer from the cumulative distribution functions of two
alternatives A and B that A will be preferable to B regardless of the degree to which the
decision-maker is risk-averse [Howard, R. A., 1998]; in such cases, it might thus be
unnecessary to assess the decision-maker's risk-aversion coefficient y with great
precision. The situation arises in instances of:
• deterministic dominance: when the worst prospect achievable under alternative A
is better than the best prospect achievable under alternative B;
• first-order probabilistic dominance: when the difference between the cumulative
distribution functions of A and B keeps a negative sign over the entire range;
• or of second-order probabilistic dominance: when the integral of the difference
between the cumulative distribution functions of A and B keeps a negative sign
over the entire range.
In the example shown above, there is no dominance relationship between A and B. It is
not a surprising conclusion when one considers that B offers a better mean profit than A,
23
100 -$50 $150 $0 $50 $100
Net Present Value ($ Million)
Figure 1.7- Cumulative distribution functions for two alternatives
$200
but at the cost of a substantially larger downside: to a risk-neutral decision-maker, since
the mean is all that matters, B will thus appear as the better alternative, whereas to a
sufficiently risk-averse decision-maker, the magnitude of B's potential downside will
seem so large that A will earn his favors overall.
24
Appraisal Phase
The decision analyst identified the best alternative during the evaluation phase and
computed its certain equivalent. But that does not mark the end of the analysis; in order
to achieve complete clarity of action, it is helpful to answer two additional questions:
• Under what conditions should we make a different decision?
• Before choosing the alternative which currently appears to be the best available,
can the decision situation be improved by gathering further information, or by
exerting some influence over some of the uncertainties involved? How much
should the decision-maker be willing to pay for each of those activities?
Sensitivity Analysis
Sensitivity analysis [Howard, R. A., 1968] addresses the first of the two issues: by
perturbing the variables involved in the model, one by one and over reasonable ranges of
values, and by tracking the impact that such changes have on the certain equivalent of
each alternative, the analyst can assess the relative importance of different variables;
some may be recognized as being incapable of altering the decision-maker's choice,
while others may trigger spectacular shifts in the decision. At the term of the analysis, the
decision-maker will thus be able discern the elements which are worthy of his attention
from those which are not.
For instance, if there is no dominance between two alternatives up to the second order, it
might prove helpful to perform a sensitivity analysis on the decision-maker's risk-
aversion coefficient, y. An example follows; a decision-maker needs to choose between
four alternatives, A, B, C and D. He follows the delta property over the entire range of
prospects involved and during the assessment process for his risk-aversion coefficient, his
answers all implied values of y comprised between 0.07 and 0.11. At first, that might
seem to be a wide interval of possible values, and the analyst might think that it will be
indispensable to ask more questions in order to assess y more accurately. However,
inspection of the sensitivity analysis chart below reveals that alternative B is the best
alternative within this entire range by a comfortable margin, and its certain equivalent is
25
contained in a relatively narrow interval: [$112,000; $121,000]. All of a sudden, what
seemed to be an uncomfortably wide range of possible values for y appears as no more
than an inconsequential annoyance.
$160
$140
$120
$100
c a V3 S o
J=
H © $80 =
4>
$60 '3 a-w S $40
$20
Alternative A
Alternate e B
^«. Alternative C
$0 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.02
Y
Figure 1.8 — Sensitivity to the risk-aversion coefficient
Sensitivity analysis also helps the decision-maker understand how the best course of
action might change given slightly different circumstances. This is especially important
in business settings, since the results of a market study, an important announcement made
by a competitor or a large-scale economic event often leave little time to react, and it
would not be to the decision-maker's advantage if he had to commission a fresh
evaluation of his decision situation every time he receives a new piece of information.
Sensitivity analysis is usually conducted slightly differently in the case of the
uncertainties which were modeled probabilistically during the evaluation phase. As a
reminder, typically, three values are elicited from the decision-maker for each of those
26
uncertainties: a low, a base and a high, which respectively correspond to the 10 , 50 and
90th percentiles of the distribution. Those three values can then be used to perform what
is called open-loop and closed-loop sensitivity analyses [Howard, R. A., 1968]; more
specifically, if we denote by A the alternative which was shown to be the best at the term
of the evaluation phase, the analyst computes for uncertainty U with fractiles uiow, Ubase
and Uhigh:
• the certain equivalents of A given that U = uiow, given that U = Ubase, and given
that U = Uhigh- This is called an open-loop sensitivity analysis: the alternative is
permanently set at A, and the analyst simply computes the fluctuations in its
certain equivalent as U varies;
• the certain equivalent of the alternative which is preferred when U = uiow, the
certain equivalent of the alternative which is preferred when U = Ubase, and the
certain equivalent of the alternative which is preferred when U = Uhigh. Those
alternatives need not be the same, nor does any of them need to be A. This is
called a closed-loop sensitivity analysis: this time, the alternative is not rigidly
anchored on A; as the analyst varies U, he also allows himself to change his
decision in favor of a better alternative, and it is the certain equivalent of that
alternative which is recorded.
The results of the open-loop and closed-loop sensitivity analyses can be plotted as
illustrated below. It should be noted that the open-loop curve should not surpass the
closed-loop curve at any point.
27
$300
1? $250 -.2
§ % $200 j -**
s
1 $150 -'3 e $100 '3 V.
o W $50
$0 -U i o w Ubase ^high
Figure 1.9 — Open loop and closed loop sensitivity analyses
Open-loop and closed-loop sensitivity analyses are computationally intensive, since they
require three separate and complete reevaluations of the certain equivalent of each
alternative - one reevaluation for U = uiow, one for U = Ubase and one for U = Uhigh-
However, the results they produce are remarkably compact, since six data points on a
chart are enough to capture them in their entirety, and admirably powerful: in our
illustrative example, the analyst can conclude that A would remain the best alternative
even if the decision-maker were to obtain additional information which led him to believe
that U = uiow with certainty, or that U = Ubase; indeed, at those two values, the results of
the open-loop and closed-loop sensitivity analyses coincide. Conversely, the best
alternative would change if the decision-maker became sure that U = Uhigh- In that event,
A is outperformed by at least one alternative by $56 million.
We will soon see that a few more significant insights can be extracted from the open-loop
and closed-loop curves; but before that, we should mention as a conclusion to our
overview of sensitivity analysis procedures that in many decision situations, it can also be
illuminating to evaluate the sensitivity of the decision to the relevance of various
uncertainties to one another [Lowell, D. G., 1994]. This helps distinguish the relevance
relationships which need to be elicited and explicitly modeled as conditional probability
distributions from those which can be left as marginal distributions without impairing the
28
quality of the recommendation. Sensitivity analysis to relevance is especially critical in
decision situations with large and intricate probabilistic structures, since it is in those
instances that simplifying the model in order to avoid unessential conditional probability
assessments will tend to prove most valuable.
Value of Information and Value of Control
As announced earlier, a second objective of the appraisal phase is to determine whether
the decision situation can be improved by gathering further information, or by exerting
influence over some of the uncertainties involved.
The value of information [Howard, R. A., 1966b] addresses the first half of the question;
for any information gathering scheme, be it a clinical test in a medical decision setting or
market studies and R&D experiments in a major corporation, the value of information is
defined as the decision-maker's PIBP for the additional information. In other words, it is
the price P at which he is just indifferent between facing his present decision situation as
is, and facing the same decision situation with the help of the additional information but
also with a bank account which was reduced by P.
For decision-makers who follow the delta property over the entire range of prospects
involved, we observed earlier that the value of information on some uncertainty U can be
computed as the difference between the value of the deal with free and perfect
information on U, and the value of the deal without any further information on U. Both of
those quantities can easily be calculated based on the results of the open-loop and closed-
loop sensitivity analyses on U:
• The value of the deal with free and perfect information on U corresponds to what
we might call the certain equivalent of the closed-loop curve; it is equal to the
certain equivalent of an uncertain deal in which there are three monetary
prospects, Qow, Cbase and Chigh, as identified on the previous figure, with
respective probabilities {uiow | &}, { i w I &}, and {uhigh | &}.
• On the other hand, the value of the deal with no additional information on U
corresponds to what we might call the certain equivalent of the open-loop curve;
29
it is equal to the certain equivalent of an uncertain deal in which there are three
monetary prospects, Oiow, Obase and Ohigh, with respective probabilities {uiow | &},
{ubase I &}, and {uhigh | &}. It should be noted that the analyst already calculated
the quantity in question earlier in the analysis - it is equal to the certain equivalent
of A, as computed during the evaluation phase.
Similarly, the value of control over some uncertainty U is defined as the decision-maker's
PIBP for being able to choose the outcome of U [Matheson, J. E., 1990]. In a business
setting, it is for example possible to try and influence a product's market share by
launching a marketing campaign which will increase the product's visibility. In the high
technology and pharmaceutical sectors, firms might also choose to devote more resources
to a particular project or molecular compound in order to increase the likelihood of
technical success.
Just like in the case of the value of information, there exists an easy procedure to
compute the value of control for decision-makers who follow the delta property over the
entire range of prospects: then, the value of control is equal to the difference between the
value of the deal with free and perfect control over the outcome of U, and the value of the
deal without any control over U. Those two numbers can again be calculated with the
help of the open-loop and closed-loop chart:
• The value of the deal with free and perfect control over U corresponds to the
highest point of the closed-loop curve; in other words, it is equal to the maximum
of the values Ciow, Cbase and Chigh- That can be explained by the fact that this time,
the decision-maker is free to select the outcome of U he prefers, as well as the
best alternative under those conditions.
• The value of the deal with no control over U corresponds once again to the certain
equivalent of the open-loop curve, or more explicitly to the certain equivalent of
A as computed during the evaluation phase.
30
Conclusion of the Appraisal Phase
By the end of the appraisal phase, the decision-maker has clarity on the following issues:
• He has identified the course of action which is the best given his current
preferences, the alternatives that are available to him and his current beliefs.
• He has identified the variables which may cause him to modify his decision if
they were to change. He also knows for which revised values of those variables
his choice should change, and how large of an impact such an event would have
on his certain equivalent.
• He knows how much he should be willing to pay for information on each one of
the uncertainties involved in the decision situation, and how much he should be
willing to pay for the ability to exert control over them.
If there exists an information gathering scheme whose cost would be inferior to the value
of information it would provide to the decision-maker, then it should be pursued. The
first three steps of the decision analysis process would then be repeated with the help of
the additional information, iteratively, until it eventually becomes sensible to stop the
analysis and act. In Ronald A. Howard's words [Howard, R. A., 1988]:
At some point, the appraisal step will show that the recommended alternative is
so right for the decision-maker that there is no point in continuing the analysis
any further.
31
5. Hedging in the Financial Literature
In this part of the dissertation, I will provide a succinct account of the views on hedging
which are most commonly expressed in the financial literature. We will first examine
conventional definitions of hedging, before moving on to an exposition of the key points
of the two prevalent normative theories on hedging - the family of mean-variance
approaches, and the family of utility-based approaches.
I will not, however, review the descriptive literature on hedging. This is not for lack of
research on the subject; indeed, a large body of literature studies questions such as the
extent to which hedging is used by corporations [Guay, W. R., and Kothari, S. P., 2003]
or the degree to which corporate hedging is effective [Nance, D. R., 1993, Smith, C. W.,
and Smithson, C. W., 1993]. But this dissertation exclusively focuses on normative
perspectives, and on issues which are almost entirely disconnected from the descriptive
research.
32
a) Traditional Definitions of Hedging
In much of the financial literature, "hedging" refers to the use of financial derivatives
such as futures and forwards in order to mitigate the risk that exists in another investment.
The following definition by Kandice H. Kahl [Kahl, K. H., 1983] illustrates that view:
The traditional literature on commodity futures markets defined a hedge as a
futures market position which is equal but opposite to the individual's cash
market position.
The fact that trading derivatives and hedging have become synonyms for many in the
financial community has to do with the long common history that those two ideas share.
It is indeed with the inception of commodity futures that the concept of hedging first
emerged: in late 17th century Japan, a futures market in rice was developed at Dojima,
near Osaka, to help suppliers protect themselves against the risks imposed by nature
(poor weather) or by man (war and pillage). It was the first recorded instance in which
such a market was created. In the U.S., derivatives trading also started with commodity
futures, with the Chicago Board of Trade, which had been created in 1848, allowing
investors to trade wheat, pork belly and copper futures as early as in the 1860s.
That history probably explains, at least in part, why there has been so much interest in
hedging in a field such as agricultural economics, and more importantly why in academic
circles the term has often been understood to mean the use of forwards or futures. But
more recently, a more inclusive definition of hedging began to surface: in the financial
literature of the last two decades, more and more frequently, hedging has referred to an
activity which helps reduce or cancel out the risk imposed by another investment.
Shehzad L. Mian, for example, proposes the following definition [Mian, S. L., 1996]:
Corporations are exposed to uncertainties regarding a variety of prices. Hedging
refers to activities undertaken by the firm in order to mitigate the impact of these
uncertainties on the value of the firm.
In that broader definition, the reference to derivatives was progressively relegated to the
role of mere example. For instance, Deana R. Nance and al. observe that there is more to
hedging than the use of financial derivatives, and they make an explicit distinction
between "off-balance-sheet hedging", in other words the use of those derivatives, and
33
"on-balance-sheet hedging" [Nance, D. R., Smith, C. W., and Smithson, C. W., 1993],
which involves the use of more standard investment mechanisms with the intention of
reducing risk:
Corporate hedging refers to the use of off-balance-sheet instruments - forwards,
futures, swaps, and options - to reduce the volatility of firm value.
Hence, if the value of an American manufacturing firm that faces competition in
its U.S. markets from foreign manufacturers is inversely related to the value of
the dollar, it could employ off-balance-sheet instruments to hedge that exposure.
This exchange rate-induced volatility can be hedged by (1) selling a forward
contract on the foreign currency, (2) selling foreign exchange futures on the
foreign currency, (3) entering into a currency swap in which it receives cash
flows in dollars and pays cash flows in the foreign currency, (4) buying a put
option on the foreign currency, or (5) writing a call option on the foreign
currency. Alternatively, the firm could hedge through an on-balance-sheet
strategy; it might relocate production facilities abroad or fund itself in the
foreign currency.
It should be noted that in spite of the rising popularity of that broader definition of
hedging as an activity which helps reduce the risk of an investment, even beyond
academic circles, many technical papers on the subject still focus exclusively on the
pricing of derivatives. But the definition of hedging we will present and defend in this
dissertation has more in common with the more inclusive of the two definitions.
34
b) Mean- Variances Approaches
In the 1950s, Harry M. Markowitz developed the first theory on portfolio allocation in
which there was an explicit treatment of risk [Markowitz, H. M., 1952 and 1959]. The
theory taught investors how to identify optimal mean-variance portfolios - portfolios in
which no added diversification could lower the portfolio's risk for a given expected
return, and in which no gain in expected return could be achieved without simultaneously
consenting to an increase in the risk of the portfolio.
Markowitz' work, which earned him the Nobel Prize in Economics in 1990, has had a
profound and lasting influence on financial mathematics. Portfolios which offer the
highest expected return for each level of risk are still called Markowitz efficient
portfolios, and at a philosophical level, much of financial theory remains grounded on the
premise that investors should make trade-offs between a given reward level, as measured
by the mean return of the portfolio, and a given risk level, as captured by the variance of
the portfolio.
It is thus no surprise that mean-variance thinking gave rise to a normative theory of
hedging, and a popular one at that. In what is perhaps its most basic and most radical
form, the method recommends that the investor completely eliminate the variance in his
investment by taking an equal and opposite position in the futures market [Luenberger, D.
G., 1997]; if it is infeasible in practice, for example because there exists no future
contract for the asset whose value needs to be hedged, the investor should identify a
future contract related to a different commodity, but such that the price of the future
contract is probabilistically relevant to that of the asset that requires hedging. The
investor can then compute the number of future contracts he should buy in order to
minimize the variance of the resulting portfolio [Brown, S. L., 1985; Luenberger, D. G.,
1997]. For that reason, the approach is often called "minimum-variance hedge".
I will now illustrate the minimum-variance approach through a simple example, which is
loosely based on one discussed by David G. Luenberger [1997]. A U.S. firm will receive
100,000 euros in a month; they decide to hedge the value of that contract, in U.S. dollars,
by trading future contracts on the Japanese yen: more specifically, the firm will enter an
35
agreement now to sell h yens in a month at an exchange rate of 0.88 dollar for 100 yen.
The current exchange rate between U.S. dollars and euros is 1.34 dollars for every euro:
$/€ = 1.21
0.3
$/€ = 1.34 O
0.4
$/€ = 1.42
0.3
$/100¥ = 0.82
0.7
$/100¥ = 0.88
0.2
$/100¥ = 0.94
0.1
$/100¥ = 0.82
o
0.3
$/100¥ = 0.88
0.4
$/100¥ = 0.94
0.3
$/100¥ = 0.82
<y
o.i
$/100¥ = 0.88
0.2
$/100¥ = 0.94
0.7
$121,000 /i(0.88-0.82)
$121,000 A(0.88-0.88)
$121,000 /i(0.88-0.94)
$134,000 A(0.88-0.82)
$134,000 /i(0.88-0.88)
$134,000 6(0.88-0.94)
$142,000 /i(0.88-0.82)
$142,000 £(0.88-0.88)
$142,000 /»(0.88-0.94)
Figure 1.10 - Minimum-variance hedging for foreign currency
The above tree shows the possible evolution of exchange rates over the coming month, as
well as the profits the company would be making from the euro contract and the yen
contract under different scenarios as a function of h. Minimizing the variance of the
36
entire portfolio, we obtain h = -8,750,000 as the optimal value of h. But the answer
raises some fundamental questions: for example, why would the solution not depend on
the decision-maker's risk attitude? If we believe that some investors are more risk-averse
than others, why would they all choose the exact same hedging strategy?
In order to remedy to that weakness of the minimum-variance method, some authors from
the financial literature advocate replacing as the purpose of hedging the minimization of
variance by the optimization of an objective function which encodes the investor's mean-
variance trade-offs [Heifner, R. G., 1972 and 1973; Kahl, K. H., 1983; Frechette, D. L.,
2000]. For example, Kandice H. Kahl elects to optimize the following function, where n
designates profit:
<7T | &> - A. V<7l|&)
In that function, A. is a positive number which captures the risk-aversion of the investor.
As X increases, the influence of the variance of the portfolio on the investor's decision's
will also increase - in other words, larger values of X correspond to more risk-averse
investors.
While the use of such objective functions allows the investor to take into account his
personal risk attitude as he makes hedging decisions, it can be argued that such an
approach is still not entirely satisfactory. In reality, it can sometimes be misleading to
evaluate an investment opportunity based solely on the first two moments of its
probability distribution over profits.
For instance, let us consider the situation of a risk-averse investor who owns a portfolio
with uncertain returns, and who is given the chance to receive for free another financial
instrument which, if added to his present investments, would leave both the mean and
variance unchanged, but would also significantly improve the third central moment of his
profits - by shifting the downside and the upside of the distribution to the right, for
example, and the central region of the distribution to the left. Why would a risk-averse
investor necessarily dismiss such an investment opportunity as uninteresting? Would he
not appreciate this opportunity to reduce the magnitude of the potential downside,
without altering the mean or variance of his investment?
37
We will come back to that criticism of mean-variance methods much later in the
dissertation: we will show that moments of order three and above can be of considerable
importance, and we will even be able to quantify their weight in the investor's hedging
decisions.
38
c) Utility-Based Approaches
In the late 1980s, interest in another family of normative theories for the study of hedging
started to surface in the financial literature. Those new approaches, which I will call
utility-based approaches, were originally inspired by a paper written by Stewart D.
Hodges and Anthony Neuberger [1989], in which they adapted the expected utility
framework from decision theory to options pricing: essentially, Stewart D. Hodges and
Anthony Neuberger were computing the price at which the investor would be indifferent
between having a given option in his portfolio and not having it.
The theory, which was soon extended to a larger class of transactional cost structures
than the one it was originally developed for [Davis, M. H. A., Panas, V. G., and
Zariphopoulou, T., 1993], became especially popular for the pricing of derivatives in
what are called "incomplete markets" in the financial literature. In such markets, the cash
flows which an option yields cannot always be replicated by a suitable combination of
other assets in the market; therefore, the traditional financial paradigms of portfolio
replication and risk-neutral valuation are of little help to an investor who is trying to put a
price tag on an option, and that explains the popularity of utility-based approaches to
solve such problems [Cvitanic, J., Schachermayer, W., and Wang, H., 2001; Delbaen, F.,
Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., and Strieker, C , 2002;
Henderson, V., 2002; Musiela, M , and Zariphopoulou, T., 2004]. For the same reasons,
utility-based approaches have also been used recently to price highly sophisticated assets
such as volatility derivatives [Friz, P., and Gatheral, J., 2005; Carr, P., Geman, H., Madan,
D., and Yor, M., 2005; Grasselli, M. R., and Hurd, T. R., 2007].
In most of those papers, the utility function which is used is the exponential form which
we already discussed in our review of decision analysis concepts; Matheus R. Grasselli
and Thomas R. Hurd for example suggest [Grasselli, M. R., and Hurd, T. R., 2007]:
U(x) = -e - y x
Power utility functions have also drawn some interest from the financial community
[Henderson, V., 2002], but their use has been more sporadic:
39
x1_R
U(x) = ,whereR>0 1-R
Of all types of power utility functions, the quadratic form is one of the most commonly
encountered in the literature [Luenberger, D. G., 1997]. However, the use of quadratic
utility functions should probably be regarded as a bridge between mean-variance and
utility-based approaches rather than as a utility-based approach stricto sensu, since
maximizing expected utility then becomes equivalent to optimizing a function which only
depends on the mean and variance of the profit distribution. The fact that quadratic-
utility-function maximizers seldom use the vocabulary of indifference pricing is further
proof that their thinking is more characteristic of the mean-variance school of thought
rather than its utility-based counterpart.
The normative theory of hedging which I will present and defend in this dissertation is
much closer in its philosophy to the views developed by the proponents of utility-based
approaches. This is not surprising if we consider the fact that the financial community's
utility-based approaches to hedging are built on a few principles which are similar to
those of decision analysis, such as indifference pricing.
40
Chapter 2 - Definition & Valuation of Hedging
"We think only through the medium of words; languages are true
analytical methods. Algebra, which is adapted to its purpose in every
species of expression, in the most simple, most exact, and best manner
possible, is at the same time a language and an analytical method. The
art of reasoning is nothing more than a language well-arranged. "
Antoine-Laurent de Lavoisier (1743-1794),
Traite Elementaire de Chimie
Antoine-Laurent de Lavoisier's words should remind us that employing a well-
constructed language is indispensable if we want to think clearly about an issue. Before
we can make any serious attempt at ascribing a monetary value to hedging, it is thus
imperative that we define it in the most precise terms - there is little point in trying to
build a solid edifice on a weak foundation.
Formulating a clear definition of hedging will be our first ambition in this chapter. We
will then see that with that definition in place, it becomes easy and natural to ascribe a
monetary value to hedging.
41
1. Definition of Hedging
As noted in the previous chapter, hedging is now commonly understood to refer to an
investment "that is taken out specifically to reduce or cancel out the risk in another
investment" [Wikipedia]. Such a definition, unfortunately, is not entirely clear, because it
leaves one important question unanswered - what do we mean by "risk"? Is the word
used as shorthand for "variance", as suggested by supporters of the mean-variance family
of normative theories on hedging?
As we seek to define hedging, we should avoid relying on terms which would only add to
our interlocutor's confusion, such as "risk". In fact, we will abandon the idea of a risk
reduction characterization of hedging altogether, to focus instead on the effect hedging
produces on the decision-maker's preferences with respect to a specific deal: we will
speak of hedging if the decision-maker finds a deal he does not own more attractive when
he values it with his current portfolio in mind than when he values it without considering
his existing portfolio; more formally:
Definition 2.1 - Hedging
Suppose that the decision-maker has wealth w and owns an unresolved portfolio of deals
X; suppose also that he is considering purchasing a deal Y.
We will say that Y provides hedging with respect to X given w if B~(Y | w, X) is greater
than (Y | w + ~(X | w)). In other words, if we compare purchasing deal Y in the
following two states:
• The decision-maker owns portfolio X and wealth w (State 1),
• The decision-maker owns wealth w + S~(X | w), but does not own X (State 2),
then purchasing Y is regarded as more valuable by the decision-maker in the first state
than in the second.
Conversely, we will say that Y provides negative hedging with respect to X given w if B~(Y | w, X) is less than B~(Y | w + S~(X | w».
42
We are comparing the decision-maker's PIBP for Y under two different sets of
circumstances. Interestingly, those two states of the world, (1) and (2), are so defined that
the decision-maker is just indifferent between them; state (2) is nothing else than state (1),
following the sale of portfolio X for the exact price at which the decision-maker was
indifferent to parting with it, S~(X | w). And yet, nothing guarantees that the decision
maker would also be indifferent between adding Y to his portfolio in state 1, and adding
Y to his portfolio in state 2.
It is the possible difference in the decision-maker's valuation of Y, in the presence or
absence of X but all else being equal, which determines whether there is hedging or not.
The decision-maker is just indifferent between
State 1 and State 2...
State 2
}S~<X|w>
... but would he prefer to add Y to his portfolio in State 1 or in State 2?
Figure III - Hedging as the comparison of two PIBPs
43
Example 2.1:
A decision-maker, who states that he is comfortable following the delta property
within the [-$20,000, $20,000] range and who has a risk tolerance of $10,000,
already owns deal X and is thinking of acquiring deal Y as shown below:
•o
0.6
S2
$1,000 -$100
$0 $300
S~<X I 0) = $388.08 B~<Y | 0> = $138.07
Figure II. 2 - Hedging with complete relevance between X and Y
Then: B~<Y | 0) =
But: B~<Y | X)
= $138.07
= $147.61
> B~<Y | 0)
In this example, by our definition, Y does provide hedging with respect to X
given w.
It is reassuring that such a conclusion is not at odds with our intuition - if we had
had nothing to guide us but the vague notion that hedging qualifies an investment
which reduces the risk of another investment, then we would also have declared
that Y provides hedging with respect to X. Indeed, adding Y to X reduces the
magnitude of the possible downside in the decision-maker's prospects, from $0 to
44
$300, at the cost of a slight reduction in the magnitude of the possible upside,
from $1,000 to $900.
Furthermore, this example shows that in situations of hedging, not only do we
have the inequality B~(Y | w, X) > B~(Y | w + S~(X | w)>, but it sometimes even
turns out that B~(Y | w, X) > (Y | &): a risk-averse decision-maker can thus be
willing to pay more for a deal Y than a risk-neutral decision-maker would pay for
it, even if they both agree on the probabilities and magnitudes of the prospects,
and even if they both own the exact same portfolio X and the same wealth w.
Here:
B~(Y|w,X> =$147.61
><Y|&)(=$140). J
Example 2.2:
Hedging does not only arise in situations in which the monetary prospects of the
deals under consideration are determined by the exact same set of uncertainties, as
in Example 2.1. In order to demonstrate it, let us consider a decision-maker who
has the exact same u-curve as in our first example, and owns the same unresolved
deal X; this time, he is contemplating acquiring a different deal Y, as described by
the next figure.
We now have:
B~(Y | 0> = $82.02
But the PIBP for Y when we take the existing portfolio X into account is larger:
B~(Y | X) = $87.71
> B~(Y | 0 )
45
Si
0.4
-0
0.6
s2
$1,000
$0
S'l
0.9
•o
•o
0.1
s'2
S'l
0.3
0.7
~sV
-$100
$300
-$100
$300
b~<X | 0 ) = $388.08 B~(Y | 0> = $82.02
Figure II. 3 - Hedging with partial relevance between Xand Y
In spite of the fact that the monetary prospects of X and Y are determined by
different uncertainties, Y provides hedging with respect to X given w. Also, just
as in our first example, that conclusion matches our intuition's predictions, since
Y is structured in such a way that it is more likely than not to compensate for
some of the potential losses imposed by X. 3
46
To the first time reader of our definition of hedging, it may seem puzzling that we chose
to define it by comparing the decision-maker's PIBPs for Y in states 1 and 2:
• The decision-maker owns portfolio X and wealth w (State 1),
• The decision-maker owns wealth w + S~(X | w), but does not own X (State 2),
instead of comparing the decision-maker's PIBPs for Y in the following states 1 and 2':
• The decision-maker owns portfolio X and wealth w (State 1),
• The decision-maker owns wealth w +J^P^i*w7, but does not own X (State 2').
In more formal terms, why compare B~(Y | w, X) to B~(Y | w + S~(X | w)), instead of
comparing it to B~(Y | w)? Would the latter not be more convenient, and yet yield the
exact same result?
Sadly, problems with that second possible definition of hedging would arise for u-curves
which do not satisfy the delta property. For such u-curves, B~(Y | w + S~(X | w)> is not
necessarily equal to B~(Y | w). The discrepancy has to do with a phenomenon which, in
decision analysis, has often been called the "wealth effect": for a risk-averse individual,
an increase in wealth tends to cause an increase in their PIBP for an uncertain deal.
With that in mind, let us come back to the comparison between states 1 and 2' as a
possible definition of hedging. Consider an existing portfolio X which is deterministic
and offers a sure profit equal to x: one might conclude from applying that erroneous
definition of hedging that since the decision-maker's PIBPs for Y in states 1 and 2' are
different, Y provides hedging or negative hedging with respect to X given w. And yet, it
seems absurd to think that one can hedge a portfolio which involves no uncertainty.
The next example demonstrates the same point numerically.
47
Example 2.3:
A decision-maker owns $1,000 as his initial wealth, as well as an unresolved deal
X which has an assured outcome, equal to $1,000. Let us suppose that the
decision-maker can choose to purchase deal Y from Example 2.1. Finally, we will
suppose that his u-curve is encoded by u(x) = ln(w + x), where w denotes his
initial wealth - so that the decision-maker does not follow the delta property for
the range of prospects under consideration. Then:
f B~(Y | w,X) = $130.16
L B~(Y | w> = $119.93
The two PIBPs for Y are different, so if we were to define hedging based on a
comparison of B~(Y | w, X) and B~(Y | w) we would then infer that there is
hedging in this example. Yet, it seems preposterous to argue that Y compensates
for any of X's risk.
We can explain that the difference between B~(Y | w, X) and B~(Y | w) with wealth
effects. In our example, the potential negative consequence of acquiring Y,
namely, the possibility of losing $100, is not as much of a concern to a decision
maker who has an initial wealth of $2,000 (i.e., in our example, owns w = $1,000
as well as deal X) as it is for a decision-maker who simply has an initial wealth of
$1,000.
It should be noted, however, that applying the correct definition of hedging which
we proposed earlier in this chapter would lead to the intuitively acceptable
conclusion that there is no hedging provided by Y with respect to X:
f B~(Y | w,X) = $130.16
I B~(Y|w + s~(X | w» = $130.16.3
48
2. Definition of the Value of Hedging
Now that we have a better understanding of what constitutes hedging, we are ready to
move on to the question of how to ascribe a monetary value to it. There are three reasons
why we should be keen on defining a value of hedging:
• The value of hedging concept would add to our understanding of the possible
merits of a deal which the decision-maker is considering acquiring -
The value of hedging would help us distinguish between what we might call the
intrinsic value of the deal, as measured by the decision-maker's PIBP for it
without taking his existing portfolio into account, and the merits of the deal in
terms of the hedging it provides, as measured by the value of hedging. We will
come back to that point in greater detail later, but in the meantime, here is an
example to ponder: if the PIBP for Y without taking the existing portfolio into
account is equal to zero, but Y is found out to provide a positive value of hedging
with respect to the existing portfolio, the notion of value of hedging will have
helped the decision-maker understand why Y should be of any interest to him.
• The value of hedging would also allow us to compare several deals based on how
well or how poorly they complement the existing portfolio -
Having a common monetary scale on which we can measure the hedging that
each of those deals provides enables us to sort them, from the best complement of
the portfolio to the worst.
• Using such an ordered list, we can then attempt to identify some common
characteristics that the best hedges might share -
If we succeed in recognizing such patterns, we can start looking for other deals,
outside of the original list, which would also possess the characteristics we
identified. Some of those deals might prove even more valuable as additions to
the portfolio than any of the acquisitions which we had considered prior to that.
49
a) The Value of Hedging as a Difference of Two PIBPs
How can we quantify the value that a deal provides through hedging, though? The very
form of Definition 2.1 gives us the answer - since Y is said to provide hedging with
respect to X given w if B~(Y | w, X) is greater than B~(Y | w + S~(X | w», we could simply
define the value of hedging as the difference between those two quantities:
Definition 2.2 — Value of Hedging of an Uncertain Deal Y
We will define the value of hedging provided by Y given w and X, which we will denote
by VoH(Y | w, X), as the difference of two PIBPs:
VoH(Y | w, X) = B~(Y | w, X) - B~(Y | w + S~<X | w» [2.1]
Consequently, Y provides hedging with respect to X given w if and only if VoH(Y | w,
X) > 0, and negative hedging if and only if VoH(Y | w, X) < 0.
Example 2.4:
Coming back to Example 2.1:
VoH(Y | X) = B~(Y | X> - B~(Y | 0>
= $147.61-$138.07
= $9.54.
In this example, the value of hedging appears to be relatively small compared to
the intrinsic value of Y, B~(Y | 0 ) . But it is still substantial enough that a decision
maker who omits the value of hedging in his valuation of Y might end up
declining to buy the deal even though it was in his interest to do so; this would
arise for instance if the price at which the decision-maker can purchase Y was
$140 . •
50
As a reminder, B~(Y | w, X) and (Y | w + ~(X | w» can both be computed using
equations involving the decision-maker's u-curve.
To show that, let us first introduce more explicit probabilistic notations: suppose the
decision-maker has wealth w and owns an unresolved portfolio of deals X = {(pi, x ^ i ,
n]} whose outcomes are determined by the set of state variables S. He is considering
purchasing a deal Y = {(qi, yOie[i,m]}, whose prospects are a function of another set of
state variables 5". For convenience, we will denote by q^ the conditional probability that
the decision-maker assigns to s'j given Sj.
Then, B~(Y | w, X) and B~(Y | w + S~(X | w)) are the quantities which satisfy the following
two equations:
f x Pi u(w+xs) = x Pi Z % u ( w + x i + y j - B~<Y Iw' x » i2-2] ie[l,n] ie[l,n] je[l,m]
u(w + S~<X | w» = X Qj u ( w + S~<x I w> + Yj" B~<Y I w + S~(X | w») [2.3] je[l,m]
We can already see that the value of hedging is intimately connected with the decision
maker's u-curve and, thereby, with his risk attitude. We will come back to this point later
on in order to determine the exact nature of that relationship between hedging and risk
attitude.
51
b) The Value of Hedging as the PIBP of a Translated Deal
Equation [2.1] may give the misleading impression that the value of hedging provided by
Y with respect to w and X can only defined as the difference between two PIBPs, and
that it is not necessarily possible to express it as the PIBP of just one deal. In reality, this
is quite feasible, and I will show how in this part of the dissertation. This will provide
another perspective on the value of hedging concept, which will prove especially useful
in deriving some of the results we will encounter later in this chapter of the dissertation.
So how can we define the value of hedging as the decision-maker's PIBP for one deal?
All we need to do is define a deal Y' by subtracting ~(Y | w + ~(X | w» from all
prospects of Y; Y' = {(qis y'i)ie[i,m]} = {(qit y; - B~<Y | w + S~(X | w»)iG[1,m]}. Then it
turns out that the value of hedging that Y provides with respect to w and X is equal to the
decision-maker's PIBP for Y':
B~/ VoH(Y | w, X) = B~(Y' | w, X) [2.4]
Indifference
Figure II.4 — An equivalent definition of the value of hedging
52
To use a mathematical metaphor, we could say that the value of hedging provided by Y is
equal to the PIBP for a deal which is the result of translating Y by an amount -B~(Y | w +
S~<X|w».
[2.4] can also be written in terms of u-values as:
£ PiUCw + x,) i*,n] [2A,}
= S Pi Z % u ( w + x i + y > - B ~ < Y i w + s~<x i w » - V o H ( Y i w ' x ) ) ie[l,n] je[l,m]
But why are expressions [2.1] and [2.4] equivalent? Why is B~(Y' | w, X) equal to B~(Y |
w, X) - B~(Y | w + S~(X | w»? The equality follows from a much more general result:
Lemma 2.1 — PIBP for a Deal Translated by an Amount S
Let us consider a decision-maker with wealth w and an existing portfolio of unresolved
deals X. For any deal Z and any amount 8 (positive or negative), his PIBP for a deal Z' in
which all prospects of Z were augmented by the same amount 8 is equal to his PIBP for
Z, augmented by 8:
B~(Z' | w, X) = B~(Z | w, X) + 8
The result holds whether the decision-maker's u-curve satisfies the delta property or not.
Proof:
We start by writing the definition of the decision-maker's PIBP for Z', denoting by
(pOiefi, n] the probability distribution over the prospects of the existing portfolio X,
and by (rk|i)ke[i, S] the conditional probability distribution over the prospects of Z'
given those of X:
X p,u(w + X i )= Z P. Z ^uCw + X i + z ' k - ^ Z ' l w ) ) ie[l,n] ie[l,n] ke[l,s]
= Z P. Z rk|lu(w + xi + (zk+S)-B~<Z' |w» ie[l,n] ke[l,s]
53
We also have, this time by definition of the PIBP for Z:
2 Piu(w + X i)= X Pi Z rk|iu(w+xi + z k- B ~< z l w » ie[l,n] ie[l,n] ke[l,s]
Comparing the last two equations, we can conclude that B~(Z' | w, X) = B~(Z | w, X) +
8.D
If we then apply this lemma to the particular case of deals Y and Y' as defined earlier,
with 8 = - B~<Y | w + S~<X | w», we can conclude that [2.1] and [2.4] are indeed
equivalent. [2.4] occasionally turns out to be more convenient to use as a definition of
VoH(Y | w, X) than [2.1]. In fact, many of the proofs in this dissertation will be based on
the former rather than the latter.
Example 2.5:
We will reuse Example 2.2 in order to illustrate our alternative definition of the
value of hedging as the PIBP for just one deal through a practical situation.
As a reminder, we previously computed B~(Y | 0) to be equal to $82.02. We first
transform deal Y into Y' by translating it by - B~(Y | 0): we subtract $82.02 from
all of its prospects, and we obtain the deal which is represented in the next figure.
The PIBP for this new deal Y' given X is equal to:
B~(Y' | X) = $5.69
Finally, let us compare that PIBP to the value of hedging for Y. We observe that
the two quantities are equal, as expected:
VoH(Y | X) = B~(Y | X) - B~<Y | 0)
= $87.71-$82.02
= $5.69. D
54
Si
0.4
-0
0.6
S2
$1,000
$0
-0
•o
S'l
0.9
)
0.1
S'2
S'l
0.3
0.7
-$182.02
$217.98
-$182.02
$217.98
~<X | 0 ) = $388.08
Figure II. 5- Hedging with partial relevance between Xand Y
55
3. Influence Diagram Representation
I have already mentioned in the first chapter that influence diagrams are among the most
effective communication tools available to a decision analyst, and that it is even possible
to encode probabilities and value measures into them in order to solve them and identify
the best course of action and its associated certain equivalent.
Those features of influence diagrams are the reason I would now like to present an
influence diagram representation of the value of hedging computation. This will help
consolidate our understanding of hedging as defined in this dissertation, and we will see
that the value of hedging can be computed by means of an influence diagram evaluation.
More specifically, the value of hedging can be computed by evaluating and comparing
the two influence diagrams shown below:
Influence Diagram 1 Influence Diagram 2
Figure II. 6 - Using influence diagrams to compute the value of hedging
In both diagrams, the decision-maker would like to decide whether he should buy deal Y
or not, given its cost. The difference between the two situations lies in the fact that in the
first, the decision-maker also owns initial wealth w and an uncertain portfolio X, whereas
in the second, he owns initial wealth w + S~(X | w), but does not own X.
56
The value of hedging can then be computed using the following algorithm:
Step 1: Adjust the value of the node "Cost of Y" in Influence Diagram 1 until the
decision-maker is indifferent between the two alternatives available at the
decision node, i.e. buying Y and not buying Y. Once that is achieved, we
know that the value of the node "Cost ofY" is exactly ~(Y | w, X).
Step 2: Similarly, adjust the value of the node "Cost ofY" in Influence Diagram 2
until the decision-maker is indifferent between buying Y and not buying Y
in that second diagram as well. Once that is achieved, we know that the
value of the node "Cost ofY" is exactly B~<Y | w + S~<X | w».
Step 3: The value of hedging is equal to the difference between the results
obtained at the end of steps 1 and 2 of the algorithm:
VoH(Y | w, X) = B~<Y | w, X) - B~(Y | w + S~(X | w».
In practice, a binary search would allow for a fast and efficient evaluation of B~(Y | w, X)
and B~(Y | w + S~(X | w)) in steps 1 and 2. For example, here is a detailed view of a
possible evaluation procedure for step 1, for a chosen level of accuracy e:
Step 1-a: Solve Influence Diagram 1 when "Cost of Y" is set to the highest dollar
prospect achievable in deal Y, which we will call ymax. We start from there
because ymax constitutes a natural upper bound on the decision-maker's
PIBPforY.
If the difference between the certain equivalents of the "Buy Y" and the
"Do Not Buy Y" alternatives is less than the desired level of accuracy e,
then return ymax as the approximate value of B~(Y | w, X). Otherwise,
proceed to Step 1-b.
57
Step 1-b: Solve Influence Diagram 1 when "Cost of Y" is set to the lowest dollar
prospect achievable in deal Y, which we will call ymin. We know that ymin
constitutes a lower bound on the decision-maker's PIBP for Y.
If the difference between the certain equivalents of the "Buy Y" and the
"Do Not Buy Y" alternatives is less than 8, then return ym;n as the
approximate value of B~(Y | w, X). Otherwise, create two variables A and
B and initialize them with values ymjn and ymax respectively, and proceed
to Step 1-c.
Step 1-c: Solve Influence Diagram 1 again when "Cost ofY" is set to C = lA (A+B).
If the difference between the certain equivalents of the "Buy Y" and the
"Do Not Buy Y" alternatives is less than s in that case, then return C as the
approximate value of B~(Y | w, X).
Otherwise, if the best decision is to "Buy Y", it implies that we need to
increase the cost of Y for the decision-maker to be indifferent between
buying it or not: B~(Y | w, X) is thus larger than C. We should then set A to
C, keep B as is, and go back to Step 1-c with those new values of A and B.
In the last possible case, if the best decision is to "Not Buy Y", we can
infer from it that B~(Y | w, X) is less than C; we should then keep A as is,
set B to C, and go on to Step 1-c again with those new values of A and B.
58
4. Basic Properties of the Value of Hedging
a) Hedging in the Risk-neutral Case
We observed during our review of the minimum-variance approach that one of the most
common criticisms against the theory is the fact that it leaves us with no possibility to
take into account the decision-maker's risk attitude - the solution we would arrive at
would invariably be the same, irrespective of how risk-averse the decision-maker is.
The definition of hedging we have formulated and studied does not suffer from that flaw.
In fact, the value of hedging for a deal depends upon two elements of the decision basis:
• the decision-maker's information, as encoded for instance in the probabilities he
assigns to the prospects of all the deals involved in his decision situation,
• and the decision-maker's preferences, since his risk preference will have a direct
effect on the calculation of PfflPs B~<Y | w, X) and B~<Y | w + S~<X | w», which
are the two quantities we need to compute in order to form the value of hedging.
An interesting risk preference case to examine is that of a decision-maker who is risk-
neutral within the range of prospects involved: if we consider Example 2.1, for instance,
we will realize that the decision-maker would ascribe the same value to deal Y, equal to
the e-value of its prospects ($140), irrespective of whether he has taken his existing
portfolio X into account or not. Hence, the value of hedging will be equal to zero, a
conclusion which appears quite sensible given that a risk-neutral decision-maker will not
see any value in the fact that Y compensates for some of the potential downside imposed
byX.
The result actually turns out to be true in general - to a risk-neutral decision-maker,
hedging considerations are simply irrelevant:
59
Property 2.1 - Value of Hedging and Risk-neutrality
VoH(Y | w, X) = 0 for a risk-neutral decision-maker, no matter what deals X, Y and
wealth w are.
Proof:
Let us start from equation [2.4], for the particular case of a risk-neutral u-curve:
VoH(Y | w, X) = B~(Y' | w, X)
= <Y'|&>
= <Y-B~(Y|w + s~<X|w»|&>
= <Y-(Y|&) |&)
= <Y | &> - «Y | &> | &>
= <Y|&>-<Y|&>
= 0. D
60
b) Existence of Negative Hedging
I will continue this overview of some of the elementary properties of hedging with an
important reminder - our definition of hedging allows for the existence of situations in
which the value of hedging is negative. Just as it can be valuable for some deals to
incorporate them into our portfolio in order to mitigate some of its risk, for other deals it
can be detrimental to acquire them because they would only add to the risk.
Property 2.2- Sign of the Value of Hedging
It is important to note that VoH(Y | w, X) can in some cases be negative.
Example 2.6:
Let us suppose that the decision-maker who already owns deal X (as in Example
2.1) needs to decide whether he should buy a second copy of the very same deal.
The PIBP for this new deal, if we were to omit the existence of the first deal,
would be $388.08. If we do take it into account, then the PIBP is equal to $364.62.
It implies that the value of hedging provided by the second copy of X is negative:
it is equal to -$23.46.
This should come as no surprise: when adding a second deal X to the current
portfolio, the monetary prospects involved in both deals reveal a bias in the same
direction; the best outcome occurs for si for both the deal that is already owned,
and the one that the decision-maker is contemplating acquiring. Therefore, by
adding the second copy of deal X to the one he already possesses, our risk-averse
decision-maker would be increasing his exposure to risk, instead of neutralizing
some of it. U
61
c) Sign and Monotonicity of the Value of Hedging in the Risk-Averse Case
Property 2.1 drew our attention to the fact that hedging considerations can never be of
any use to a risk-neutral decision-maker, because the value of hedging is always equal to
zero for such a choice of u-curve. It is only for other types of risk attitude that the value
of hedging can take on non-zero values.
This insight can lead us to investigate a few related and interesting questions. For
simplicity, we will assume that the decision-maker whose situation we are discussing is
risk-averse and follows the delta property over the entire range of prospects involved,
although the conclusions we will reach can be extended beyond the delta case. A valid u-
curve for such a decision-maker is u(x) = 1 - e"yx. Then:
• For two given investment opportunities X and Y, does the value of hedging
provided by Y with respect to X retain the same sign for all possible values of the
risk-aversion coefficient y? In other words, does the value of hedging have the
same sign for all such decision-makers, irrespective of the degree to which they
are risk-averse?
• Is the value of hedging provided by Y a monotonic function of y?
Even in as simple a situation as the delta case, the answer to both questions is no. Here is
a simple example which will help us convince ourselves of it.
62
Example 2.7:
A decision-maker wishes to follow the delta property over the [-$1,000,000;
$1,000,000] range. He owns some deal X, and is considering acquiring Y; both
are shown on the next figure:
Sl
•o
0.3
S2
0.4
S3
0.3
$100,000
$40,000
-$20,000
$18,000
-$12,000
-$2,000
Figure II. 7 -An example for which VoH(Y \ w, X) is not monotonic in y
What we will do here is plot VoH(Y | w, X) for different values of y. The results
of that sensitivity analysis are displayed below:
0.00001 0.0001 0.001 0.01 0.1
y (for Deals Expressed in $ Thousand)
Figure II. 8 - Sensitivity of VoH(Y \w,X)toy
63
The chart shows unequivocally that VoH(Y | w, X) does not keep a constant sign
in the risk-averse range, and that it is not monotonic either.
How can that be explained? We will see in chapter IV that the value of hedging
essentially corresponds to the value which can be derived from reengineering the
moments of the original portfolio X. More explicitly, when choosing to add a deal
Y to a portfolio X, a decision-maker will first and foremost be interested in
augmenting the mean of X, secondly in reducing its variance, and thirdly in
enhancing its third moment.
The reason for the peculiar change of sign in VoH(Y | w, X) we can observe in
the present example is that adding Y to the decision-maker's portfolio has
contradicting effects on its moments; starting with the low-order moments, Y has
the undesirable effect of increasing the variance, but it also has the advantage of
increasing the value of the third central moment. For some values of y the positive
effects will outweigh the negative, so that overall Y will provide positive hedging
with respect to X, while for other values of y the negative effects will outweigh
the positive. H
64
d) Value of Hedging for Two Deals Which Differ by a Constant
Another important feature of the value of hedging is the fact that if we look at two deals
whose prospects are determined by the same set of uncertainties but differ by a constant,
then we are assured that the two deals provide the exact same value of hedging.
The result is intuitive: after all, two such deals should offer the same benefits or dangers
from a hedging perspective, since the constant by which they differ can neither mitigate
nor add to the risk of the existing portfolio. What is perhaps most surprising about this
insight is the fact that it holds for any possible u-curve, and not just for those which
satisfy the delta property.
Theorem 2.1 — Hedging Provided by Two Deals Which Only Differ by a Constant
Suppose that the decision-maker is considering purchasing a deal Y = {(q;, yi)i<=[i, m]}
whose prospects are fully determined by a set of uncertainties 5', or a deal Z = {(q;, Zi)je[i>
m]} whose prospects are also fully determined by S'.
Furthermore, suppose that the prospects of Y and Z only differ by a constant 8: Zj = yj + 5
for all values of i. Then,
VoH(Z | w, X) = VoH(Y | w, X)
Proof:
By virtue of [2.4], if we define deal Y' by subtracting B~(Y | w + S~(X | w» from all
prospects of Y: Y' = {(qi, y'i)ie[i,m]} = {(qi, y; - B~<Y | w + S~(X | w»)ie[,,m]}, then:
VoH(Y | w, X) = B~<Y' | w, X)
Similarly, if we define deal Z' by subtracting B~(Z | w + S~(X | w)> from all prospects
of Z: Z' = {(qi, z'i)i6[1,m]} = {(qi, Zi - B~(Z | w + S~<X | w»)je[i,m]}, then:
VoH(Z | w, X) = B~(Z' | w, X)
65
But:
z'i = Zi - B~(Z | w + S~<X | w»
= (y; + 8) - B~(Z | w + S~(X | w»
= y; + 8 - (B~<Y | w + S~(X | w» + 8) (by virtue of Lemma 2.1)
= yi - B~<Y | w + S~(X | w»
= y'i
It follows that B~(Y' | w, X) = B~<Z' | w, X) and therefore that VoH(Y | w, X) =
VoH(Z | w, X). n
Example 2.8:
Let us go back to Example 2.2; this time, however, we will analyze the situation
from the point of view of a decision-maker whose u-curve is ln(w + x), where w =
$2,000 denotes his present wealth.
We will first compute the value of hedging provided by Y. We have:
B~(Y | w + S~(X | w» = B~(Y | w + $259.82)
= $75.30
But the PIBP for Y when we take portfolio X into account is larger:
B~<Y | w, X) = $97.24
> B~<Y | w + S~<X | w»
Therefore, Y provides hedging with respect to X, of a value equal to:
VoH(Y|w,X) =$21.94
Let us now consider Z, which differs from Y only by a positive constant of $100:
66
Si
0.4
-0
0.6
S2
$1,000
$0
-0
S'l
0.9
)
0.1
S'l
S'l
0.3 ) 0.7
"sV
$0
$400
$0
$400
S~<X | w) = $259.82 B~(Z | w + S~(X | w» = $175.30
Figure II.9- Hedging provided by a deal which differs from Y by a constant
We repeat the same calculations for Z:
B~<Z | w + S~(X | w» =$175.30
B~(Z|w,X) =$197.24
Therefore:
VoH(Z|w,X) =$21.94
The values of hedging provided by Y and Z are indeed equal. •
67
e) PIBPfor an Uncertain Deal and Value of Hedging
In some situations, it is helpful to reverse the chain of inference implied by our definition
of the value of hedging - instead of computing PIBPs B~(Y | w, X) and B~<Y | w + S~(X |
w» in order to obtain the value of hedging, we might want to infer the value of B~(Y | w,
X) from the value of the other PIBP and from the value of hedging that Y provides.
Such reasoning can be of practical use if we have access to an approximation of the value
of hedging provided by Y, or to an upper bound on it. This might arise, for example, if
we can easily detect that Y is inferior to another deal Z for hedging purposes, and if we
have already computed the value of hedging provided by Z. That property will also be
useful in the next chapter of this dissertation: we will then see that when the decision
maker's u-curve satisfies the delta property, an upper bound on the value of hedging can
be computed based on the risk premiums of the current portfolio and of the new deal Y.
Property 2.3 - PIBP for an Uncertain Deal and Value of Hedging
As a direct consequence of Definition 2.2, a convenient relationship links the decision
maker's PIBP for deal Y to the value of hedging provided by deal Y:
B~<Y | w, X> = B~(Y | w + S~<X | w» + VoH(Y | w, X) [2.5]
In its philosophy [2.5] resembles some of the laws of physics, in which the total energy of
a system is the result of a combination of the effects of some variables which are
inherently related to the system itself, as well as of other variables which capture the
interaction of the system with its environment. In the case of [2.5], we decomposed the
PIBP for a deal Y into the sum of two terms; the first, B~(Y | w + S~(X | w)>, measures the
intrinsic worth of Y, while the second, VoH(Y | w, X), is an interaction term which
captures the reciprocal effects of Y and X on each other.
68
5. Extension of the Value of Hedging Concept to the Sale of Deals
Ever since we introduced our definitions of hedging and of its value, we have been
focusing on situations in which the decision-maker is contemplating acquiring a new deal
which he could incorporate into his existing portfolio. This might give the impression that
we have neglected to address the symmetric issue, in which he is wondering whether he
should sell a deal he currently owns, bearing in mind what the rest of his portfolio is.
Yet, it seems that hedging should play just as large of a role in selling situations as in
purchasing situations. Let us think of an investor who owns a diversified portfolio of
stocks, including stocks of oil companies and stocks of airline companies, which he
believes to mitigate each other's risks; or of a conglomerate with activities that are
thought by the management to compensate for each other's possible downturns. Loss of
hedging should certainly be part of their preoccupations as they contemplate selling one
of their stocks, or ceding one of their business activities.
Can our approach to hedging be extended to selling situations? Fortunately, it can - an
elementary transformation allows us to regard any selling decision situation as an
equivalent buying situation:
Theorem 2.2 -Extension of our Results to Selling Deals
Suppose that the decision-maker owns wealth w, a portfolio X and a deal Y, which the
decision-maker is considering selling. Then,
s~<Y|w,X) = - B ~<-Y |w,X,Y>
This implies that all of our definitions and results on the value of hedging are applicable
to the case in which a decision-maker is contemplating selling an asset, if we only regard
the sale of Y as equivalent to buying -Y.
69
Proof:
By definition, S~(Y | w, X) satisfies the following equation:
Z piu(w + x i+s~(Y|w,X»= X Pi S qjiiUCw + Xi+Yj)
is[l,n] ie[l,n] je[l,m]
As for B~( -Y | w, X, Y), it satisfies:
Z Pi Z q j i i u ( w + x i+y j -y r B ~<- Y l w > x > Y »= Z Pi Z qji iu(w + x i+yj) ie[l,n] Ml,m] is[l,n] je[l,m]
After simplification of the left-hand side, what remains is:
Z P lu(w + x-B~<-Y|w,X,Y» = Z ^ Z qjiiUCw + x .+y^ is[l,n] ie[l,n] je[l,m]
This proves that S~<Y | w, X) = B~< -Y | w, X, Y). •
Example 2.9:
We will take yet another look at Example 2.2, this time in the case in which the
decision-maker starts from a situation in which he owns both deal X and deal Y
and is considering selling Y. Both deals are shown on the next page as a reminder.
The decision-maker's u-curve satisfies the delta property within the [-$20,000,
$20,000] range, with a risk tolerance of $10,000.
We have: S~(Y | 0) = $82.02
We should recall, however, that Y provides hedging with respect to X; in other
words, selling Y would leave the decision-maker more exposed to risk. It hence
seems sensible that the decision-maker should actually place a higher value on Y
than (Y | 0) initially seemed to suggest. The actual computation of (Y | X)
confirms this prognosis:
S~(Y | X) = $87.71
70
This example thus illustrates our observation that hedging can be just as important
a consideration when selling a deal as it is when acquiring a deal. On a side note,
it is also interesting to remark that the numeric values of S~(Y | X) and B~(Y | X)
(which we computed in Example 2.2) match; this is due to the fact that the
decision-maker's u-curve satisfies the delta property. •
Sl
0.4 $1,000
0.6 $0 S2
S'l
0.9
-0 0.1
s'2
S'l
0.3
•O
0.7
S'2
-$100
$300
-$100
$300
~<X I 0> = $388.08 ~(Y | 0) = $82.02
71
Chapter 3 - Value of Hedging in the Delta Case
"All through history, simplifications have had a much greater long-
range scientific impact than individual feats of ingenuity. The
opportunity for simplification is very encouraging, because in all
examples that come to mind the simple and elegant systems tend to be
easier and faster to design and get right, more efficient in execution,
and much more reliable than the more contrived contraptions that have
to be debugged into some degree of acceptability. "
Edsger Dijkstra (1930-2002)
The delta property often leads to considerable simplifications in a decision analysis. One
of its most notable implications is the fact that the value of any information gathering
process is easier to compute for a decision-maker whose u-curve satisfies the delta
property than for one who does not. In this chapter we will see that the same can be said
of the value of hedging, and we will investigate many of the special properties it exhibits
in the delta case.
Throughout this part of the dissertation, we will work with a u-curve of the form:
u ( x ) = l - e - y x ,
where y > 0 denotes the decision-maker's risk-aversion coefficient. The sign of y
indicates that the decision-maker is risk-averse. We will also write VoH(Y | X) as a
shorthand for VoH(Y | w, X), due to the fact that the decision-maker's initial wealth does
not impact his valuation of uncertain deals in the delta case.
72
1. A Simpler Formula to Compute the Value of Hedging
In the delta case, the value of hedging provided by Y with respect to X is equal to the
difference between the certain equivalent of the portfolio (X u Y) and the sum of the
certain equivalents of X and Y when considered individually:
Theorem 3.1 -Hedging as the Comparison of Joint and Individual Certain Equivalents in
the Delta Case:
If the decision-maker follows the delta property over the entire range of prospects
involved, then we have:
VoH(Y | X) = S~(X, Y | 0> - S~(X | 0 ) - S~<Y | 0 ) [3.1]
Proof:
Starting from [2.4], VoH(Y | X) is the dollar amount which makes the decision-maker
indifferent between the following two states:
By definition of the certain equivalent for deal X, the above indifference statement
implies that:
73
s~ (X | 0 )
DealX
+ Deal Y
- S~(Y | 0)
VoH(Y | X)
By definition of the certain equivalent for deals X and Y together, we also have:
s~ < X , Y | 0 )
Due to the fact that the decision-maker follows the delta property, the indifference
statement above remains valid if we add a constant, - S~<Y | 0 ) - VoH(Y | X), to both
sides of this indifference statement:
Finally, we can conclude from the second and fourth indifference statements we have
derived that:
s~ '<X | 0>
This completes the proof of [3.1]. •
74
Like many of the relationships we discussed in the second chapter, [3.1] illustrates the
fact that the value of hedging measures the direction and magnitude of the interaction that
exists between the portfolio of deals X and the deal which the decision-maker is
considering acquiring, Y. In a sense, we could say that the value of hedging bears the
same relationship to the interaction of two monetary deals X and Y as does the
covariance to the relevance of two random variables; there is indeed a perceptible
similitude between:
VoH(Y | X) = S~(X, Y | 0) - S~(X | 0) - S~(Y | 0)
and:
Cov(X , Y | &) = (X, Y | &) - (X | &> (Y | &)
Also, the fact that the formula for computing the value of hedging can be greatly
simplified in the delta case makes for another striking resemblance, this time between
value of hedging and value of clairvoyance. We will see over the course of this
dissertation that the two concepts actually have many more features in common.
Example 3.1:
We can apply [3.1] to the case of Example 2.2, since the decision-maker follows
the delta property:
VoH(Y | X) = S~(X, Y | 0) - S~(X | 0) - S~(Y | 0)
= $475.79 - $388.08 - $82.02
= $5.69
This matches the result we had obtained in Example 2.2, at a time when all we
could use to compute the value of hedging was its original definition as B~(Y | w,
X)-B~(Y|w + s~<X|w». •
75
2. Special Properties of the Value of Hedging in the Delta Case
a) Symmetry
The value of hedging provided by Y to a decision-maker who already owns X is the same
as that which is provided by X if he already owns Y, provided by his u-curve satisfies the
delta property. In the delta case it would thus be more adequate to speak of the value of
hedging between two deals, rather than of the value of hedging provided by one deal with
respect to the other.
Theorem 3.2 - Symmetry of the Value of Hedging in the Delta Case:
The value of hedging is symmetric when the decision-maker follows the delta property:
VoH(Y | X) = VoH(X | Y) [3.2]
Proof:
Relationship [3.1] is symmetric in X and Y; consequently, VoH(X | Y) = VoH(Y | X)
when the decision-maker's u-curve satisfies the delta property. •
Many will find such a conclusion intuitively pleasing; after all, we defined the value of
hedging with the intention of quantifying the interaction between two portfolios, and it
only seems fitting that the interaction would be symmetric at least in some situations.
76
b) Upper Bound on the Value of Hedging
The value of hedging between two deals is dominated by the sum of the risk premiums of
the two deals:
Theorem 3.3 -- Uvper Bound on the Value
If the decision-maker follows the delta
involved and is risk-averse, then we have:
VoH(Y | X) < RP(X |
of Hedging in the Delta Case:
property over
0) + RP(Y 0)
the entire range
[3.3]
of prospects
Proof:
From [3.1]:
VoH(Y | X) = S~(X, Y | 0) - S~(X | 0> - S~<Y | 0>
Since the decision-maker is risk-averse, we also know that:
S~(X, Y | 0> < (X + Y | &)
Combining these two results:
VoH(Y | X) < <X + Y | &> - S~<X | 0> - S~(Y | 0)
< «X | &) - S~(X | 0 » + «Y | &> - S~(Y | 0 »
< RP(X | 0) + RP(Y | 0). 3
77
Example 3.2:
We will see that the tightness of the upper bound on the value of hedging depends
on the specific probabilities and monetary prospects of the deals involved.
For that purpose, let us consider the deal X which was common to Examples 2.1
and 2.2; with deal Yi which was regarded as a potential addition to X in Example
2.1, upper bound [3.3] yields:
VoH(Yi | X) < RP(X | 0) + RP(Y, | 0)
< ((X | &) - S~<X | 0 » + «Y! | &) - S~<Y, | 0 »
< ($400 - $388.08) + ($140 - $138.07)
< $13.84
As a reminder, we computed earlier the exact value of hedging to be equal to
$9.54. In that instance the difference between the upper bound and the actual
result is thus reasonable.
As for deal Y2 which was combined with X in Example 2.2:
VoH(Y2 I X) < RP(X I 0) + RP(Y2 | 0)
< ((X I &) - S~(X I 0 » + «Y2 I &> - S~<Y2 I 0 »
< ($400 - $388.08) + ($84 - $82.02)
< $13.90
This time the comparison is less impressive, since the actual value of hedging
between Y2 and X is $5.69.
We can find in the probabilistic structures of Yi and Y2 the reason why the upper
bound yields a better answer in the case of the former than in the case of the
latter: Yi and Y2 are similar in some respects (the magnitude of their monetary
prospects is the same), but due to the better relevance between Yi and X
compared to the one between Y2 and X, Yi and X will compensate for more of
each other's risk than Y2 and X will.
That observation leads us to another question - can the upper bound from [3.3] be
achieved? In other words, is there a deal Y* such that the value of hedging
78
between it and X will be equal to, and not less than, the sum of the risk premiums
of the two deals?
The answer is yes. Intuitively, we might think that what we need to discover is the
best possible complement to deal X, and what better hedge is there than one
which completely eliminates uncertainty from the portfolio? We thus examine the
interaction between X and Y , which we define as follows:
r^ $1,000 $0 0.4
O
0-6 1 $0 $1000 S2
S~<X | 0> = $388.08 B~<Y* | 0) = $587.92
Figure III.l — Best hedge for X
Then:
VoH(Y* | X) < RP(X | 0) + RP(Y* | 0)
< «X | &> - S~<X | 0 » + «Y* | &> - S~(Y* | 0 »
< ($400 - $388.08) + ($600 - $587.92)
< $23.99
Interestingly, but not surprisingly, the actual value of hedging provided by Y* is
equal to that same number:
VoH(Y* | X) = S~<X, Y* | 0) - S~(X | 0) - S~<Y* | 0)
= $1000-$388.08-$587.92
= $23.99.
79
It should also be noted, by virtue of Theorem 2.1, that the value of hedging would *
have turned out to be identical for any deal which differs from Y by a constant -
for example, if the monetary prospects of Y corresponding to s\ and S2 had
respectively been {-$500; $500}, or {-$2,000; -$1,000}. This means that there
exists an infinity of deals Y such that VoH(Y | X) = RP(X | 0) + RP(Y | 0). J
In chapter II, we introduced the idea that the value of hedging concept can also be used to
infer the value of the decision-maker's PIBP for a deal which he does not own; we had
observed that:
B~(Y | w, X) = B~<Y | w + S~<X | w» + VoH(Y | w, X), [2.5]
and we had announced that the property would prove especially valuable in the delta case,
because we would then have an upper bound on the term which corresponds to the value
of hedging. It is now time to fulfill that promise: we will combine [2.5] with the results of
the theorem we just discussed to obtain a general upper bound on the decision-maker's
PIBP for any deal which he does not already own:
Theorem 3.4- Upper Bound on the PIBP for any Deal Y:
If the decision-maker follows the delta property over the entire range of prospects
involved and is risk-averse, then for any deal Y which he does not own:
B~(Y | X) < RP(X | 0) + (Y | &) [3.4]
80
Proof:
We first rewrite [2.5] for a decision-maker whose u-curve satisfies the delta property:
B~(Y | X) = B~(Y | 0) + VoH(Y | X)
By virtue of Theorem 3.3, the second term of the sum is dominated by RP(X | 0) +
RP(Y | 0):
B~(Y | X) < B~<Y | 0) + (RP(X | 0) + RP(Y | 0))
< RP(X | 0) + (B~<Y | 0) + RP(Y | 0))
< RP(X | 0) + <Y | &). 3
A decision-maker should thus never pay more for a deal than the sum of the risk premium
of his existing portfolio and the mean of the possible acquisition.
It shows that as decision-makers, we would be well-advised to write down on a piece of
paper the value of the risk premium of our present portfolio and carry it with us:
occasionally, that knowledge will be sufficient to quickly rule out a potential investment
as being unworthy of our attention; all we have to do is add to the risk premium number
the mean of the new investment, which is a simple task compared to the full computation
of a certain equivalent, and compare the sum to the price of the investment. If the price
exceeds RP(X | 0) + (Y | &), no further analysis is required and we should walk away
from the investment opportunity.
Example 3.3:
The decision-maker from Examples 2.1 and 2.2 only holds deal X in his current
portfolio; the associated risk premium is:
RP(X|0) =$11.92
The decision-maker is contemplating acquiring deal Y shown below, for a price
of$260:
81
S'l
$1,000 0.1
^ ~ $ 6 5 0
&— $200 0.4
^ $100 0.2
f j -$500 o.i
S~(X I 0) = $388.08 <Y | &) = $240.00
Figure III. 2 - Computation of an upper bound on the PIBP ofY
No conditional probability distribution over the prospects of Y given those of X
has been assessed yet; all we have elicited from the decision-maker is a marginal
distribution. Still, based on [3.4], this will be enough to compute an upper bound
on his willingness to pay for Y:
B~(Y | X) < RP(X | 0) + <Y | &>
< $11.92+ $240
< $251.92
Assessing a complete conditional probability distribution over the prospects of Y
given those of X would thus be a waste of time: the decision-maker is already
assured that he should reject the offer to buy Y at a price of $260. •
82
Sl
0.4 $1,000
S '
0.6 $0 S2
c) Value of Hedging for Multiples of a Deal with Respect to Itself
In this part, for any deal X, we will denote by AX the deal whose monetary prospects
{x'i}ie[i,n] are determined by the same set of uncertainties as the prospects {xi}ie[i>n] of
X, and such that x'j = XXJ for all values of i. We will call such deals multiples of X.
What is the value of hedging between two deals which are multiples of each other? We
have already encountered an instance in which VoH(X | X) was negative (Example 2.6);
but is that true in general?
Theorem 3.5 -- Hedging of a Deal With Respect to Itself:
If the decision-maker follows the delta property over the e
involved and if he is risk-averse, then we have:
For any scalar X
In particular:
Equivalently,
f VoH(X |
\ VoH(-X
in terms of risk
f VoH(X |
L VoH(-X
mtire range of prospects
, VoH(A,X X) is of the opposite sign of X.
X) = S~(2X | 0 ) - 2 S~<X | 0 ) < 0
| X) = - S~(X | 0 ) - S~(-X | 0 ) > 0
premium:
X) = 2 RP(X | 0 ) - RP(2X | 0 ) < 0
[3.6]
[3.7]
[3.8]
| X) = RP(X | 0 ) + RP(-X | 0 ) > 0 [3.9]
[3.5]
Proof
We will prove that VoH(X | X) < 0 for simplicity, but the same reasoning can be
extended to the general case of VoH(A,X | X):
• Proof that VoH(X | X) < 0 for binary deals X where the decision-maker
assigns p chance to receiving x and 1 - p chance to receiving 0:
83
For now, we are only considering deals of the form detailed above. By
virtue of [3.1], we know that:
VoH(X | X) = S~<2X | 0) - 2 S~<X | 0>
Also, by definition of the certain equivalent, we have:
u(s~<X|0» = pu(x) + (l-p)u(O)
u(s~(2X | 0 » = p u(2x) + (1 - p) u(0)
Let us choose u(z) = 1 - e"yz for all z as a u-curve, to simplify those
equations; then u(0) = 0 and:
u(s~<X|0» = pu(x)
u(s~<2X | 0 » = p u(2x)
This leads to:
s ~(X|0>=-- ln( l -p + pe-yx)
s~ (2X10) = —- ln(l - p + pe"2l,x)
Forming the difference S~(2X | 0) - 2 S~<X | 0):
f s~(2X|0)-2 s~<X|0>=-ln
Y
1, f a - p + pe"^)2^ 1 - p + pe -2yx
We see that the sign of the difference depends on the sign of
(1-p + pa)2 - ( 1 - p + pa2), where a = e"7*. That function of a is always
less than or equal to 0; in fact, the only case in which it is not negative but
equal to zero is the case in which a = 1, or equivalently x = 0.
Consequently, VoH(X | X) = S~(2X | 0) - 2 S~<X | 0> < 0 for such a deal X.
84
• Proof that VoH(X | X) < 0 for any binary deal X:
Let us now examine VoH(X | X) for deals X where the decision-maker
assigns p chance to receiving x and 1 - p chance to receiving y:
I $x
o
1-p t 1 $y
We define deal X' as follows:
r — $x-y
o
Then, by virtue of the delta property, S~(2X | 0) - 2 S~<X | 0) = (S~<2X' | 0) + 2y) - 2 (S~(X' | 0) + y)
= S~<2X' | 0) - 2 S~<X' | 0)
<0
The last inequality is a mere application of the result we proved earlier for
deals where the decision-maker assigns p chance to receiving some
amount x and 1 - p chance to receiving 0. We have thus shown that
VoH(X | X) < 0 holds for any binary deal X.
85
Proof that VoH(X | X) < 0 for any deal X:
Any deal can be regarded as a combination of binary deals. It is by
applying the result derived above successively to all of those binary deals
that we will demonstrate that VoH(X | X) < 0 holds for any deal X.
More formally, we will reason by induction. Let Pn be the following
proposition: "For any deal X with exactly n different monetary prospects,
VoH(X\X) <0". Then:
• P2 is true, as shown earlier. As a side note, it may also be
interesting to note that Pi is true, since VoH(X | X) = 0 in that
case. However, the validity of Pi is not significant for the rest
of this proof.
• Let us suppose that for some value n, P2, ..., Pn-i are all true;
we will show that this implies that Pn is true as well. For that,
we will consider a deal X with exactly n different monetary
prospects. We will label them X], ..., xn, and we will call pi, ...,
pn their associated probabilities:
Pi
-0
Pn-l
Pn
$x,
$xn-l
$ X n
Equivalently, the last two branches of deal X can be rearranged
as follows:
86
Pi $ X j
We will call Xn_i the subdeal which consists of the two
branches of the tree which were highlighted above. Then (2X
| 0 ) is equal to the certain equivalent of the following deal:
Pi
• 0
Pn-2
Pn-l + Pn
$2x,
3> 2 x n .
2X„,
By the substitution rule, that in turn is equal to the certain
equivalent of the deal shown below:
87
Pi $2x.
•o
Pn-2 q> 2xn.
Pn-l + Pn "<2 Xn., | 0>
By P2, s~<2Xn.i | 0) < 2 s~(X„.i | 0); therefore, S~(2X | 0) is less
than the certain equivalent of:
Pi $2x i
-0
Pn-2
Pn-l + Pn
j> 2xn_
2 s~<Xn., | 0 )
The deal shown above now comprises exactly n-1 branches,
which allows us to make use of Pn_i; its certain equivalent will
be less than twice the certain equivalent of:
Pi $ X i
0
Pn-2 $ xn.
Pn-l + Pn ~<X„-, I 0 )
Finally, to complete the proof, we observe that the certain
equivalent of the deal shown above is equal to the certain
88
equivalent of X. It demonstrates that s (2X | 0) < 2 S~(X | 0),
and therefore that Pn is valid if P2, ..., Pn-i are valid. :
We have successively proved that P2 is true, and that the
validity of Pn is hereditary: VoH(X | X) < 0 thus holds for any
deal X, no matter how many monetary prospects are involved
in it. 3
Example 3.4:
A decision-maker, who wishes to follow the delta property within the [-$20,000,
$20,000] range and has a risk tolerance of $10,000, holds deal X shown below:
Sl
O
0.3
S2
0.4
S3
0.3
$3,000
$0
-$2,000
We can then plot VoH(A,X | X) for different values of X (see next figure):
89
DJD
*M $1,000
X
Figure III. 3 - Sensitivity of VoH(XX \X) to X
Two insights can be derived from that sensitivity analysis:
• the value of hedging has the sign predicted by Theorem 3.5;
• it also appears to be a monotonic decreasing function of X over the range
which we studied. D
90
d) Value of Hedging and Irrelevance
We already observed, in the previous chapter, that the impact of risk attitude on the value
of hedging can be hard to predict and can pose a challenge to our intuition, even in
situations in which the decision-maker's u-curve satisfies the delta property (see Example
2.7). In contrast, we will now see that the absence of probabilistic relevance between two
deals produces an effect which we could have easily foretold: the value of hedging then
becomes equal to zero.
But before we show and discuss this result in greater detail, we should first precisely
define what it means for two monetary deals to be relevant or irrelevant. Stricto sensu,
probabilistic relevance is a concept which qualifies the relationship of two uncertainties,
and not of monetary deals; in order to extend its applicability to the latter, we would be
well inspired to find a way to associate with any given deal an uncertainty which, in a
manner of speaking, would be probabilistically equivalent to it. The solution is evident as
long as the uncertain deal under consideration has prospects which are all distinct:
Definition 3.1 - Minimal Underlying Uncertainty Associated with an Uncertain Deal:
Given an uncertain monetary deal Y = {(p;, yi)ie[i,n]}5 whose prospects are all distinct, we
will call "minimal underlying uncertainty associated with Y" an uncertainty S with n
degrees st such that:
For all i e [ l , n], (Y = y; o S = s.)
Another way to think about this is to notice that minimal underlying uncertainties satisfy
{Y = yj | S = sh &} = 1 as well as {S = s,•• \ Y = ys, &} = 1 for all values of i. But what
about uncertain deals whose prospects are not all distinct? In their case, we cannot apply
the above definition directly, because it would make us create an uncertainty with several
degrees which in fact all correspond to one same monetary prospect of Y. But all we need
to do is collapse Y by successively recombining every pair of equal prospects (y,, yj) into
91
a single prospect, whose probability will be equal to the sum pi + p,, and we can then
apply Definition 3.1 to the uncertain deal which is produced by that transformation.
From this point on, we will thus say that two uncertain deals are probabilistically relevant
given a specific state of information if their respective minimal underlying uncertainties
are themselves probabilistically relevant given that state of information. With this
foundation in place, it then becomes possible to examine the important result which we
had announced earlier - in the delta case, the value of hedging between two deals which
are irrelevant given & is equal to zero:
Theorem 3.6- Irrelevance and Value of Hedging:
If the decision-maker follows the delta property over
involved, and if X and Y are irrelevant given &, then:
VoH(Y | X) = 0
the entire range of prospects
Proof:
We will start with an alternate form of equation [3.1]:
VoH(Y | X) = S~<Y | X) - S~(Y | 0)
By definition of certain equivalent S~(Y | X):
X P iu(x i + S~<Y|X»= X Pi Z qjiUCxj+yj)
ie[l,n] ie[l,n] je[l,m]
We then make use of the irrelevance between X and Y:
X Piu(x1 + S~<Y|X»= S P, Z qjU(xi+yj)
ie[l,n] ie[l,n] je[l,m]
Since the decision-maker follows the delta property, the right hand side can be
simplified as follows:
X Pi u(x, + S~<Y|X»= X p. u(x, + S~<Y|0» ie[l,n] ie[l,n]
92
We can then conclude that S~(Y | X) = S~<Y | 0>, and therefore that VoH(Y | X) = 0. •
Why is this result not necessarily true outside of the delta case? In order to understand
that, let us remember that the value of hedging is an entity by which we place a monetary
value on the interaction between X and Y. There are only two ways in which that
interaction can manifest itself:
• if there exists a probabilistic relevance between X and Y,
• and through wealth effects: even if there is no probabilistic relevance between
X and Y, for a decision-maker who does not follow the delta property, the
value which he places on deal Y might be affected by the change in wealth
triggered by receiving some reward or penalty Xj from deal X. At a more
fundamental level, we could say that because X has yet to be resolved, its
presence in the portfolio and its possible outcomes can still influence the
decision-maker's valuation of Y.
Only in the delta case is it guaranteed that such wealth effects need not be considered;
therefore, it is only in the delta case that the absence of probabilistic relevance between X
and Y necessarily implies that the value of hedging between them will be equal to 0.
93
Example 3.5:
Let us examine the situation of the same decision-maker as in Example 2.2, who
is contemplating buying a deal Y which is irrelevant to the unresolved portfolio X
he already owns:
Sl
0.4
-0
0.6
S2
$1,000
$0
S'l
0.3
-0 0.7
s'2
S'l
-0
0.3
0.7
s'2
-$100
$300
-$100
$300
S~<X | 0) = $388.08 B~(Y | 0) = $178.31
Figure III.4 - Value of hedging for two irrelevant deals
We can then verify that the value of hedging between X and Y is indeed equal to
zero:
VoH(Y | X) = S~<X, Y | 0) - S~(X | 0) - S~(Y | 0>
= $566.40 - $388.08 -$178.31
= $0.3
94
3. The Chain Rule for the Value of Hedging and its Implications
a) Chain Rule
So far we have examined many important properties of the value of hedging which a
single deal Y provides with respect to the decision-maker's existing portfolio X - but
what can be said of the value of hedging which two deals Y and Z jointly provide with
respect to X? Is the joint value of hedging, VoH(Y, Z | X), related in some way to the
individual values of hedging VoH(Y | X) and VoH(Z | X)?
The idea that there might exist a chain rule connecting those entities is not incongruous.
Indeed, such a chain rule exists in the case of the value of clairvoyance; for a risk-neutral
decision-maker, who is facing a decision situation with two uncertainties, A, with degrees
{ai}ie[i,n], andB:
VoC(A, B) = VoC(A) + ]T {a; | &} VoC(B | a,) i=l
In this equation, VoC(B | a;) stands for what we might call the conditional value of
clairvoyance on B given a\: it is, the decision-maker's personal indifferent buying price
for perfect information on B once he has already acquired the certainty that A = aj.
We will now see that a chain rule which is similar in its spirit can be derived for the value
of hedging. However, we will discuss some tempting but incorrect guesses at what the
chain rule might be before we present the correct answer; indeed, there is much to be
learnt from the reasons why those guesses are incorrect. Here is what might seem the
most evident conjecture:
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X).
But we have seen that the relevance relationships between the distinctions involved in the
decision situation can have a great impact on the value of hedging; since the sum VoH(Y
| X) + VoH(Z | X) completely ignores any information pertaining to the relevance or
irrelevance of Y and Z given X, we can dismiss the conjecture as highly unlikely to be
correct. Bearing that objection in mind, this next attempt would certainly seem a more
fitting candidate:
95
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y)
The structure of that equation echoes that of the chain rule for the value of clairvoyance;
in fact, that second conjecture even turns out to be true in a number of cases - but it still
does not hold in the general case.
This time, the intuitive explanation for the invalidity of our conjecture is more subtle. Let
us recall the interaction metaphor we have used many times as a guide for our intuition -
for two deals A and B, VoH(A | B) captures the interaction between those two deals. The
term VoH(Z | X, Y) thus captures the interaction between Z on the one hand, and deals X
and Y on the other. Incidentally, VoH(Z | X, Y) will occasionally be affected by the
nature of the interaction between Y and Z. But the latter should not have any influence on
the value of the number we ultimately seek to evaluate, VoH(Y, Z | X), which should
solely depend upon the interaction between X on the one hand, and the deal formed by
the union of Y and Z on the other.
To put it in another way, in the expression VoH(Y, Z | X), Y and Z are already regarded
as a bundle, and thus their interaction does not need to be accounted for anymore; but in
the term VoH(Z | X, Y), the mere fact that Y and Z are now dissociated and sitting on
opposite sides of the vertical bar makes their interaction germane to the value of the
right-hand side VoH(Y | X) + VoH(Z | X, Y). This is the root of the issue with our second
conjecture. Interestingly, the actual chain rule for the value of hedging is nothing other
than a version of our second conjecture in which that same issue was rectified.
96
Theorem 3.7' - Chain Rule for the Value of Hedging:
If the decision-maker follows the delta property over the entire range of prospects
involved, then VoH(Y, Z | X) can be decomposed into a sum of other terms involving the
values of hedging provided by Y and Z individually:
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y) [3.10]
Naturally, it is also possible to start the chain rule decomposition with VoH(Z | X)
instead:
VoH(Y, Z | X) = VoH(Z | X) + VoH(Y | X, Z) - VoH(Y | Z) [3.11]
Proof:
We will evaluate VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y) by first applying [3.1] to
each one of the three terms in that expression: r
VoH(Y | X) = S~(X, Y | 0) - S~(X | 0> - S~(Y | 0>
< VoH(Z | X, Y) = S~(X, Y, Z | 0) - S~(X, Y | 0) - S~<Z | 0>
VoH(Z | Y) = S~<Y, Z | 0) - S~(Y | 0) - S~<Z | 0)
When we then form VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y), we can observe that
many terms cancel out; all we are left with is:
VoH(Z | X) + VoH(Y | X, Z) - VoH(Y | Z) = S~(X, Y, Z | 0) - S~(X | 0) - S~(Y, Z | 0)
Finally, we can recognize in the right-hand side of the above equation the
decomposition of VoH(Y, Z | X) using [3.1]:
VoH(Y, Z | X) = S~<X, Y, Z | 0> - S~<X | 0) - S~(Y, Z | 0)
Therefore, VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y). U
97
As announced ahead of Theorem 3.7, our second guess was not far off the mark - all we
needed to do was to subtract the term VoH(Y | Z) from the sum VoH(Z | X) + VoH(Y | X,
Z) in order to compensate for the problem we had detected in that conjecture.
It is also interesting to note that it becomes even easier to remember [3.10] if one draws a
Venn diagram in which X, Y and Z are represented by three intersecting circles, and if,
for any deals A and B, one interprets VoH(A | B) as jl(A fl B), the area of A fl B:
VoH(Y | X) = n * +
+ VoH(Z | X, Y) = 1 + +
_ VoH(Z | Y) = 1 +
VoH(Y, Z | X) + +
Figure III. 5 - Mnemonic for the chain rule for the value of hedging
We will now study a concrete example in which the chain rule can be applied, and see
which kinds of practical insights it allows us to derive.
98
Example 3.6:
A decision-maker, who wishes to follow the delta property within the monetary
range [-$5,000, $5,000] with a risk tolerance of $2,000, owns deal X and is
contemplating acquiring deals Y and Z as described below:
0.4 $1,000
-0
0.6 $0
0.8
•0
0.2
0.3
-O 0.7
-$200
$500
-$200
$500
-0 o.i 0.9
•0. 0.3 0.7
•O 0.4 0.6
0.3 '0.7
$600
-$100
$600
-$100
$600
-$100
$600
-$100
s~, (X I 0) = $342.50 ~<Y | 0) = $119.53
Figure III.6- A chain rule example
~(Z | 0) = $55.94
We will decompose VoH(Y, Z | X) using [3.10] first and [3.11] afterwards:
VoH(Y|X) =$39.21
+ VoH(Z|X,Y) =$10.52
-VoH(Z|Y) = $5.08
VoH(Y, Z | X) $54.82
99
Here is the second application of the chain rule to VoH(Y, Z j X):
VoH(Z|X) =$13.64
+ VoH(Y | X, Z) = $36.09
-VoH(Y|Z) = $5.08
VoH(Y, Z | X) = $54.82
There are a few conclusions that we can draw from this analysis:
• The decision-maker should have a positive PIBP for Y and Z taken
individually, and for Y and Z when they are regarded as a bundle: B~(Y |
X) = $158.74; B~<Z | X) = $69.59; B~<Y, Z | X) = $225.21.
• Also, both Y and Z provide hedging with respect to the current portfolio X,
and that again is true whether Y and Z are considered in isolation or
jointly: VoH(Y | X) > 0; VoH(Z | X) > 0; VoH(Y, Z | X) > 0.
• However, it is also important to realize that Z provides less value than Y
in itself, B~(Z | 0) < ~(Y | 0), and it also provides less value than Y
through hedging with respect to X. In fact, a significant portion of VoH(Y,
Z | X) is attributable to deal Y alone.
Those remarks should help the decision-maker realize that, while Y and Z both
constitute valuable additions to his portfolio, Y is the stronger choice of the two.
Perhaps it would be worthwhile for the decision-maker to look for a better
acquisition than Z. •
100
b) Toward an Irrelevance-Based Value of Hedging A Igebra
We mentioned earlier that VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y) actually turns
out to be true in some situations; for that, all we need is for VoH(Z | Y) to be equal to
zero. Our earlier discussion of the effect of irrelevance on the value of hedging (see
Theorem 3.6) showed us when such a situation might arise:
Theorem 3.8- Chain rule for the Value c fHedsins when Y' J.Z &:
If the decision-maker follows the delta property over the entire range
involved, and if Y and Z are irrelevant given &, then:
VoH(Y, Z | X) = VoH(Y X) + VoH(Z X,Y) [3.12]
of prospects
Proof:
The general chain rule for the value of hedging was given by equation [3.10]:
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y)
By Theorem 3.6, the fact that Y and Z are irrelevant given & implies that VoH(Z | Y)
= 0. The chain rule thus simplifies into:
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y). •
Theorem 3.8 gives us a first glimpse of the analytical power that we get access to once
we start combining the irrelevance properties of a decision situation with the chain rule.
As the decision-maker contemplates a new acquisition, applying the chain rule in a way
that will let him exploit the irrelevance structure of the resulting portfolio can help him
compute the value of hedging much faster. Such thinking will be of greatest value to
decision-makers who own particularly complex portfolios of unresolved deals.
In a way, we can speak of an irrelevance-based algebra for the value of hedging, an
algebra in which the theorems of probability on irrelevance are transposed into sister
101
rules on the value of hedging. Another example follows, in which we start from a classic
rule from probability called the contraction property to derive more results on hedging:
Theorem 3.9 - Contraction Property for the Value of Hedging:
If the decision-maker follows the delta property over the entire range of prospects
involved, and if he believes that X L Y | & and that X _L Z | Y, &, then:
f VoH(Y, Z | X) = 0
1 VoH(Z | X, Y) = VoH(Z | Y)
Proof:
Let us first write the contraction property from probability in its most general form -
for any uncertainties or sets of uncertainties A, B, C and D, if A J. C | B, D, & and A
1 B | D, &, then A 1 B, C | D, &.
We will apply this result to the particular case in which A = X, B = Y, C = Z and D =
&; since the decision-maker believes that X 1 Y | & and that X 1 Z | Y, &, he should
also believe that X 1 Y, Z | &.
By Theorem 3.6,
X 1 Y | & implies that VoH(Y | X) = 0
X I Y, Z | & implies that VoH(Y, Z | X) = 0
This already proves the first clause of the conclusion of Theorem 3.9. For the second
part, let us recall the chain rule for the value of hedging given by equation [3.10]:
VoH(Y, Z | X) = VoH(Y | X) + VoH(Z | X, Y) - VoH(Z | Y)
Two terms in the equation are equal to zero; therefore, it simplifies into:
VoH(Z | X, Y) = VoH(Z | Y).
This proves the second clause of Theorem 3.9. D
102
We will now study a practical illustration of the contraction property for the value of
hedging in order to show how useful it can be:
Example 3.7:
The same decision-maker as in Example 3.6 owns two uncertain deals, X and Y,
which he believes to be probabilistically irrelevant given &. He is contemplating
acquiring a third deal Z, which is irrelevant to X given Y and & but relevant to Y
given X and &. For instance, some investors might well have such beliefs about a
portfolio in which X was the stock of a pharmaceutical company, Y the stock of
an oil company, and Z that of an airline company. All three assets are shown on
the figure below.
Let us help the decision-maker compute his PIBP for deal Z, given that he
presently owns X and Y. There are two ways in which we can do this:
• First, we could compute B~(Z | X, Y) directly. This would require that we
take into account the full probabilistic structure of the problem.
With this approach, we obtain:
B~(Z | X, Y) = $408.96
• Secondly, we could also choose to exploit the irrelevance properties of the
portfolio. We can start by applying Property 2.3:
B~<Z | X, Y) = B~<Z | 0> + VoH(Z | X, Y)
Next we can invoke the contraction property for the value of hedging and
observe that:
VoH(Z | X, Y) = VoH(Z | Y)
Combining the two results:
B~<Z | X, Y) = B~<Z | 0 ) + VoH(Z | Y)
103
= $378.10+ $30.86
= $408.96
S~(X | 0> = $342.50 S~(Y|0> =-$14.47 S~(Z | 0) = $378.10
Figure III. 7 — Applying the contraction property for the value of hedging
What is particularly interesting about this second approach is that it enables us to
derive the result much more rapidly and efficiently than the first. In general, if X
has nx degrees, Y ny degrees and Z nz degrees, the first approach requires that we
evaluate a tree of size nx x ny x nz. As for the second approach, it requires two
evaluations: one of a tree of size nz (to compute B~(Z | 0)) and one of a tree of
size ny x nz (to compute VoH(Z | Y)).
104
In other words, the contraction property for the value of hedging eliminates
altogether the need to take the irrelevant component X of our portfolio into
account. The gain in efficiency will be all the larger as X is more complex. •
105
Chapter 4 - Hedging as Moment Reengineering
"Cause and effect are two sides of one fact. "
Ralph Waldo Emerson (1803-1882),
Circles
The question I will address in this chapter of the dissertation is perhaps more fundamental
and theoretical than any of the other issues we have examined so far - what probabilistic
phenomena lie at the source of hedging? My ambition is to identify some universal
characteristics of the situations in which hedging arises, in order to be able to detect
hedging more easily.
In this entire chapter, we will again assume that the decision-maker is risk-averse and
follows the delta property over the range of prospects involved. We will denote his risk-
aversion coefficient by y. A possible u-curve for him is thus given by:
u(x) = 1 - e-yx
106
1. General Principle
Ronald A. Howard showed that, for people who follow the delta property over some
range, a convenient decomposition of their certain equivalent for a deal involving
prospects within that range is given by:
"<Z|0) = X ^ ^ Y n - 1 K „ n!
[4.1]
where K„ denotes the nl cumulant of the distribution {Z | &} [Howard, R. A., 1971]. For
the reader who is not familiar with the concept of cumulants, it is sufficient for the
purposes of our discussion to know that the values of those cumulants can be computed
recursively based on the values of the moments of equal or lower order:
K n = U n - E k=l
n-1
k-1 Kk IV:
In particular:
Ki=<Z|&>
K2 = V(Z | &)
K3 = < ( Z - < Z | & » 3 | & >
(first cumulant)
(second cumulant)
(third cumulant)
By truncating the series in [4.1], we can then obtain approximations of the certain
equivalent which will be more or less precise depending on the term at which we stopped.
For example, one of the most commonly used approximations of the certain equivalent in
decision analysis is:
s~<Z|0)«<Z|&)-ly v (Z|&) [4.2]
The approximation turns out to be exact if the probability distribution over Z is normal.
Otherwise, it will be all the more precise as:
V ( Z | & ) « ^
y
If we instead consider the first three terms of [4.1], we obtain:
107
s~<Z|0>*<Z|&>-^yv<Z|&> + V<(Z-<Z|&>)3 |&>, [4.3] 2 o
which implies that if two deals have the same mean and variance, but one of them is more
skewed toward the right, a risk-averse decision-maker who follows the delta property will
tend to prefer that deal over the other.
Such decompositions of the certain equivalent should remind us that if we are offered an
uncertain deal of which we know nothing, and if we are allowed to ask one question and
only one about the moments of the deal in question, we should inquire about its mean; if
it is negative, we should reject the offer. If we are allowed to ask two questions, then we
should ask about the mean and variance of the deal, so that we can use approximation
[4.2] to assess our certain equivalent for it.
Likewise, if we own a deal and wish to make it more valuable, our best efforts should be
aimed at increasing its mean, then at decreasing its variance, then at increasing the value
of its third central moment, etc. We will now examine some concrete examples in order
to demonstrate that such a process is exactly what hedging is about - in other words,
some deal Y has a positive value of hedging with respect to a deal X which we already
own if and only if Y helps us reengineer the cumulants of X of order 2 and above in a
helpful manner.
108
2. Variance Reengineering
We saw in the first chapter that the idea that hedging can be used to reduce the variance
of an existing deal is already widespread in the financial literature, to the point that
hedging and variance reduction have often been regarded as complete synonyms. In this
part we will for once also focus entirely on the variance; we will see that in many
situations the certain equivalent approximation
s~(Z|0>*(Z|&>-iyv(Z|&>
can help us compute an approximate PIBP for any deal with a respectable accuracy,
because in those cases a large part of the value that hedging provides comes from a
variance reduction effect.
Example 4.1:
The same decision-maker as in Example 2.1 owns some unresolved deal X and is
considering buying deal Y. The two deals are shown below:
Sl
0.4
O
0.6 1 $0 $200
Figure IV.l - Example of variance reengineering
$1,000 -$300
109
As with some of our earlier examples, at first sight, it might be tempting to
dismiss Y as a bad investment opportunity; indeed, S~(Y | 0) = -$3.01 < 0, which
is not surprising since the decision-maker is risk-averse and since Y has a mean of
0. However, closer examination reveals that VoH(Y | X) = $11.94, which more
than compensates for the low value of ~(Y | 0) and suffices to make Y worth
buying for a price of up to B~<Y | X) = $8.93.
Where does the positive value of hedging come from? In order to understand it a
little better, let us recall [3.1]:
VoH(Y | X) = S~(X, Y | 0) - S~<X | 0) - S~<Y | 0)
Let us also recall [2.5]:
S~<Y | X) = S~<Y | 0) + VoH(Y | X)
Therefore:
S~<Y | X) = S~<X, Y | 0) - S~(X | 0)
We will decompose that difference along the first two cumulants as in [4.2]:
s~(Y|X) = (<X,Y|&>-iY V(X,Y|&) ]-( <X|&)-^y V(X|&)
V 2 ) \ 2 )
Rearranging the terms, so that the terms corresponding to first moments are
separated from those which correspond to second moments:
s~(Y|X> = (<X,Y|&>-<X|&>)-iY(v<X,Y|&>-v<X|&))
The following table shows the values of the first and second moments of the
original portfolio, X, and of the portfolio in which X and Y are combined, (X u
Y):
110
Table IV.l - Decomposition of certain equivalents along the first two cumulants
Several interesting observations can be made based on those results:
1) The main reason why S~(X, Y | 0 ) is greater than S~(X | 0 ) is the fact that
Y helps reduce some of the variance of X. The variance of deals X and Y
combined is 60,000, versus 240,000 for the original deal X. In other words,
even though Y does not add to the e-value of the monetary prospects of
our portfolio, it might be worth acquiring because it helps us reduce its
variance.
2) Our approximation of the difference (X, Y | 0 ) - (X | 0) , even when
truncated after only two moments, is accurate enough to help us compute
the decision-maker's PIBP for Y with an error of less than 1%. That can
be explained by the fact that the portfolios we are examining here, be it the
original one, X, or the one resulting from the combination of X and Y, all
have variances small enough for [4.2] to be precise:
V ( X | & > « - ^ and V<X,Y| & > « - ! -Y T
3) The approximation would have been even more accurate if we had
stopped the decomposition after three cumulants instead of two, as shown
below:
111
Table IV.2 - Decomposition of certain equivalents along the first three cumulants
This time the error is of less than a cent. However, the two-cumulant
decomposition was already so good that it is doubtful that the slight
increase in accuracy obtained as a result of using a three-cumulant
decomposition instead of a two-cumulant decomposition really warrants
the extra computational effort.
The next chart represents the evolution of the certain equivalents of the
original and new portfolio, S~(X | 0 ) and S~(X, Y | 0 ) , as one takes into
account more and more cumulants in summation [4.1]. As shown by that
study, the fourth and fifth cumulants have even less of an impact on the
accuracy of our approximation than the third. D
112
$410.00
$405.00
$400.00
$395.00
$390.00
$385.00
$380.00
1 Cumulant 3 Cumulants 5 Cumulants 2 Cumulants 4 Cumulants
Figure IV.2 - Accuracy of approximation depending on the number of cumulants
S-KHUMI
S400.0o\
-
i
sw.oo ^ ^ - - - B - • -8
\ * PIBP(Y | X)
$388.00
i i
- • - S~(X | 0) -m- S~(X, Y | 0)
„ „ , , ! § - , . . . „ _ „ . . . «
($8.93)
A A
" ' " " ' ' " :"""
Example 4.2:
As mentioned in the first part of this chapter, there is one situation in which the
certain equivalent approximation based on only two cumulants [4.2] is exact -
that is when the probability distribution over the deal Z whose certain equivalent
we wish to evaluate is normal.
Let us consider the case of a firm, which currently owns a single business division,
X, and is contemplating acquiring a division Y from another company. The profits
generated by X are modeled as a normal distribution, with a mean of $200 million
and a standard deviation of $100 million, whereas those of Y are modeled as
another normal distribution, with mean $50 million and standard deviation $20
million. The company's management also believe that the probabilistic relevance
between X and Y is best captured as a correlation coefficient rXy = -0-5, and they
state that the company is comfortable using an exponential u-curve with a risk-
tolerance of $200 million for this particular decision situation.
113
What is the most that the firm should be willing to pay for Y? If we neglect the
probabilistic relevance between the two business divisions, and directly apply
[4.2] to Y, we obtain:
s~(Y|0) = ( Y | & ) - - y v(Y|&)
s~<Y 10) - 5 0 - - x 0.005x400 2
s~ <Y 10)= $49 million
The situation looks quite different once we take into account the fact that the
company already owns X and that Y would compensate for some of its risk:
S~<Y | X) = S~<X, Y | 0) - S~<X | 0}
As in the previous example, we will decompose that difference along the first two
cumulants:
S ~ ( Y | X ) = ( ( X , Y | & ) - < X | & > ) - | Y ( V ( X , Y | & > - V ( X | & > )
S~<Y|X) = ( Y | & ) - | Y ( V < X , Y | & > - V ( X | & ) )
s~(Y|X) = (Y|&)-^Y(^Y+2rX YaxaY)
s~<Y|X>=50--0.005 (400-2x0.5x20x100)
Finally:
s~ <Y | X>= $54 million
The company should thus be willing to pay more for the new business division Y
than the mean of the profits it would generate, because Y would mitigate some of
the risk associated with X and reduce the variance of the whole portfolio. It is not
the first time in this dissertation that we encounter a situation in which hedging
114
effects justify that a risk-averse decision-maker value an uncertain deal at more
than its mean: we had already observed the same phenomenon in Example 2.1. H
115
3. Reengineering Moments of Higher Order
In spite of the accuracy of the results we obtained on our first example with a two-
cumulant decomposition, it would be a mistake to systematically associate hedging with
variance reduction. [4.1] should remind us that even if we do not have any way to
enhance the e-value of the monetary prospects, nor any way to reduce the variance, there
are still many more cumulants on which we can have a valuable effect.
Here is an example which will demonstrate the potential importance of those higher order
cumulants.
Example 4.3:
The decision-maker's risk tolerance is assessed as being equal to $1,000; his
current portfolio X and the deal he is considering acquiring, Y, follow:
si
-O
0.3
S2
0.4
S3
0.3
$1,000
$400
-$200
s~ (X | 0) = $294.49
$99.52
-$217.71
$190.7
~(Y|0) =-$16.63
Figure IV.3 - Hedging as reengineering of moments of order 3 and above
It turns out that Y, if acquired and used as an adjunct to the portfolio, preserves its
mean as well as its variance. In spite of that, Y does provide hedging with respect
116
to X; a closer examination of the first five cumulants of deal X, versus those of
deals X and Y when considered together, shows why:
S~(X | 0> S~(X, Y | 0 ) ^(\, Y | 0 ) - S~(X | 0 )
$400.00
-$108.00
$0.00
$2.59
$0.00
$294.59
$400.00
-$108.00
$13.00
$2.45
-$1.02
$306.43
$0.00
$0.00
$13.00
-$0.14
-$1.02
$11.84
Table IV. 3 - Decomposition of certain equivalents along the first Jive cumulants
The next figure presents the exact same information as the table, but in a
graphical manner:
$400.00 "
$380.00 "
$360.00 "
$340.00 "
$320.00 "
$300.00 -
$280.00 H
MOU.IMI
S400.0o\
i
V ' W (10
S292.00 I
s;ov(K i
« \ >
S292.00 1
PIBP(Y | X) ($11.84)
1 Cumulant 3 Cumulants 5 Cumulants 2 Cumulants 4 Cumulants
Figure IV.4 - Accuracy of approximation depending on the number of cumulants
117
The decision-maker's PIBP for Y, S~(Y | X), turns out to be positive. It is
approximately equal to $11.84, which means that the value of hedging provided
by Y is in the region of $11.84 - (-$16.63) = $28.48.
Therefore, Y would be a valuable addition to the decision-maker's portfolio,
despite the fact that it does not affect its first two moments in any way. Here, most
of the value of hedging seems to be attributable to the fact that adding Y to X has
a beneficial impact on the third moment of the portfolio, with the skewness
shifting into the positive realm. Another, perhaps slightly more graphic way to
look at the effect of Y on the third moment of the portfolio is shown below; we
plot the probability mass function for deal X considered individually (dotted
lines) and then for deals X and Y combined into one portfolio (solid lines):
| 9 r4§-
0.4
0.35
* 0 . 3 i
0.25
'0 .2 I 0.15 I
,0 .1
O.05
L^_
i
i
-$500.00 $0.00 $500.00 $1,000.00 $1,500.00
Figure IV. 5 - Probability mass functions of X and (X uY)
The chart confirms the favorable effect that Y has on the decision-maker's
portfolio - while preserving its mean and variance, it provides a welcome boost to
its skewness. The resulting portfolio has the same center of gravity and the same
spread as the original portfolio, but we have successfully limited its downside
118
while improving its upside. Both changes will be of great interest to our risk-
averse decision-maker.
The key to analyzing this second example correctly was to include the third
cumulant in our study. If we had assessed S~(Y | X) = S~(X, Y | 0) - S~(X | 0>
based on a three-cumulant decomposition of the two certain equivalents on the
right hand side, we would have obtained S~(Y | X) = $13.00, which is not far from
the result we obtain with five cumulants ($11.84) and not far either from the
actual value of S~<Y | X) ($11.99).
Conversely, we would have made a grave mistake if we had only analyzed the
effects of Y on the mean and variance of the portfolio: we would have rejected a
proposition which in reality was quite attractive given the decision-maker's risk
attitude and current assets.
This example should remind us of the limitations of the two-cumulant
approximation of the certain equivalent; the approximation will only yield
satisfactory results if the variance of the deal is negligible compared to the square
of the decision-maker's risk tolerance. Here the square of the decision-maker's
risk tolerance is equal to 1,000,000, whereas the variances of X and (X u Y) are
equal to 216,000: the risk involved in those portfolios is so close to the decision
maker's risk tolerance that it is not safe to truncate the decomposition of the
certain equivalent after the mean and variance. •
119
Chapter 5 - The Value of Perfect Hedging
"To not plan ahead is to whimper already. "
Leonardo Da Vinci (1803-1882)
In earlier parts of this dissertation, we discussed how we can think of hedging in the
context of a specific deal Y with respect to a portfolio of unresolved deals which the
decision-maker owns, X.
We will now examine a different but related situation: suppose that the decision-maker
owns a portfolio of unresolved deals which is fairly complex. He would like to take a
proactive stance and identify deals which might help him hedge his portfolio - however,
he just does not have the time and resources to analyze and evaluate the merits of the
myriad of financial deals which are available to him on the markets. How can he
determine which uncertainties are most worthy of his attention? For which uncertainties
should he seek hedging opportunities, and how much of his resources should he devote to
that research?
120
1. Definition
The value of perfect hedging is the concept which will help us answer those questions:
Definition 5.1 — Value of Perfect Hedging
Suppose the decision-maker has wealth w and owns an unresolved portfolio of deals X =
{(Pi> Xj)ie[i> n]}, for which S is a minimal underlying uncertainty.
For any uncertainty or set of uncertainties S', we will call perfect hedge on S' given X
and w the deal Y = {(q;, yi)iG[i, m]} which has the highest PIBP of all the deals which
satisfy the following two conditions:
• Its prospects are solely determined by the outcomes of S' (in the sense that S' is a
minimal underlying uncertainty for that deal Y),
• <Y | &> = 0.
We will define the value of perfect hedging on S' given w and X as the decision-maker's
PIBP for that choice of Y. We will denote it by VoPH(S' | w, X).
More formally, VoPH(5" | w, X) is defined as follows:
VoPH(S'|w,X) = max B~<Y | w, X) [5.1] Y s.t. (Y | &> = 0
Y determined by S'
Or, more explicitly:
VoPH(S' | w, X) = max <p [5.2]
s.t. f Y is chosen so that S' is a minimal underlying uncertainty for it and (Y | &) = 0
| Z p iu(w+x i)= X Pi Z q j i i u ( w + x i+y j -^ ) L ie[l,n] ie[l,n] je[l,m]
121
It is important to understand that our perspective has undergone a radical change
compared to what it was when we were studying the value of hedging concept. In
previous chapters, the inputs to our problem were the decision-maker's current wealth,
his existing portfolio of unresolved deals and a pre-identified potential adjunct to his
portfolio; the goal of our analysis was to quantify the interaction between the existing
portfolio and that possible adjunct, and to place a monetary value on it. In contrast, the
inputs to the value of perfect hedging are current wealth, the existing portfolio of
unresolved deals, X, and a set of uncertainties S' which will serve as a support to build
new deals; the goal of our investigation is to identify the value provided by the best
possible hedging deal of mean zero which one can construct based on that support 5".
In other words, the first situation was one in which our behavior was reactive: the
decision-maker had been given a choice of a deal or several which he could add to his
portfolio, and those were the only alternatives he was considering. From now on, our
approach will be more proactive and open-ended: it is not an external stimulus which has
triggered our decision to look for valuable additions to the decision-maker's portfolio, but
our own resolve to improve it.
Example 5.1:
An oil company is just about to start operating an oil field which it recently
acquired. They believe that their profit will be entirely determined by two
uncertainties - the volume of oil that they will be able to extract, and the price at
which they will be able to sell the oil. They believe the two uncertainties to be
irrelevant given &. Finally, the board also state that they are comfortable
following the delta property for the range of prospects involved, and their risk
tolerance is assessed to be $500 million.
The following figure provides an overview of their current decision situation and
certain equivalent; all monetary prospects are expressed in millions of dollars. We
will denote by X the deal that they own at present:
122
High Price
0.4
-0
0.6
Low Price
High Volume
0.3 ) 0.7
Low Volume
High Volume
•o 0.3
0.7
Low Volume
$300 M $a
$50 M $b
$120 M $c
-$50 M $d
~<X | 0 ) = $38.99 M
Figure V.l — The value of perfect hedging: an example
We can then compute the value of perfect hedging on the oil price uncertainty and
on the volume uncertainty, in order to help the company ascertain the variable on
which it is most important that they seek hedging.
To compute the value of perfect hedging on oil price, we need to identify the best
hedge such that its prospects are only determined by oil price, which implies that
a = b and c = d on the picture shown above, and such that the mean of its
monetary prospects is equal to 0, which implies that .4 a + .6 c = 0. We thus need
to solve the following optimization problem:
max S~(Y | X)
s.t. f a = b
c = d
.4a + .6c = 0
123
The solution we obtain is:
a = b =-$70.55 M
c = d = $47.03 M
V6PH(Oil Price | X) = $3.26 M
As was to be expected, the perfect hedge is a deal which provides a benefit if the
situation which is least favorable to the oil company arises (low oil prices), but
leads to a loss if the situation which is most favorable arises (high oil prices).
We then compute the value of perfect hedging on the volume uncertainty by a
similar process; the solution to that second optimization problem is:
a = c =-$137.77 M
b = d = $59.04 M
VoPU(Volume | X) = $7.68 M
The analysis thus shows that perfect hedging on volume is more valuable to the
oil company than perfect hedging on price.
It is also possible to look for a perfect hedge which is determined by none of the
uncertainties which are directly involved in the existing portfolio, but is
probabilistically relevant to at least one of them. Let us suppose for example that
it is not possible for the company to hedge directly on oil price, but instead they
could hedge based on the price of another commodity, gold. They believe gold
price to be irrelevant to the volume of oil they will extract from their field given
&, but relevant to oil price. Given a high oil price, they assign a .8 probability to a
high gold price and .2 to a low gold price, and given a low oil price, they assign
a .9 probability to a low gold price and . 1 to a high gold price.
We then solve:
max S~(Y|X)
s.t. r Y's prospects are solely determined by the gold price uncertainty
L <Y|&) = 0
124
The best hedge entails a profit of $31.68 million if gold prices are low, and a loss
of $51.69 million if they are high; furthermore, Vo?U(Gold Price | X) = $1.61
million. Interestingly, the value of perfect hedging on gold price is lower than that
on the uncertainty to which it is relevant and which is directly involved in the
determination of the monetary prospects of X, namely the oil price. Also, in spite
of the strong relevance between the two uncertainties Oil Price and Gold Price,
the difference between the values of perfect hedging that they provide is
considerable: one is twice as much as the other. G
125
2. Basic Properties
a) Risk Attitude and Sign of the Value of Perfect Hedging
We will soon discuss the behavior of VoPH(5" | w, X) as a function of various
characteristics of S'. But before that, it is worth spending a few minutes on some of the
most basic properties of the value of perfect hedging - for example, the relation between
its sign and the decision-maker's risk attitude. The first clause of Property 5.1, which
states that the value of perfect hedging is always equal to zero for a risk-neutral decision
maker and can therefore never be a useful consideration to him, echoes a similar
observation we made earlier about the value of hedging.
Property 5.1 - Sign of the Value of Perfect Hedging and Risk Attitude
For a risk-neutral decision-maker, VoPH(5" | w, X) is always equal to zero.
For a risk-averse decision-maker, VoPH(5" | w, X) is always greater than or equal to zero.
Proof:
For a risk-neutral decision-maker, B~(Y | w, X) = (Y | &) for any deal Y. Since the
value of perfect hedging is computed as the maximum ofB~<Y|w,X> over the set of
deals Y which have a mean of zero, VoPH(5" | w, X) = 0.
For a risk-averse decision-maker, all we need to do is notice that the deal Y° whose
prospects are all determined by S' and are all equal to 0 belongs to the set of deals
over which we maximize B~(Y | w, X) in [5.1]. Consequently, VoPH(S" | w, X) > B~(Y° | w, X) = 0. 3
126
b) PIBPfor an Uncertain Deal and Value of Perfect Hedging
Now that we know of situations in which the value of perfect hedging is not an
illuminating concept, let us turn to those in which it is and ask ourselves what makes it
helpful. Why should we ever be interested in computing VoPH(5" | w, X)? What can it
add to our understanding of a decision situation? The following theorem presents one of
the most persuasive reasons:
Theorem 5.1 — PIBPfor an Uncertain Deal and Value of Perfect Hedging
The value of perfect hedging allows us to calculate an upper bound on our PIBP for any
deal Y whose outcomes are solely determined by some set of uncertainties iS":
B~<Y | w, X) < (Y | &) + VoPH(S' | w, X) [5.3]
Proof:
Let us construct a deal Y' which is the exact copy of Y up to a constant: more
specifically, we will define Y' so that y'j = yj - (Y | &) for all values of j .
Let us recall that for any deal Z, any wealth w and any amount 8, the PIBP for a deal
Z' in which all prospects of Z were changed by the same amount 8 is equal to the sum
of the PIBP for Z, augmented by 8 (Lemma 2.1); in particular, that result implies that:
B~<Y | w, X) = (Y | &} + B~(Y' | w, X)
But by virtue of Definition 5.1, we also have:
B~(Y' | w, X) < VoPH(5' | w, X)
Therefore:
B~(Y | w, X> < <Y | &) + VoPH(S' | w, X). •
[5.3] is a remarkable relationship because it provides us with an upper bound on B~(Y | w,
X), a quantity which, in general, is not easy to compute, especially if the probabilistic
127
structure of the current portfolio of deals X is elaborate. The upper bound consists of the
sum of (Y | &>, which is easy to evaluate and does not depend at all on the decision
maker's existing portfolio X or even on his exact risk preference, and VoPH(5" | w, X),
which is a more obscure notion at this point of our story but for which we will soon be
able to derive some upper bounds and approximations.
Example 5.2:
Let us go back to Example 5.1, and consider deal Y which is defined as follows
and whose prospects are determined by the Gold Price uncertainty:
High Gold Price
0.38
0.62
Low Gold Price
-$50 M
$100 M
B~(Y | 0) = $37.59 M
Figure V.2- Upper bound on the PIBPfor Y based on the value of perfect
hedging
We directly apply [5.3] in order to compute an upper bound on the decision
maker's PIBP for Y:
B~<Y | w, X) < <Y | &) + VoPU(Gold Price | w, X)
< $43.00 M +$1.61 M
< $44.61 M
128
Based on that simple analysis, the oil company already knows that it should not
buy deal Y if its price exceeds $44.61 million.
It should also be noted that in this example, the actual value of the oil company's
PfflP for Y is:
B~(Y | w, X) = $43.56 M
The difference between upper bound and actual value is small, but we should be
aware that the tightness of the bound can vary considerably from example to
example. If we swap the two monetary prospects of the deal Y we have just
studied, and call the resulting deal Y', then the upper bound gives: B~(Y' | w, X) < <Y' | &) + VoPH(GoldPrice | w, X)
< $7.00 M +$1.61 M
<$8.61M
The actual value of the PIBP is:
B~(Y'|w,X> =-$3.65M
The bound gives disappointing results in the case of Y'; the fact that unlike Y, Y'
provides negative hedging with respect to X explains it. •
129
c) Joint Value of Perfect Hedging
It is a tempting mistake for many new students of decision analysis to believe that there is
a general rule connecting the joint value of clairvoyance on two uncertainties S' and S" to
the sum of the individual values of clairvoyance on S' and S". In reality, it turns out that
in some situations the joint value of clairvoyance is greater than the sum of the individual
values of clairvoyance, and in others it is lower; all that can be said with certainty is that:
VoC(5", S") > max (VoC(S'), VoC(5"))
In other words, having perfect information on both 5" and S" is at least as valuable as
having perfect information on S' alone, and at least as valuable as having perfect
information on S" alone. We will now see that the same can be said of the value of
perfect hedging:
Theorem 5.2 -Joint Value of Perfect Hedging
For two sets of uncertainties S' and S", the joint value of perfect hedging on S' and S" is
greater than or equal to the higher of the two individual values of perfect hedging on S'
and S":
VoPH(S', S" | w, X) > max (VoPH(S' | w, X), VoPH(S" | w, X)) [5.4]
Proof:
We will first prove that VoPH(S', S" | w, X) > VoPH(S' | w, X). For that, let us
denote by Y the deal which helps achieve perfect hedging on 5":
Y* = argmax B~(Y | w, X>
Y s.t. (Y | &> = 0
Y determined by S'
It implies that:
B~(Y* | w, X) = VoPH(S' | w, X)
130
We can remark that a fortiori, Y* also belongs to the set of deals Y whose monetary
prospects are solely determined by S' and 5"', and whose mean is equal to 0. That is
precisely the set of deals over which we maximize ~(Y | w, X) when we wish to
compute VoPH(S', S" | w, X):
VoPH(S ' ,S" |w,X)= max B~(Y | w, X)
Y s.t. (Y | &> = 0
Y determined by S',S"
Therefore:
VoPH(S', S " | w, X) > B~<Y* | w, X)
>VoPH(S' |w,X)
By the same reasoning, we can prove that VoPH(5", S" | w, X) > VoPH(5"' | w, X).
We thus conclude that VoPH(S', S" | w, X) > max (VoPH(S' | w, X), VoPH(S" | w,
X)).I!
Example 5.3:
We will again refer to Example 5.1; in that situation, the set S of the uncertainties
which completely determine the monetary prospects of X comprises two
elements: Oil Price and Volume.
We will compute the value of perfect hedging on S, and then compare it to the
individual values of perfect hedging on Oil Price and Volume, which we already
computed. Using the same notation {a, b, c, d} as in Example 5.1 to designate the
monetary prospects of the perfect hedge Y we wish to construct, we start by
solving the optimization problem shown below:
131
max S~<Y | X>
s.t. 0 = <Y|&> = . 1 2 a + . 2 8 b + .18c + .42d
The solution we obtain is:
C a =-$249.40 M
b = $0.60 M
{ c = -$69.40 M
d = $100.60 M
, VoPH(5|X) = $11.61M
For clarity, X and its perfect hedge Y are represented below:
High Volume
High Price
0.4
0.3 ) 0.7
-o Low Volume
High Volume
0.6 •0
Low Price
0.3 l
0.7
Low Volume
$300 M + -$249.4 M
$50 M + $0.60 M
$120 M + -$69.40 M
-$100 M + $100.60 M
$50.60 M
$50.60 M
$50.60 M
$50.60 M
s~ (X | 0 ) = $38.99 M
Figure V.3 — Joint perfect hedging on Oil Price and Volume
132
We then compare the joint value of perfect hedging to the values of perfect
hedging on Oil Price and Volume:
f VoPH(S|X) = $11.61M
VoPH(CW Price | X) = $3.26 M
Vom(Volume | X) = $7.68 M
It proves that VoPH(S', S" | X) > max (VoPH(S' | X), VoPH(S" | X)), as
announced by Theorem 5.2. Incidentally, let us also make a note of the fact that
the value of perfect hedging on S in this example is less than the sum of the values
of perfect hedging on the individual uncertainties which compose S:
VoPH(0/7 Price | X) + Vo?H(Volume | X) = $3.26 M + $7.68 M
= $10.94 M
< VoPH(S | X)
It illustrates the interesting parallel between the value of perfect hedging and the
value of perfect information which we mentioned previously. The value of
clairvoyance on two uncertainties is not necessarily equal to the sums of the value
of clairvoyance on the individual uncertainties, even when the uncertainties are
irrelevant. Now we know that the same can be said of the value of perfect hedging.
It is also thought-provoking that the perfect hedge which we constructed on Oil
Price and Volume simultaneously would transform the oil company's uncertain
portfolio into a deterministic one, whose monetary prospects are all equal to
$50.60 million (as shown by the previous figure). A point of further interest is that
$50.60 million happens to be precisely equal to the mean of the decision-maker's
original portfolio. We will soon determine that none of this happened by
coincidence. 3
133
3. Impact of Relevance on the Value of Perfect Hedging
In the course of our investigation of hedging, we have already gathered a few clues here
and there as to the nature of its connection with probabilistic relevance. As we studied the
special properties of the value of hedging in the delta case, for example, we remarked that
it is equal to zero for two deals which are irrelevant given &.
We will examine the connection between relevance and hedging more systematically and
more in depth over the next few pages; we will see that the value of perfect hedging on
any uncertainty S' is determined to a large extent by the relevance relationship between
S' and the set S of the uncertainties which completely determine the monetary prospects
of the existing portfolio. We will examine the following three cases separately:
• Complete relevance: S and S' are identical, or S and S' are deterministic given
each other. In other words, the probability distributions {S | S' = s ) , &} as well as
{S' | S - Sj, &} only consist of ones and zeros.
• Complete irrelevance: S A. 5" | &.
• Incomplete relevance: S and S' are relevant, but not to the extent that S is
deterministic given S'.
134
a) Complete Relevance
Theorem 5.3 - Value of Perfect Hedging when S' = S
VoPH(,S | w, X) = RPS(X | w). [5.5]
Furthermore, equality of B~(Y | w, X) and VoPH^ | w, X) is achieved if and only if Y is
the deal which transforms every single prospect of X into (X | &).
The result also holds for an uncertainty S' such that S and S' are deterministic given each
other: then again, VoPH(5" | w, X) = RPS(X | w).
Proof:
Proof that VoPH(£ | w, X) > RPS(X | w):
For each prospect Xj of X, define y; = (X | &> - x;. Then (Y | &> = 0 and by
definition of the value of perfect hedging, we have VoPH(5' | w, X) > B~(Y
| w, X). Let us compute B~(Y | w, X) for that particular choice of Y. By
definition of the PIBP, we know that:
£ P iu(w + X i)= X P. Z qj|iU(w + x i +y j -B -<Y|w,X»
ie[l,n] ie[l,n] je[l,m]
X P iu(w + x,)= X Pi Z qjiUCw + x ^ X I & ^ x J - ^ Y I w . X ) ) ie[l,n] ie[l,n] je[l,m]
X PiUCw + x ^ - J ] P iu(w + <X|&)-B~<Y|w,X» ie[l,n] ie[l,n]
Y, p iu(w + x i) = u(w + (X|&)-B~(Y|w,X» i£[l,n]
Let us now remember what the definition of the certain equivalent S~(X |
w) for X is:
X piu(w + xi) = u(w+s~(X|&» ie[l,n]
Combining the last two equations, we obtain:
135
b~<X I w) = (X ] &) - B~(Y | w, X) B~(Y | w, X> = <X | &> - S~(X | w)
B~(Y | w, X) = RPS(X | w)
This proves that VoPH(5 | w, X) > RPS(X | w).
• Proof that VoPH(S | w, X) < RPS(X | w):
Let us now prove that VoPH(S | w, X) < RPS(X | w). For that let us
consider any hedging deal Y whose outcomes are solely determined by S
and for which (Y | &> = 0. We will show that we necessarily have B~(Y | w,
X) < RPS(X | w).
Consider the situation in which the decision-maker has acquired deal Y for
exactly B~(Y | w, X). He then owns w - B~(Y | w, X), as well as X and Y
which are still unresolved. The mean of those dollar prospects is equal to
w + (X | &) - B~(Y | w, X), since Y itself has a mean of zero.
Since the decision-maker is risk-averse, we have:
£ Pi u(w + x, + y, - B~(Y | w, X)) < u(w + (X | &) - B~(Y | w, X» ie[l,n]
By definition of PIBP B~(Y | w, X), we also have:
£ PiU(w + Xi)= £ PiUCw + Xi+yi -^YIw.X)) ie[1,n] ie[l,n]
Therefore:
£ p iu(w + x i)<u(w + <X|&)- B~<Y|w,X» ie[l,n]
The left-hand side of that inequality can be rewritten as u(w + S~(X | w))
by definition of the PISP for deal X; doing so, we obtain:
u(w + S~<X | w» < u(w + (X | &) - B~(Y | w, X»
u(w + (X | &) - RPS(X | w)) < u(w + (X | &> - B~(Y | w, X»
136
•
That in turn shows that B~(Y | w, X) < RPs(X | w), and since it is true for
any deal Y whose prospects are solely determined by S and for which (Y |
&> = 0, VoPH(5 | w, X) < RPS(X | w).
It should also be noted that for a strictly concave u-curve, and for a
hedging deal Y which is not defined by y; = (X | &) - xi, the concavity
inequality we based ourselves upon to prove that VoPH(5' | w, X) < RPs(X
| w) becomes strict:
£ Pi u(w + x; + y, - B~(Y | w, X» < u(w + (X | &> - B~(Y | w, X» ie[l,n]
VoPH(,S | w, X) < RPs(X | w) would then also have become a strict
inequality.
Proof that VoPH(5" | w, X) = RPS(X | w) if S' is an uncertainty such that S is
deterministic given S', and vice versa:
In Theorem 5.5, we will show that for a general uncertainty S', VoPH(5" |
w, X) < RPS(X | w).
As for proving the reverse inequality, i.e. VoPH(5" | w, X) > RPs(X | w) if
5" is an uncertainty such that S is deterministic given 5" and vice versa, we
can simply notice that the deal Y which we defined by taking y; = (X | &)
- Xj in the first part of the proof also belongs to the set of uncertain deals
which have prospects which are solely determined by S' and which have a
mean of 0.
We already computed the PIBP for that deal Y as being equal to the
selling risk premium of X. Therefore, VoPH(S' | w, X) > B~(Y | w, X) =
RPS(X | w). J
Theorem 5.3 shows exactly what becomes of deal X once it is perfectly hedged based on
distinction S: to perfectly hedge X, we use a new deal Y which is defined as the exact
opposite of X, plus a constant equal to (X | &>. Adding Y to the current portfolio thus
137
transforms every single one of its monetary prospects into (X | &), thus removing any
uncertainty from it.
As a result, the portfolio will gain RPs(X | w) in value because its risk, which was
initially captured by RPs(X | w), is now entirely compensated for by the addition of deal
Y. In light of that explanation, not only does Theorem 5.3 appear as remarkably simple in
its conclusions, but it also seems quite intuitive.
Example 5.4:
Here we will take another look at perfect hedging on the set S = {Oil Price,
Volume}. We have already naively computed the value of perfect hedging on S in
the previous example, by solving an optimization problem as we have always
done it so far; we will now compare that answer to the predictions of our theorem
on perfect hedging when S = S'.
The first clause of Theorem 5.3 announces that the perfect hedge should
transform the oil company's uncertain portfolio into a deterministic one, by
making all of its monetary prospects equal to $50.60 million, which is the mean of
the oil company's original portfolio. If we examine Figure V.4, we will see that it
is indeed the result we had reached when we naively computed the value of
perfect hedging on S.
Next, let us verify the second clause of the theorem; it states that the value of
perfect hedging on S should be equal to the risk premium of the original portfolio:
RPs(X | 0) = (X | &) - S~(X | 0)
= $50.60 M-$38.99 M
= $11.61M
The risk premium matches the value of perfect hedging on S we had computed in
the previous example: RPs(X | 0) = VoPH^ | X) is verified. H
138
b) Irrelevance
We continue our investigation of the connection between value of perfect hedging and
relevance with the second case in our nomenclature - the case in which S and S' are
irrelevant given &:
Theorem 5.4- Value of Perfect Hedging on S' when S J. S' | &
For any set of state variables 5" which is irrelevant to S given &:
VoPH(S' | w, X) = 0. [5.6]
Furthermore, equality of B~(Y | w, X) and VoPH(5" | w, X) is achieved if and only if Y is
the deal which has all of its prospects set to be equal to 0.
Proof:
Proof that VoPH(S' | w, X) > 0:
We already proved in Property 5.1 that the value of perfect hedging is
non-negative for a risk-averse decision-maker.
Proof that VoPH(S' | w, X) < 0:
Let us now prove that VoPH^ | w, X) < 0. For that let us consider any
hedging deal Y whose outcomes are determined by S'. We will show that
we necessarily have B~(Y | w, X) < 0.
By definition of B~(Y | w, X):
£ P iu(w + X i)= X Pi Z qj i i u(w + x i + yj-B~<Y lw> x» ie[l,n] ie[l,n] je[l,m]
Exploiting the irrelevance between S and S':
^ piu(w + x i)= Z Pi Z qju(w + x i+y j-B^<Y|w,X»
ie[l,n] ie[l,n] je[l,m]
139
We can now use the fact that the decision-maker is risk-averse to
transform the summation over index j on the right hand side:
X P iu(w + Xl)< X p1u(w + x1+(Y|si,&>-B~<Y|w,X» ie[l,n] is[l,n]
Due to the irrelevance of S and 5", (Y | Sj, &) = (Y | &) = 0 for all values of
i:
£ PiUCw + x,)^ X P iu(w + X i-B~<Y|w,X»
ie[l,n] ie[l,n]
We can finally conclude that B~(Y | w, X) < 0. Also, using the same series
of arguments as in the end of the proof of Theorem 5.3, we can notice that
the inequality VoPH^ | w, X) < 0 would have been strict for any choice of
Y other than the deal whose prospects yj were equal to zero for all values
ofj.D
This theorem shows that it is pointless to hope to hedge a portfolio X based on a set of
uncertainties S' which is probabilistically irrelevant to it: the acquisition of any deal Y
with a mean of zero and at least one non-zero payoff would then have no chance to
compensate for any of X's risk, and could only add to the overall level of risk of the
portfolio.
The theorem should also remind us of an analogous result, which we encountered when
we studied the properties of the value of hedging: we had seen that VoH(X | Y) = 0 if the
decision-maker's u-curve satisfies the delta property and if X and Y are irrelevant given
&. There is an important difference between the two theorems, though: the claims of the
first result were restricted to the delta case, whereas those of the one we just derived are
not.
140
Example 5.5:
We return to our oil field hedging example, but this time, the oil company wishes
to hedge based on Copper Price. They believe Copper Price to be irrelevant to
both Oil Price and Volume given &. More specifically, they assign a probability
of 0.6 to a high copper price and a probability of 0.4 to a low copper price,
irrespective of the outcomes of Oil Price and Volume.
In order to compute the value of perfect hedging on Copper Price, we need to
solve the following maximization problem:
max S~<Y | X>
s.t. r Y's prospects are solely determined by the copper price uncertainty
1 (Y|&> = 0
The solution we obtain is the one we expected based on Theorem 5.4: it is
impossible to create a hedge Y with a mean of zero but a positive certain
equivalent. The best we can do is to set all of the monetary prospects of Y to zero,
which yields a certain equivalent of zero:
Vo?U(Copper Price | X) = 0. J
141
c) Incomplete Relevance
When the decision-maker believes S and 5" to be relevant given &, but not completely,
the value of perfect hedging can be proved to be less than or equal to what it would have
been in the case of complete relevance, but higher than or equal to what it would have
been in the case of irrelevance:
Theorem 5.5 - Value of Perfect Hedging on S' in the General Case
For any set of state variables S\ 0 < VoPH^' | w, X) < RPS(X | w). [5.7]
Proof:
• Proof that VoPH(S' | w, X) > 0:
We can again refer to Property 5.1 to find the proof of that result.
• Proof that VoPH(S' | w, X) < RPS(X | w):
We will slightly adapt to the case of a general S' the second part of the
proof we wrote for Theorem 5.3.
Let us prove that VoPH(5" | w, X) < RPS(X | w). For that let us consider
any hedging deal Y whose outcomes are solely determined by S' and for
which (Y | &) = 0. We will show that we necessarily have B~(Y | w, X) <
PvPs(X | w).
Since the decision-maker is risk-averse, we have:
Z P. £ qjpu(w + x i + y j -B ~<Y|w,X»
ie[l,n] je[l,m]
< X piu(w + x i+(Y|s i,&) -B~<Y|w,X» ie[l.n]
Applying one more concavity inequality to the right hand side, we obtain:
142
I Pi ^qi]iu(VJ + xi+yJ-B~(Y\w,X))
ie[l,n] M ' . m ]
<u(w + <X|&) + « Y | S i , & ) | & ) - B ~ ( Y | w , X »
Since «Y | si? &) | &) = <Y | &) = 0:
Z Pi Z qj|iu(w + x 1 +y J - B ~<Y|w,X»<u(w + <X|&> -B~<Y|w,X» ie[l,n] je[l,m]
Let us now recall the definition of ~(Y | w, X):
Z p iu(w + x i )= Z Pi Z q j i ^ w + X i + y j - ^ Y I w . X ) ) ie[l,n] ie[l,n] je[l,m]
If we combine this equation with the previous inequality, we find that:
Z p,u(w + x i)<u(w + <X|&> - B ~(Y |w,X» ie[l,n]
But the left-hand side of that inequality is also equal to u(w + (X | w)) by
definition of the PISP for deal X. Consequently, we have:
u(w + S~<X | w» < u(w + <X | &> - B~<Y | w, X))
u(w + (X | &) - RPS(X | w)) < u(w + (X | &) - B~(Y | w, X»
This in turn shows that B~(Y | w, X) < RPs(X | &). Since it is true for any
deal Y whose outcomes are solely determined by S' and for which (Y | &)
= 0, VoPH(S' | w, X) < RPS(X | w).
Finally, we should observe that our concavity inequalities and our final
inequality VoPH(5" | w, X) < RPS(X | w) all become strict when S and S'
are not completely relevant. 3
143
Example 5.6:
In Example 5.1, we computed the value of perfect hedging on Gold Price, an
uncertainty which was believed by the firm to be relevant to the set S = {Oil Price,
Volume} given &, and yet not completely relevant to it.
We had then computed that:
V6PH(Gold Price | X) = $1.61 M
More recently, we established that:
VoPH(S|X) = R P s ( X | 0 )
= $11.61M
We can thus observe that the value of perfect hedging on gold price is indeed
comprised between zero and the risk premium of the original portfolio, as
predicted by Theorem 5.5. However, that value of perfect hedging is much lower
than the total risk premium of X. It is not surprising once we consider the
probabilistic structure of the problem, which is captured by the influence diagram
below:
Figure V.4- Gold Price is only relevant to the value through Oil Price
The probabilistic relevance between the Gold Price uncertainty and the original
portfolio X is entirely channeled through the set S = {Oil Price, Volume}, more
precisely through Oil Price, which is also the one of the two uncertainties in S on
which the value of perfect hedging is the lowest ($3.26 million). Therefore, it
144
seems intuitive that there should be at most as much to be gained from perfect
hedging on Gold Price as there is to be gained from perfect hedging on Oil Price.
The next theorem expresses the same result in greater generality. D
145
d) Relevance-Based Dominance of one Hedge over Another
Let us suppose that our decision-maker needs to choose to hedge based on only one of
two uncertainties, S' and S". Both S' and 5"' are incompletely relevant to S given &. Is
there any way to determine that the value of perfect hedging on one uncertainty will be
greater than the value of perfect hedging on the other, without explicitly computing those
two quantities? Or, to put it differently, can we exploit the relevance structure of a
decision situation in such a way as to infer that some value of perfect hedging will
necessarily dominate another?
The following theorem shows that it is possible indeed:
Theorem 5.6-
1fS"±S\S',
of perfect hed
- Value of Perfect Hedging on S" z/S"
&, then the value of perfect hedging
ging provided by S':
J.S\S\ &:
provided by
VoPH(S" | w, X) < VoPH(S' | w, X)
S" is
[5.8]
less than the value
Proof:
We know from Theorem 5.2 that for two sets of uncertainties S' and S", the joint
value of perfect hedging on S' and 5"' is greater than or equal to the higher of the two
individual values of perfect hedging on S' and S":
VoPH(S', S" | w, X) > max (VoPH(S' | w, X), VoPH(5" | w, X))
Here we will aim at proving that S" 1 S | S', & implies that VoPH(5", S" \w,X)<
VoPH(5" | w, X), because that result, combined with the inequality on the joint value
of perfect hedging stated above, would allow us to conclude that VoPH(5"' | w, X) <
VoPH(5'|w,X).
We will thus consider a deal Y whose prospects are solely determined by the
uncertainties S' and S", and such that (Y | &) = 0. If for any such deal Y, B~(Y | w, X)
146
< VoPH(S' | w, X), then we will have proved that VoPH(S', S" | w, X) < VoPH(S' | w,
X), as desired.
By definition of B~(Y | w, X), it satisfies the following equation involving the
decision-maker's u-curve:
£ P iu(w+X i)= X Pi Z % Z rk|1 ,Ju(w + x i+yJ ,k-B~(Y lw,x», ie[l,n] ie[l,n] je[l,m] ke[l,l]
where qj|j denotes the conditional probability distribution over S' given S and &, and
rk|ij the conditional probability distribution over S" given S, S' and &. Since S" J_ S \
S\ &, rk|ij = rky, the equation above then simplifies into:
£ P iu(w + x,)= X Pi Z % Z rk|ju(w + x,+yJ,k-B~<Y|w,X»
ie[l,n] ie[l,n] je[l,m] ke[l,l]
Next, we decompose each one of the prospects y^ of deal Y as follows:
yj* = «j + pj,k,
where each a was defined as the conditional mean of Y given S' = s'j and &:
Oj = <Y | S" = s'j, &>
We substitute aj + Pjk for y^ in the definition of B~(Y | w, X) and obtain:
£ piu(w + x i)= Z Pi Z Ijli Z rkij^w + Xi+Oj+p^-^YIw.X) ) ie[l,n] ie[l,n] je[l,m] ks[l,l]
Since the decision-maker is risk-averse:
£ P iu(w + Xi)< S P, Z qj|lu(w + x i + a J + < p | y = s'j,&)-B~<Y|w,X» iefl.n] ie[l,n] je[l,m]
Because of the way we defined the a's, the conditional mean of p given S' = s'j and &
is equal to 0:
£ PiUCw + x,)^ X P. Z qjpuCw + Xi+oij-^YIw.X))
ie[l,n] ie[l,n] je[l,m]
The inequality above implies that:
B~<Y | w, X) < B~<a | w, X)
147
We can then remark that a is a deal whose mean is equal to 0, and whose prospects
are solely determined by 5"; therefore, the decision-maker's PIBP for that deal is less
than his value of perfect hedging on 5":
B~(Y | w, X) < B~(a | w, X) < VoPH(S' | w, X)
This proves that B~(Y | w, X> < VoPH(S' | w, X), as desired. 3
Theorem 5.6 is yet another result which highlights the strong resemblance between value
of perfect hedging and value of clairvoyance: for the value of clairvoyance as well, if a
decision-maker has a choice between free and perfect information on 5" and 5"' when S"
± S | S\ &, his preference should go to the perfect information on S'.
In more formal and more mathematical terms, we could say that the partial ordering of
uncertainties based on the conditional irrelevance of one of them with respect to S given
the other and & induces an equivalent partial ordering of the same uncertainties based on
the value of perfect hedging that they provide.
148
Example 5.7:
Again we will refer to Example 5.1, and more specifically to the Gold Price
uncertainty: by its very definition, it is irrelevant to S= {Oil Price, Volume} given
Oil Price and &. This is also illustrated by Figure V.4.
By Theorem 5.6, the value of perfect hedging on Gold Price should thus be lower
than or equal to the value of perfect hedging on Oil Price. And indeed, we have:
Vo?U(Gold Price | X) = $1.61 M
VoPH(Oz7Price | X) = $3.26 M. •
149
e) Summary - Impact of Relevance on the Value of Perfect Hedging
The following figure offers a qualitative graphical summary of the results we have just
derived on the impact of the relevance relationship between S and S' on the value of
perfect hedging:
RP, g(X | w)'
= PM o
>
Best hedging deal: y, = <X|&)-X i
S"±S\S',&
SIS' Dependence between S and S' S=S'
Figure V.5- Impact of the relevance between S and S' on VoPH^ \ X, w)
The diffuse area between the two extreme cases S 1 S' and S = S' should remind us that
in order to determine the exact value of perfect hedging when S and S' are relevant given
& but not identical, we first need to assess all the probability distributions and monetary
prospects involved; all we can be assured of prior to computing an exact value is that the
value of perfect hedging is comprised between zero and the selling risk premium of X.
150
f) Influence of the U-Curve on the Choice of a Perfect Hedge
There is another remarkable observation that we can make based on Theorems 5.3, 5.4
and 5.5:
Property 5.2 - Perfect Hedging Strategies and U-Curves
The deals Y which allow us to achieve perfect hedging in the case of complete relevance
between S and 5" (Theorem 5.3) or complete irrelevance (Theorem 5.4) do not depend on
the decision-maker's u-curve.
In other words, only the value of perfect hedging depends on the decision-maker's u-
curve in those cases; his choice of a preferred hedging strategy does not. Very rarely do
we come across recommendations in a decision analysis which are so universal that they
would be equally valid for any decision-maker, regardless of his u-curve, so the event is
certainly worth remembering.
On the other hand, if there is incomplete relevance between S and S', then the selection of
a perfect hedge might be affected by the decision-maker's u-curve. Here is an example
which will demonstrate it.
Example 5.8:
We will study the behavior of the value of perfect hedging on Gold Price as a
function of the oil company's risk tolerance.
For the risk tolerance figure of $500 million which we have been using so far, we
already know what the perfect hedge and the value of perfect hedging are:
151
YHigh Gold Price ~ $ 3 1 . 6 8 M
J YLow Gold Price = "$51 69 M
Vo?H(GoldPrice | X) = $1.61 M \-
We then repeat the analysis for two other values of the risk tolerance, one being
higher than the value we have used so far (p = $1 billion), and the other being
lower (p = $200 million). The results are shown in the next table, with the first
two rows indicating what the prospects of the best hedge are for each possible
value of the risk tolerance, and the last row indicating what the corresponding
value of perfect hedging is:
p = $200 M p = $500 M p = $1 B
$29.33 M
-$47.86 M
$3.38 M
$31.68 M
-$51.69 M
$1.61 M
$32.61 M
-$53.21 M
$0.86 M
Table V.l -Impact of the risk tolerance on the selection of a perfect hedge
We can thus remark that the decision-maker's risk attitude has an influence on his
choice of a perfect hedge on Gold Price. This is no surprise since
Also, in this example at least, it seems that the value of perfect hedging increases
with risk-aversion over the range of risk tolerance values we have considered. In
addition, we see here that a more risk-averse decision-maker will tend to select a
perfect hedge with a narrower spread between its monetary prospects. •
152
4. General Upper Bound on the PIBP of an Uncertain Deal
We mentioned earlier, in Theorem 5.1, that the decision-maker's PIBP for any uncertain
deal whose monetary prospects are solely determined by a set of uncertainties S' will be
less than or equal to the sum of the mean of the deal and the value of perfect hedging on
5". Since we also proved that the value of perfect hedging on 5" is itself less than or equal
to the risk premium of the existing portfolio, we can conclude that:
Theorem 5.7 - Upper Bound on the PIBP for an Uncertain Deal
(Y | &) + RPs(X | w) gives us an easy to compute upper bound on the decision-maker's
PIBP for any deal Y.
Proof:
This is a direct consequence of combining Theorems 5.1 and 5.5. •
The result, which is similar to Theorem 3.4 except for the fact that it is valid beyond the
delta case, further demonstrates the importance to a decision-maker of knowing the value
of the risk premium of his existing portfolio.
As mentioned in our discussion of Theorem 3.4, what is particularly appealing about this
upper bound is that it relies on two quantities of which one, RPs(X | w), can be computed
ahead of time by the decision-maker, and the second, (Y | &), can be assessed without so
much as assessing the relevance relationship between the potential acquisition Y and the
existing portfolio. Indeed, in order to compute (Y | &), assessing a marginal probability
distribution over Y is enough; there is absolutely no need to elicit a conditional
probability distribution over Y given X from the decision-maker, which could be a
daunting task if the existing portfolio was complex.
153
Example 5.9:
Let us suppose that one of the oilfields bordering the one that the oil company
presently owns is available for sale, at a price of $25 million.
The management of the company believes that since the two oilfields are adjacent,
there is some relevance between the volume of oil that they would be able to
extract from one and what they would be able to extract from the other. More
specifically, they believe that the two fields would connect to the same reservoir;
if the first location allows them to pump out a high volume of oil, it probably
means that the additional amount of oil that can be extracted from the second
location will be low, and vice versa.
However, at present the management are only comfortable assessing a marginal
probability distribution over the volume of oil for the second field, and they are
not ready yet to assess a conditional probability distribution over it given the
volume they will extract from the field they already own: they believe that they
would need to consult experts in the geology of the area before they can think
about such a distribution with clarity.
The next tree shows the distribution over profits for the second oilfield; naturally,
the company's beliefs as to oil prices are the same as in Example 5.1:
154
High Price
0.4
-6
0.6
Low Price
High Volume
-6 0.2 ) 0.8 Low Volume
High Volume
$140 M
$30 M
$70 M
- -$40M
<Y|&> = $10M
Figure V.6- Distribution over profit for the second oilfield.
-0 0.2
i
0.8
Low Volume
Even though the company has not yet fully assessed the relevance that they
believe exists between this new opportunity Y and their existing portfolio X, we
can compute an upper bound on their PIBP for Y as:
B~(Y | X) < <Y | &> + RPS(X | 0)
< $10.00 M +$11.61 M
< $21.61 M
In this example, we can thus spare the company the cost and the trouble of having
to make any conditional probability assessments: the marginal distribution over
the profits generated by Y was enough to establish that to them, $25 million is too
high a price for that second oilfield. •
155
5. Approximation of the Value of Perfect Hedging in the Delta Case
The upper bound on the value of perfect hedging which we just discussed has one flaw -
nothing guarantees that it will be tight. In fact, our earlier inspection of the effects of
relevance on hedging has taught us that the value of perfect hedging on 5" can only be
equal to the risk premium of the existing portfolio if S' is completely relevant to S given
&. If instead the relevance between S and S' is weak, we might suspect that the risk
premium of the existing portfolio might make for a poor approximation of the actual
value of perfect hedging.
And yet, we should also bear in mind that in order to compute the exact value of perfect
hedging on S', in the general case in which S' is relevant to S given & but not completely,
we need to solve the optimization problem
VoPH(S' | w, X) = max B~<Y | w, X),
Y s.t. <Y | &> = 0
Y determined by S'
which can be a challenging and computationally intensive task if the probabilistic
structure of the decision situation is complex.
Hence, it would be of considerable practical importance to have access to an
approximation of the value of perfect hedging whose accuracy, unlike that of the risk
premium upper bound we have already derived, would be assured at least as long as some
reasonable conditions are met. Fortunately, such an approximation exists if the decision
maker's u-curve satisfies the delta property:
156
Theorem 5.8 - Approximation for the Value of Perfect Hedging
For a risk-averse decision-maker who follows the delta property, B~(Y* | X) where Y is
defined such that y j = (X | &) - (X | S' = s'j, &) is an approximation of VoPH(5" ] X).
The approximation is an underestimate of the actual value of perfect hedging. Also, the
higher the square of the risk tolerance (p2) is compared to the variance of the deals
involved, the better the approximation will be.
Finally, it should be noted that the approximation happens to give an exact answer when
S and iS" are completely relevant or when S and S' are irrelevant given &; unlike the first
part of this theorem, that last result holds no matter what u-curve the decision-maker has.
Proof:
• Proof that y*j = (X | &> - (X | s'j, &) yields an approximation of VoPH(5" | X) in
the general case:
[4.2] reminded us that for small enough values of the risk-aversion
coefficient y, we should devote our best efforts to minimizing the variance
of the combined portfolio made of X and Y together before we focus on
optimizing any of the higher order moments of that portfolio. More
precisely, [4.2] stated that for a decision-maker who follows the delta
property, his certain equivalent for a deal Z can be approximated by:
s ~ ( Z | 0 ) » ( Z | & ) - i y v ( Z | & ) ,
as long as deal Z has a small enough variance compared to the square of
the decision-maker's risk tolerance:
V ( Z | & ) « ^ Y
Consequently, let us consider the problem of the minimization of the
variance of X and Y, where X is a deal whose prospects are determined by
157
the outcomes of S and Y another deal whose prospects are determined on
the outcomes of S':
( V
Minv~(X,Y|&)=X Z p ^ x . + y j ) 2 - Z E p ^ x . + y j ) ie[l,n] je[l,m] V'6!1 '"] M1.1"]
s.t. <Y|&> = 0,
where we denoted by pij the joint probability {S = Sj, S' = s'j | &}. We can
first notice that the problem is equivalent to the following:
Min £ £ p u (x; +yj)2
y j = o
ie[l,n] je[l,m]
/ \ ^ Z SPM
je[l,m] Vie[l,n] J
In order to find the minimum, we can form the Lagrangian and set its
derivatives to 0 - for all values of k in [1, n], we have:
d
dyk Z IM*.^)2-*! IP,J
ie[l,n] je[l,m] je[l,m] V,ie[l,n] J
= 0
This leads to:
2 Z P u ^ + yJ-X Z Pu=0, ie[l,n] is[l,n]
and to the value of the monetary prospect yk:
Z Pux i _ _ ^ _ ie[l,n]
y k " 2 " S P i ie[l,n]
We can then find the value of X by substituting those values of the
prospects yk into our original constraint, (Y | &) = 0:
f \ ^ = 2 Z Z PiJxi
je[l,m] l^ietl.n] ,
It shows that X = 2 (X | &). Therefore, y k = (X | &) - (X | s'k, &) is indeed
the solution to our variance minimization problem.
158
• Proof that B~< Y* | w, X) = VoPHCS' | X) when S and S' are completely relevant, or
when S15":
When S = S',(X\S'= s'j, &) = Xj, which yields y*j = (X | &) - XJ. This is
the same hedging deal as the one defined in Theorem 5.3 to achieve
perfect hedging if 5 and 5" are completely relevant.
When S1 S\ (X | S' = s'j, &) = (X | &), which yields y*j = 0. This is the
same hedging deal as the one defined in Theorem 5.4 to achieve perfect
hedging if S IS'.
The approximation is thus exact when S and 5" are completely relevant, or
when S and S' are irrelevant given &.
• Proof that the approximation always gives an underestimate of VoPH(5" | X):
One simply needs to notice that the deal Y we defined belongs to the set
of deals such that <Y | &) = 0. Therefore, by definition of VoPH(S' | X),
we have VoPH(S" | X) > B~(Y* | X). 3
159
Example 5.10:
We will demonstrate how good the approximation is based on a simple example:
Sl
0.4
-o
0.6
S2
$100,000
$0
S'l
0.8
-0 0.2
S'l
S'l
0.3
-0 0.7
S 2
$yi
$y2
$yi
$y2
Figure V.7 -Accuracy of our value of perfect hedging approximation: an
example
A decision-maker currently owns deal X, which has a mean of $40,000. He is
considering acquiring a deal Y, whose monetary prospects are determined based
on the outcomes of a distinction iS" which is relevant, but not identical, to the S
distinction.
The decision-maker is interested in knowing which choice of deal Y, among all
possible deals that have 0 as their mean, constitutes the best match for his current
portfolio X - more specifically, which deal Y provides the highest PIBP B~(Y | X).
Before we look for the exact answer to that question, let us consider the
approximation we would obtain based on Theorem 5.8. It would require building
160
a hedging deal Y* such that y*j = (X | &} - (X | S' = s'j, &); we can compute those
two prospects as being y \ = -$24,000 and y 2 = $24,000 respectively.
Let us now numerically solve the problem of selecting the best possible
complement Y to the current portfolio X for different values of y. For that, let us
first notice that Y is fully specified by the choice of yi; indeed, since we want (Y |
&) to be equal to zero, any choice of yi imposes that we choose -yi as y2.
The curves on the following figure show B~(Y | w, X) as a function of yi, for six
different values of the risk-aversion coefficient y. The large dots mark the value of
yi for which the maximum value of B~( Y | w, X) is attained, again for each of the
six values of y. Several insights can be derived from a careful examination of this
figure:
• As noted earlier, Property 5.2 does not stand any longer once we consider
cases in which we have neither complete relevance or irrelevance between S'
and S: the selection of a best hedging deal Y does depend on risk attitude
apart from those extreme cases. For example, B~(Y | X) is maximized for yi ~
-$23,500 when y = 0.001, and for y, = -$20,500 when y = 0.01.
161
$4,000
$3,000
$2,000
$1,000
$0
-$1,000
-$2,000
-$3,000
-$4,000
-$5,000
-$6,000
k^jm^
^ Value of y given by approximation
$50 -$46 --442 -$38 -254 -$30 -$261 -$22 -$18 -$14 -$>6 -$6 -$2
I 'v=0.01
y=0.02
yi $ Thousand)
Figure V.8- Influence of risk-aversion on the accuracy of the approximation
The approximation suggested in Theorem 5.8, as predicted, seems to yield
more accurate results for low values of y; this is potentially problematic, in
a sense, because on the other hand, hedging will not be as valuable an
alternative to someone whose risk attitude is close to risk-neutral.
That being said, the approximation appears to be quite reliable for values
of y up to 0.02, as shown by the following figure:
162
0.001 0.005 0.01 0.02
$4,000
$2,000
$0
-$2,000
-$4,000
-$6,000
-$8,000
-$10,000
-$12,000
-$14,000
Figure V.9 - Influence of risk-aversion on the accuracy of the approximation (2)
i VoPH Estimated Using Our Approximation
I Actual VoPH
Why does the accuracy of the approximation break down as risk-aversion
increases? Consider the hedging deal Y suggested by Theorem 5.8;
incorporating it into the decision-maker's portfolio will have two effects:
o It does provide value by reducing the variance of the combined portfolio
of deals X and Y to its minimum.
o It also increases the size of the range of prospects the decision-maker will
be facing, and as a result it has a detrimental impact on some of the higher
order moments of the portfolio (such as decreasing the third central
moment).
The first effect is desirable from the point of view of a risk-averse decision
maker, whereas the second is a negative side effect from hedging with the
variance-minimizing deal Y ; for higher values of y, risk-averse decision
makers will grow increasingly concerned about the second effect.
163
However, a natural solution exists in order to somewhat immunize oneself
against that second effect - it consists in choosing a lower value of yi than the
one suggested by Theorem 5.8. By doing so, a decision-maker renounces to
part of the benefits of the variance reduction effect, while curbing the negative
impact of the second effect more drastically. On the whole, this can turn out to
be a wiser decision than blindly choosing deal Y as defined in Theorem 5.8.
This explains why, on Figure 5.8, the value of yi for which the value of
perfect hedging is achieved increases with y.
• VoPH(5" | X), which we already knew to be equal to 0 for risk-neutral
decision-makers, is also equal to 0 for extremely risk-averse decision-makers.
The value of perfect hedging also seems to reach a maximum for some
specific value of y. In our example, that maximum is reached for y ~ 0.02. For
that value of y, the optimal choice of yi is yi ~ -$16,000, and the value of
perfect hedging is roughly equal to $2,650. •
The PIBP for the deal Y which minimizes the variance of the portfolio gives us, at best,
an accurate approximation of the value of perfect hedging, and at worst, an underestimate
for it. We can thus use that PIBP in conjunction with the risk premium of the original
portfolio, and thereby gain access to both an upper and a lower bound on the value of
perfect hedging. The next example will illustrate that approach.
164
Example 5.11:
We return to our Gold Price hedging example; to make matters more interesting
we will consider that the oil company's risk tolerance is equal to $150 million
(instead of the $500 million from Example 5.1), so that the approximation from
Theorem 5.8 is not quite as robust as it could be otherwise.
We first need to compute the values of the monetary prospects which will help
minimize the variance of the decision-maker's portfolio; Theorem 5.8 tells us
that:
f y*Low Gold Price = (X | &> - (X | Low Gold Price, &)
L y*High Gold Price = <X | &> - <X | High Gold Price, &)
Therefore:
f y*Low Gold Price = $50.60 M - $17.00 M = $33.6 M
\ y*High Gold Price = $50.60 M - $105.42 M = -$54.82 M
We can then compute the PIBP for deal Y as we defined it:
B~(Y* | X) = $3.99 M
Next, we compute the risk premium of the existing portfolio:
RPs(X | 0 ) = $32.09 M
From that we can infer that the value of perfect hedging on Gold Price is
comprised between $3.99 million and $32.09 million. The exact value of perfect
hedging is equal to $4.15 million, which is much closer to the lower bound than to
the upper bound of our interval, since even with a risk tolerance of $150 million,
the oil company is not risk-averse enough for the approximation from Theorem
5.8 to lead to very inaccurate conclusions. •
165
6. Similarities between Value of Perfect Hedging and Value of
Clairvoyance
As a summary of our study of the value of perfect hedging and its properties, I will
enumerate the similarities it shares with one of the best known and most useful concepts
of traditional decision analysis: the value of clairvoyance.
The following table presents some of the most striking resemblances; all of them are
supported by various theorems and examples shown throughout this chapter. The last row
of the table, which discusses the impact of risk attitude on the value of clairvoyance and
on the value of perfect hedging, should remind us that in disciplines such as probability
or decision analysis, many of the conjectures which might initially to our intuition as
tempting to believe in are in fact often wrong.
There is an important difference between the properties of the value of information and
those of the value of perfect hedging - in a decision situation, the value of information on
some uncertainty is positive if and only if the decision-maker's preferred alternative
changes for at least one possible report on the value of the uncertainty. Conversely, the
value of perfect hedging on an uncertainty can be positive even if the best alternative
remains the same with or without hedging. The examples we have discussed throughout
this chapter demonstrates this well: the value of perfect hedging was positive even though
the availability of hedging did not modify the decision-maker's choices in any way;
hedging only made his portfolio more valuable in his eyes by limiting its risk.
166
Definition
Sign
Usefulness
Joint value of
clairvoyance
or of perfect
hedging
Impact of
relevance
Impact of
risk attitude
VALUE OF CLAIRVOYANCE ON S' VALUE OF PERFECT HEDGING ON S'
The most we should be willing to pay
for the clairvoyant's services.
>0
1) Provides an upper bound on the
value of any experiment
2) Helps prioritize uncertainties
based on how valuable it is to
gather additional information on
them
VoCOS", S") > max (VoC(S"),
VoC(S"))
No general rule between joint VoC
and sum of individual VoCs
= 0ifS'±S\&
Increases with relevance otherwise in
the sense that S" 1S | S', & =>
VoC(5") < VoC(S"), until it reaches
VoC on S when S' = 5
Consider two risk-averse decision
makers who each own a deal X in
which they believe they face the exact
same chances of the same prospects;
then, it is not necessarily true that the
most risk-averse person of the two
will be willing to pay more for
clairvoyance on 5".
The most we should be willing to pay
for a deal whose prospects are
determined solely by 5 ' and whose
mean is equal to 0.
>0
1) Provides an upper bound on our
PIBP for any deal we do not own
2) Helps prioritize uncertainties
based on how valuable it is to
hedge on them
VoPH(5", 5") > max (VoPH(5"),
VoPH(5"))
No general rule between joint VoPH
and sum of individual VoPHs
= 0 i f S " ± S | &
Increases with relevance otherwise in
the sense that S" 1S | S', & =>
VoPH(5" | w, X) < VoPH(5' | w, X),
until it reaches RPs(X w) when S' =
S
Consider two risk-averse decision
makers who each own a deal X in
which they believe they face the exact
same chances of the same prospects;
then, it is not necessarily true that the
most risk-averse person of the two
will be willing to pay more for perfect
hedging on S'.
Table V.2 - Comparison of value of clairvoyance and value of perfect hedging
167
7. The Value of Perfect Hedging as a New Appraisal Tool
Of all the steps of the decision analysis cycle, the appraisal phase is probably, with the
formulation phase, the step in which the decision analyst has the most freedom in
choosing how to conduct the process. As a result, it is in the appraisal phase that some of
the analyst's choices will most significantly affect the quality of his final
recommendation to the decision-maker.
What we will advocate in this part of the dissertation is that one more tool be added to
that critical part of the decision analysis cycle - the calculation of the value of perfect
hedging. Just as the value of clairvoyance will help the decision-maker focus his
information gathering activities in the next iteration of the cycle on the uncertainties
which matter the most, the value of perfect hedging will help him decide whether he
should pursue any hedging opportunities, and if so, which uncertainties he should seek to
hedge on.
Formulation Evaluation Appraisal
c . . . Deterministic . Probabilistic . , Structure • , . • , . • Appraisal p. ngpjeJQri
t Analysis Analysis
• Sensitivity analysis • Value of information • Value of control • Value of perfect hedging
Figure V.10 - The value of perfect hedging in the decision analysis cycle
168
In some cases, the availability of hedging can greatly affect the final recommendation of
an analysis by causing a reversal in the preference order between two alternatives.
Generally speaking, hedging is most likely to have such an effect in situations in which
the decision-maker's risk-aversion prevents him from selecting the alternative with the
highest mean, A, and forces him to settle for an alternative B with a lower mean, but a
narrower risk profile and a higher certain equivalent overall. Perfect hedging might then
enable the decision-maker to redeem more of the risk premium of A than of the risk
premium of B, to the extent that he may eventually prefer A to B.
In that respect, the concept of the value of perfect hedging fills the same need as what is
sometimes called risk-sharing in the decision analysis literature: if an uncertain deal has a
positive mean, but the decision-maker's certain equivalent for it is negative because of
the risk the deal carries, a possible alternative consists in bringing in another individual
and possible investor in this deal, who assigns the same probabilities as the first
individual to the monetary prospects of the deal; we then split the uncertain deal between
the two of them by giving a fraction f of every prospect to the first investor, and a
fraction 1 - f to the other. If the second decision-maker's risk tolerance is sufficiently
high, it will be possible to find a value of f such that both decision-makers have a positive
certain equivalent for the portion of the deal that they are left with.
Example 5.12:
The following example shows the impact that the value of perfect hedging can
have on the final recommendation of a decision analysis.
Several years ago, Zeta, a small biotechnology company, launched a drug for the
treatment of Parkinson's disease called Nemesis. A small study conducted in a
European university was recently published, showing that the compound might
have a neuroprotective quality which no one had suspected up to that point. Zeta's
management would like to convince the Food and Drug Administration (FDA) of
granting them an official neuroprotection claim on the label of Nemesis: if they
succeed, Nemesis will become the first drug with such a claim, and its
profitability will be greatly enhanced. Unfortunately, Zeta's board believes that
169
the data from the European study would be insufficient to convince the FDA, due
to the small scale of the experiment; Zeta is thus considering funding a phase IV
study, with the hope that it will provide enough evidence to persuade the FDA.
Endpoint 1
Phase IV Study
0.1
Endpoint 2
CE: -$4 M Mean: $286M
0.3
Endpoint 3
•0 0.6
No Phase IV Study
Neuroprotection
0 0.6
0.4
No Neuroprotection
Neuroprotection
0.15
0.85
No Neuroprotection
Neuroprotection
0 0.05
0.95 No Neuroprotection
$3.4 B
-$200 M
$3.4 B
-$200 M
$3.4 B
-$200 M
$0
Figure V.ll — Example for the use of perfect hedging as an appraisal tool
The figure above captures the most important elements of Zeta's decision
situation. The company's board also states that they wish to follow the delta
property for the range of prospects involved, with a risk tolerance of $1.5 billion.
We observe that at present, the best decision for Zeta is to renounce to the phase
IV study: conducting the study yields a negative certain equivalent of -$4 million.
170
However, we can also notice that the mean of the alternative is $286 million,
which implies that the only reason why it is not the best course of action is that it
currently involves too much risk for the company's taste. That remark should
prompt us to investigate the effect of the value of perfect hedging on the final
recommendation.
When asked about possible hedges, the company's board states that it might be
possible to hedge based on whether or not any drug at all will receive the FDA's
approval for a neuroprotection claim before 2010. Concretely, this could take the
form of an insurance contract. We will denote this uncertainty by NP < 2010.
Naturally, if Nemesis itself were to conduct a phase IV study and receive the
FDA's approval, then NP < 2010 would automatically be true; on the other hand,
if Nemesis were to be denied the claim, the board would assign a 20% chance to
NP < 2010, and an 80% chance to No NP < 2010.
We can then compute VoPH(JVP < 2010 | X): in doing so, we identify as the
perfect hedge on that uncertainty a deal which associates a loss of $528 million to
NP < 2010, and a profit of $235 million to No NP < 2010. The corresponding
value of perfect hedging for the Phase IV Study alternative is:
VoPH(JVP < 2010 | X) = $39 million
Hence, we can conclude that perfect hedging on NP < 2010 is indeed susceptible
of altering the final recommendation in favor of performing a phase IV study.
With perfect hedging, the certain equivalent of the study increases to $35 million.
We can also note that in order to obtain the same certain equivalent of $35 million
for the basic decision situation of Figure V.l 1, in the absence of any hedging, the
company would have needed to have a risk tolerance of about $1.95 billion
instead of $1.5 billion. In a sense, perfect hedging on NP < 2010 can thus be said
to be equivalent to a $450 million increase in the company's risk tolerance for the
purposes of this decision. •
171
Chapter 6 - Conclusions & Future Work
"It was our use of probability theory as logic that has enabled us to do
so easily what was impossible for those who thought of probability as a
physical phenomenon associated with 'randomness'. Quite the
opposite; we have thought of probability distributions as carriers of
information. At the same time, under the protection of Cox's theorems,
we have avoided the inconsistencies and absurdities which are
generated inevitably by those who try to deal with the problems of
scientific inference by inventing ad hoc devices instead of applying the
rules of probability theory. "
Edwin Thompson Jaynes (1922-1998),
Probability Theory: the Logic of Science
1. Contributions
Over the course of this dissertation, I have argued in favor of adding a study of possible
hedging alternatives to the appraisal phase of the decision analysis cycle. We have
observed on many examples that like additional information, hedging can be of
considerable value to a decision-maker.
I have also shown how a decision analysis of hedging can be conducted in practice. The
concept of the value of perfect hedging helps us identify the uncertainties on which it is
most valuable to seek hedging; it should act as our compass when we do not know where
to start and which uncertainties to hedge on. As for situations in which the decision
maker needs to choose from an already established shortlist of available hedging
alternatives, we have learnt to compute or approximate the value of hedging which an
uncertain deal provides with respect to the existing portfolio. That value of hedging then
172
helps us determine the decision-maker's personal indifferent buying price for the
uncertain deal in question.
In this dissertation, we have also proposed a new definition of hedging: we have spoken
of hedging in situations in which the decision-maker's valuation of a deal differs
depending on whether he owns his present wealth w and his current portfolio X, or his
present wealth w augmented by his certain equivalent for X. This new definition has the
merit of being broad enough that it does not restrict our characterization of hedging to the
use of financial derivatives, and yet precise enough for it to pass the clarity test.
In addition, we have covered the topic of the probabilistic origins of hedging and shown
that hedging boils down to moment reengineering. It is probable that our findings on that
front are further removed from the practical concerns of most decision-makers than are
some of the other concepts we discussed, such as the value of perfect hedging; still,
knowledge of the probabilistic origins of hedging might appear as quite useful to a
practicing decision analyst, whose profession requires that he should always be keen on
learning new ways of identifying valuable alternatives for his client.
We have thus addressed all the research questions which I had originally listed in the first
chapter and which I proposed to examine.
173
2. Limitations
The research presented in this thesis has an important fundamental limitation: in all of our
findings and illustrative examples, we have only considered situations in which the
decision-maker's preferences could be expressed in monetary units. We should bear in
mind that there are decision situations in which it is not natural to do so - most notably in
medical settings, and more infrequently in engineering settings. But those are perhaps
also the decision situations in which it is hardest to think of alternatives which would
provide hedging with respect to the decision-maker's current situation. It thus remains a
relatively modest weakness of this research that its scope is limited to uncertain deals
whose prospects can be evaluated with a monetary measure.
A second limitation of the thesis is that in spite of our efforts to study hedging in its
largest generality, there is one situation which we have not explicitly addressed:
occasionally, hedging can have side effects. By that I mean that it can entail
consequences for the decision-maker which the monetary prospects of the existing
portfolio and of the hedging deal would not suffice to explain.
Some of the best examples can be found in the gold-mining industry, where some
companies are proud to announce that they do not hedge their gold production at all.
There are several reasons for their behavior. The first is that many of those mining
companies appear to be convinced that there is a tremendous upside to gold prices in the
current economic environment, and that hedging might lead them to agree to prices which
would be inferior to the actual future price of gold. The second argument which gold
companies offer to explain why they do not hedge is more atypical: they believe that
many of the shareholders who invest in their company do in fact desire to be exposed to
evolution of the price of gold1, be it because those investors also believe that there is a
great upside to gold prices, or because they regard gold as a safe haven in the event of a
major economic crisis. In either case, gold-mining companies believe that if they began
to hedge their gold positions, their stock would lose some of its appeal to shareholders.
1 See for example a 2006 interview of Peter Marrone, CEO of Yamana Gold, by Bill Mann:
http://www.fool.com,/investing/small-cap/2006/09/13/a-brazilian-gold-mine.aspx
174
It would be fascinating to see whether the same argument, when applied to other
industries than gold-mining, can work in the opposite direction, that is, in favor of
hedging. Perhaps some companies attach a greater value to hedging practices than our
concept of the value of hedging would detect, because they believe that by hedging, they
also make their stock more attractive to certain kinds of shareholders.
While it is true that I have not explicitly addressed issues of side effects to hedging in the
dissertation, we should also realize that we only need to make a few modifications to the
framework we have proposed before it can lend itself to the successful study of those
situations as well. Those adjustments are best illustrated by an influence diagram such as
the one below; it captures the sort of dilemma which a gold-mining company might face,
if it believes that there would be a negative side effect attached to hedging:
Figure VI. 1 - Hedging with negative side effects
If we compare the influence diagram above with the general influence diagram for
hedging which we presented in Figure II.6, we can see that there is only one important
difference: we added a new uncertainty, Loss of Safe Haven Status, and a new path
leading through it from the decision to buy the hedge to the value node. The result is an
influence diagram with a richer and more complex probabilistic structure, but with the
same overall philosophy as the diagram from Figure II.6. We can then use this
175
representation of the decision situation to determine the gold-mining company's PIBP for
the hedge as the price at which they would just be indifferent between buying the hedge
and not buying it.
176
3. Directions for Future Work
The two limitations of the thesis which we mentioned can also be envisaged as promising
research opportunities.
We have already given a few hints as to how our hedging framework can be adapted to
the study of situations in which hedging has side effects, but we have not yet discussed
any possible solutions to the framework's inadequacy to cope with instances in which
preferences are not captured by a monetary value measure.
In that respect, the similarities we have so often detected between value of information
and value of hedging can give us an idea which might be worth investigating. Just like
the value of hedging, the value of information can only be computed if the prospects are
measured in monetary units; however, a related concept, called "probability of knowing"
or "equivalent probability of clairvoyance" [Howard, R. A.], can help us reason on the
merits of an information gathering scheme in the absence of a monetary value measure.
The equivalent probability of clairvoyance is defined as the probability p such that the
decision maker would be just indifferent between using the information gathering scheme
in question, and receiving either full clairvoyance on his decision situation with
probability/' or no additional information at all with probability l—p:
Receive full clairvoyance
Information gathering ^ ^ ^ 6 scheme
1 - P Receive no further information
Figure VI.2 — Equivalent probability of clairvoyance
111
It is then possible to sort information gathering schemes from most valuable to least
valuable by comparing their respective equivalent probabilities of clairvoyance. Perhaps
it is possible to accomplish the same thing for hedging alternatives, by defining a concept
of equivalent probability of hedging.
I would finally like to suggest the search for additional evaluation methods, for the value
of hedging as well as for the value of perfect hedging, as a third direction for future work
on hedging. We have often remarked throughout this dissertation that the computation of
those quantities can be an intensive task, which is why we have so frequently stressed the
significance of the approximations and bounds we were able to derive. However, it is
probable that we can do much more to complete our arsenal of hedging valuation
techniques. For cases in which the decision-maker's u-curve satisfies the delta property,
we have just laid out the foundations of an irrelevance-based algebra for the value of
hedging, and it might be worthwhile to look for further results in that direction. But it is
the situations in which the decision maker's u-curve does not satisfy the delta property
that we are least equipped to handle: only a small number of the approximations and
bounds we have identified are valid in that case, and it would be of great practical use to
have access to a few more results for at least the most commonly used u-functions, such
as the logarithmic and power u-curves.
178
List of References
Probability, Bavesian Methods and Decision Analysis:
[1.1] Abbas, A. E., 2003, Entropy Methods in Decision Analysis, dissertation submitted
to the department of Management Science & Engineering of Stanford University.
[1.2] Bayes, T., 1763, "An Essay Towards Solving a Problem in the Doctrine of
Chances", Biometrika, 45, 1958, pp. 293-315. Reprinted from Philosophical
Transactions of the Royal Society, 53, pp. 370-418.
[1.3] Bickel, E. J., 1999, The Corporate Contractual System and Normative Corporate
Risk Attitude, dissertation submitted to the department of Engineering Economic
Systems and Operations Research of Stanford University.
[1.4] Cover, T. M., and Thomas, J. A., 1991, Elements of Information Theory, Wiley,
New York.
[1.5] Cox, R. T., 1946, "Probability, Frequency and Reasonable Expectation",
American Journal of Physics, 14:1-13.
[ 1.6] De Finetti, B., 1970, Theory (/Probability, Wiley, New York.
[1.7] Fishburn, P. C , 1964, Decision and Value Theory, Publications in Operations
Research No. 10, Wiley, New York.
[1.8] Fishburn, P. C , 1981, "Subjective Expected Utility: a Review of Normative
Theories", Theory and Decision, 13:139-199.
[1.9] Howard, R. A., 1964, "The Science of Decision-making", Readings on the
Principles and Applications of Decision Analysis, Vol. 1, R. A. Howard and J. E.
Matheson (eds.), Strategic Decisions Group, Menlo Park, California, 1984, pp.
159-176.
179
[1.10] Howard, R. A., 1965, "Bayesian Decision Models for Systems Engineering",
IEEE Transactions on Systems Science and Cybernetics, SSC-1:1, pp. 36-40.
[1.11] Howard, R. A., 1966a, "Decision Analysis: Applied Decision Theory",
Proceedings of the Fourth International Conference on Operational Research, D.
B. Hertz and J. Melese (eds.), Wiley-Interscience, New York, pp. 55-71.
[1.12] Howard, R. A., 1966b, "Information Value Theory", IEEE Transactions on
Systems Science and Cybernetics, SSC-2:1, pp. 22-26.
[1.13] Howard, R. A., 1968, "The Foundations of Decision Analysis", IEEE
Transactions on Systems Science and Cybernetics, SSC-4:3, pp. 211-219.
[1.14] Howard, R. A., 1970, "Decision Analysis: Perspectives on Inference, Decision
and Experimentation", Readings on the Principles and Applications of Decision
Analysis, Vol. 2, R. A. Howard and J. E. Matheson (eds.), Strategic Decisions
Group, Menlo Park, California, 1984, pp. 821-834.
[1.15] Howard, R. A., 1971, "Proximal Decision Analysis", Management Science,
Theory, Vol. 17 (May 1971), No. 9, pp. 507-541.
[1.16] Howard, R. A., 1980, "On Making Life and Death Decisions", Readings on the
Principles and Applications of Decision Analysis, Vol. 2, R. A. Howard and J. E.
Matheson (eds.), Strategic Decisions Group, Menlo Park, California, 1984, pp.
483-506.
[1.17] Howard, R. A., 1988, "Decision Analysis: Practice and Promise", Management
Science, 34:6, pp. 679-695.
[1.18] Howard, R. A., 1992, "In Praise of the Old Time Religion", Utility Theories:
Measurements and Applications, Kluwer Academic Publishers, Norwell,
Massachusetts, pp. 27-55.
[1.19] Howard, R. A., 1998, The Fundamentals of Decision Analysis, manuscript in
progress, unpublished.
[1.20] Howard, R. A., 2004, "Speaking of decisions: Precise decision language",
Decision Analysis, 1(2), pp. 71-78.
180
[1.21] Howard, R. A., and Matheson, J. E., 1981, "Influence Diagrams", Readings on the
Principles and Applications of Decision Analysis, Vol. 2, R. A. Howard and J. E.
Matheson (eds.), Strategic Decisions Group, Menlo Park, California, 1984, pp.
719-762.
[1.22] Jaynes, E. T., 1976, "Confidence Intervals vs. Bayesian Intervals", Foundations
of Probability Theory, Statistical Inference, and Statistical Theories of Science, W.
L. Harper and C. A. Hooker (eds.), D. Reidel, Dordrecht, pp. 175.
[1.23] Jaynes, E. T., 2003, "Probability Theory: the Logic of Science", Cambridge
University Press, England.
[1.24] Jeffreys, H., 2003, "Theory of Probability", Oxford University Press, England.
[1.25] Keeney, R. L., 1982, "Decision Analysis: an Overview", Operations Research,
Vol. 30, pp. 803-838.
[1.26] Keeney, R. L., and Raiffa, H., 1976, Decisions with Multiple Objectives:
Preferences and Value Trade-Offs, Wiley, New York.
[1.27] Lowell, D. G., 1994, Sensitivity to Relevance in Decision Analysis, dissertation
submitted to the department of Engineering Economic Systems of Stanford
University.
[1.28] Matheson, D., and Matheson, J. E., 1998, The Smart Organization: Creating
Value Through Strategic R&D, Harvard Business School Press, Massachusetts.
[1.29] Matheson, J. E., 1983, "Managing the Corporate Business Portfolio", Readings on
the Principles and Applications of Decision Analysis, Vol. 1, R. A. Howard and J.
E. Matheson (eds.), Strategic Decisions Group, Menlo Park, California, 1984, pp.
719-762.
[1.30] Matheson, J. E., 1990, "Using Influence Diagrams to Value Information and
Control", Influence Diagrams, Belief Nets and Decision Analysis, R. M. Olivier
and J. Q. Smith, eds., proceedings of the conference entitled "Influence Diagrams
181
for Decision Analysis, Inference, and Prediction", Berkeley, California, May 9-11,
Wiley, Chichester, England, pp. 427-455.
[1.31] Matheson, J.E., and Howard, R. A., 1968, "An Introduction to Decision Analysis",
Readings on the Principles and Applications of Decision Analysis, Vol. 1, R. A.
Howard and J. E. Matheson (eds.), Strategic Decisions Group, Menlo Park,
California, 1984, pp. 17-55.
[1.32] McNamee, P., and Celona, J., 1987, Decision Analysis for the Professional,
Scientific Press, Redwood City, California.
[1.33] McNamee, P., and Celona, J., 1990, Decision Analysis with Supertree, Scientific
Press, Redwood City, California.
[1.34] Owens, D. L., 1978, "The Use of Influence Diagrams in Structuring Complex
Decision Problems", Readings on the Principles and Applications of Decision
Analysis, Vol. 2, R. A. Howard and J. E. Matheson (eds.), Strategic Decisions
Group, Menlo Park, California, 1984, pp. 763-771.
[1.35] Pearl, J., 1988, Probabilistic Reasoning in Intelligent Systems, Morgan Kaufmann,
San Mateo, California.
[1.36] Pratt, J. W., 1964, "Risk-aversion in the Small and in the Large", Econometrica,
32, pp. 122-136.
[1.37] Pratt, J. W., Raiffa, H., and Schlaifer, R., 1964, "The Foundations of Decision
under Uncertainty: an Elementary Exposition", Journal of the American
Statistical Association, Vol. 59, pp. 353-375.
[1.38] Raiffa, H., 1968, Decision Analysis: Introductory Lectures on Choice under
Uncertainty, Addison-Wesley, Reading, Massachusetts.
[1.39] Savage, L. J., 1951, The Theory of Statistical Decision, Journal of the American
Statistical Association, 46:55-67.
[1.40] Savage, L. J., 1954, The Foundations of Statistics, Wiley, New York.
182
[1.41] Shachter, R. D., 1986, Evaluating Influence Diagrams, Operations Research, 34
(Nov.-Dec. 1986), pp. 871-882.
[1.42] Shachter, R. D., 1988, Probabilistic Inference and Influence Diagrams,
Operations Research, 36 (July-Aug. 1988), pp. 589-605.
[1.43] Shachter, R. D., 1997, Examples without Numbers, unpublished manuscript.
[1.44] Spetzler, C. S., and Stael von Holstein, A. S., 1975, "Probability Encoding in
Decision Analysis", Management Science, Vol. 22, No. 3 (Nov., 1975), pp. 340-
358.
[1.45] Tribus, M. T., 1969, Rational Descriptions, Decisions and Designs, Pergamon
Press Inc., Maxwell House Fairview Park Elmsford, New York.
[1.46] Tversky, A., and Kahneman, D., 1974, "Judgment under Uncertainty: Heuristics
and Biases", Science, New Series, Vol. 185, No. 4157 (Sept. 27, 1974), pp. 1124-
1131.
[1.47] Tversky, A., and Kahneman, D., 1983, "Extensional vs. Intuitive Reasoning: the
Conjunction Fallacy in Probability Judgment", Psychological Review, 91, pp.
293-315.
[1.48] Von Neumann, J., and Morgenstern, O., 1947, Theory of Games and Economic
Behavior, Princeton University, Princeton, New Jersey.
183
Finance and Hedging:
[2.1] American Heritage® Dictionary of the English Language, Fourth Edition.
Houghton Mifflin Company, 2004.
[2.2] Black, F., 1989, "Equilibrium exchange rate hedging", Journal of Finance, 43, pp.
899-908.
[2.3] Black, F. and Scholes, M., 1973, "The Pricing of Options and Corporate
Liabilities", Journal of Political Economy, 81 (1973), pp. 163-175.
[2.4] Boyle, P. P., and Emanuel, D., 1980, "Discretely Adjusted Option Hedges", J.
Financial Economics, 8 (1980), pp. 259-282.
[2.5] Brandt, K., "An Answer to Climate Change - Weather Hedging for Everybody",
available online: http://www.weathereconomy.com/Weather%20Hedging.pdf.
[2.6] Brown, S. L., 1985, "A Reformulation of the Portfolio Model of Hedging",
American Journal of Agricultural Economics, Vol. 67, No. 3 (Aug. 1985), pp.
508-512.
[2.7] Carr, P., Geman, H., Madan, D., and Yor, M., 2005, "Pricing Options on Realized
Variance", Finance andStochastics, pp. 453-475.
[2.8] Castelino, M. G., 1992, "Hedge Effectiveness: Basis Risk and Minimum-Variance
Hedging", Journal of Futures Markets, 12 (April 1992), pp. 187-201.
[2.9] Cvitanic, J., Schachermayer, W., and Wang, H., 2001, "Utility Maximization in
Incomplete Markets with Random Endowment", Finance and Stochastics, 5(2),
pp.259-272.
[2.10] Davis, M. H. A., Panas, V. G., and Zariphopoulou, T., 1993, "European Option
Pricing with Transaction Costs", SI AM Journal of Control and Optimization, 31
(1993) (2), pp. 470-493.
[2.11] Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., and
Strieker, C , 2002, "Exponential Hedging and Entropic Penalties", Mathematical
Finance, 12(2), pp. 99-123.
184
[2.12] Frechette, D. L., 2000, "The Demand for Hedging and the Value of Hedging
Opportunities", American Journal of Agricultural Economics, Vol. 82, No. 4.
(Nov. 2000), pp. 897-907.
[2.13] Friz, P., and Gatheral, J., 2005, "Pricing Volatility Derivatives as Inverse
Problem", Quantitative Finance, Vol. 5, No. 6, (2005), pp. 531-542.
[2.14] Glen, J., and Jorion, P., 1993, "Currency Hedging for International Portfolios",
Journal of Finance, Vol. 48, No. 5. (Dec. 1993), pp. 1865-1886.
[2.15] Grasselli, M. R., and Hurd, T. R., 2007, "Indifference Pricing and Hedging for
Volatility Derivatives", Applied Mathematical Finance, Vol. 14, No. 4 (Sept.
2007), pp. 303-317.
[2.16] Guay, W. R., and Kothari, S. P., 2003, "How Much Do Firms Hedge with
Derivatives?", Journal of Financial Economics, 70 (2003), pp. 423-461.
[2.17] Heifner, R. G., 1972, "Optimal Hedging Levels and Hedging Effectiveness in
Cattle Feeding", Agricultural Economic Research, 24 (1972), pp. 25-36.
[2.18] Heifner, R. G., 1973, "Hedging Potential in Grain Storage and Livestock
Feeding", Washington DC: U.S. Department of Agriculture Agricultural
Economic Report No. 238.
[2.19] Henderson, V., 2002, "Valuation of Claims on Nontraded Assets Using Utility
Maximization", Mathematical Finance, 12(4), pp. 351-373.
[2.20] Hodges, S. D., and Neuberger, A., 1989, "Optimal Replication of Contingent
Claims under Transaction Costs", Review of Futures Markets, 8, pp. 222-239.
[2.21] Howard, C. T., and D'Antonio, L. J., 1984, "A Risk-Return Measure of Hedging
Effectiveness", Journal of Financial and Quantitative Analysis, 19 (March 1984),
pp. 101-112.
[2.22] Johnson, L. L., 1959, "The Theory of Hedging and Speculation in Commodity
Futures", Review of Economic Studies, 27(1959-60), pp. 139-51.
185
[2.23] Kahl, K. H., 1983, "Determination of the Recommended Hedging Ratio",
American Journal of Agricultural Economics, Vol. 65, No. 3. (Aug. 1983), pp.
603-605.
[2.24] Lewent, J. C , and Kearney, A. J., 1990, "Identifying, measuring and hedging
currency risk at Merck", Journal of Applied Corporate Finance, 2, pp. 19-28.
[2.25] Luenberger, D. G., 1997, Investment Science, Oxford University Press, New York.
[2.26] Luenberger, D. G., 2004, "Pricing a Nontradeable Asset and Its Derivatives",
Journal of Optimization Theory and Applications, Vol. 121, No. 3, June 2004, pp.
465-487.
[2.27] Markowitz, H. M., 1952, "Portfolio Selection", Journal of Finance, 7(1), pp. 77-
91.
[2.28] Markowitz, H. M., 1959, Portfolio Selection: Efficient Diversification of
Investments, New York.
[2.29] Mello, A. S., and Neuhaus, H. J., 1998, "A Portfolio Approach to Risk Reduction
in Discretely Rebalanced Option Hedges", Management Science, Vol. 44, No. 7.
(Jul. 1998), pp. 921-934.
[2.30] Mian, S. L., 1996, "Evidence on Corporate Hedging Policy", Journal of Financial
and Quantitative Analysis, Vol. 31, No. 3. (Sept. 1996), pp. 419-439.
[2.31] Musiela, M., and Zariphopoulou, T., 2004, "An Example of Indifference Prices
under Exponential Preferences", Finance and Stochastics, 8(2), pp. 229-239.
[2.32] Nance, D. R., Smith, C. W., and Smithson, C. W„ 1993, "On the Determinants of
Corporate Hedging Policy", Journal of Finance, Vol. 48, No. 1. (Mar. 1993), pp.
267-284.
[2.33] Smith, C. W., and Stultz, R. M., 1985, "The Determinants of Firms' Hedging
Policies", Journal of Financial and Quantitative Analysis, Vol. 20, No. 4. (Dec.
1985), pp. 391-405.
[2.34] Stultz, R. M., 1984, "Optimal Hedging Policies", Journal of Financial and
Quantitative Analysis, Vol. 19, No. 2. (June 1984), pp. 127-140.
186
[2.35] Stulz, R. M., 1996, "Rethinking Risk Management", Journal of Applied
Corporate Finance, 9(3), pp. 8-24.
[2.36] Wikipedia.org on hedging: http://en.wikipedia.org/wiki/Hedging
187
Index of Terms
Certain equivalent xi, 12
Chain rule
Chain rule for the value of hedging 95
Chain rule for the value of information 95
Clarity of action 7
Contraction property from probability 102
Cumulant 107
Cumulative distribution function (CDF) 22
Decision analysis 7
Decision analysis cycle 15
Appraisal phase 25
Evaluation phase 20
Formulation phase 16
Decision diagram See Influence diagram
Decision hierarchy 16
Delta property 13
Deterministic analysis 20
Deterministic dominance 23
Elements of decision quality 8
Equivalent probability of clairvoyance See Probability of knowing
Five rules of actional thought 9
Choice rule 10
Equivalence rule 9
Order rule 9
Probability rule 9
Substitution rule 9
Framing 16
HedgeStreet 3
Hedging 33,42
188
Minimum-variance hedge 35
Negative hedging 61
Side effects of hedging 174
Incomplete markets 39
Influence diagram 17
Decision node 17
Deterministic node 17
Functional arrow 18
Influence arrow 18
Influence diagram for hedging 56
Informational arrow 18
Relevance arrow 18
Uncertainty node 17
Value node 17
Irrelevance xi
Judgmental bias 22
Mean-variance approaches 35
Minimal underlying uncertainty associated with an uncertain deal 91
Multiple of a deal 83
Normative discipline 7
PIBP xii, 11
PISP xi, 11
Preference probability 9
Probabilistic analysis 22
Probabilistic dominance
First order probabilistic dominance 23
Second order probabilistic dominance 23
Probability encoding 22
Probability of knowing 177
Risk premium
Buying risk premium xii
Selling risk premium xi
Risk-averse 12
189
Risk-seeking 12
Risk-sharing 169
Sensitivity analysis 25
Open loop / closed loop sensitivity analysis 27
Tornado diagram 20
U-curve 11
Utility-based approaches 39
U-value 11
Value of control 30
Value of hedging
Value of hedging in the delta case 73
Value of perfect hedging 121
Dominance for the value of perfect hedging 146
Joint value of perfect hedging 130
Value of perfect information 29
Venn diagram 98
Wealth effect 47
190