spectral measures of risk coherence in theory and practice
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Subject of the talk: only finance (and a bit of statistic)
Risk Management Questions
Financial Statistical ProbabilisticComputation
al
What do I measure ?
How do I estimate it ?
What hypotheses
should I make ?
How can I carry out the computation
(in time) ?
Our investigation will be completely devoted to financial and statistical
questions.
The results will be
however absolutely general
The qualitative concept of “risk” and “risk premium”
Everybody has an innate feeling for financial risk ....
... more or less this ......
How to define risk in a quantitative fashion ? ...
fundamental shared
principles
requirements (axioms) on the risk measure
Concept of Risk Risk Measure?
test
The risk diversification principle
The aggregation of portfolios has always the effect of reducing or at most leaving unchanged the overall risk.
+
=
Port
folio
A
Port
folio
B
Port
folio
A +
B
Risk of ( A + B )
is less or equal to
Risk of (A) + Risk of (B)
Coherent Measures of Risk
(Monotonicity) if then
(Positive Homogeneity) if then
(Translational Invariance)
(Subadditivity)
)()(
)()()(
0 )()( aa )()(
In the paper “Coherent measures of Risk” (Artzner et al. Mathematical Finance, July 1999) a set of axioms was proposed as the key properties to be satisfied by any “coherent measure of risk”.
The diversification principle goes here
Value at Risk (VaR): how it works
To compute VaR, we need to specify
A time horizon: for instance one day. It represents the
future period over which we measure the risks of a portfolio
A confidence level: for instance a 5% probability. It
represents the fraction of future worst case scenarios of the
portfolio that we want to single out.
The definition of VaR is then:
“The VaR of a portfolio is the minimum loss that a portfolio can suffer in one day in the 5% worst cases”
Strange as it may seem, this is the question
most frequently asked to risk managers
worldwide today
Or equivalently:
“The VaR of a portfolio is the maximum loss that a portfolio can suffer in one day in the 95% best cases”
The formidable advantages introduced by VaR
Since its appearance, VaR turned out to be a more flexible instrument w.r.t. more traditional measures of risk such as the “greeks” or “sensitivities”, because VaR is
1. Universal: VaR can be measured on portfolios of any type (greeks on the contrary are designed “ad hoc” for specific risks)
2. Global: VaR summarize in a single number all the risks of a portfolio (IR, FX, Equity, Credit, …) (while we need many greeks to detect them all)
3. Probabilistic: VaR provides a loss and a probability occurrence (while greeks are “what if” measures, which tell us nothing on the probabilities of the “if”)
4. Expressed in Lost Money: VaR is expressed in the best of possible units of measures: LOST MONEY. Greeks have peculiar and less transparent u.o.m.
A VaR-based portfolio risk report is exceedingly clearer than a greeks-based one
No practitioner in 2003 would ever give up to these advantages anymore
The deadly sin of VaR
Unfortunately however VaR
1. Violates the subadditivity axiom and so is not coherent
Or equivalently
2. Violates the diversification principle and so for us it is not a risk measure at all
In other words it may happen that …
+ =VaR = 2
VaR = 3
VaR = 10
The source of all VaR’s troubles: neglecting the tail
VaR doesn’t care what’s beyond the
threshold.I do care !
Subadditivity and capital allocation
BANK
business unit: Fixed
Income
business unit:
Equities
business unit: Forex
Due to the lack of subadditivity, VaR appears to be unfit for determining the capital adequacy of a bank.
In a financial institution made of several branches, it is common (or it might be unavoidable for practical reasons) to perform the risk measurements in each branch separately, reporting the results to a central Risk Management dept.
VaR = 5
VaR = 3
VaR = 2
Capital reserves as if VaR = 10 ?
What is the concept of risk of VaR ?
From an epistemologic point of view however,
the main problem of VaR is not its lack of subadditivity
but the very lack of any associated consistent set of axioms
We still wonder what concept of risk Value at Risk has in mind !
A natural question
Is it possible to find coherent measures which are as versatile and flexible as VaR ?
The answer is fortunately YES
(… and they are also infinitely many …)
Expected Shortfall as an improvement of VaR
Definition of Expected Shortfall:
“The ES of a portfolio is the average loss that a portfolio can suffer in one day in the 5% worst cases”
Remember that
“The VaR of a portfolio is the minimum loss that a portfolio can suffer in one day in the 5% worst cases”
ES = the average of worst cases
VaR = the best of worst cases
Is the Expected Shortfall coherent ?
)()()( )( XVaRXXEXESXTCE OLD
The original definition of Expected Shortfall (also known as Tail Conditional Expectation TCE) is
This measure is also NON - SUBADDITIVE in general and so NON - COHERENT.
2001 : new definition of Expected Shortfall
duXFES u )(1
0
This measure is SUBADDITIVE and in fact COHERENT with no hypotheses on the pdf
When TCE and ES differ
0 10 200
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 10 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
while TCE is the average of the pdf defined by all the first two
columns (>5% worst)
In this case ES is the average of the pdf defined by the darkened
area only (5% worst)
Estimating Expected Shortfall
][
1:
)(
][
1)(
N
iNi
N XN
XES
One can show that ES is indeed estimable in a consistent way as the “Average of 100% worst cases”.
Ordered statistics
(= sorted data from worst to best)
)()()( XESXESN
N
Example 1: a subadditivity violation of VaR
Consider a Bond A and suppose that, at maturity, there are three possible cases:
1) No default: it redeems the nominal (100 Euro) and the
coupon (8 Euro)
or
2) Soft default: it redeems only the nominal (100 Euro) but not
the coupon
or
3) Hard Default: it pays nothing
A subadditivity violation of VaR
Consider another Bond B perfectly identical to A, but issued by a different issuer
Suppose now that the risks of the two bonds happen to be mutually
exclusive, in the sense that if issuer A defaults, B does not, and vice-
versa.
Typical case:
ANTICORRELATED RISKS =
RISK REDUCTION IN CASE OF DIVERSIFICATION
Risk Measurement
Final Event Probability Bond A Bond B Bond A + Bond BHard default B 3% 108 0 108Soft Default B 2% 108 100 208Hard default A 3% 0 108 108Soft Default A 2% 100 108 208
No default 90% 108 108 216
Final Redeem
Bond A Bond B Bond A + Bond B
104.6 104.6 209.2
Initial Value
Risk Variable Bond A Bond B Bond A + Bond B Subadditivity5% VaR 4.6 4.6 101.2 violated5% ES 64.6 64.6 101.2 not violated
Risk Measurement
Risk Variable Bond A Bond B Bond A + Bond B Convexity1000 Euro VaR 44 44 484 violated1000 Euro ES 618 618 484 not violated
Risk Measurement on a fixed size portfolio (1000 Euro)
VaR dissuades from diversification !ES advises diversification
Risk surfaces
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
100% A 100% B50%-50%
VaR
ES
Coherent risk measures display always convex risk surfaces with a
unique global minimum and no local minima
Non-coherent measures display in general risk surfaces affected by
multiple (local) minima
Example 2: a simple prototype portfolio
Consider a portfolio made of n risky bonds all of which have a 3% default probability and suppose for simplicity that all the default probabilities are independent of one another.
Portfolio = { 100 Euro invested in n independent identical distributed Bonds }
Bond payoff = Nominal (or 0 with probability 3%)
Question: let’s choose n in such a way to minimize the risk of the portfolio
Let’s try to answer this question with a 5% VaR, ES and TCE (= ES (old)) with a time horizon equal to the maturity of the bond.
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
ESTCEVaR
“risk” versus number of bonds in the portfolio
VaR and TCE suggest us NOT TO BUY the 13th, 28thor 47th
bond because it would increase the risk of the portfolio .... (?)
The surface of risk of ES has a single global minimum at n= and no fake local minima.
ES just tell us: “buy more bonds you can”
Are things better for large portfolios ???...
...but for large n the pdf should be normal and VaR coherent ... (!!!)
Notice that the pdf really becomes normal-shaped for
large N
... or not ?
0 50 100 150 200 250 300 350 4000
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1ESTCEVaR
But convexity problems still remain !!!
A natural question: ... other coherent measures ?
Is the Expected Shortfall an “isolated exception” or does it belong to a large class of coherent measures ?
Is it possible to create new coherent measures
starting from some given known ones?
The answer is simple and allows to create a wide CLASS of coherent measures.
Given n coherent measuresof risk 1, 2,... n
any convex linear combination
= 1 1 + 2 2 + ...+ n n ( with k k = 1 and k>0 )
is another coherent measure of risk
Geometrical interpretation
If any point represents a given coherent measure ...
... Then any other point in the generated “convex hull” is a new coherent measure of risk
Given n coherent measures, their most general convex combination is any of the points contained in the generated “convex hull”
Our strategy ....
Set of all Expected Shortfalls with (0,1]
Convex hull =
New space of coherent measures
We already know infinitely many coherent measures of risk, namely all the possible -Expected Shortfalls for any value between 0 e 1
In this way we can generate a new class of coherent measures.
This class is defined
“Spectral Measures of Risk”
Spectral measures of risk: explicit characterization
Definition: Spectral measure of risk with spectrum
1
0
)()()( dppFpXM X
)( p
Theorem: the measure M(X) is coherent if and only if
1. is positive
2. is not increasing
1.
)( p
)( p
1)(1
0
dpp
The “Risk Aversion Function” (p)
Any admissible (p) represents a possible legitimate rational attitude toward risk
A rational investor may express her own subjective risk aversion through her own subjective (p) which in turns give her own spectral
measure M
(p): Risk Aversion Function
Best cases Worst cases
It may thought of as a function which
“weights” all cases from the worst to the best
“(p) decreasing” explains the essence of coherence
...a measure is coherent only if it maps
“bigger weights to worse cases”
Risk Aversion Function (p) for ES and VaR
Expected Shortfall:
Step function
• positive
• decreasing
• 1)(1
0
dpp
Value at Risk:
Spike function
(Dirac delta)
• positive
• not decreasing
• 1)(1
0
dpp
Estimating Spectral Measures of Risk
N
iiNi
N XXM1
:)( )(
It can be shown that any spectral measure has the following consistent estimator:
Discretized function
Ordered statistics
(= data sorted from worst to best)
)()()( XMXMN
N
Tailoring Risks !
The Expected Shortfall is just one out of infinitely many possible Spectral Measures
ES expresses just a specific risk aversion
But is there a spectral measure which is optimal for all portfolios ?
NO
DIFFERENT PORTFOLIOS
DIFFERENT SPECTRAL MEASURES
What are the distinguishing properties of Spectral measures ?
A characterization of spectral measures among coherent measures via additional properties (axioms) would give us not only more information on Spectral Measures, but also useful information on NON-spectral measures.
Coherent Measures
Spectral Measures
Coherent but not Spectral
A fifth axiom ? A sixth one ?
One can show that the Spectral Measures M are all the coherent
measures which satisfy two additional conditions: (Kusuoka
2001, Tasche 2002)
The first condition may be expressed alternatively as
The second condition is:
c. (“Comonotonic additivity”)
If X and Y are comonotonic risks, then (X+Y) = (X) + (Y)
a. (Monotonicity w.r.t. “First Stochastic Dominance”)
If Prob(X a) Prob(Y a), aR then (Y) (X)
b. (“Estimability from empirical data” or “law invariance”)
It must be possible to estimate (X) from empirical data of X
If X and Y are “perfectly correlated”, then the risk of X+Y must be the sum of the risks of X and Y.
(X+Y) = (X) + (Y)
If “X is worse than Y in probability”, then its risk must be biggerThe measure of risk depends ONLY from the probability distribution of X and it is therefore estimable from empirical data of X.
A coherent measure of risk which is NOT estimable is WCE (Artzner et al. 1997)
A coherent measure of risk which is NOT Comonotonic Additive is semivariance
(Fischer 2001)
Liquidity Risk: when coherency violations make sense
When an asset position in a portfolio has a size which is comparable with the capacity of the market (market depth) of absorbing a sudden sell off, we are in presence of liquidity risk. Selling large asset amounts moves market bids downwards.
In this case clearly
Risk (A+A) = Risk (2 A) > 2 Risk (A) = Risk(A) + Risk (A)
Subadditivity fails )()()(
Positive Homogeneity fails 0 )()(
Convex Measures of Risk (or Weakly Coherent Measures of Risk)
(Monotonicity) if then
(Translational Invariance)
)()( aa )()(
Heath, Follmer et al., Frittelli et al. define a larger class of measures which allow for possible coherency violations due to liquidity risk.
(Positive Homogeneity) if then
(Subadditivity) )()()( 0 )()( (Convexity)
)()()(
10,
Weaker Condition
Our point of view: some care is needed
We believe however that in absence of liquidity risk, coherency violations are completely undesired for a measure of risk. The “small size limit” of a measure of risk should therefore be a (strongly) coherent measure of risk.
coherentstronglyXX
)(lim0
This observation
• Rules out measures of risk which are intrinsically non coherent in their analytical dependence from pdf’s.
• Forces a convex measure to carry possible coherency violations only through dimensional constants (typically the market depth di of each market’s asset Ai)
When each asset’s position is much smaller than its market depth we want the measure to be strongly coherent
Convex measures: a step forward ?
We are persuaded that convex measures of risk may represent a significant step forward in risk market practice provided that they respect the “small size coherent limit”. Otherwise, trying to take liquidity into account we may jeopardize the properties of coherency where it should hold in a strong sense.
A convex measure “beyond coherency” is therefore typically NOT a smarter formula which allows coherency violations, because it should be sensitive to positions sizes.
A convex measure “beyond coherency” is more probably a measure with a coherent analytical structure PLUS a database of each assets’ market depths to which the position sizes have to be compared in the search for illiquidities.
A natural solution
A natural way to define a convex measure satisfying the small size coherent condition is adding a coherent measure a liquidity charge
)()()( XCXX liquiditycoherentconvex
The liquidity charge C
• Apply to illiquid assets only and contain their dimensional market depths.
• Goes to zero in the liquid limit when all position becomes much smaller of its market depth.
We do not propose any specific modelling of the liquidity charge
Coherency and Convexity in short
Coherency of the Risk Measure
Convexity of the “Risk Surface”
Absence of local minima / Existence of a unique global minimum
Minimizing the Expected Shortfall
][
1:
)( )(][
1min))((min
N
iNi
w
N
wwX
NwXES
Let a portfolio of M assets be a function of their “weights” wj=1....M and let X=X(wi ) be its Profit & Loss. We want to find optimal weights by minimizing its Expected Shortfall
In the case of a N scenarios estimator we have
0
)( )(1
min))((min dppFwXES wXww
Notice: also in the case of non parametric VaR a SORTING operation is needed in the estimator and the same problem appears
PROBLEM ! A SORTING operation on data makes the dependence NOT EXPLICITLY ANALYTIC. Serious problems for any common optimizator.
The Pflug-Uryasev-Rockafellar solution
Pflug, Uryasev & Rockafellar (2000, 2001) introduce a function which is analytic, convex and piecewise linear in all its arguments. It depends on X(w) but also on an auxiliary variable
In the discrete case with N scenarios it becomes
)(1
)),(( wXEwX
N
ii
N wXN
wX1
)( )(1
)),((
Notice: the SORTING operator on data has disappeared. The dependence on data is manifestly analytic.
Properties of : the Pflug-Uryasev-Rockafellar theorem
Minimizing in its arguments (w,) amounts to minimizing ES in (w) only
Moreover the parameter in the extremum takes the value of VaR(X(w)).
))((min)),((min,
wXESwXww
))(()),((minarg wXVaRwX
The auxiliary parameter in the minimum becomes the VaR
(w) and ES(w) coincide but just in the minimum !
Properties of - linearizability of the optimization problem
A convex, piecewise linear function is the easiest kind of function to minimize for any optimizator. Its optimization problem can also be reformulated as a linear progamming problem
It is a multidimensional faceted surface ... some kind of multidimensional diamond with a unique global minimum
Minimizing a general Spectral Measure M
N
iNii
w
N
wwXwXM
1:
)( )(min))((min
The “SORTING” problem appears in the minimization of any Spectral Measure
1
0
)( )()(min))((min dppFpwXM wXww
Generalization of the solution of Pflug-Uryasev-Rockafellar
Acerbi, Simonetti (2002) generalize the function of P-U-R to any spectral measure. Also in this case it is analytic, convex and piecewise linear in all arguments. In general it depends however on N auxiliary variables i
In the discrete case it becomes
1
0
)(1
)())(),(( XEd
ddwX
N
iijj
N
jj
N wXj
jwX11
)( )(1
)),((
Properties of the generalized
Minimizing in all parameters (w,) amounts to minimizing M in (w)
Moreover, in the extremal, k takes the value of VaR(X(w)) associated to the quantile k/N.
))((min)),((min,
wXMwXw
w
))(()),((minarg wXVaRwXN
kk
Coherent Measures – based internal models ?
Spectral measures are
1. Universal
2. Global
3. Probabilistic
4. Expressed in lost money
But also at last
5. Coherent (namely real risk measures)
Are there still good reasons to use VaR ?
Let us see some issues more related to practical risk management aspects
Spectral Measures: worse statistical properties ?
Definitely not
1. The statistical error of Spectral Measures is easily computable and is by no means worse in general than VaR’s (Acerbi, Meucci, Tasche: in preparation).
2. It is not true that “ES is Extreme Value Theory and VaR not”. They are probably both EVT in most cases, and this depend mostly on the size of the chosen confidence levels.
More difficult probability modelling ?
No, identical situation to VaR’s.
One uses for spectral measures exactly the same models used in a VaR engine (parametric techniques, montecarlo, historical bootstrap, subjective scenarios, stress tests, … )
The risk measure computation is just the very last step of a long assembly line and requires the same input data.
Probabilistic hypotheses
Portfolios distributions
VaR
Spectral Measures
More serious computational problems ?
No, essentially the same situation as with VaR, however
subadditivity allows to split the “firmwide” capital allocation problem into subpreblems in separate subportfolios, which can drastically reduce the computational complexity but giving nevertheless prudential extimates.
BANK
business unit: Fixed
Income
business unit:
Equities
business unit: Forex
Spectral Measure
Spectral Measure
Spectral Measure
Optimization is more difficult ?
The contrary is certainly true !
Coherence in optimization problems implies convexity of risk measures.
Optimizing VaR on a portfolio of 100 assets may turn out to be a formidable NON-CONVEX problem of huge complexity.
Optimizing a Spectral Measure on a portfolio of thousands of assets is a problem that can always been faced since it is
1. CONVEX
2. LINEARIZABLE
Pflug, Rockafellar, Uryasev, (2000 –2001), Acerbi, Simonetti (2002)
Different Risk Measures in any bank ?
Regulating a bank industry where any bank could in principle use a different risk measure would prompt complications that are beyond the scope of our talk.
We note however that
1. No measure of risk fits any portfolio of a generic financial institution. The risk aversion function is a precision instrument which allows to better detect the risks of portfolios in different business styles (e.g.: insurance portfolios from bank portfolios …).
2. The possibility of providing market players with different risk measures could cancel the suspect that a unique measure of risk could prompt “flock effects” in the case of market crises.
However, we personally don’t believe that VaR has been a source for systemic risk in recent crises. Panic selling had been invented long
before !!
Conclusions
The space of Spectral Measures M provides the representation of a huge class of coherent measures suitable for applications. Coherent measures which are not spectral display properties which are undesirable in real life finance.
Any coherent measure of this space is in one-to-one correspondence with any possible rational aversion to risk of an investor.
For any spectral measure M a consistent estimator is available based on empirical data.
The practical application of any spectral measure is elementary and the conversion of a VaR engine straightforward.
Optimization of spectral measures is a convex and linearizable problem.
There’s nothing special with ES. It is just one out of infinitely many other spectral measures.
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