Electronic copy available at: http://ssrn.com/abstract=1410555

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Principal Measures of Investment Risk:The Price Density Function-a tool for finding

Volatility, Sensitivity, Duration, Convexity

Tinashe Mangoro*

(May 2009)

Abstract:This essay explores the link between the exponential probability density function and the present value function coupled with moment theory to derive important non-probabilistic parameters from the Present value function in which are then used to derive a measure of the volatility of interest rate and also that of the Prices. This is achieved by exploiting the mean-reverting characteristic of interest-rates in order to come up with a relationship between the two volatilities. The paper also looks at deriving a direct formula for the mean average of the term structure in relationship to its cash flow structure. We also look at measuring the price elasticity (PE) of the interest rate or yield accurately especially in the wake of price sensitive securities. In doing this we take great care in avoiding computationally complex methods, difficult mathematical derivations and discourse and thus try to come up with simple results which are highly applicable, tractable and also spot-on accuracy levels.

Keywords:Volatility, Duration, Price density function, Price distribution function, Convexity, Price generating function, Term structure of interest rates, Zerorising the price.

INTRODUCTIONThis is a paper which tries to cover some of the most important aspects of interest rate risk measurement and its effect on the value of assets and liabilities, cash flows in general. The risk-reward aspect of an investment is a very important element of finance and because of this, much focus is on returns (and interest rates) and also the time structure of the present value, hence we try to derive some important formulae such as volatility measures of both interest-rates and relative prices of given securities. We also touch on traditionally important parameters especially in interest-rate risk measurement these include Duration, Convexity and Yield Elasticity.

The paper structure is as follows Section I looks at the derivation of a density function for the present value of a cash flow stream and then defining the moments identified by this density function. Section II involves extending the theory obtain a volatility measure for the give term structure of interest rates. We then look at a measure of the weighted average term to maturity which has a dimension of time units. In Section III and IV we derive the relative volatility of the Price or Present value function and a sensitivity measure or price elasticity measure respectively.

Included also is an Appendix which will see us derive a new measure for Duration and Convexity which is dimensionless, we conclude with a summary of the main results of the paper. Again I reiterate on the computationally simple and tractability of the main results of the paper which makes it very user friendly and understandable to every one with interest in finance. The paper is not confined to valuation of bonds only but all securities which are valued using (interest rates) continuously compounded to find the present value on a finite infinite time scale.__________________________* tinashemangoro@gmail.com, 4 ebony rd, Rhodene, Masvingo, Zimbabwe, +2639(0)39 264 547

Electronic copy available at: http://ssrn.com/abstract=1410555

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I. The Price/PV density function

Let the Price of an investment of 1 today as at time t as:

tetP ),( (1)

Where δ = ln(1+i) the force of interest and t is the time to maturity.

From this price function we can derive the price density function of P by applying and solving simple differential equations. This is how we can derive the price density function. Given that the price is a two variable equation we can extract two derivatives:

(i) te

t

P , and (ii)

tteP

(2)

Let us solve (i) Assuming that both δ and t take values from 0 to ∞

teP tt (3)

Solving the differential equation gives us

teP tt

teP tt

(4a)

This also gives us a solution for (ii)

tteP (4b)

Taking limits t (for PVt) and δ (for PVδ) from 0 to ∞ we get,

1},0{

ttPV , and 1

},0{

PV

As you can see the total mass of this price measure(s) is –1, and define the presiding negative density functions (neg(.)) as:

tt enegf , and ttenegf

},0{, t

In short we can define Price Density Function (pdf) to be:-

t

tt eInegIf ,, )(

(5)

Where It,δ is an indicator function such that if the random variable is taken to be time (t) then interest rate (δ) becomes the parameter and vice-versa. This is identical to the exponential

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probability density function though ours is a price density function; let us see how they differ mathematically.

Now what is the effect of the negative sign to the valuation of P using the negative Price exponential density function? Here is an explanation. What we have done here is we have induced the negative sign into the limits, that is, instead of integrating from 0 to ∞ we are now integrating from ∞ to 0 and the effect to the solution of the differential equation of this reversevaluing is that, PV(X = x) is now valued from x to ∞ instead of from 0 to x.

Since (5) is identical to the exponential probability density function all other properties and definitions are also identical we are not going to dwell much into that, but what we are going to do is to derive important first and second moments of the interest rate and time to maturity using this price density function.

Finding WAT and DMF by Density Mapping

Suppose we multiply each price density function buy a cash flow ck , also (t, δ)→(tk, δk) and by the linearity property of the exponential function and the fundamental theorem of calculus that the sum of differentiable functions is also differentiable we sum them over k thus we will have:

n

k tkktk IFcM

1 ,, )( k = 1, 2…, n

Where F(It,δ) is the price distribution function such that )()( ,, tt IFIf , actually )( ,tk IF

describes the distribution of value of cash flows {ck} from initial date to maturity date ,that is, the intrinsic value of the investment being paid out at times {tk} valued at spot interest rates {δk}.

Using moment theory and the concept of weighted averaging we get the mean of time (tk) and (δk)

n

k

tkk

n

k

tkkk

nkkfc

ftctE

1

1},1{

)( , and

n

k kk

n

k kkk

nkkfc

fcE

1

1},1{

)(

(7)

Where kktk

tk ef and kk t

kk etf .

we then end up with

kk

kk

tk

n

k k

n

k

tkkk

nkkec

etctE

1

1},1{

)( , and 1 (8a)

n

k

tkk

n

k

tkkk

nkkkk

kk

etc

etcE

1

1},1{

)(

(8b)

__________________

1 see appendix A for a discussion on how we get to E(tk) and E(δk)

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Now define },1{

)(nkktE

as the WAT, the Weighted Average Mean Term to Maturity and

},1{)(

nkkE

the discounted mean force of interest-rate/ Yield (DMF). Before we discuss much

about these two quantities let us take a quick non-descriptive pick at the second moment about these pdf’s.

Suppose we denote DMF = F and WAT =T, then we can extend the above moments to have the variances as:

2

1

1

2

},1{)( T

ec

etctV

kk

kk

tk

n

k k

n

k

tkkk

nkk

(9a)

And

2

1

1

2

},1{)( F

etc

etcV

kk

kk

tk

n

k k

n

k

tkkk

nkk

, which from now on will be termed DYV the Discounted

Yield Variance. (9b)

This paper shall mainly focus on the interest-rate related issues so we will not get that much into the time random variable but we will deal with it later in the paper, as for now let us look at DMFand DYV since they are significant in interest-rate risk measurement.

The Discounted Mean Yield

We have derived the mean of the interest rate structure {δk = ln(1 + ik): k = 1, 2…n} where ik is the interest rate prevailing at time (tk).

n

k

tkk

n

k

tkkk

kk

kk

ect

ectF

1

1

, that is

},1{)(

nkkE

(8b)

F is itself a force of interest. This can be seen intuitively from the formula.

Hence if you substitute k with F in the equation kktn

k kkk ectP 1

),( that is ),( ktFP you

will get the present value. In short F will give you the yield to maturity for a bond or the IRR for a project or the mean rate of return, mean-WACC and so forth, whatever type of interest-rate or discount rate you will be using it will give you a most precise mean average of that.

No matter how complex the structure of the PV/ price basis’ structure might be, F will give you very accurate measures of 99-100% depending on the flactuality of the rates, cash flow structure and time structure. F has no limitations to this layout, meaning that there is an allowance for;

Cash flows to be the same or different by any magnitude between points (tk) in time. Time periods (tk) to be consecutive or non consecutive. Interest rates may be constant or variable to any degree between time periods.

This allows us to compute easily mean rates for very complex ),( kk tP functions which usually

require iteration processes or interpolation or a computer software or some algorithm, one can see

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how this formula for F greatly (or conveniently) short cuts these method. For Example (i)suppose we have a cash flow stream with c1 = 2, c3 = 6, c5 = 8, and c7 = 3 where the year on year rates of return are year1 = 0.01, y3 = 0.03, y5 = 0.02 and y7 = 0.01 how would one calculate the annual mean rate of return for the seven year period?

Solving for this would call for any of the above methods which are usually not user friendly if done manually and may require solving a polynomial which yields seven solutions (some of which complex or negative ) of which we all know no straight formula exists for the solution of polynomials of order 4 and above. Yet a simple application of F (formula 8b) will give you the answer as: F = 0.019015 the corresponding iF = eF – 1 = 0.019289 which gives you ),( kk tP =

),( kF tiP =$17.52 a 100% accurate figure done on a handheld scientific calculator within a

minute.

So this is the formula for mean interest rate for a given interest rate structure, now you do not have to use interpolation of guessed figures or trial and error methods, do it directly with F,furthermore it incorporates the cash flows and time structure. They are many advantages of using this formula and also many of its uses but we are not going to dwell much into that. Next let us look at the DVY.

II. The Discounted Yield Variance (DYV)

Statistically this means we will be trying to find the average or mean distance of each spot rate from the mean, that is, the scale of deviance. Mathematically the square-root of this quantity is called the standard deviation and in finance can be described as the volatility (in this case of the interest-rate structure k for this given cash flow system). Defined

n

k

tkk

n

k

tkkk

kk

kk

ect

ectFDYV

1

1

2)(

Where F=DMF

Simplifying the formula is an easy process which will find you with this expression:-

2

1

1

2

DMFeCt

eCtDYV

n

k

tkk

n

k

tkkk

kk

kk

(9b)

You see how nicely it conforms to statistical theory. The square root of DYV is a standard deviation and will surely give us a measure of the variability of interest-rate over t1 up to tn the given investment period. At a snapshot the higher this value is, the more interest rates fluctuated meaning the more risky the investment and vice-versa. Let’s denote this standard deviation as σr

then:

2

1

1

2

DMFeCt

eCtn

k

tkk

n

k

tkkk

rkk

kk

(10)

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For the example we have looked at before we will have DYV = 0.5136%% and the standard deviation of the interest rates is σr = 0.7167%. Much literature agrees on the fact that the standard deviation of returns is a measure of the volatility of the returns hence we can use this fact to define σr above as a measure of interest rate volatility pertaining to a particular PV-space. The importance of this value is well known and ranges from modelling interest rates to pricing of interest sensitive securities, portfolio structuring not to mention assertion of the riskiness of an investment and so forth.

Of course we can use this method of moments on the pdf to easily calculate the skewness or excess kurtosis of interest rates (which are very important in downside risk measurement) but this is beyond the scope of this paper. Most importantly we are going to use this formula to derive a measure of the volatility of the present value of cash flows for which we have derived DMF and DYV to do this we need one more parameter which we are going to look at next. As you have already seen all the parameters we have derived (and those coming forth later) are not conformed to bonds only but cover all securities valued by the PV function ),( kk tP .

The Weighted Average Term to Maturity (WAT)

We have already defined WAT as:

n

k

tkk

n

k

tkkk

kk

kk

ec

ectT

1

1

, that is

},1{)(

nkktE

(8a)

WAT measures the average payback period of a set of cash flows. What this formula tells us is simple, it only tells us that if the price or present value of any cash flow set {ck} no matter how complex valued by a term structure r(t) giving a force of interest k with a term {tk: k = 1, 2…n}

then the price or present value can be correctly approximated by replacing each tk with T -the WAT. Mathematically we are saying:-

Given a cash flow stream { ),( kk tc ; k = 1, 2…, n}, with the present value given by:

n

k kkkk cbtP1

)( , where kktk eb

Then

),()( TPtP kkk , where T=WAT

Like the P(F, tk) approximation, the P(δk, T) approximation is also very accurate with accuracy levels ranging also from 99-100%, the modal level being a 100%. If you do a check with the previous example where we calculated F, you will find WAT to be T = 4.0159 years givingP(δk, T= 4.0159) = $17.52 a 100% accurate Price.

We are not going to look at the variance of tk since it is not of real significance to this paper. We shall look now at how both T and F are very important tools in determining the sensitivity space of the present value. Here we are going to look at a simple concept I will call zerorising (or zeroing) the cash flows ck which will in turn lead us to finding a measure for the volatility of

),( kk tP (the price/present value) and also its elasticity measure due to interest rate movement.

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III.The Present Value or Price Volatility

Now that we have the three tools needed to derive a measure for the price volatility namely the Discounted Mean Force of interest/Yield (F), the Weighted Average Term to Maturity (T) and the Discounted Yield Variance (DYV) we are almost there, what now is left is the zerorising part.

In definition it is a term that can be used to describe the transforming of

n

k kkkk cbtP1

),(into an equation of value with a single cash flow A and a standard singular and constant time to maturity N valued with an interest rate Ij which is variable. For example suppose a Bond has a

price of

n

k

tkkk

kkectP1

),( then (zerorising/zeroing) it would be transforming it to a zero

coupon bond-hence the term zerorising. Zerorising is important for the obvious reasons with the main one being to achieve computational simplicity achieved by the smoothed exponential graph of the zerorised function.

The Zerorised position

Now here is how we achieve a zerorised position for:

n

k kkkk cbtP1

),( , where kktk eb

We have seen in the previous sections (page 1 and 3) that ),( kk tP can be approximated by

),( ktFP and also by ),( TP k to a very accurate degree for any set of cash flows and term

structure this is because F and T are both means ( or averages) to the time and interest-rate variables defining ),( kk tP .

Now consider the ),( TP k approximation, this is defined as:

n

k

Tkk

kecTP1

),(

In order to achieve what we want we need to make a simple improvisation on this equation. We know that F is an average mean of kk so we can safely substitute k with F=fj but instead of

taking it as a constant we induce it into the equation as a variable fj representing the mean force of interest for term structure j or state j. So we have:

n

k jkj TfcTFP1

)exp(),(

)exp(...)exp()exp( 21 TfcTfcTfc jnjj

)...()exp( 21 nj cccTf

n

k kj cTf1

)exp(

)exp( TfA j Where

n

k kcA1

(11)

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Hence we have found the zerorised version of ),( kk tP that is ),( TfP j a smooth exponential

function by which we can find statistics such as Macaulay duration, convexity, and so forth-we will look at this later in the appendix. Now we are going to use the { ),( TfP j ; fj} graph to try

and come up with a simple volatility measure for ),( kk tP . From the graph we can establish the

relationship between the variance of interest-rate and that of the corresponding present value (or price).

Note that ),( TfP j describes the PV of a set of cash flows {ck} which have duration or weighted

term to maturity T whereby the spot rate structure {ik: k = 1, 2…n} is defined by F = fj in such a way that if any of the spot rates changes in such a way that F alters say to fm, T remains the same.

Now we look at the DYV which is measures the variance of interest-rate in relation to the cash flow stream and the term structure to which its present value is valued over. The standard

deviation from the mean is DYVr this means that the mean distance of each k from F

is r now if try to map this to the { ),( TfP j ; fj} curve, and if the average (mean rate) for k is

F then the price is approximated by P(F,T) lets use P(F) for short ,(see exhibit (i)). In the context of DYV, F is bounded above and below by a distance of r , that means P(F) is also bounded by

P(F+ r ) and P(F – r ).

Lets define d1 = P(F) – P(F+ r ) and d2 = P(F) – P(F – r ) as the distances or deviations of

P(F+ r ) and P(F- r ) from P(F) ,see exhibit (i) below.

Exhibit (i)The graph of a zerorised PV

Price/PV

P(f) = Aexp(-f ×T)

P(F- r )

d1

P(F)

d2

P(F+ r )

F- r F F+ r DMF (f)

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We can see from the curve that due to the exponential effect of the curve, the standard deviation from the mean rate F is symmetric but the implied d1 and d2 are asymmetric about the standardized present value P(F). The next step is establishing a connection between d1 and d2

with the variance of interest rate, to do this we need to define a quotient λ which is derived from the following statement of variation.

“The product of the distances a1 = F – (F– r ) and a2 = F – (F+ r ) varies directly with the product of the distances d1 and d2.”

Mathematically:

d1 × d2 a1 × a2

This means that a constant λ such that:

d1 × d2 = λ (a1 × a2) (12)

Solving for λ we get:

λ 21

21

aa

dd

For computational reasons let us deal in absolute values, ignoring the signs,

2

2

1

1

a

d

a

d

Taking into consideration the concept of mean reversion of interest-rates that is the tendency of interest rates to revert to a long term mean leads us to take limits of λ as 0r hence:

)(

)()(

)(

)()(00

r

r

r

r

FF

FPFp

FF

FPFpim

(13)

)(

)()(

)(

)()(00

r

r

r

r

FF

FPFpim

FF

FPFpim

=

2)(

df

fdP

We know that )exp()(

TfTdf

fdP hence

2)exp( TfT

λ = (T×P(F))2 (14)

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Note: we can take divide through by P(F)2 and be left with λ = T2 to work with percentage/relative figures.

Now that we have established a relativity quotient for the ratio between the volatility (or variance) of the interest-rates (that is a1 × a2) with that of the relative/ implied product of the distances d1

and d2 (Note that as σ→0 then d1 – d2 →0 also, tempting us to define that the product d1 × d2 → a price/PV- variance as σ→0).

Now since we have found λ by taking limits as σr→0 we can denote d1 × d2 as a volatility measure but not as a variance (though I am very tempted to do so), hence denoting d1 × d2 as we substitute them into equation (12) and get:

λ 21

21

aa

dd

rr

Solving for γ we get,

22 ))(( fPTr (15)

Or better still by dividing both sides by P( f )2 we get a relative figure ,

2)( Tr , or DYV×WAT2. (See Appendix B for Convexity = WAT2) (16)

r -Standard deviation of interest rates and T - WAT (The Duration).

So there we have it a measure of Price Volatility constructed from the present value of a set of cash flows, in most cases if not all practitioners use the standard deviation as the volatility measure so its more refined if we use the square-root of γ as the volatility measure. This formula agrees with the known fact of the relationship between Price Volatility and Duration, that is, “longer-term bond prices fluctuate more than prices for short-term ones”.

If you look at the volatility formula γ = (σr×T) 2 you will see that if the WAT increases then γ also increases, the reverse is also true. It is also the same relationship with interest rate variance σr of which it is logically true. So if an investor is concerned with controlling or subsidizing γ in an environment where interest rates are highly volatile then he could go for short-term securities to offset the high σr similarly in long term investments stable interest rates may lower the price/PV volatility (γ).

This formula, in summary, will work well especially in risk management whereby one may want to describe the relationship between Price or Value risk with factors like Term to Maturity, level of variability of the term structure, and so forth, portfolio selection-probably the financial manager would prefer projects or investments with a higher/lower γ depending on whether (s)he is risk averse or not, it can be used as a parameter in Asset and Security pricing, the list goes on.

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One advantage or merit of this (and other formulas in this paper) is its tractability and computational simplicity.

Since volatility is usually given as a standard deviation we can use the square root of γ as the implied volatility measure that is:

= σr×WAT (17)

This can be confirmed by the use of stochastic pricing, which I will not dwell much into but will briefly describe in order to make the reader see how this volatility measure confirms with existing theory.

Consistency with the Cox-Ingersoll-Ross (1979) Framework

Now if one is familiar with stochastic mathematics and traditional one factor models for interest rate modeling (s)he will agree with me that if the entire movements of the term structure are governed by the continuously compounded short-term interest rate, r (.), which evolves according to the stochastic differential equation tdWtrtdttrtmtdr ))(,())(,()( . The market price of

risk or r-risk is a function λ(t,r), such that the risk adjusted short rate drift is ),(),(),(),(~ rtrtrtmrtm .The price of a zero-coupon bond (though ours is not

necessarily a zero-coupon bond but any set of cash-flows whose present value has been zerorised) is a function P(t, T, r) of the short rate and, by using Itồ’s Lemma you will find that the relative (Price) volatility v(t, T, r) is:

),(),,(

),,(

1),,( rt

r

rTtP

rTtPrTt

(18)

You can readily see that

r

rTtP

rTtP ),,(

),,(

1, is a proxy for WAT (see Appendix B) and

),( rt , is the volatility proxy for σr in our formula for γ.

So one can see how efficient and significant the γ-volatility measure is in determining the relative volatility of the present value due to underlying interest rate volatility, even stochastic diffusion pricing methods agrees with our theory.

Again let us use our example to determine the price volatility, therefore γ = WAT2×DYV =4.01592 ×0.00005136 giving us 0.000828 and using equation (17) we have the volatility v = 0.02878 and in monetary value we have $17.52 × 0.02878 = $ 0.504 and thus we have a price volatility of about 50 cents for that particular investment.

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IV. Price Elasticity Conventionally it is known that the percentage change in price = duration change in yield:

DdP

dP , or 1)1( iDdi

P

dP (note the constant interest-rate)

Greater precision of this bond’s responsiveness to yield shift (dδ) by also accounting for convexity .Using the Fundamental Property of calculus which states that any mathematical function can be approximated by a Taylor(or Maclaurin) series.

This will bring us to:

2)(2

1)( dCdD

P

dP (19)

Where k

k

tn

k k

tk

n

k k

ec

ectD

1

1 , andk

k

tn

k k

tk

n

k k

ec

ectC

1

1

2

kk ,

Duration and Convexity respectively, by taking account of both quantities in calculating this bond responsiveness one would have assumed the non-linearity of this change in price due to change in yield hence giving better and more accurate approximations than just using duration alone.

Now without discrediting this method, lets look at an alternative method of finding this % change in the PV due to shift in interest-rate. With this method one does not have to calculate convexity or use the Taylor expansion its very user friendly and will give you a direct description of the curvature structure of the bond price (and responsiveness/sensitivity) and its yield-besides the method is just as accurate as equation (18) only easier to use.

The Alternative: the , φ method.

This method stems out of zerorising the PV function (see page 7), such that we can easily find this change. The zerorised estimate for price or PV is:

TFF iAiP )1()( , with T=WAT

We are using iF=eF-1 where F = DMF

If the interest-rate moves from iF to j then the % change in Price / PV is given by:

)(

)()(

F

F

iP

jPiP

)(

)(1

FiP

jP

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T

Fi

j

1

11

-see how the method describes the exponential convexity

T

F

j

i

1

11 (20)

That’s the formula for the % change in P due to the change in interest-rate change from i to j.The actual change factor () is given by:

1

T

F

j

i

1

1 (21)

So we simply have P(j) = × P(iF) (22)

Let us look at this numerical example for price elasticity so as to see how accurate this method is:

Example (ii)

For a bond with a coupon rate of 8% payable in arrears, redeemable at par after 10 years, valued at 5% an interest rate. Create a table to show the accuracy levels of the two methods of sensitivity measures of Price to yield shift.

Solution

The fair price of the Bond is:

1010 1008)05.0( vaP

= $ 123.17

Since i= 0.05 is constant throughout the term Duration = WAT = 7.54 years and C = 51.98.Lets denote Duration = D and WAT = T and Convexity = C.

Actual % change in price = )05.0(

)()05.0(

P

jPP

Duration Method (d) = -(j − 0.05)D (1.05)-1+0.5C×(j − 0.05)2

WAT Method ( ) = 11

05.1

T

j

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*Accuracy level is in bracketsTable for % change from i = 0.05 to j

j Actual % Change D-C Method (d) WAT Method (φ)0.2 -59.65 -49.24 (82.5% accurate) -63.46 (94% accurate)0.09 -24.02 - 24.97 (96.2%) -24.57 (97.8%)0.08 -18.81 -19.20 (98.1%) -19.14 (98.3%)0.07 -13.11 -13.33 (98.4%) -13.26 (98.9%)0.06 -6.86 -6.92 (99.1%) -6.89 (99.6%)0.05 0 0 (100%) 0 (100%)0.04 7.53 7.44 (98.8%) 7.48 (99.3%)0.03 15.82 15.41 (97.4%) 15.60 (98.6%)0.02 24.95 23.88 (95.7%) 24.43 (97.9%)0.01 35.02 32.88 (93.9%) 34.02 (97.1%)

From the table you can see how the WAT method maintains its accuracy levels even in the wake of an extreme value like in this example a change from 5% to 20% was approximated real good (94%), as you can see the φ method loses its accuracy at a much slower rate than the duration-convexity method in this example it is most apparent when interest rates are falling.

This is probably because of the linearity of the duration-Convexity method though the introduction of convexity forces some curvature approximation to this price-shift, one can use higher order factors of the Taylor (or Maclaurin) series to get better approximations but these tend to make the computation of this sensitivity more rigorous. Hence a good alternative would be to use the φ method which is more accurate and very simple to calculate, let us conclude this section with an illustrative example (using Example (i) we have been dealing with previously).

Suppose shocks in year 3 and 7 result in the relative spot rates (of those years) shifting in such a way that the resulting DMF (Fnew) differs from the DMF in (ii) by +165 basis points, calculate the resulting percentage change in the Present Value of the Assets.

Now if on average interest rates move up 165 basis points then it means we are calculating effect of change from iF to iF + (165/1002). Hence effect of movement of interest rates fromiF = 0.019289 to inew = 0.035789 where (T is WAT ) is:

T

new

F

i

i

1

11

06245.0

035789.1

019289.11

0159.4

So we expect a PV decrease of 6.25%.due to that increase in interest-rate

nashfiles 15

V.CONCLUSION

This brings us to the end of this short note on measuring the Volatility of both the term structure and the Present /Prices of securities. We also looked at the mean averages of both the time variable and the interest rate variable which apart from being very important in their own right have also been very useful in deriving the volatility measure for price arising from interest-rate fluctuations. A point to note is the intersection use of the price density function and centre of moment theory in the first section of the paper in deriving the formulae which form the basis of this paper-and thus the results of this paper are of very accurate design and exposition, easy to accept and use (though one may attempt successfully to use calculus to derive WAT, DMF andDYV) .This paper has been non-descript with regards to most of the results in this paper becausethe mathematics involved are not very discrete and definitely not very complex with every measure taken to ensure that its understandable by everyone at every level and also practically employable. In addition to this conclusion let us summarize the results derived in this essay.

The exponential price density function ttt eInegIf ,, )(

2

The Discounted Mean Yield/Force of interest

n

k

tkk

n

k

tkkk

kk

kk

ect

ectDMF

1

1

The Weighted Average Term to Maturity

n

k

tkk

n

k

tkkk

kk

kk

ec

ectWAT

1

1

The Discounted Yield Variance 2

1

1

2

DMFeCt

eCtDYV

n

k

tkk

n

k

tkkk

kk

kk

The γ-volatility measure for price 2)( Tr Or DYV×WAT2

The φ sensitivity/Elasticity measure

WAT

new

old

DMF

DMF

1

11

_______________________

2 f(It,δ) = neg(It,δe-δ×t) means negative of exponential density function.

nashfiles 16

Appendix A

Finding the mean averages from the price density function

We have derived the price density function as:

ttt eInegIf ,, )(

And by reverse valuing due to the negative sign which makes us value the pdf from ∞ to 0 we

have the following property:

t

sdsetSPVtSPV )()( , and is the price generating

function G(It,δ), define F(It,δ) the distribution function, hence we calculate the mean:

0 ,,

0 ,,,

,

)(

)()(

tt

ttt

t

dIIF

dIIFIIE , but 1)(

0 ,,

tt dIIF

Let us take the special case in which (t, δ) → (tk, δk) for all k = 1, 2…, n hence summing kkt

ttktk eIIfIF

,,, )()( over k becomes a discrete function. Such that if we include a cash

flow ck for each kth-density function we get a sum-price density function Mk :

n

k kk

n

k kkk fcFcM11

And the price/PV becomes the sum-price generating function P

n

k

tk

n

k tkkkkkecIGcP

11 , )(

It follows that the mean Ek(It,δ) becomes

n

k tkt

n

k tkt

k

tk

tIFI

IFI

M

MIE

1 ,,

1 ,,,

,)(

)()(

, where

,

,,

)()(

tk

tkktk Id

IFdIF

Since )(')(' ,, tt IGIF

n

k

tkk

n

k

tkkk

nkkfc

ftctE

1

1},1{

)( , and

n

k kk

n

k kkk

nkkfc

fcE

1

1},1{

)(

(7)

Where kktk

tk ef and kk t

kk etf .

kk

kk

tk

n

k k

n

k

tkkk

nkkec

ecttE

1

1},1{

)( , and

n

k

tkk

n

k

tkkk

nkkkk

kk

etc

ectE

1

1},1{

)(

Thus we have derived the mean time and mean force of interest rate.

nashfiles 17

Appendix B

Duration and Convexity in the context of sensitivity

We have established that the WAT (the Weighted Average Term To Maturity) is a quantitywhich is absolutely measured in time units and in its own right can not be used to measure the elasticity of Price due to shift in interest rates. Now let us derive a measure of this elasticity which is dimensionless and can be used very much like the Macaulay Duration DM or the Fisher-Weil Duration DF , since in definition these two (and many other duration types) measure the responsiveness of a security’s market price (intrinsic, or present value ) to a change in the underlying interest rates.

In short given ),( kk tP then the Duration in found by:

n

k

tk

n

k

tkk

k

kk

kkF

kk

kk

ec

ectk

d

tdP

tPD

1

1),(.

),(

1

(23)

This is known as the Fisher-Weil Duration, and if δk = δ k then DF becomes DM that is:

n

k

tk

n

k

tkkk

kM

k

k

ec

ectk

d

tdP

tPD

1

1),(.

),(

1

(24)

Now let us use the same technique of using derivatives of calculus to find our duration. Now we have found out that ),( kk tP can be accurately approximated by zerorising it to

)exp(),( TfATfP jj , where

n

k kcA1

a constant

With fj chosen such that fj = F makes ),( TFP the nucleus Price for a neighborhood of prices bounded within F ± σr we can find duration:

j

j

jT df

TfdP

TfPD

),(.

),(

1

)exp(.),(

1TfTA

TfP jj

TTfA

TfAT j

)exp(

)exp(

Since T = WAT we then have:

nashfiles 18

n

k

tkk

n

k

tkkk

Tkk

kk

ec

ectD

1

1

(25)

as a measure of duration, there are some points to note though:1. DT is dimensionless.2. If δk = δ k then it becomes identical to the Macaulay Duration.3. It is a measure of PV elasticity due to interest rate movements.

Now let us look at the convexity of ),( kk tP which we shall denote, is an approximation to the

curvature of the PV-yield curve. It describes the incremental price change misestimated by the duration and is defined by:

2

2 ),(.

),(

1

k

kk

kkF d

tPd

tPC

As before for CM we just replace δk by the constant δ. Now we want to derive a convexity measure, CT in the context of the zerorised ),( kk tP hence as before we have.

2

2 ),(.

),(

1

j

j

jT df

TfPd

TfPC

)exp(.),(

1 2 TfTATfP j

j

This eventually becomes:

2TCT

Hence we have derived a duration and convexity measure for the present value function ),( kk tP , which are DT and CT that is:

n

k

tkk

n

k

tkkk

Tkk

kk

ec

ectD

1

1

, the duration

2

1

1

n

k

tkk

n

k

tkkk

Tkk

kk

ec

ectC

, the convexity (26)

Hence the identity CT = (DT)2

nashfiles 19

Deriving duration using calculusWe can also confirm the above duration (in the context of dimensionless duration that is as a sensitivity measure) using pure calculus. The idea behind this derivation is treating the present value as a function of two variables (of time and interest-rate). So we say:

Change in PV due to small shift in interest rate ≈ relative Change in PV due to small shift in time and interest rate divided by the relative change in PV due to small shift in time only.

Using partial derivatives we have:

t

P

p

t

P

PP k

1

1 2

, this gives us

n

k

tkk

n

k

tkkk

Tkk

kk

ec

ectD

1

1

Hence we come to the same result. One may also be interested in the modified duration version of this formula since it is more commonly used in interest risk management strategies. This is how we can derive it:

Say instead of using P(δk, tk) we use straight interest rates that is, P(ik tk) that is :

n

k

tkkk

kictiP1

)1(),(

So we need kk

kk

ti

tiP

),(2

and k

kk

t

tiP

),(

Let us start by findingk

kk

t

tiP

),(

, to do this we have to employ a simple trick.

Let ktkkk icD )1( such that

n

k kk DtiP1

),(

kkkk tcD lnln , where )1ln( kk i

Hence

kk

t

D

ln

→ kkk D

t

D

This means

n

k

tkkk

n

k kkkicD

t

P10

)1(

nashfiles 20

Also:

kkkk

kk

t

P

iti

tiP ),(2

This gives us:

n

k

tkkkk

kk

kk kictti

tiP1

)1(2

)1(),(

Since we now have the two derivatives their ratio gives us modTD as:

n

k

tkkk

n

k

tkkkk

Tk

k

ic

ictD

1

1

)1(

mod

)1(

)1(

(27)

Hence we now have our durations in the context of sensitivity measurement for spot-rates.

Change in WAT due to change in interest rate

The change in WAT due to change in interest rate is a measure of convexity for the present value function in relationship to its term structure and is very important in risk analysis especially when comparing two or more securities. This value is very important in immunization of funds since it gives a measure of how the price will respond in the wake of a non-parallel shift in the interest rate structure, it will appeal more to an investor who is worried about slope changes in the term structure. Thus if we expect slope changes in the term structure and cannot determine which direction they will move then a high value will mean greater risk-exposure. This is how we can come up with it.

In addition to that, one can use it as a test to see how much interest rates will have to shift before we can have to adjust WAT in the P(δk, T) approximation, a small value is preferable since it means a greater range of fj’s can be used (the set of DMF’s) can be used before the approximation starts to loose accuracy.

n

k

tkk

n

k

tkkk

ttkk

kk

eC

eCtWAT

1

1

Simple derivative calculation gives us:

2

1

1

2

WATeC

eCtn

k

tkk

n

k

tkkk

kk

kk

, where

22 )()(

k

kt

k

k

kt

k

P

PP

P

PP 3 (28)

_________________________

3Pk = ∑cke-tδ i.e. the PV and Pk

t = ∑tkcke-tδ i.e. sum-moment of PV about time.

nashfiles 21

References

-Berk, Jonathan and Peter DeMarzo. 2007. Corporate Finance. New York: Pearson/Addison Wesley.

- Bierwag, Gerald O. 1987. Duration Analysis: Managing Interest Rate Risk. Cambridge:

Ballinger.

- Bodie .Z, A .Kane and A. J. Marcus (2002).Investments, 5th Edition. New York, McGraw Hill/Irwin

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- Frank j Fabozzi, Bond Markets, Analysis and Strategies, New Jersey, Prentice-Hall

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- Jasper Lund, Dynamic Models of Term Structure of Interest Rates.

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- Richard .F .Bass, Probability Theory Notes.

-Timothy. F. Crack and Sanjay. K. Nawalkha, Common misunderstandings concerning duration and convexity, Journal of Applied Finance.