The price density function, a tool for measuring investment risk,volatility and duration

7/28/2019 The price density function, a tool for measuring investment risk,volatility and duration

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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 presentvalue function coupled with moment theory to derive important non-probabilistic parametersfrom the Present value function in which are then used to derive a measure of the volatility ofinterest rate and also that of the Prices. This is achieved by exploiting the mean-revertingcharacteristic 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 inrelationship to its cash flow structure. We also look at measuring the price elasticity (PE) of theinterest 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 mathematicalderivations 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, Pricegenerating 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 riskmeasurement 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 wetry to derive some important formulae such as volatility measures of both interest-rates andrelative prices of given securities. We also touch on traditionally important parameters especiallyin 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 thepresent value of a cash flow stream and then defining the moments identified by this densityfunction. Section II involves extending the theory obtain a volatility measure for the give termstructure of interest rates. We then look at a measure of the weighted average term to maturitywhich has a dimension of time units. In Section III and IV we derive the relative volatility of thePrice 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 Convexitywhich is dimensionless, we conclude with a summary of the main results of the paper. Again Ireiterate on the computationally simple and tractability of the main results of the paper whichmakes it very user friendly and understandable to every one with interest in finance. The paper isnot 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.__________________________* [email protected], 4 ebony rd, Rhodene, Masvingo, Zimbabwe, +2639(0)39 264 547


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I. The Price/PV density function

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

tetP ),( (1)

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

From this price function we can derive the price density function ofPby applying and solvingsimple differential equations. This is how we can derive the price density function. Given that theprice is a two variable equation we can extract two derivatives:

(i)

tet

P, and (ii)

tteP

(2)

Let us solve (i)Assuming that both 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(forPVt) and (forPV) from 0 to we get,

1},0{

tt

PV , and 1},0{

PV

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

tt enegf , and ttenegf },0{, t

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

ttt eInegIf ,, )( (5)

Where It, is an indicator function such that if the random variable is taken to be time (t) theninterest rate () 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 differmathematically.

Now what is the effect of the negative sign to the valuation ofP using the negative Priceexponential density function? Here is an explanation. What we have done here is we haveinduced 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 reversevaluingis that,PV(X =x) is now valued fromx to instead of from 0 tox.

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

Finding WAT and DMF by Density Mapping

Suppose we multiply each price density function buy a cash flow ck , also (t, )(tk, k) and bythe linearity property of the exponential function and the fundamental theorem of calculus that the

sum of differentiable functions is also differentiable we sum them overkthus we will have:

n

k tkk

t

k IFcM 1 ,, )( k= 1, 2, n

Where F(It,) 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, theintrinsic value of the investment being paid out at times {tk} valued at spot interest rates {k}.

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

nk

t

kk

n

k

t

kkk

nkk

fcftctE

1

1

},1{)( , and

nk kk

n

k kkk

nkk

fcfcE

1

1

},1{)(

(7)

Where kkt

k

t

k ef

and kktkk etf

.

we then end up with

kk

kk

t

k

n

k k

n

k

t

kkk

nkk

ec

etctE

1

1

},1{)( , and 1 (8a)

n

k

t

kk

n

k

t

kkk

nkkkk

kk

etc

etcE

1

1

},1{)(

(8b)

__________________

1see appendixA for a discussion on how we get to E(tk) and E(k)


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Now define},1{

)(nkk

tE

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 aboutthesepdfs.

Suppose we denoteDMF = Fand WAT =T, then we can extend the above moments to have thevariances as:

2

1

1

2

},1{)( T

ec

etctV

kk

kk

t

k

n

k k

n

k

t

kkk

nkk

(9a)

And

2

1

1

2

},1{)( F

etc

etcV

kk

kk

t

k

n

k k

n

k

t

kkk

nkk

, which from now on will be termed DYVthe Discounted

Yield Variance. (9b)

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

The Discounted Mean Yield

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

n

k

t

kk

n

k

t

kkk

kk

kk

ect

ectF

1

1

, that is},1{

)(nkk

E

(8b)

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

Hence if you substitute k with Fin the equationkkt

n

k kkkectP

1),( that is ),( ktFP youwill get the present value. In short Fwill give you theyield to maturity for a bond or theIRR for aproject or the mean rate of return, mean-WACC and so forth, whatever type of interest-rate ordiscount 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, Fwill give you

very accurate measures of 99-100% depending on the flactuality of the rates, cash flow structureand time structure. Fhas 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 yearrates of return areyear1 = 0.01, y3 = 0.03, y5 = 0.02 andy7 = 0.01 how would one calculate theannual 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 ofwhich complex or negative ) of which we all know no straight formula exists for the solution ofpolynomials of order 4 and above. Yet a simple application ofF(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 nothave 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 usingthis formula and also many of its uses but we are not going to dwell much into that. Next let us

look at theDVY.

II. The Discounted Yield Variance (DYV)

Statistically this means we will be trying to find the average or mean distance of each spot ratefrom the mean, that is, the scale of deviance. Mathematically the square-root of this quantity iscalled 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

t

kkk

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

t

kk

n

k

t

kkk

kk

kk

(9b)

You see how nicely it conforms to statistical theory. The square root ofDYV is a standarddeviation and will surely give us a measure of the variability of interest-rate overt1 up to tn the

given investment period. At a snapshot the higher this value is, the more interest rates fluctuatedmeaning the more risky the investment and vice-versa. Lets denote this standard deviation as rthen:

2

1

1

2

DMFeCt

eCt

n

k

t

kk

n

k

t

kkk

rkk

kk

(10)


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For the example we have looked at before we will have DYV = 0.5136%% and the standarddeviation of the interest rates is r = 0.7167%. Much literature agrees on the fact that the standarddeviation of returns is a measure of the volatility of the returns hence we can use this fact todefine r above as a measure of interest rate volatility pertaining to a particularPV-space. Theimportance of this value is well known and ranges from modelling interest rates to pricing ofinterest 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 orexcess kurtosis of interest rates (which are very important in downside risk measurement) but thisis beyond the scope of this paper. Most importantly we are going to use this formula to derive ameasure of the volatility of the present value of cash flows for which we have derived DMFandDYVto do this we need one more parameter which we are going to look at next. As you havealready seen all the parameters we have derived (and those coming forth later) are not conformed

to bonds only but cover all securities valued by thePVfunction ),( kk tP .

The Weighted Average Term to Maturity (WAT)

We have already defined WATas:

n

k

t

kk

n

k

t

kkk

kk

kk

ec

ectT

1

1

, that is

},1{)(

nkktE

(8a)

WATmeasures the average payback period of a set of cash flows. What this formula tells us issimple, 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, 2n}

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 kkkkcbtP

1)( , where kktk eb

Then

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

Like the P(F, tk) approximation, theP(k, T) approximation is also very accurate with accuracylevels 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 WATto be T= 4.0159years givingP(k, T= 4.0159) = $17.52 a 100% accurate Price.

We are not going to look at the variance oftk since it is not of real significance to this paper. Weshall look now at how both Tand Fare very important tools in determining the sensitivity spaceof the present value. Here we are going to look at a simple concept I will call zerorising(orzeroing) 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 theDiscounted Mean Force of interest/Yield (F), the Weighted Average Term to Maturity (T) andthe 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 kkkkcbtP

1),(

into an equation of value with a single cash flow A and a standard singular and constant time tomaturity Nvalued with an interest rate Ijwhich is variable. For example suppose a Bond has a

price of

n

k

t

kkkkkectP

1),( 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 themain one being to achieve computational simplicity achieved by the smoothed exponential graphof the zerorised function.

The Zerorised position

Now here is how we achieve a zerorised position for:

n

k kkkkcbtP

1),( , 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

T

kkkecTP

1),(

In order to achieve what we want we need to make a simple improvisation on this equation. We

know that Fis an average mean of kk so we can safely substitute k with F=fjbut instead of

taking it as a constant we induce it into the equation as a variablefj representing the mean force ofinterest for term structurej or statej. So we have:

n

k jkjTfcTFP

1)exp(),(

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

)...()exp( 21 nj cccTf

n

k kjcTf

1)exp(

)exp( TfA j Where n

k kcA

1(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 (orprice).

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

term to maturity Twhereby the spot rate structure {ik: k= 1, 2n} is defined by F=fj in such away that if any of the spot rates changes in such a way that Falters say tofm, Tremains the same.

Now we look at the DYVwhich is measures the variance of interest-rate in relation to the cashflow 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

Fthen the price is approximated byP(F,T) lets useP(F) for short ,(see exhibit (i)). In the contextofDYV, Fis bounded above and below by a distance of r , that meansP(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 ) andP(F- r ) fromP(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 deviationfrom the mean rate F is symmetric but the implied d1 and d2 are asymmetric about thestandardized present value P(F). The next step is establishing a connection between d1 and d2with the variance of interest rate, to do this we need to define a quotient which is derived fromthe 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 such that:

d1 d2 = (a1 a2) (12)

Solving forwe get:

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 ofas 0r hence:

)(

)()(

)(

)()(

00r

r

r

r

FF

FPFp

FF

FPFpim

(13)

)(

)()(

)(

)()(

00r

r

r

r

FF

FPFpim

FF

FPFpim

=

2

)(

df

fdP

We know that )exp()(

TfTdf

fdP hence

2)exp( TfT

= (TP(F))2 (14)


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Note: we can take divide through by P(F)2 and be left with = T2 to work withpercentage/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 d1and d2 (Note that as 0 then d1d2 0 also, tempting us to define that the product d1 d2 a

price/PV- variance as 0).

Now since we have found by taking limits as r0 we can denote d1 d2 as a volatility measurebut 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:

21

21

aa

dd

rr

Solving for we get,

22 ))(( fPTr (15)

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

2)( Tr , orDYVWAT2. (SeeAppendix 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 ofcash flows, in most cases if not all practitioners use the standard deviation as the volatilitymeasure so its more refined if we use the square-root of as the volatility measure. This formulaagrees 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 = (rT)2 you will see that if the WATincreases then also

increases, the reverse is also true. It is also the same relationship with interest rate variance r ofwhich it is logically true. So if an investor is concerned with controlling or subsidizing in anenvironment where interest rates are highly volatile then he could go for short-term securities to

offset the high rsimilarly in long term investments stable interest rates may lower the price/PVvolatility ().

This formula, in summary, will work well especially in risk management whereby one may wantto describe the relationship between Price or Value risk with factors like Term to Maturity, levelof variability of the term structure, and so forth, portfolio selection-probably the financialmanager would prefer projects or investments with a higher/lower depending on whether (s)heis 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 andcomputational simplicity.

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

= rWAT (17)

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

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

Now if one is familiar with stochastic mathematics and traditional one factor models for interestrate modeling (s)he will agree with me that if the entire movements of the term structure aregoverned by the continuously compounded short-term interest rate, r(.), which evolves according

to the stochastic differential equation tdWtrtdttrtmtdr ))(,())(,()( . The market price ofrisk or r-risk is a function (t,r), such that the risk adjusted short rate drift is

),(),(),(),(~ rtrtrtmrtm .The price of a zero-coupon bond (though ours is notnecessarily a zero-coupon bond but any set of cash-flows whose present value has been

zerorised) is a functionP(t, T, r) of the short rate and, by using Its Lemma you will find that therelative (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 forWAT(seeAppendix B) and

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

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

Again let us use our example to determine the price volatility, therefore

= WAT

2

DYV =4.0159

2

0.00005136giving us 0.000828 and using equation (17) we have thevolatility v = 0.02878 and in monetary value we have $17.52 0.02878 = $ 0.504 and thus wehave 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

, or1

)1(

iDdiP

dP(note the constant interest-rate)

Greater precision of this bonds responsiveness to yield shift (d) by also accounting forconvexity .Using the Fundamental Property of calculus which states that any mathematicalfunction can be approximated by a Taylor(or Maclaurin) series.

This will bring us to:

2)(2

1)( dCdD

P

dP (19)

Wherek

k

tn

k k

t

k

n

k k

ec

ectD

1

1 , andk

k

tn

k k

t

k

n

k k

ec

ectC

1

1

2

kk ,

Duration and Convexity respectively, by taking account of both quantities in calculating this bondresponsiveness one would have assumed the non-linearity of this change in price due to change inyield 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 % changein thePVdue to shift in interest-rate. With this method one does not have to calculate convexityor use the Taylor expansion its very user friendly and will give you a direct description of thecurvature structure of the bond price (and responsiveness/sensitivity) and its yield-besides themethod is just as accurate as equation (18) only easier to use.

The Alternative: the , method.

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

T

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

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

If the interest-rate moves from iFtoj then the % change in Price /PVis 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

ji

111 (20)

Thats the formula for the % change inPdue to the change in interest-rate change from i toj.

The actual change factor () is given by:

1

T

F

j

i

1

1(21)

So we simply haveP(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, valuedat 5% an interest rate. Create a table to show the accuracy levels of the two methods of sensitivitymeasures of Price to yield shift.

Solution

The fair price of the Bond is:

10

10 1008)05.0( vaP

= $ 123.17

Since i= 0.05 is constant throughout the termDuration = WAT = 7.54 years andC= 51.98.Lets denoteDuration =D and WAT= Tand 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 toj

j Actual % Change D-C Method (d) WAT Method ()

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 WATmethod maintains its accuracy levels even in the wakeof an extreme value like in this example a change from 5% to 20% was approximated real good(94%), as you can see the 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 theintroduction of convexity forces some curvature approximation to this price-shift, one can usehigher order factors of the Taylor (or Maclaurin) series to get better approximations but these tendto make the computation of this sensitivity more rigorous. Hence a good alternative would be touse the method which is more accurate and very simple to calculate, let us conclude this sectionwith an illustrative example (using Example (i) we have been dealing with previously).

Suppose shocks in year3 and 7 result in the relative spot rates (of those years) shifting in such away that the resulting DMF(Fnew) differs from the DMF in (ii) by +165 basis points, calculatethe 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 effectof change from iFto iF+ (165/100

2). 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

06245.0035789.1

019289.11

0159.4

So we expect aPVdecrease of6.25%.due to that increase in interest-rate


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nashfiles 15

V.CONCLUSION

This brings us to the end of this short note on measuring the Volatility of both the term structureand the Present /Prices of securities. We also looked at the mean averages of both the timevariable 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 ofmoment theory in the first section of the paper in deriving the formulae which form the basis ofthis paper-and thus the results of this paper are of very accurate design and exposition, easy toaccept and use (though one may attempt successfully to use calculus to derive WAT, DMFandDYV) .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 everymeasure taken to ensure that its understandable by everyone at every level and also practicallyemployable. 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

t

kk

n

k

t

kkk

kk

kk

ect

ectDMF

1

1

The Weighted Average Term to Maturity

n

k

t

kk

n

k

t

kkk

kk

kk

ec

ectWAT

1

1

The Discounted Yield Variance2

1

1

2

DMF

eCt

eCtDYV

n

k

t

kk

n

k

t

kkk

kk

kk

The -volatility measure for price2)( Tr OrDYVWAT

2

The sensitivity/Elasticity measure

WAT

new

old

DMF

DMF

1

11

_______________________

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


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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 valuingdue to the negative sign which makes us value thepdffrom to 0 we

have the following property:

t

sdsetSPVtSPV )()( , and is the pricegenerating

function G(It,), defineF(It,) 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, ) (tk, k) for all k = 1, 2, n hence summingkkt

ttktk eIIfIF

,,, )()( overkbecomes a discrete function. Such that if we include a cash

flow ckfor each kth-density function we get a sum-price density functionMk:

n

k kk

n

k kkkfcFcM

11

And the price/PVbecomes the sum-pricegeneratingfunctionP

n

k

t

k

n

k tkkkkkecIGcP

11 ,)(

It follows that the mean Ek(It,) becomes

n

k tkt

n

k tkt

k

t

kt

IFI

IFI

M

MIE

1 ,,

1 ,,,

,

)(

)()(

, where

,

,

,

)()(

tk

tkk

tkId

IFdIF

Since )(')(' ,, tt IGIF

n

k

t

kk

n

k

t

kkk

nkk

fc

ftctE

1

1

},1{)( , and

n

k kk

n

k kkk

nkk

fc

fcE

1

1

},1{)(

(7)

Where kkt

k

t

k ef

and kktkk etf

.

kk

kk

t

k

n

k k

n

k

t

kkk

nkk

ec

ecttE

1

1

},1{)( , and

n

k

t

kk

n

k

t

kkk

nkkkk

kk

etc

ectE

1

1

},1{)(

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


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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 whichis dimensionless and can be used very much like the Macaulay Duration DM or the Fisher-WeilDuration DF , since in definition these two (and many other duration types) measure theresponsiveness of a securitys market price (intrinsic, or present value ) to a change in theunderlying interest rates.

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

n

k

t

k

n

k

t

kk

k

kk

kk

Fkk

kk

ec

ectk

d

tdP

tPD

1

1),(.),(

1

(23)

This is known as the Fisher-Weil Duration, and ifk= kthenDFbecomesDM that is:

n

k

t

k

n

k

t

kkk

k

Mk

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

kkcA

1

a constant

With fj chosen such that fj = Fmakes ),( TFP the nucleus Price for a neighborhood of prices

bounded within F rwe can find duration:

j

j

j

Tdf

TfdP

TfPD

),(.

),(

1

)exp(.),(

1TfTA

TfPj

j

TTfA

TfAT j

)exp(

)exp(

Since T= WATwe then have:


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nashfiles 18

n

k

t

kk

n

k

t

kkk

Tkk

kk

ec

ectD

1

1

(25)

as a measure of duration, there are some points to note though:

1. DTis dimensionless.2. If k = kthen it becomes identical to the Macaulay Duration.3. It is a measure ofPVelasticity 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-yieldcurve. It describes the incremental price change misestimated by theduration and is defined by:

2

2 ),(.

),(

1

k

kk

kk

Fd

tPd

tPC

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

2

2 ),(.

),(

1

j

j

j

Tdf

TfPd

TfPC

)exp(.),(

1 2TfTA

TfPj

j

This eventually becomes:

2TCT

Hence we have derived a duration and convexity measure for the present value function ),( kk tP

, which areDTand CT that is:

n

k

t

kk

n

k

t

kkk

Tkk

kk

ec

ectD

1

1

, the duration

2

1

1

n

k

t

kk

n

k

t

kkk

Tkk

kk

ec

ectC

, the convexity (26)

Hence the identity CT= (DT)2


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nashfiles 19

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

Change inPVdue to small shift in interest rate relative Change inPVdue to small shift in time

and interest rate divided by the relative change inPVdue to small shift in time only.

Using partial derivatives we have:

t

P

p

t

P

PP k

1

1 2

, this gives us

nk

tkk

n

k

t

kkk

Tkk

kk

ec

ectD

1

1

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

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

n

k

t

kkkkictiP

1)1(),(

So we needkk

kk

ti

tiP

),(2and

k

kk

t

tiP

),(

Let us start by findingk

kk

t

tiP

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

Let kt

kkk icD )1( such that

n

k kkDtiP

1),(

kkkk tcD lnln , where )1ln( kk i

Hence

kk

t

D

ln kkk Dt

D

This means

nk

t

kkk

n

k kkkicD

t

P10

)1(


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nashfiles 20

Also:

kkkk

kk

t

P

iti

tiP ),(2

This gives us:

nk

t

kkkk

kk

kk kictti

tiP1

)1(2

)1(),(

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

n

k

t

kkk

n

k

t

kkkk

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 valuefunction in relationship to its term structure and is very important in risk analysis especially whencomparing two or more securities. This value is very important in immunization of funds since itgives a measure of how the price will respond in the wake of a non-parallel shift in the interestrate structure, it will appeal more to an investor who is worried about slope changes in the termstructure. Thus if we expect slope changes in the term structure and cannot determine whichdirection they will move then a high value will mean greater risk-exposure. This is how we cancome up with it.

In addition to that, one can use it as a test to see how much interest rates will have to shift beforewe can have to adjust WAT in the P(k, T) approximation, a small value is preferable since itmeans a greater range offjs can be used (the set ofDMFs) can be used before the approximationstarts to loose accuracy.

n

k

t

kk

n

k

t

kkk

ttkk

kk

eC

eCtWAT

1

1

Simple derivative calculation gives us:

2

1

1

2

WATeC

eCt

n

k

t

kk

n

k

tkkk

kk

kk

, where

22 )()(

k

k

t

k

k

k

t

k

P

PP

P

PP 3 (28)

_________________________

3Pk= cke-t i.e. thePVandPk

t= tkcke-t i.e. sum-moment ofPVabout time.


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References

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

- 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, McGrawHill/Irwin

- Cox J.C.,J. E. Ingersoll and S. A. Ross.Duration and Measurement of Basis Risk. Journal ofBusiness, 52 (1979), 51-61.

- Edwin .J .Elton, Martin. J. Gruber, and Roni Michaeli. The Structure of Spot Rates andImmunization. Journal of Finance.

-Fisher .L. and R. L. Weil. Coping with the Risk of interest-rate fluctuations: Return to BondHolders from Nave and Optimal Strategies. Journal of Business, 44 (1971), 408-431.

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

- Ho, Thomas (1992),Key Rate Durations: Measures of interest-rate Risk, The Journal of FixedIncome (Sept 1992,) 29-43

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

- Klaffky, Thomas E, Y.Y.Ma and Adavarn Nozari,Managing Yield Curve Exposure IntroducingReshaping Durations. Journal of Fixed Income (1992)

- Peter Albrecht / Thomas Stephan, Single-Factor Immunizing Duration of an Interest Rate Swap,Journal of Finance.

- Richard .F .Bass,Probability Theory Notes.

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

The price density function, a tool for measuring investment risk,volatility and duration

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