Distribution of the Present Value of Future Cash Flows - · PDF fileDISTRIBUTION OF THE PRESENT VALUE OF FUTURE CASH FLOWS GARY PARKER ABSTRACT The present value of future cash flows

DISTRIBUTION OF THE PRESENT VALUE

OF FUTURE CASH FLOWS

GARY PARKER

ABSTRACT

The present value of future cash flows with random interest rates is stud- ied. The first three moments of the present value of future cash flows and an approximation of its cumulative distribution function are presented. Particu- lar results for three stochastic processes that may be used to model the force of interest are presented. The three processes considered are the White Noise, Wiener and Ornstein-Uhlenbeck processes. A n-year certain annuity-immediate is used to illustrate the results.

KEY WORDS: Cash Flows, Force of Interest, Present value, White Noise process, Wiener process, Ornstein-Uhlenbeck process, Annuity-immediate.

1. INTRODUCTION

In actuarial science and finance, the present value of future cash flows is obtained by discounting the cash flows by appropriate discount factors. When considering future interest rates that vary in a stochas- tic fashion, the present value of future cash flows becomes a random variable.

Stochastic interest rates and application has been the subject of many publications. Some papers are mainly concerned with the moments of the present value of certain or contingent cash flows. Examples are Boyle (1976), Panjer and Bellhouse (1980), Waters (1978), Wilkie (1976), Giacotto (1986), Dhaene (1989), Beekman and F’uelling (1990), Parker (1992b)).

Other papers using stochastic models for the interest rates present applications to investment or immunization theory (see, for example, Boyle (1980), Wilkie (1987), Beekman and Shiu (1988)).

Recently, Dufresne (1990), Frees (1990), Parker (1992c) studied the density function or the cumulative distribution function of present val-

832 Gary Parker

ues.

The aim of this paper is to present a general approach for finding the moments of the present value of cash flows and to suggest a method for approximating its distribution.

The layout of the paper is the following. Section 2 gives a proba- bilistic definition of P, the present value of future cash flows. Section 3 describes how one can obtain the first three moments about the origin of P. These moments are later used to find the standard deviation and the coefficient of skewness of P. Three stochastic processes for the force of interest are studied in section 4. They are the White Noise process, the Wiener process and the Ornstein-Uhlenbeck process. Section 5 sug- gests a method for approximating the cumulative distribution function of P. Illustrations for n-year certain annuity-immediate are presented in section 6.

2. PRESENT VALUE OF FUTURE CASH FLOWS

Consider a stream of future cash flows of deterministic amounts CFi,i=1,2 ,..., n where the ith cash flow, CFi , is payable at time ii. Let P be the present value of these cash flows. Then P may be defined as:

[ll p = 2 CFi . exp { - L!l(ti)) .iTl

where

J t Y(t) = 6, . ds

0

and 8, is the force of interest at time s. So, if the force of interest varies stochastically, P is a random variable.

We will assume that the same realization of future forces of interest applies to any common period of time involved in the discounting of the cash flows. In other words, the value that the force of interest will actually take at a futluc time s will be used (for time s) to discomit any cash flow payable at or after time s.

Distribution of the presente value of future cash flows 833

3. MOMENTS OF P

The expected value of P is simply the ~~111 of the expected values of the cash flows. That is:

[3] clpI=E[~CFj.exp{ -y(ti)}] =cCFi.E[eXp( -y(ti)}]. i=l i=l

The second moment about the origin of P is given by:

WI E[P2] = E [F, F CFi . CFj exp { - y(ti) - y(Q)}] i=l jzl

151 E[P”] = f-&TF; .CF, -E[exp{ -g(ti) -&)}] i=l j=l

The third moment, about the origin of P is given by:

[6] E[P3]=E f+k [

CFi.CFj’Cl;i;‘eXp { -?,(ti)-y(tj)-y(tk)} i-1 jzl k=l 1

[7] E,P”,=j:j:jl: CFi.CFj.CFk’E exp { -y(tx)-g(t,j)-?J(tk)} i=l j=l k=l 1 Parker (1992b, sections 4 and 5) suggests using recursive equations

for the numerical evalllation of equations similar to 151 and [7]. These recursive equations are easily adapted for finding the second and third moments of P and such recursive equations were llsed to obtain the illustrated results presented in section 6 of this paper.

The expected values involved in equations [3], [5] and [7] are still to be determined. If the forces of interest arc gaussian, t,hcn the func- tion y(l) is normally distributed. Consequently, expressions of the form w { - A?.) - Y(S) - v(t)) are lognormally distriblitetl, i.e.

PI exp { - g(r) -g(s) - y(f)} - A(p. P)

with parameters

834 Gary Parker

and

PI P = V[dr)] + V[ds)] + V[dt)] + 2~ [Y(r), y(s)]

+ 2~ [y(r), y(t)] + 2cov [Y(s), y(t)] .

The expected value of the lognormal variable in [8] is then:

WI E [exp { - y(r) - Y(S) - y(t)}] = expb + .5. PI

(see, for example, Aitchison and Brown (1963, p.8)).

Finally, [ll] can be used, with the fact that by definition y(0) = 0, to obtain the needed expected values in [3], [5] and [7]. Note that the parameters p and p depend on the particular gaussian stochastic process used to model the force of interest.

In the following section we present three gaussian processes that may be used to model the force of interest and derive the specific results needed for finding p and p for each model.

4. FORCE OF INTEREST

4.1. WHITE NOISE PROCESS

The first model to be considered is the White Noise process. Let the force of interest be defined as:

where Wt is a standard Wiener process.

The forces of interest are therefore gaussian and i.i.d. The function y(t) is then a Wiener process with drift 6. Its expected value is:

1131 E [y(t)] = 6 . t

and its autocovariance function is:

P41 cov [y(s), y(t)] = a2 . min(s,t)

(see, for example, Arnold (1974, section 3.2)).

Distribution of the presente value of future cash flows 835

4.2. WIENER PROCESS

The second model considered for the force of interest is the Wiener process. Let the force of interest be defined as:

PI d6, = CT . dW, CT 20.

From section 8.2 of Arnold (1974) we know that conditional on 60 being the known current value of the process, the mean and autocovari- ante function of this process are:

and

P71 Cov[6,, St] = f12 . min(s, t)

Then, from the definition of y(t), it follows that y(t) is normally distributed with mean:

P81 JqYW] = 60 . t

and autocovariance function:

WI

WI

cov [y(s), y(t)] = 1’ s’cov,&, &,]dudv 0

cov [Yb), YW] = ,“z . (s2t/2 - s3/6) s 5 t

4.3. ORNSTEIN-UHLENBECK PROCESS

The third model that we will consider for the force of interest is an Ornstein-Uhlenbeck process. Let the force of interest be defined such that:

WI d& = -a(& - S) . dt + o . dW, a 2. o, o 2 o

with initial value 60 (see, for example, Arnold (1974, p.134)).

836 Gary Parker

Then, it can bc shown that:

1221 E[&] = s + eFat . (6” - 6)

P31 cov[6,, d^t] = e-a(t+s) k e 7 ( h5 _ 1) s<t _

and that the function y(t) is a gaussian process with mean:

I241 E[y(t)] = 6. t f (So - 6) ___

and autocovariance function:

1251 COV [y(s), y(t)] = g9 min(s, t)+

+ $ [ -2 + 2eBas + 2,c-at _ e-alt-sl _ e -u(t+s) 1 (see, for example, Parker (1992b, section 6)).

The parameters ,LL and /3 given by equations [9] and [lo] respectively may be evaluat.ed for a specific stochastic process for the force of interest by using [13] and [14] for t,l~ White Noise process, (181 and [20] for the Wiener process, or [24] and [25] for the Ornstein-1Jlllenl,eck process.

5. AIVROSIM~YIWG THE DISTRIBUTION OF P

Parker (1992c) suggests a method for approximating the lirniting distribution of the average cost per policy of a portfolio of temporary contracts. This method can be adapted to approximate the distribution of P.

Let P(j) be the present, value of t,he first j cash flows, that is:

WI P(j) = &xi . cxp { - Y(li)} i=l

and let the function yj ($1, y) be defined as:

Distrilmtion of the presente due of frhure cash flouts 837

The distribution function of P is theu given by:

PI Fp(p) = .I-= S7a (14 T/)4/ . --cv Adapting the reslllts derived and justified in Parker (1992c, sections

4 and 5), the fimction CJj((I’% YJ) may be apI,roximated llsirig the followiug recursive integral equat,ion:

[29] g&y) ” / .fy(j) (:,I& - 1) = x) . !IJ&’ - CF, . e-y, z)d5 J-CC

with the starting value:

7J - q!/(l)] PI Yl(P,Y) == (1’[YU)l).” > if p > CF1 . e-y

otherwise

where 4( .) denotes the probability density function of a zero mean and unit variance normal random variable.

And if the forces of illtcrest are gaussian, y(j) given that y(j - 1) equal 5 is uormally distributed with mean:

E[y(j)Ig(j - 1) = X] = z+(j)] -I- (‘ov [Y(j)! Y(.i - l)] .

VI v [!/(A] {cc - E[y(j - l)]}

and with variance:

P21 v [y(j) Iw(.i - 1) = :c] = v [l/W] - cm2 [l,(j), y(j - l)] Q(j - 111

(see, for exaiiiplc, Morriso1i (1990. 11.92)).

0. II,LUSTR~\TI~NS

Consider a ‘n-year certain annuity-immctliate with payment,s equal to one. The prescut value of this annuity is given by P with GF, = 1 andt;=i for i=1,2 ,..., 72.

838 Gary Parker

Figures 2, 3 and 4 illustrate the expected value, standard deviation and coefficient of skewness (see, for example, Mood, Graybill and Boes (1974, pp.68,76)) respectively of the present value, P, for the annuity under consideration. Each figure presents the results for the three models discussed in section 4 for the force of interest.

18

6

0 10 20 30 40

n

Fig. 1 Expected value Present value of a n-year certain annuity-immediate _ White Noise: 6 = 0.06 CT = 0.01 ___ Wiener: 60 = 0.06 CT = 0.01

. . . Omstein-Uhlenbeck: 6 = 0.06 60 = 0.06 Q = 0.1 u = 0.01

Note that although the parameters of each of the three processes used for the force of interest were chosen so that their expected values are the same (.06), the expected responses of the processes to a specific situation are different. This explains their different expected values for the present value P.

To illustrate this. let us assume that the force of interest at time

Distribution of the presente value of future cash flows 839

s is .08. Then the expected future forces of interest beyond time s is unchanged at .06 for the White Noise process. It changes to .08 and remains constant at this level for the Wiener process. For the Ornstein- Uhlenbeck process, the expected force of interest starts at .08 at time s and decreases exponentially to .06.

10

8

6

n

Fig. 2 Standard deviation Present value of a n-year certain annuity-immediate _ White Noise: 6 = 0.06 CT = 0.01 ___ Wiener: 60 = 0.06 (T = 0.01

. . Ornstein-Uhlenbeck: 6 = 0.06 60 = 0.06 c1 = 0.1 CT = 0.01

From figure 2 we can see that the standard deviation of the present value P varies considerably with the process involved. The White Noise process has independent forces of interest and this leads to the lowest standard deviation. The Wiener process has positively correlated forces of interest with no mechanism to bring the process back to any given value, thus giving the largest standard deviation. Finally, the Ornstein-

840 Gary Parker

Uhlenbeck process has positively correlated forces of interest with a friction force bringing the process back to a long term mean.

It should be pointed out that fitting the parameters of these three processes based on past data is unlikely to give the same values for C. The estimated c for the White Noise process might be the largest. Tllis would increase the standard deviation of P for the \Vhite Noise process relative to the other two.

,' 6- , _ , I I - I I I I I 4- I /

,I ,I

I' I / I , 2- , , I' /' ,'

.*I c- xX -* _- -.-... . . . . . . . . .' .,,._,,...,, ,. ,,.,._..... -..- .... -......'..". _ .,.,.._._.._ _...- ......'.' r ;

O- -,_. *..-. <..r. ....

I I I1 t I I I I I I I I I I I 0 10 20 30 40

Fig. 3 CoeRicient of Skewness Present value of a n-year certain annuity-imrnediale _ White Noise: 6 = O.OF IT = 0.01 _ __ Wiener: 60 = 0.06 0 = 0.01

. . . Ornsteill-Ulllenbeck: 6 = 0.06 50 = 0.06 a = 0.1 0 = 0.01

From figure 3, it appears that the distribution of P generated by White Noise forces of interest is fairly symmetric as the coefficient of skewness is close to 0. With a Wiener process, P has a highly asymmet- ric distribution. For reasons discussed above, the Ornstein-Ulllen~~eck

Distribution of the presente due of future ctash flows 841

generates a distribution of P with a cocflicient of skewness between the other two.

Figures 4 and 5 present the crumulative distrihltion functions of 5 and 25 years annuity-immediate. The integral equations [28] and [29] were evaluated by an arbitrary discretisation.

0.8

0.6

0.4

0.2

0

3.5 3.8 4.1 4.4 4.7

Fig. 4 Cumulative distribution function Present value of a 5.year ccrt,ain an~llrit,y-immediate _ White Noise: 6 = 0.06 IT = 0.01

\Niener: 60 = 0.06 0 = 0.01 . Ornstein-Uhlellbeck: ii = 0.06 So = O.OG (Y = 0.1 c = 0.01

842 Gary Parker

1

0.8

0.6

l- I I I”I’- ’ r I’ ,“I”@ ‘(“I 1 I1 “‘I I I I”I’- ’ r I’ ,“I”@ ‘(“I 1 I1 “‘I

,,,.,.___............... ,,,.,................... - -

,:’ ,:’ /.” /.” **-- **--

_ _ _ _ _ _ _ _ - - _ _ _ _ _ _ _ _ - -

.:’ , .:’ , ,* ,* / I / I

,.; I ,.; I j ’ I j’ I : I : I

0.8 - i I i I ,: ,’ ,: ,’

i , i , i, i,

i , i , i , i , i/ i/

0.6- i, i, ir ir

ir ir i/ i/ I I

I I 0.4-

i i I! I!

1; 1; 1; 1;

1: 1: 1; 1;

‘ j ‘j I ; I ;

0.2 - I i I i I i I i

I ! I ! I ; I ;

I : I : I I I ; I ;

, ! , ! O- - _ - - - - _ _ _._-__ ,..< ._._.. . . _ - - - - - - _._-__ ,..< ._.... . . .:’ .:'

I I I, I I, I, I I I I t; I I I I t I s & I I I I I I I I_ I,,, I I,,,; I I I I, I, & I I I I I I I I_

0 0 5 5 10 15 20 25 30 10 15 20 25 30

0.4

0.2

0

n

Fig. 5 Cumulative distribution function Present value of a 25-year certain annuity-immediate _ White Noise: 6 0.06 = c 0.01 = ___ Wiener: 60 = 0.06 CT = 0.01 . . . Ornstein-Uhlenbeck: 6 = 0.06 60 = 0.06 a = 0.1 0 = 0.01

7. CONCLUSION

The present value of future cash flows is treated as a random vari- able. A general approach for finding its moments when the force of interest is modeled by a gaussian process is presented. A method for approximating its distribution is suggested. The illustrations indicate that the choice of a stochastic process for the force of interest has a significant impact on the distribution of the present value of future cash flows.

Distribution of the presente value of future cash flows 843

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(1) J. AITCHISON, J.A.C. BROWN, The Lognormal Distribution, Cambridge University Press, 1963, 176~~.

(2) L. ARNOLD (1974), Stochastic Difierential Equations: Theory and Appli- cations, John Wiley & Sons, New York, 1974, 228~~.

(3) J.A. BEEKMAN, C.P. FUELLING, Interest and Mortality Randomness in Some Annuities, Insurance: Mathematics and Economics 9, 1990, p.185- 196.

(4) J.A. BEEKMAN E.W.S. SHIU, Stochastic models for bond prices, func- tion space integrals and immunization theory, Insurance: Mathematics and Economics 7, 1988, p.163-173.

(5) P.P. BOYLE, Rates of Return as Random Variables, Journal of Risk and Insurance XLIII, 1976, p.693-713.

(6) P.P. BOYLE, Recent Models of the Term Structure of Interest Rates with Actuarial Applications, Transactions of the 21st International Congress of Actuaries, 1980, p.95-104.

(7) J. DHAENE, Stochastic Interest Rates and Autoregressive Integrated Mov- ing Average Processes, ASTIN Bulletin 19, 1989, p.131-138.

(8) D. DUFRESNE, The Distribution of a Perpetuity, with Applications to Risk Theory and Pension Funding, Scandinavian Actuarial Journal, 1990, p.39- 79.

(9) E.W. FREES, Stochastic Life Contingencies with Solvency Considerations, Transactions of the Society of Actuaries XLII, 1990, p.91-148.

(10) C. GIACOTTO, Stochastic Modelling of Interest Rates: Actuarial vs. Equi- librium Approach, Journal of Risk and Insurance 53, 1986, p.435-453.

(11) A.M. MOOD, F.A. GRAYBILL, D.C. BOES, Introduction to the Theory of Statistics, McGraw-Hill, 1974, 564~~.

(12) D.F. MORRISON, Multivariate Statistical Methods, McGraw-Hill., 1990, 495pp.

(13) H.H. PANJER, D.R. BELLHOUSE, Stochastic Modelling of Interest Rates with Applications to Life Contingencies, Journal of Risk and Insurance 47, 1980, p.Sl-110.

(14) G. PARKER, An Application of Stochastic Interest Rates Models in Life Assurance, Ph.D. thesis (submitted), Heriot-Watt University, 1992a.

(15) G. PARKER, Moments of the present val,ue of a portfolio of policies, (Sub- mitted), 1992b.

(16) G. PARKER, Limiting distribution of the present value of a portfolio, (Sub- mitted), 1992c.

(17) H.R. WATERS, The Moments and Distributions of Actuarial Functions, Journal of the Institute of Actuaries 105, Part l., 1978, p.61-75.

(18) A.D. WILKIE, The Rate of Interest as a Stochastic Pmcess -Theory and Applications, Proc. 20th International Congress of Actuaries, Tokyo, 1, 1976, p.325-338.

(19) A.D. WILKIE, Stochastic Investment Models - Theory and Applications, Insurance: Mathematics and Economics 6, 1987, p.65-83.