Stochastic interest rates with actuarial applications

APPLIED STOCHASTIC MODELS AND DATA ANALYSIS

Appl. Stochastic Models & Data Anal. 14, 335}341 (1998)

STOCHASTIC INTEREST RATES WITH ACTUARIALAPPLICATIONS

GARY PARKER*

Genesis Development Company Ltd., 6523 Mississauga Road, Mississauga, Ont., Canada L5N 1A6.

SUMMARY

This paper presents recursive double integral equations to obtain the distribution of the discounted value oraccumulated value of deterministic cash #ows. The double integrals have to be evaluated numerically at eachiteration. Those distributions are useful when studying the investment risk of portfolios of insurancecontracts. The methods suggested take advantage of the Markovian property of the Gaussian process usedto model the future rates of return. We start with the "rst cash #ow and successively add the other cash #owswhile keeping track of the latest information about the rate of return in order to update the distribution ateach step. Various means and covariances of bivariate normal distributions which are required if one wantsto apply the results in practice are given. In the paper, the Ornstein}Uhlenbeck process is chosen tomodel the rate of return but the results could be extended to a second order di!erential equation.Copyright ( 1998 John Wiley & Sons, Ltd.

KEY WORDS: insurance cash #ows; discounted value; accumulated value; Ornstein}Uhlenbeck process

1. INTRODUCTION

The value of future cash #ows is undoubtedly of interest to insurance companies. The long-termnature of life insurance and annuity contracts makes it particularly important to value the futurepayments to be received or to be made when pricing or determining the reserve for these products.

One approach used in practice to price insurance contracts is to determine the expectedinsurance cash #ows and study their value at time of issue or at the end of the contract term in anenvironment with random rates of return. Once contracts are sold, actuaries need to determine anappropriate amount of reserve for the future liabilities of the company. This is another situationwhere companies are particularly interested in the interest or investment risk. Insurance cash#ows have been studied by Norberg1 and Parker2 among others.

In this paper we consider an insurance portfolio and assume that the expected cash #owsgenerated by this portfolio have been determined. We are interested in the distribution of theexpected insurance cash #ows.

In Reference 3, an approximation method is presented for the distribution of the discountedvalue of the insurance cash #ows of a portfolio of temporary insurance contracts. These results

*Correspondence to: Gary Parker, Genesis Development Company Ltd., 6523 Mississauga Road, Mississauga, Ont.,Canada, L5N 1A6. E-mail: [email protected]

CCC 8755}0024/98/040335}07$17.50 Received 8 June 1997Copyright ( 1998 John Wiley & Sons, Ltd. Revised 23 April 1998

are now extended to o!er an exact recursive method for "nding the distribution of the discountedand accumulated values of insurance cash #ows.

For reasons that are explained in Reference 2, we present our results using the Or-nstein}Uhlenbeck process as a model for the instantaneous rate of return that we denote by d

t.

We could use other Gaussian processes like, for example, the second-order stochastic di!eren-tial equation studied in Reference 4.

Speci"cally, we will use the stochastic process

ddt"!a(d

t!d) dt#pd=

t. (1)

That is, the instantaneous rate of return is pulled towards a long-term mean of d with a frictionforce a and a di!usion coe$cient p.

To simplify the notation, the cash #ows are assumed payable at regular intervals but the resultscould easily be adapted to situations with irregular timing of cash #ows.

2. DISCOUNTED AND ACCUMULATED VALUE OF CASH FLOWS

The discounted value, fn, of a series of deterministic future cash #ows, CF

t, payable at time

t"1, 2,2, n is given by: (Note that the cash #ows here correspond to the expected cash #ows inSection 6.1 of Reference 2.)

fn"

n+t/1

CFtexp G!P

t

0

dsdsH"f

n~1#CF

nexp G!P

n

0

dsdsH . (2)

For convenience, we will de"ne

y (t)"Pt

0

dsds. (3)

The accumulated value at time n, mn, of the same cash #ows payable at time t"1, 2,2, n is

mn"

n+t/1

CFtexp GP

n

t

dsdsH"m

n~1exp GP

n

n~1

dsdsH#CF

n. (4)

The distribution of fnand m

nwhen d

tis modelled by a Gaussian process like the one de"ned in

(1) is the subject of the next two sections. The approach used is similar to the one found inReference 3.

3. DISTRIBUTION OF fn

In order to evaluate the distribution of the discounted value of cash #ows, we will introduce thediscounted value of each cash #ow one at a time while making sure we keep the relevantinformation about the process for the rate of return at each step. That is, at any step we need thelatest value of y (t) as it will be used for discounting the next cash #ow and we need the latest rateof return, d

tfor updating the function y (t).

Consider the function

gj(z, y, x)"f

y(j),dj (y, x) .P (fj)z Dy ( j)"y, d

j"x) (5)

336 G. PARKER

Copyright ( 1998 John Wiley & Sons, Ltd. Appl. Stochastic Models Data Anal. 14, 335}341 (1998)

from which we can obtain the distribution of fnby the double integral

Ffn(z)"P=

~=P

=

~=

gn(z, y, x) dy dx. (6)

The function gn(z, y, x) can be obtained recursively using

gj(z, y, x)"P

=

~=P

=

~=

fy(j),dj (y, x Dy( j!1)"v, d

j~1"w)

]gj~1

(z!CFjexpM!yN, v, w) dv dw (7)

with the starting value

gj(z, y, x)"G

fy(1),d1 (y, x)

0

if z*CF1expM!yN

otherwise(8)

where fy(j),dj (y, x) is the joint density function of the bivariate normal vector (y( j), d

j) at (y, x).

To derive the above recurrence equation, we start by noting that from (2),

P (fj)z Dy( j)"y, d

j"x)"P (f

j~1)z!CF

jexpM!yN Dy ( j)"y, d

j"xN. (9)

Then, we can write g as

gj(z, y, x)"P (f

j~1)z!CF

jexpM!yN) f

y(j),dj (y, x D fj~1

)z!CFjexpM!yN). (10)

The conditional probability density function of (y ( j), dj) in the above equation can be expressed

as (see Reference 5, p. 98)

fy(j),dj (y, x D f

j~1)z!CF

jexpM!yN)"P

=

~=P

=

~=

fy(j),dj (y, x Dy ( j!1)"v, d

j~1

"w, fj~1

)z!CFjexpM!yN)]f

y(j~1),dj~1(v, w D f

j~1)z!CF

jexpM!yN) dv dw. (11)

From the de"nition of g, we have

fy(j~1),dj~1

(v, w D fj~1

)z!CFjexpM!yN)"

gj~1

(z!CFjexpM!yN, v, w)

P (fj~1

)z!CFjexpM!yN)

(12)

and from the Markovian property of (y ( j), dj) we have


j~1"w, f

j~1)z!CF

jexpM!yN)

"fy(j),dj (y, x Dy ( j!1)"v, d

j~1"w). (13)

STOCHASTIC INTEREST RATES WITH ACTUARIAL APPLICATIONS 337


We can therefore write (11) as

fy(j),dj (y, x D f

j~1)z!CF

jexpM!yN)"P

=

~=P

=

~=


j~1"w)

]gj~1

(z!CFjexpM!yN, v, w)

P (fj~1

)z!CFjexpM!yN)

dv dw (14)

which in (10) gives us (7).Note that other recursive equations could be derived but the one given by (7) is the most

convenient and numerically stable in our opinion.

4. DISTRIBUTION OF mn

We can "nd the accumulated value of a series of cash #ows at a future time n in essentially thesame way.

Consider the function

hj(z, y, x)"f

y(j),dj (y, x) )P (mj)z Dy ( j)"y, d

j"x). (15)

The distribution of mnis then given by the double integral

Fmn(z)"P=

~=P

=

~=

hn(z, y, x) dy dx. (16)

The function hn(z, y, x) can be obtained recursively from

hj(z, y, x)"P

=

~=P

=

~=

fy(j),dj (y, x Dy ( j!1)"v, d

j~1"w)

]hj~1A

z!CFj

expMy!v), v, wB dv dw (17)

with the starting value

h1(z, y, x)"G

fy(1),d1 (y, x)

0

if z*CF1

otherwise.(18)

The steps required to obtain (17) follow closely the ones we used to show (7). Note that in thiscase, we must have m

j~1"(z!CF

j) expMv!yN whenever y( j)"y, y ( j!1)"v and m

j"z and

this fact is used in deriving (17).

5. JOINT DISTRIBUTION OF (dt, y (t))

Equations (7) and (17) can be evaluated accurately by various numerical methods provided wecan "nd the conditional joint density function which appears inside the double integrals. In otherwords, we need to determine f

y(t),dt(y, x Dy (t!1)"v, dt~1

"w).

338 G. PARKER


It can be shown that the y (t)'s and dt's have a multinormal distribution. Therefore, we know

that (see Reference 6, p. 92)

(ds, y (s))@ D (d

t, y (t))@&N (k

s#&

12&~122

((dt, y(t))@!k

t), &

11!&

12&~122

&@12

) (19)

where

ks"E[(d

s, y (s))@] (20)

kt"E[(d

t, y(t))@] (21)

and

&11"<[(d

s, y (s))@] (22)

&22"<[(d

t, y (t))@] (23)

&12"covAA

ds

y(s)B , Adt

y (t)BB . (24)

If we can "nd each of (20)}(24), we will be able to use the bivariate normal distribution (19) inthe recursive relationships (7) and (17) in order to obtain the distribution of f

nand m

n. Approxima-

tions of the bivariate normal distribution can be found in Chapter 36, Section 4 of Reference 7.The basic results needed to determine the parameters of the bivariate normal in (19) can be

obtained from the properties of the bivariate stochastic di!erential equation

dAdt!dy (t) B"A

!a1

0

0B )Adt!dy (t) B dt#A

p0B d=

t. (25)

De"ning the matrix M as

M"A!a1

0

0B (26)

it can be shown that its exponential is

expMMtN"AexpM!atN1~%91M!atN

a

0

1B . (27)

The mean and autocovariance function of this bivariate normal vector are (see, for example,Reference 8, p. 131)

E Cdt

y (t)D"AddtB#expMMtN A

d0!d

y(0) B"A

d#(d0!d) expM!atN

dt#(d0!d) (1~%91M!atN

a )B (28)



and, for s)t,

covAAds

y (s)B , Adt

y (t)BB"eMsCPs

0

(eMu)~1 Ap0B (p, 0) ((eMu)~1)@ duD (eMt)@ . (29)

Evaluating (29), we "nd the following covariances for s)t

cov(ds, d

t)"

p2

2a(expM!a (t!s)N!expM!a(t#s)N) (30)

cov(y(s), dt)"P

s

0

cov(dr, d

j) dr"P

s

0

p2

2aexpM!a(r#t)N (expM2arN!1) dr

"

p2

2a2[expM!a(t!s)N!2 expM!atN#expM!a (t#s)N], (31)

cov(y(t), ds)"P

t

0

cov(dr, d

s) dr"P

s

0

p2

2aexpM!a(r#s)N (expM2arN!1) dr

#Pt

s

p2

2aexpM!a (r#s)N (expM2asN!1) dr

"

p2

2a2[expM!a(t#s)N!2 expM!as )!expM!a(t!s)N#2 (32)

and

cov(y(s), y (t))"p2

a2s#

p2

2a3[!2#2 expM!asN#2 expM!atN

!expM!a (t!s)N!expM!a (t#s)N]. (33)

See Reference 6 for the derivation of (30) and (33).Using various combinations of (28) and (30)}(33) we can now determine the parameters of the

conditional bivariate normal distribution (19) which in turn can be used in (7) and (17) to evaluatethe distributions of f

nand m

n, respectively.

It is possible to avoid the numerical evaluation of bivariate normal distributions by writing theconditional density in (7) and (17) as a product of univariate normal distributions. The parametersof the univariate distributions would be given by expressions similar to (19) and would require thesame basic results.

6. CONCLUSIONS

We have presented two methods for obtaining the distribution function of the discounted valueand accumulated value of a series of deterministic future cash #ows in an environment where the

340 G. PARKER


rates of return are random. This would be useful when pricing or valuing portfolios of insurancecontracts.

Those two distributions could also be obtained by simulating many paths of future rates ofreturn and using them to discount or accumulate the cash #ows (see the discussion by Dr. Cairnsin Reference 2). It would be interesting to compare a Monte Carlo approach and the approxima-tion suggested in Reference 3 with the approach of this paper.

REFERENCES

1. R. Norberg, &Bonus in life insurance: principles and prognoses in a stochastic environment',=orking Paper No 142,Laboratory of Actuarial Mathematics, University of Copenhagen, 1996.

2. G. Parker, &Stochastic analysis of the interaction between investment and insurance risks', North Am. Actuarial J. 1(2),55}84 (1997).

3. G. Parker, &Limiting distribution of the present value of a portfolio', AS¹IN Bulletin 24(2), 167}81 (1994).4. G. Parker, &A second order stochastic di!erential equation for the force of interest', Insurance: Math. Economics, 16(3),

211}24 (1995).5. J. L. Melsa, and A. P. Sage, An Introduction to Probability and Stochastic Processes, Prentice-Hall, Englewood Cli!s,

NJ, 1973.6. D. F. Morrison, Multivariate Statistical Methods, McGraw-Hill, New York, 1990.7. N. L. Jonhson and S. Kotz, Distributions in Statistics: Continuous Multivariate Distributions. Wiley, New York, 1972.8. L. Arnold, Stochastic Di+erential Equations: ¹heory and Applications, Wiley, New York, 1974.9. G. Parker, &Moments of the present value of a portfolio', Scandinavian Actuarial J. 1, 47}60 (1994).



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Stochastic interest rates with actuarial applications