DISTRIBUTION
OF THE
OF FUTURE
GARY
PRESENT
CASH
VALUE
FLOWS
PARKER
ABSTRACT
The present value of future cash flows with random interest rates is studied. The first three moments of the present value of future cash flows and an
approximation
of its cumulative
distribution
function are presented.
Particular 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:
process, Wiener
Cash Flows, Force of Interest, Present value, White
process, Ornstein-Uhlenbeck
process, Annuity-immediate.
1.
Noise
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),
Giacotto
(1986), Dhaene (1989), Beekman
(1992b)).
Other
papers
using
stochastic
models
Waters (1978),
and F’uelling
for the interest
Wilkie
(1990),
(1976),
Parker
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
density function
or the cumulative
distribution
function
of present
the
val-
Gary Parker
832
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 probabilistic 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 suggests 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
.iTl
where
. exp
{ - L!l(ti))
Y(t)
=Jt6,.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
applies to any common
the cash flows. In other
actually take at a futluc
any cash flow payable at
the same realization of future forces of interest
period of time involved in the discounting of
words, the value that the force of interest will
time s will be used (for time s) to discomit
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 ~~111of the expected values
of the cash flows. That is:
[3] clpI=E[~CFj.exp{
i=l
-y(ti)}]
-y(ti)}].
=cCFi.E[eXp(
i=l
The second moment about the origin of P is given by:
WI
E[P2]
151
E[P”]
= E [F,
F
exp { - y(ti) - y(Q)}]
CFi . CFj
i=l jzl
= 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
[7] E,P”,=j:j:jl:
f+k
[ i-1
CFi.CFj’Cl;i;‘eXp
jzl
{ -?,(ti)-y(tj)-y(tk)}
k=l
i=lj=lk=l
CFi.CFj.CFk’E
exp { -y(tx)-g(t,j)-?J(tk)}
1
1
Parker (1992b, sections 4 and 5) suggests using recursive equations
for the numerical evalllation of equations similar to 151and [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 function y(l) is normally distributed. Consequently, expressions of the form
w { - A?.) - Y(S) - v(t)) are lognormally distriblitetl, i.e.
PI
with parameters
exp { - g(r) -g(s)
- y(f)} - A(p. P)
834
Gary Parker
and
P = V[dr)] + V[ds)] + V[dt)] + 2~
PI
+ 2~
[Y(r), y(s)]
[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
the force of interest be defined as:
where Wt is a standard
Noise process. Let
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
and its autocovariance
P41
E [y(t)]
= 6.t
function is:
cov [y(s), y(t)] = a2 . min(s,t)
(see, for example, Arnold
(1974, section 3.2)).
Distribution
4.2. WIENER
of the presente
value of future
cash flows
835
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 autocovariante function of this process are:
and
P71
Cov[6,, St] = f12 . min(s, t)
Then, from the definition
distributed with mean:
P81
of y(t), it follows that y(t) is normally
JqYW]
and autocovariance
= 60. t
function:
WI
cov [y(s), y(t)]
WI
cov[Yb),YW] = ,“z . (s2t/2
= 1’
s’cov,&,
&,]dudv
0
4.3. ORNSTEIN-UHLENBECK
- s3/6)
s 5 t
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,
with initial value 60 (see, for example, Arnold
a 2. o,
o 2 o
(1974, p.134)).
Gary Parker
836
Then, it can bc shown that:
1221
E[&] = s + eFat. (6” - 6)
P31
cov[6,, d^t] = e-a(t+s)
k 7 ( eh5 _ 1) s<t_
and that the function y(t) is a gaussian process with mean:
E[y(t)]
I241
and autocovariance
= 6. t f (So - 6)
___
function:
[y(s), y(t)] = g9 min(s, t)+
COV
1251
+ $
[ -2 + 2eBas + 2,c-at _ e-alt-sl _ e -u(t+s)
1
(see, for example, Parker (1992b, section 6)).
The parameters ,LLand /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:
of the presente due
Distrilmtion
The distribution
of frhure
cash flouts
837
function of P is theu given by:
Fp(p) =
PI
S7a
(14
T/)4/
.
.I-=
--cv
Adapting the reslllts derived and justified in Parker (1992c, sections
4 and 5), the fimction
YJ)may be apI,roximated llsirig the followiug
recursive integral equat,ion:
CJj((I’%
.fy(j)(:,I& - 1)= x) . !IJ&’ - CF, . e-y, z)d5
[29] g&y) ” /
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:
VI
E[y(j)Ig(j
- 1) = X] = z+(j)]
{cc - E[y(j
-I-
(‘ov [Y(j)! Y(.i - l)] .
v [!/(A]
- l)]}
and with variance:
- 1)= :c]= v [l/W]P21 v [y(j)Iw(.i
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.
Gary Parker
838
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 OrnsteinUhlenbeck 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-
Gary
840
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-
4-
,I
2-
I'
,' /'
O0
,,
/
,I
I I'
/I
,'
,I, _
I II I
II
,I
c- .*I
xX
_ .,.,.._._.._
_...- ......'.'
-*
,.,,.,._.....
-......-......'..".
_-.........'.,,._,,...,,
r
;
-.-...
-,_.*..-.<..r.....
I I I1 t I I I I I I I I I I I
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 asymmetric distribution.
For
reasons discussed above, the Ornstein-Ulllen~~eck
Distribution
of the
generates
a distribution
other two.
presente
of P with
due
of future
a cocflicient
ctash flows
of skewness
Figures 4 and 5 present the crumulative
distrihltion
and 25 years annuity-immediate.
The integral
equations
were evaluated
by an arbitrary
discretisation.
0.8
0.6
0.4
0.2
0
3.5
3.8
4.1
4.4
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
4.7
841
between
the
functions
of 5
[28] and [29]
842
Gary Parker
II II I”I’I”I’-
1
l-
0.8
’ rr I’I’
’
,“I”@
-
0.6
0.6-
0.40.4
0.2
-
‘(“I
‘(“I
11 I1
I1 “‘I“‘I
,,,.,...................
,,,.,.___...............
__ __ __ __ __ __ __ __ -- -/.”
/.”
**-**-,:’
,:’
.:’ ,, ,*,*
.:’
/,.;,.;/ II II
j’j’
II
::
II
ii II
,:,: ,’,’
ii ,,
i,i,
ii ,,
ii ,,
i/i/
i,i,
irir
irir
i/i/
II
,“I”@
--
I
i
I!
1;
1;
1:
1;
I‘j;
Ii
I i
I !
I ;
I :
I ;
, I .:'!
I
i
I!
1;
1;
1:
1;
‘j
I ;
I i
I i
I !
I ;
I :
I ;
,I !
.:’
_ - - - - _ _ _._-__ ,..< ._._....
I I I,
I I,,,
I,
I,
I I I t;
I I,,,;
..
O0 - _ - - - - - - _._-__,..<._....
0
5
10
I
I
I
15
I,I t
I,I s
& I
20
I
I
25
I
I
I
I
I_
30
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 variable. 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
843
flows
BIBLIOGRAPHY
(1)
J. AITCHISON,
J.A.C.
BROWN,
University Press, 1963, 176~~.
(2)
L. ARNOLD
(1974),
Stochastic
Difierential
cations,
John Wiley & Sons, New York,
J.A. BEEKMAN,
C.P.
FUELLING,
Interest
Some Annuities,
Insurance: Mathematics
(3)
196.
J.A.
Lognormal
Cambridge
Distribution,
Equations:
Theory
1974, 228~~.
and Mortality
and Economics
and
Appli-
Randomness in
9, 1990, p.185-
Stochastic
models for bond prices,
funcimmunization
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
Moving 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.3979.
(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. Equilibrium
Approach,
Journal of Risk and Insurance 53, 1986, p.435-453.
(4)
tion
(11)
(12)
BEEKMAN
E.W.S.
space
integrals
and
The
SHIU,
A.M.
MOOD,
F.A.
GRAYBILL,
Statistics,
McGraw-Hill,
1974,
D.F.
MORRISON,
Multivariate
495pp.
(13) H.H. PANJER,
with
Applications
D.R.
D.C.
BOES,
Introduction
to the
Theory
of
564~~.
Statistical
BELLHOUSE,
Stochastic
to Life
Contingencies,
McGraw-Hill.,
Methods,
Modelling
Journal
of
Interest
1990,
Rates
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,
(Submitted), 1992b.
(16) G. PARKER,
Limiting
distribution
of the present
value of a portfolio,
(Submitted), 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.