Starting from the moment generating function of the annuity certain with stochastic interest rate written by means of a time discretization of the Wiener process as an n-fold integral, a straightforward evaluation of the corresponding distribution function is obtained letting n tend to infinity. The advantage of the present method consists in the direct calculation technique of the n-fold integral, instead of using moment calculation or differential equations, and in the possible applicability of the present method to varying annuities which could be applied to IBNR results, as well as to pension fund calculations, etc.Distribution; Annuities; Processes; Evaluation;

De Schepper, A

Dhaene, Jan

Goovaerts, Marc

Vanneste, M

English

Research Papers in Economics

DEPARTEMENT TOEGEPASTE 
ECONOMISCHE WETENSCHAPPEN 
ONDERZOEKSRAPPORT  NR  9627 
A Straightforward Analytical Calculation of the Distribution 
of an Annuity certain with Stochastic Interest Rate 
by 
M. Vanneste 
M.J. Goovaerts 
A. De Schepper 
J. Dhaene 
Katholieke Universiteit Leuven 
Naamsestraat 69,  8-3000  Leuven ONDERZOEKSRAPPORT  NR  9627 
A Straightforward Analytical Calculation of the Distribution 
of an Annuity certain with Stochastic Interest Rate 
0/1996/2376/27 
by 
M. Vanneste 
M.J. Goovaerts 
A. De Schepper 
J. Dhaene A STRAIGHTFORWARD ANALYTICAL 
CALCULATION  OF THE DISTRIBUTION 
OF AN ANNUITY CERTAIN WITH 
STOCHASTIC INTEREST RATE! 
M. V  ANNESTEa, M.J. GOOV  AERTSa.C,  A.  DE SCHEPPERb,  J. DHAENE3 
a  K.U. Leuven, CRIR, Minderbroederstraat 5,  B-3000 Leuven 
b  RUCA, Middelheimlaan 1,  2010 Antwerpen 
c  Univ.  Amsterdam, Roeterstraat 11,  WB  NL-1018 Amsterdam 
Abstract 
Starting from the moment generating function of the annuity certain with stochastic interest rate written by 
means of a time discretization of the Wiener process as an n-fold integral, a straightforward evaluation of the 
corresponding distribution function is obtained letting n tend to infinity. The advantage of the present method 
consists  in  the  direct calculation technique of the  n-fold  integral, instead  of using moment  calculation or 
differential equations, and in the possible applicability of the present method to varying annuities which could 
be applied to IBNR results, as well as to  pension fund calculations, etc. 
I  Work performed under OT/93/5 1.  Introduction 
In the actuarial literature, the problem of how to incorporate the randomness of interest rates 
in the evaluation of actuarial quantities, has been given a lot of attention. 
Different models  for  modelling  the  interest rates,  as  well  as  other types of rates,  such  as 
growth rates, inflation rates, etc., are studied. A lot of papers only obtain results for first and 
second  order  moment of the  actuarial  quantities  under  investigation.  We  mention e.g.  the 
papers of Beeckman and Fuelling (1990,1991). 
The papers by  Yor (1992,1993), De Schepper et al  (1992,  1994),  give  also results for  the 
distribution of the annuity certain. Both authors make use of different techniques, to  obtain 
the same result for the distribution. The randomness of  the interest rates, is measured in both 
cases by a Wiener process. 
The problem dealt with in the present paper is the straightforward evaluation of  the generating 
function of the annuity certain with random interest rate 
- fe-lit  -x(t)  d-r 
all II  ~  " 
where x("C)  represents a Brownian motion, and ()  denotes the risk free interest intensity. This 
means that we aim at the direct evaluation of 
by considering a discretization of the Wiener process with respect to the time variable, in n 
subintervals and subsequently letting n tend to infinity. 
The expectation can be obtained by taking the integral over XI  =x(t) of the expression 
K( t ,x(t); °  ,x(O))  lim  1 
n-+CIJ  ~11  v2nE  f··· f 
as explained in De Schepper et al  (1992). 
rr
"-1  ~  (X-X_1)2  -O·E-t 
dx  .. exp{ -(~  1  J  +Eue  ?Ii)} 
j=1  1  j=1  2E 
The present method differs from the existing methods because straightforward evaluation of 
the  (n-I)-fold  intergral  is  presented  instead  of using  differential  equations  or  moment 
calculations. 
The  same  methods  will  also  enable  us  to  obtain  some  limiting  distributions  of GARCH 
processes, as  will be explained in a forthcoming paper. 
2 2.  Transformation of the (n-l)-fold Wiener integral 
We  start with the  linear transformation of the variables  8j  £  +  x.  =  y. in the expression of 
/  J 
K(t,x(t);O,x(O)).  This results in : 
~8'1  +8(x(l) -x(O)) 
K(t,x(t);O,x(O))  =  e 2  Ko(t,x(t) +8t;O,x(O)) 
with 
Ko(t,x(t) +8t;O,x(O)) =  lim  1 
n--+oo  ~/l 
V21t £  J  ... J 
In the notation of Feynman integrals extended to  imaginary time, the right hand side of this 
expression can be written as  follows 
(/,y{/))  I 
Ko(t,y(t);O,y(O))  =  f Dy('r) exp{-!.. f  y2dr  - u f  e-Ydt} 
(0,j.{0))  2 !  ! 
(2) 
where 
yeO)  = x(O)  and  yet)  = x(t)  +  8t 
The  multiple  integration in (1)  is  transformed by  means of the  coordinate transformation 
yet)  = -K lnq(t) or rather in the discritized version of the coordinates y.  = -K lnq  ..  For a 
./  ./ 
general coordinate transformation the interested reader is referred to Khandekar and Lawande 
(1986). 
and 
- q./.  +  q/'-I  q.  =  . 
./  2 
then  the  following  approximation  for 
lnq.  in  terms  of f::..q.  and q.  is obtained (preserving terms up to  order £2)  : 
./  J./ 
Inq  Inq. 
f::..q.  1  f::..q2  1  1  f::..q 3  1  2  0(£2)  =  +  ./  J  +  J  +  - --
./  ./  2 
- 4  2  -2  8  6  -3  q.  q.  q . 
./  ./  J 
from which an expression for part of the argument of the exponential in (1) can be given in 
terms of the new coordinate variables qj 
3 f:...q 
(~  + 
q, 
f:...q2  1 
1 
2£  - 2  q 
1 
_1  )  +  0(£3/2) 
-2  q 
1 
In order to obtain a Wiener process, a new stochastic time is defined for  the q("C)  process by 
means of 
where aj  denotes the new length of the time interval related to  the new time variate t'. For 
symmetric  reasons  with  respect  to  the  indices  the  relation  defining  the  new  time  is 
transformed as follows 
12a  (f:... qy 
£  _1 (1  +  J) +  0 (£  3/2) 
-2  -2  q.  4q 
1  } 
The continuous equivalent of this time relation gives, performing the integration of the time 
relation  expression  the  following  connection  between  the  'old  time'  t  and  the  process 
depending 'new time' t' 
As a first result of this transformation the following equality (up to terms in £3/2) is obtained: 
as well as the transformed term 
n  n 
~  V  1 ~  - >;:-2  312 
U  ~  Ee-' J  =  UK-~q.  a  +  O(E  ) 
1  1 
.1=1  i=1 
In order to be able to express the (n-l) fold integral in (1) in terms of an (n-l) fold integral 
over the q-variates, we  still have to evaluate the Jacobian of the transformation 
4 1  n  dy  = nn-I  Kdqj  nil  vqjqj-I  (1  +  0(£)) 
~11""-.I  ~  y2n£  .1=1  .1=1  qj  .1=1  Ky2ncrj 
1  11-1  11  1 
vqoqn  - II  dqj II  (1  +  0(£)) 
K  .1=1  .1=1  v 2ncr 
.I 
Consequently the following intermediate result is obtained for the integrandum in the integral 
representation of Ko  (t,y(t) ; O,y(O))  (before taking the limit) 
n  (b.qy 
exp{ -L  .I 
.1=1  2crj 
1  --II  dqj 
V 2ncr  j=1 
.I 
n-I 
n  1  (b.q)4 
+ L  24  -/ 
.1=1  q. cr. 
.I  .I 
n 
- u12L qjK-2 cr)  (1  +  0 (£)) 
.1=1 
Because still an integration over the qj  's has to be performed the second summation in the 
argument of the exponential is transformed using 
+00  k2  f  k4 e - 2<3 dk 
+00  kl  !  (2cr)2 f  e - 2<3  dk 
-00  -00 
such that 
+O'J  k2  +00  k2 
-vk' - -31'0'-_  f  e  2<3 dk  =  f  e  2<3 dk  +0  (if), as also explained in Khandekar ans Lawande (1986) 
-00  -co 
Consequently within the integrand of the (n-l) fold integral, taking into account q  2 
.I 
1  b.q4  1  cr 
the factor  exp(-__  .I  )  may be changed into the form  exp(-_.I  ). 
24  -2  8 - 2 
qFj  q, 
(I 
One still has to connect the old time t to the new time t' by means of the relation  t= f 0dr .  t q 2( -r;) 
This can be done introducing a 8-function, and we make use of following relation 
[
{I  1  +00  .  (  Ir d  K'  ) 
8  t - f  d-r;~  =  _1_ f  e  {a  (-t  t  q2(t)  da 
~  q2(-r;)  2n  -00 
Performing the  additional integration over the stochastic time 1',  we obtain an integrandum 
of the following form 
5 +00 
f dae iu, 
J 
-00 
1 
Integrating with respect to  qj  's, we obtain the following result in terms of the notation used 
in (2)  . 
We use the notation D  + q(  "C)  to indicate that the integrations over q, only have to be performed 
over R+. 
Finally, performing the additional integration over XI'  we obtain the following result for the 
generating function of an annuity certain with the random interest is described as  a Wiener 
process: 
6 3. Analytical results 
For special choices of the parameter value K,  some tractable integrations over q are obtained. 
Indeed, for the choice K=4,  a special case of theorem 1 of Vanneste et al is applicable. In this 
case, we find the following result : 
I  I  co  +00  Cf)  1  I 
M(u)=E[e -ua,lR IX(O) =XO ]  =  ~  e 8'"2+°8 L  dx,eO(x,-X,,) L  dae ia, !  dt'  e 8
x"+8-" 
(I ',exp( -x/4-01/4))  I  I 
f  D+qCt)exp{ - ~ f q2dr:  +  r d: (~ -16ia)  - 16u f  q2dr:} 
(O,expLi4))  t  t q  t 
Theorem 1 of Vanneste et al can be applied. In this case , we have 
g  =  16ia  - ~ 
8 
and 
vCt)  =  32 vCr:)  giving llCt)  sinh(4~T) and  ~(T)  =  cosh(4~  T), 
4~ 
This gives the following result for the moment generating function : 
with 
Z  =  XI  +  8t  and  T  sinh(4~t) 
and Ik(x)  denotes as usual a modified Bessel function. 
Transforming the Fourier transform with respect to a  into an inverse Laplace transform with 
respect to  s,  the  same original function is  obtained in case the imaginary argument of the 
Fourier transform is considered as the variable of  the image function in the Laplace transform. 
Hence, one obtains : 
7 The inverse Laplace transform with respect to s can be worked out based on a result given 
in De Schepper (1994) resulting in 
_8' !.... 
M(u)  4Vu  1  -- --e  2  e  1  ~r;;:l  x" 
exp{ -2 V2u  (e  2"  +  e  :)} 
{inyt  T2r;;:l 
00  -8y'  16  1  _  x" _ = 
f  dy  e-(- sinhy  sin(~) exp{ -4{i;  -e 4"  4coshy} 
~  t  T 
One is still faced with one additional Laplace inversion with respect to u  ---+  x 
with 
A  =  2{2 r;;:l (e-~ 
·F 
One finally obtains: 
5  A'  1  -- __  (A2_2x)  x  2  e  4x 
4{i 
=  ~I  : 
+  e  "2)  +  4{2  ..!.  e  4"  "4  coshy 
T 
T 
On the other hand, if  we take K  to be equal to 2, the complexity of the problem in our present 
setting diminish substantially. We have 
8 1  ."  1  8  I  '"  CI)  I  I 
JvJ(u)  = E[e-lIa'l"lx(O)=xoJ  = -;/'2+ 41, dx1eI'J...x,-t,,) L  dae ial  I  dt'e-4Z1I'e4X"+4X' 
(t ',exp{ -x/2 -0(/2»  I  1 
,f . J  D+q(T)exp{ - ~ r  q2dT + f d~ (~ -4ia)} 
(O,exp  -.,,,!_»  t  t q 
Theorem  1 of Vanneste et al can be applied with  V(T)  0  giving  11(T)  =  T and S(T)  1, 
resulting in 
where 
z  =  xt + 8  t  and  4t'  =  x 
In this case, we find immediately the inverse with respect to u.  On the other hand,  let~-' 
denote the inverse Laplace transform, then we get (transforming the Fourier transform into 
a Laplace transform) : 
from this we find the same result, as obtained by A. De Schepper (1994), using differential 
equations, and by M.  Yor (1993) : 
-fi" ~  2rr2  ex:>  Xo  = 
e  2 elf dze 8(=-xo)  e  2'  '2 
-00 
v2  4  _  .fU  _:: 
-2::.....  -_" -,- ., coshy  4rcy 
dy  e  1  e  x  sinhy sin(-_) 
t 
9 References 
Yor M.  (1980),  Loi de  l'Indice du Lacet Brownien, et distribution de  Hartman-Watson, Z. 
Wahrscheinl.  und Verwante Gebiete 53  pp.  71-95 
Yor M.  (1991),  On some exponential Functionals of Brownian Motion and the Problem of 
Asian Options, ETH, Zurich. pp.  40  - 56 
Yor M.  (1992), On some exponential funtionals of Brownian Motion, Advances in Applied 
Probability 14, pp.  509-531 
Yor M.  (1993), From Planor Brownian Windings to Asian Options, IME  13  pp 23  -34 
Vanneste M., M. Goovaerts, E. Labie (1994), The distribution of Annuities IME 15, pp.37-48 
Parker G.  (1994), A second order SDE for the force of Interest, IME, to  appear 
Dufresne, D.  (1990), The distribution of a Perpetuity, with application to Risk theory and 
Pension Funding, SAJ, pp.  39-79 
De  Schepper A.,  F.  De Vylder,  M.  Goovaerts,  R.  Kaas  (1992),  'Interest randomness  in 
annuities certain, IME 11,4, pp. 271-282 
De Schepper,  A.,  M.  Teunen, M.  Goovaerts (1994),  An analytical Inversion of a  Laplace 
Transform related to annuities certain, IME 14, pp. 33-37 
De Schepper A., M. Goovaerts, F. Delbaen (1992), The Laplace transform of  annuities certain 
with Exponential Time distribution, IME (1992), pp 291-294 
Beeckman J., C. Feulling (1990), Interest and Mortality Randomness in Some Annuities, IME 
9,  pp185-196 
Beeckman J., C.  Feulling (1991), Extra Randomness in Certain Annuity Models, IME 10, pp. 
275-287 
Khandekar D.C. and S.V. Lawande (1986), Feynman Path Integrals: Some exact results and 
applications, Physics Reports 137 , Ns. 2 and 3, pp.  115- 229 
10 

Loi de l'Indice du Lacet Brownien, et distribution

A straightforward analytical calculation of the distribution of an annuity certain with stochastic interest rate.

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A straightforward analytical calculation of the distribution of an annuity certain with stochastic interest rate.

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