Document 6493398

Tijdsclirift voor Econoiuie ei1 Management
Vol. XLVI, 4, 2001
How to Determine the Capita1
Requirement for a Portfolio of Annuity
LiabiPities
by J. DHAENE,
M. J. GOOVAERTS,
S. VANDUFFEL
aiid D. VYNCICE
ti.U.Leuven, Departiiient o[
Applied Economics, Actiiarial
Sciences
R.U.Leuven, Departinent of
Applied Economics, Actuarial
Sciences
K.U.Leuven, Departinent of
Applied Econoinics. Actuarial
Sciences
David Vyncke
K.U.Leuven, Department of
Applied Econornics, Actuarial
Sciences
ABSTRACT
This paper illustrates an analytic inethod iliat caii be used to determine the total
capita1 requirements necessary to properly provide for ilie fuhlre obligations of a
portfolio of annuity liabilities and to protect the enterprise froin the related risks it
iaces. This example is based on the work of Kaas, Dhaeiie and Goovaerts (2000).
I. INTRODUCTION
The deterinination of these requirements entails the aiialysis of the
distribution function, more specifically the tail of the distribution
function (os the catastrophic part) of tlie present value of tlie future
cash flows.
The projection of the future cash flows relating to the annuity
obligations, and the subsequeiit deterniination of tlieir present value,
may be at its simplest, when the aiiiount aiid timing of the asset aiid
liability cash flows are iiisensitive to varying economic conditions.
Eveii in these circumstances, tlie possible impact of inortality
improvemeiits and the fliture course of the reiiivestinent assumptions are important. On the otlier hand, these coinputations are most
complex, wlieii the timing and amount of the cash flows are affected
by the economic scenario (e.g. annuity payinents whose aniount and
tiining are solely, or partly, driven by economic perforiiiance of
some type as wel1 as to the policyholder reaction to such performance).
The use of inappropriately siinple (i.e. perliaps for computational
ease os convenience) metliods for these computatioiis, is liltely to
introduce hidden "surplus" (could be a deficit too). Capita1 req~iirements calculated under such a regime may iiot reflect the t-me capita1
req~iiredto support the insurer's busiiiess.
In order to coinpute the liltelihood that an insurer wil1 not be able
to meet its obligations when they fa11 due, knowledge is required of
the nature of the cash flows and any underlying stocliastic process
which drives tlieir timing and aiiio~int.The inethod illustrated in this
exainple starts froin this knowledge. It enables the actuary to determine (approximately) tlie relevant probabilities of extreme events. Tt
allows the actuary to deterinine the necessary provision, based on tlie
level safety desired.
IT. THE CASH FLOW OF THE FUTURE LIABILITIES
Firstly, the cash flows depicting the filture paynlents of the annuity
portfolio are projected int0 the fuh~re.The cash flows represent, for
each year between 2002 and 2079, the expected payiiient of that year,
adjusted with a safety niargin. The expectations of the paynients diie
are calculated usiiig realistic technica1 bases concerning disability,
n~orbidityetc. The additional inargin is a safety niargin against the
possible negative deviations and also includes costs.
111. THE DISTRIBUTION OF THE CAPITAL REQUIREMENT
We will detennine the present value at Januaiy 1, 2001, which we
will consider as the time 0. The time unit is chosen to be one year.
Let ai be the amount due at time i. The stochastic capital req~iirement, ?i is then given by:
V
= a,e-Y'
+ a,e-('i
+Y"
+ ... + ane-(?+ K- +..+Y>,)
where Y, is the return on the investnients from 2001 to 2002, Y, is the
rehirn from 2002 to 2003, etc.
We will assume that t l ~ eyearly returns Y, are normally distributed
with mean ,u and variance d.We also assume that the yearly returns
are inutually independent.
In the following Table, a number of possible investment strategies
with values of the paraineters ,u and o are given (figures fictious, for
illustrative purposes only).
P
0
100% Shares
0.10
0.15
75% 125%
0.09
0.12295
5OYo / 50%
0.08
0.09922
p
-
25% / 75%
0.07
100% Bonds
0.06
0.08173
0.075
.
p
It is clear that V is a sum of dependent random variables. Indeed,
the different discount factors wil1 be strongly positive dependeilt. The
computation of the distribution fùnction of V cannot be perforined
exactly. One could try to deterniiile the distribution fuilction of V by
siinulation, but this will lead to untsustable estimates for the tail
probabilities. Moreover, siniulation will be very time-consuming for
determiiiing the optiinal asset mix. Recent actuarial research results
allow to coinpute lower and upper bounds for tlie distribution of V. It
will be show11 that these lower and upper bounds are often close to
each other, which of course illustrates the accurateness of the approxiinatioils.
On the next pages one finds upper and lower bounds for the distribution of V for the different investinent strategies. These bounds are
upper and lower bounds in the sense of convex order for the exact but
unl<nown distribution íùnction. Tlie upper bound is denoted by CO
(dotted line), the lower bound by ES (hl1 line). We also present
approximations for the percentile, the expected shortfall and the conditional tail expectation at different levels.
Lower bound
1
upper bouild
I
Exact
Mean
4.225.360
1
4.225.360
/
4.225.360
Lower
Percentile
Expected Shortfall
Percentile
1 Expected Shortfall
I
Conditional Tail
I
1
/
Exoectation
1
/
1
Conditional Tail
Expectation
bound
I
Upper
/
1
bouild
1
1
Lower bound
4.413.501
Upper bound
4.413.501
4.413.501
StDev
1.898.995
2.267.604
1.908.791
Lower
Percentile
Expected Shortfall
bound
1
Exact
Mean
Upper
bound
Conditional Tail
Ex~ectation
Percentile
Expected Shortfall
Conditional Tail
Ex~ectation
Lower bound
4.783.853
Upper bould
4.783.853
Exact
Mean
StDev
l 660.798
1.966.5 17
1.665.608
Lower
Percentile
bouiid
Upper
bourid
4.783.853
Expectation
Percentile
Expected Shorlfall
Conditional Tail
Exoectation
1
Lower
Percentile
1
Expected Sliortfall
bound
80%
1
Conditional Tail
Expectation
6.556.404
Expected Shortfall
Conditional Tail
Expectation
1
Lower bound
Exact
Upper bouild
Mean
6.379.328
6.379.328 2.057.314
StDev
1.757.507
I ~ower /
Percentile
1
Expected Shortfall
6.379.328
1.760.055
1
bound
80%
Coilditioilal Tail
/
Expectation
276.730
7.690.022
1
9.073.672
1
Expectation
I
IV. COMPARISON BETWEEN THE DIFFERENT INVESTMENT
STRATEGIES
A. Upper Boz~nd(CO)
B. Lower Bounds (ES)
The expectations of tlie lower and upper bouild of V are always
equal. This is beca~isethe bouiids have exact expectations.
From numerical comparisons it follows tliat the best estimate for
the distribution of the present value of the cash flow under consideration is tlie lower bouild. This is confirmed by the fact that tlie exact
Standard Deviation aild tlie Staiidard Deviation of the lower bouiid
are very close to each otlier.
V. THE OPTIMAL ASSET ALLOCATION
The optimal asset allocatioii is a coinbination of the different iiivestineiit strategies (also talcing int0 accouilt the dependenties between
the yearly returns), that niini~njzesa certain criterion.
Here we wil1 assuine that the optima1 asset allocation miniinizes
the initia1 amount that has to be reserved sucli that, with probability
of 99%, al1 future obligations can be met. Hence, the optiinal asset
allocation is the one that minimizes the 99%-percentile of V.
Considering the lower bouiids, the optiinal investment strategy
tums out to iiivest 40,40% in shares aiid the remaining part in boiids,
see the following figure. Considering the upper bouilds (which are
less accurate) we find that the optiinal investment strategy is to iiivest
35,87% in shares.
ESLM
-----
,
i I"" O#>"
3 O<##,IiUU
'"O"
""O
V B10 000
--,
, _
s?OG%
cotkprr ao,o(m
10814
Ooi
>iu ~ i i a a
Lower
Lower bouiid
Upper bo~uld
4.986.280
5.095.640
Percentile
Expected Shoitfall
bound
1
Uppei
bound
Conditional Tail
Expectation
/
Perceniile
Expected Shortfall
Conditional Tail
Exoectation
K Kaas, J. Dhaene and M.J. Goovaerts, 2000, Upper aild Lower Bounds for Sums of
Random Variables, Inszirntice Matl~emnt~cs
& Econonzzcs 27, 151-168.
R. Kaas, M. Goovaerts, J. Dhaene aiid M. Denuit, 2001, Modem Actuarial Risk Theory,
(Kluwer), to appcar