1

Combined Accumulation- andDecumulation Plans with Risk-Controlled Capital Protection

13th International AFIR ColloquiumMaastricht, September 17th – 19th 2003

Peter Albrecht / Carsten WeberUniversity of Mannheim

2

Table of content

I. The Investment Problem

II. Methodology

III. Results

IV. Comments

3

I.

The Investment Problem

4

A retiree possesses a certain amount of wealth W, which he invests in investment funds F and money market funds MM during a certain time horizon T, according to the following targets:

The investment problem (I)

5

The investment problem (II)

A minimal F to achieve at least an accumulated wealth of the original W [or some fraction (1-h)W] in real terms for a defined bequest (capital protection in real terms).

The remaining MM to be withdrawn as an annual annuity due, constant in real terms, for consumption needs (annuitization in real terms).

6

Illustration of the investment problem

part of wealth Fto be minimized

investment funds

target: capital protection

in real terms

original amount ofwealth W

part of wealth MM(to be maximized)

money market funds

target:annuitization

constant in real terms

7

II.

Methodology

8

Methodology (I)

We apply the methodology of shortfall probability and Value-at-Risk respectively to an accumulated F.

Thus, risk-controlled capital protection intuitively means:

At the end of a previously fixed time horizon, the desired fraction of W may fall short merely in a maximum of out of 100 investment outcomes.

The confidence coefficient (or the degree of certainty (1- )) is defined by the retiree, e.g. = 5%, 10%.

9

Methodology (II)

Implying that the Value-at-Risk of the distribution of the accumulated F in T has then to be equal to the desired fraction of W, we find:

with Q representing the -quantile of a T-period return of a multi-asset portfolio and x representing the vector of fund allocations of the portfolio.

Condition of risk-controlled capital protection:

)x(QW)h1(

)x(FF

WF

10

level of confidence

time horizon T

risk-controlledfund investment

condition of risk-controlledcapital protection

calculation ofValue-at-Risk

stochastic process forthe accumulation of F

average investment returns,volatility and correlation of

funds of multi-asset portfolio

fund allocation x

optimal risk-controlledfund investment

minimal F

risk control

optimization

selection

Procedure of formalization

11

Application to a triple-asset portfolio (I)

We consider a portfolio of a representative stock, bond and property fund.

We assume a tri-variate geometric Brownian motion modelling the returns of the respective funds.

For each fund allocation x being analyzed, we generate the distribution of the T-period return of the triple-asset portfolio using a Monte-Carlo simulation and derive its Value-at-Risk.

12

Application to a triple-asset portfolio (II)

Investing in the fund allocation x, that delivers the highest Value-at-Risk, consistently leads to the minimal amount of F.

We only consider a representative set of fund allocations (varying each share in steps of 5%):

stock 0% 0% … 0% 5% … 5% 10% … 10% … 95% 95% 100%

bond 0% 5% … 100% 0% … 95% 0% … 90% … 0% 5% 0%

property 100% 95% … 0% 95% … 0% 90% … 0% … 5% 0% 0%

13

III.

Results

14

Identification of parameters in real terms

Average rates of return:

mstock= 8% (5%), mbond= 4%, mproperty= 3,3%

Volatility of funds:

vstock= 25% , vbond= 6%, vproperty= 2%

Correlation between funds:

pstock/bond= 0.2, pstock/property= -0.1, pbond/property= 0.6

Issue surcharge of funds:

astock= 5% , abond= 3%, aproperty= 5%

15

Numerical results (I)

First, we examine the case of mstock= 8%, assuming an original wealth of W=100.000 € and a real money market return of mmoney= 1,5%. Time horizon T (in years)

5 10 15 20 25

( 5% 0% 95% ) ( 5% 0% 95% ) (10% 5% 85% ) (10% 5% 85% ) (15% 15% 70% )

95% 94.851,07 81.533,17 69.232,59 58.189,23 48.499,59

1.060,91 1.973,80 2.273,47 2.401,72 2.451,92

( 5% 0% 95% ) (10% 5% 85% ) (15% 20% 65% ) (20% 30% 50% ) (25% 40% 35% )

90% 93.189,78 79.201,37 66.248,61 54.455,79 43.912,82 De

gre

e o

f ce

rta

inty

1.403,21 2.223,04 2.493,96 2.616,18 2.670,29

16

Numerical results (II)

Second, we examine the case of mstock= 5%, ceteris paribus.

Time horizon T (in years)

5 10 15 20 25

( 5% 0% 95% ) ( 5% 0% 95% ) (5% 0% 95% ) (5% 5% 90% ) (5% 5% 90% )

95% 95.552,87 82.774,21 71.127,18 60.889,03 51.978,41

916,31 1.841,16 2.133,47 2.246,63 2.286,29

( 5% 0% 95% ) (5% 5% 90% ) (5% 5% 90% ) (5% 10% 85% ) (10% 20% 70% )

90% 93.888,81 80.701,59 68.909,11 58.607,26 49.446,32 De

gre

e o

f ce

rta

inty

1.259,18 2.062,69 2.297,37 2.377,70 2.406,84

17

Structural results

The longer the time horizon, the larger the share of stocks (and bonds).

The longer the time horizon, the smaller the amount of F and the larger the amount of MM disposable for the annuity due.

The larger the degree of certainty, the lower the share of stocks and bonds (and the larger the share of property).

Applying a lower average stock return leads to a larger amount of F and to a lower share of stocks.

18

Comments

But, the fixed time horizon neglects the uncertainty of a retiree‘s live span.

Very practicable since only capital market data and a single risk preference parameter enter the model.

A single risk preference parameter, the degree of certainty (1- ), is much easier to communicate to retirees than utility based approaches.

Structural results are very intuitive and consistent with prior results about the attractiveness of stocks in the long-run.