1. 1 Combined Accumulation- and Decumulation Plans with Risk- Controlled Capital Protection 13th International AFIR Colloquium Maastricht, September 17th – 19th 2003 Peter Albrecht / Carsten Weber University of Mannheim

2. 2 Table of content I.The Investment Problem II.Methodology III.Results IV.Comments

3. 3 I. The Investment Problem

4. 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. 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  The remaining MM to be withdrawn as an annual annuity due, constant in real terms, for consumption needs (annuitization in real terms).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. 6 Illustration of the investment problem part of wealth F to be minimized investment funds target: capital protection in real terms original amount of wealth W part of wealth MM (to be maximized) money market funds target: annuitization constant in real terms

7. 7 II. Methodology

8. 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. 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:

10. 10 level of confidence  time horizon T risk-controlled fund investment condition of risk-controlled capital protection calculation of Value-at-Risk stochastic process for the accumulation of F average investment returns, volatility and correlation of funds of multi-asset portfolio fund allocation x optimal risk-controlled fund investment minimal F risk control optimization selection Procedure of formalization

11. 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. 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%):

13. 13 III. Results

14. 14 Identification of parameters in real terms  Average rates of return: m stock = 8% (5%), m bond = 4%, m property = 3,3%  Volatility of funds: v stock = 25%, v bond = 6%, v property = 2%  Correlation between funds: p stock/bond = 0.2, p stock/property = -0.1, p bond/property = 0.6  Issue surcharge of funds: a stock = 5%, a bond = 3%, a property = 5%

15. 15 Numerical results (I) First, we examine the case of m stock = 8%, assuming an original wealth of W=100.000 € and a real money market return of m money = 1,5%.

16. 16 Numerical results (II) Second, we examine the case of m stock = 5%, ceteris paribus.

17. 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. 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.