solvency ratio set by regulatory authorities, perform rather well in terms of expected total returns for long investment horizons. They come close to pure stock investments with respect to the average value they generate for the investor. However, it is also very important for the financial institution to keep an eye on the expected amount of regulatory capital required by a certain investment strategy. Due to the conditional change in allocation when the critical regulatory value is reached, the expected capital charge is significantly smaller than in the case of a pure equity investment.
Besides the basic type of strategy, the length of the investment horizon is an important factor for the risks and rewards of alternative strategies. In general, the longer the maturity of the plan, the lower the expected capital charge, since the critical level set by the authorities in Germany contains a discount factor, the higher the expected total return. Nevertheless, it is important to consider other risk variables as well in this. We are far from claiming that one of the strategies discussed here should be seen as uniformly superior to any other. Rather we seek to point out the benefits and risks offered by the different types of products, to provide a basis for a thorough discussion of the issues involved in product design and regulation.
Consider an investment plan where payments into an IPA are made at equally spaced points in time t=0, 1,. . .,T (e.g. months). Let Pt denote the sum of payments up to time t, T the planned terminal date of the plan (equal to the beginning of the payout phase), and q(rf, t, T − t) = (1 + rf, t)t−T the discount factor with risk-free rate rf, t and remaining time to maturity T−t. Without loss of generality we assume that the investor holds exactly one share of the fund at time t. We are interested in the solvency ratio Vt/Pt at time t, which makes sure that the uncertain market value of the shares Vt+1 at time t+1 is less than the sum of payments Pt into the plan discounted up to time T, that is, less than Pt· q(rf, t, T − t − 1), with a probability of at most ε.
To be able to quantify this shortfall risk, we have to specify a model for the random evolution of the value of the investment shares. Here we make the standard assumption that the dynamics of this value can be described by a geometric Brownian motion. This implies that the relative change in value (i.e. the log-return) ln(Vt+1)−ln(Vt) is normally distributed with mean μ und variance σ2. Formally we obtain the desired solvency ratio as the solution of the following inequality: