Current developments in German pension schemes: what are the benefits of the new target pension? (Q825284)

A pension scheme is considered where the benefits are adjusted if the funding ratio falls out of a certain interval. A fixed part $\pi$ of the surplus is invested into a risky asset, the remaining part into a riskless asset. The financial market is modelled with a Black-Scholes model. More specifically, $N_0 = n$ is the number of initial members in the scheme, all of the same age $x$. By $N_t$ we denote the number of members alive at time $t$. If the value of the fund is $X_t$ before the annuity payments, it is $X_t' = X_t - N_t L_t$ after the payments, where $L_t$ is the amount paid at time $t$. The value of the fund at time $t+1$ becomes $$X_{t+1} = X_t' \exp\{r + \pi(\mu-r) - \frac12\pi^2 \sigma^2 + \pi\sigma(W_{k+1} - W_k)\}\;,$$ where we used the usual notation for the Black-Scholes model. The funding ratio at time $t$ is now $$ F_t = \frac{\mathrm{Assets}_t}{\mathrm{Liabilities}_t} = \frac{X_k}{N_k L_{k-1} \ddot a_{x+k}}\;,$$ where $\ddot a_{x+k}$ is the present value of a life long annuity of size $1$. The amount paid at time $t$ is determined by $$L_t = \begin{cases} L_{t-1}\;, & \text{ if }F_t \in [1,1.25],\\ L_{t-1} + \frac\alpha{n} (X_t - N_t L_{t-1} \ddot a_{x+t})\;, & \text{ if }F_t > 1.25\\\max\{L_{t-1} + \frac\beta{n} (X_t - N_t L_{t-1} \ddot a_{x+t}), 0\}\;, \text{ if }F_t < 1.\end{cases}$$ The parameters $\alpha$, $\beta$ are in $(0,1)$. That is, if the performance is good, the value paid is increased, if it is bad, the value paid is decreased. It is discussed, how the values $\alpha, \beta$ have to be chosen in a risk neutral environment. The retiree now decides to invest a fraction $\phi \in [0,1]$ of his wealth in a classical pension scheme and the fraction $1-\phi$ in a target pension scheme as defined above. Then, using an expected utility approach, the payments to the customer are compared to the payments of a defined benefits scheme, to a defined contribution scheme with a varying annuity, and with a scheme where the benefit from the defined contribution scheme is consumed at time $0$. The utility is measured by $\mathbb{E} [\sum_{k=0}^\infty e^{-\rho k} \mathbb{I}_{\zeta > k} u(L_k)]$, where $\zeta$ is the remaining lifetime of the retiree. The utility function is chosen $u(y) = y^{1-\gamma}/(1-\gamma)$ where $\gamma > 0$, $\gamma \ne 1$ is the coefficient of relative risk aversion. The optimal share $\phi^*$ is calculated for several examples. It turns out that the target pension scheme is a good possibility to diversify the scheme. In my mind, $\gamma$ should be chosen in $(0,1)$ only because otherwise the utility increases if the retiree dies earlier.