Constant proportion portfolio insurance in defined contribution pension plan management

Abstract

We consider the optimal portfolio problem with minimum guarantee protection in a defined contribution pension scheme. We compare various versions of guarantee concepts in a labor income coupled CPPI-framework with random future labor income. Besides classical deterministic guarantees we also introduce path-dependent guarantees. To ensure that there is no bias in the comparison, we obtain the optimal CPPI-multiplier for each guarantee framework via using a classical stochastic control approach.

Appendix: Verifying the assumptions of the verification theorem

It is clear that our HJB equation has the same form as in Theorem 1. Therefore, to be able to apply the theorem, as a first step we need to prove that the optimal solution given by (11) is admissible. Then, as the second step, the proof of the following inequality should be given

$$\begin{aligned} {\mathbb {E}}\left( \sup _{t \in [0,T]} \left| J(t,C) \right| ^n \right) < \infty ~~\text {holds for real}~~ n \ge 1 . \end{aligned}$$

1st step: To investigate the admissibility of $m^*$, we give the next definition from Korn and Korn (2001).

Definition 1

(Definition 5.15 p. 225, Korn and Korn 2001) Let $(\varOmega , {\mathcal {F}}, {\mathbb {P}})$ endowed with the filtration $\left\{ {\mathcal {F}}_t \right\} _{t \in [0,T]}$ be a probability space. A U-valued progressively measurable process $u(t), t \in [t_0, t_1]$ is an admissible control if for all values $x \in {\mathbb {R}}^n$ the stochastic differential Eq. (13) with initial condition $X(t_0) = x$ possesses a unique solution $\{ X(t)\}_{t \in [t_0, t_1]}$ and if we have

$$\begin{aligned} {\mathbb {E}}\left( \int _{t_0}^{t_1} \left| u(s) \right| ^k ds \right) < \infty \end{aligned}$$

(14)

and

$$\begin{aligned} {\mathbb {E}} \left( \left\| X(\cdot ) \right\| ^k \right) < \infty \end{aligned}$$

for all $k \in {\mathbb {N}}$.

As $m^*$ is found to be a constant, it is bounded and inequality (14) holds. Moreover, the solution of our controlled process given in (6) is found as

$$\begin{aligned} C(t) = C_0 e^{\left( m \left( \mu _S - r\right) +r - \frac{m^2 \sigma _S^2}{2} \right) t + m \sigma _S W(t)}. \end{aligned}$$

Therefore, we have the uniqueness of the solution as well as the following inequality

$$\begin{aligned} {\mathbb {E}} \left( \left\| C(t) \right\| ^k \right) = C_0^2 e^{2\left( m \left( \mu _S - r\right) +r + \frac{m^2 \sigma _S^2}{2} \right) t} < \infty . \end{aligned}$$

(15)

Hence, the control $m^*$ is admissible.

2nd step: By (15), we have

$$\begin{aligned} {\mathbb {E}}\left( \sup _{t \in [0,T]} \left| J(t,C) \right| ^n \right) = {\mathbb {E}} \left( \sup _{t \in [0,T]} \left| \frac{C^{1-\eta }}{1 - \eta } h(t) \right| \right) , \end{aligned}$$

with h(t) given as in (12). As C(t) is bounded by (15), we obtain

$$\begin{aligned} {\mathbb {E}} \left( \sup _{t \in [0,T]} \left| \frac{C^{1-\eta }}{1 - \eta } h(t) \right| \right) < \infty , \end{aligned}$$

which completes the proof. Therefore, the solution $m^*$ is the optimal control for our optimal control problem (7).