Robust portfolio choice for a DC pension plan with inflation risk and mean-reverting risk premium under ambiguity

Abstract

This paper investigates a robust optimal portfolio choice problem for a defined contribution (DC) pension plan member. The member worries about model ambiguity and aims to seek robust optimal investment strategies. Specifically, the member has a stochastic salary, considers inflation risk and invests her pension account wealth into a financial market consisting of a risk-free asset, an inflation-indexed bond and a stock whose expected return rate follows a mean-reverting process. By using the dynamic programming approach, the robust optimal investment strategy and the corresponding value function are explicitly derived, and subsequently a verification theorem is provided. Furthermore, two special cases of our portfolio model are discussed. Finally, a numerical example is presented to reveal economic implications of our theoretical results and to illustrate the effects of the model parameters on the robust optimal investment strategy. We find that ambiguity about the dynamics of the inflation-indexed bond price and the stock price and expected return rate has different influences on the robust optimal investment strategy, and that neglecting ambiguity will always lead to utility loss.

Keywords:

• Robust portfolio choice
• DC pension plan
• inflation risk
• mean-reverting process
• model ambiguity

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Zhang and Ewald [5], Chen et al. [7], Guambe and Kufakunesu [30] and Zhang et al. [31] also use a similar stochastic price index model to describe the inflation risk.

2 The technical requirement will be stated later.

3 According to the definition of the relative entropy, we have $\begin{array}{rl}{\mathrm{E}}^{\mathbb{Q}}\left[\ln {\mathrm{\Lambda }}^{\phi }\left(t\right)\right]& ={\mathrm{E}}^{\mathbb{Q}}\left[{\int }_0^t{\phi }_1\left(s\right)\left(\mathrm{d}{W}_S^{\mathbb{Q}}\right(t)+{\phi }_1(s\left)\mathrm{d}s\right)-\frac{1}{2}{\int }_0^t\left({\phi }_1\right(s){)}^2\mathrm{d}s\right.\\ & +{\int }_0^t{\phi }_2\left(s\right)\left(\mathrm{d}{W}_I^{\mathbb{Q}}\right(t)+{\phi }_2(s\left)\mathrm{d}s\right)-\frac{1}{2}{\int }_0^t\left({\phi }_2\right(s){)}^2\mathrm{d}s\\ & +\left.{\int }_0^t{\phi }_3\left(s\right)\left(\mathrm{d}{W}_{\mu }^{\mathbb{Q}}\right(t)+{\phi }_3(s\left)\mathrm{d}s\right)-\frac{1}{2}{\int }_0^t\left({\phi }_3\right(s){)}^2\mathrm{d}s\right]\\ & ={\mathrm{E}}^{\mathbb{Q}}\left[{\int }_0^t\frac{1}{2}\left(\right({\phi }_1\left(s\right){)}^2+\left({\phi }_2\right(s){)}^2+({\phi }_3\left(s\right){)}^2)\mathrm{d}s\right].\end{array}$

4 The assumption of $\gamma >1$ is not only to avoid problems with infinite expected utility that may be caused by $0<\gamma <1$ but also to ensure the solution of our optimization problem (16(16) $\begin{array}{rl}& \underset{\pi \in \mathrm{\Pi }}{sup}\underset{\mathbb{Q}\in \mathcal{Q}}{inf}{\mathrm{E}}^{\mathbb{Q}}\left[U\left(X^{\pi }\right(T\left)\right)+{\int }_0^{\mathrm{T}}\left(\frac{\left({\phi }_1\right(s){)}^2}{2{\mathrm{\Psi }}_1(s,X^{\pi }(s),\mu (s),L(s\left)\right)}\right.\right.\\ & \left.\phantom{{\int }_0^{\mathrm{T}}}\left.+\frac{\left({\phi }_2\right(s){)}^2}{2{\mathrm{\Psi }}_2(s,X^{\pi }(s),\mu (s),L(s\left)\right)}+\frac{\left({\phi }_3\right(s){)}^2}{2{\mathrm{\Psi }}_3(s,X^{\pi }(s),\mu (s),L(s\left)\right)}\right)\mathrm{d}s\right],\end{array}$(16) ) exists globally. This assumption can also be found in, e.g. [33–35].

5 Ghirardato et al. [36] characterize axiomatically the α-maxmin expected utility model $\alpha \underset{\mathbb{Q}\in \mathcal{Q}}{inf}{\mathrm{E}}^{\mathbb{Q}}\left[U\right(X^{\pi }\left(T\right)\left)\right]+(1-\alpha )\underset{\mathbb{Q}\in \mathcal{Q}}{sup}{\mathrm{E}}^{\mathbb{Q}}\left[U\right(X^{\pi }\left(T\right)\left)\right],$ where $\alpha \in [0,1]$ represents the decision-maker's ambiguity attitude and a larger value of α represents a more ambiguity-averse attitude. In particular, $\alpha =0$ and $\alpha =1$ represent extremely ambiguity-loving attitude and extremely ambiguity-averse attitude, respectively. In this paper, we only consider the extremely ambiguity-averse situation when $\alpha =1$. We will investigate the more general situations in future.

6 This robust control problem can be seen in [37,38].

7 A similar approach to interpretation is made in [23]. They consider ambiguity about the stock dynamics and the volatility process.

8 Note that both $J(t,x,\mu ,l)$ and $\hat{J}(t,x,\mu ,l)$ are negative.

Additional information

Funding

This research is supported by the National Natural Science Foundation of China [grant numbers 71721001, 71701084] and the Natural Science Research Team of Guangdong Province of China [grant number 2014A030312003].