  In this paper, we investigate the optimal management of defined contribution
(abbr. DC) pension plan under relative performance ratio and Value-at-Risk
(abbr. VaR) constraint. Inflation risk is introduced in this paper and the
financial market consists of cash, inflation-indexed zero coupon bond and a
stock. The goal of the pension manager is to maximize the performance ratio of
the real terminal wealth under VaR constraint. An auxiliary process is
introduced to transform the original problem into a self-financing problem
first. Combining linearization method, Lagrange dual method, martingale method
and concavification method, we obtain the optimal terminal wealth under
different cases. For convex penalty function, there are fourteen cases while
for concave penalty function, there are six cases. Besides, when the penalty
function and reward function are both power functions, the explicit forms of
the optimal investment strategies are obtained. Numerical examples are shown in
the end of this paper to illustrate the impacts of the performance ratio and
VaR constraint.
