基于CRRA效用准则的资产负债管理

本文在连续时间不完备市场框架下，考虑了投资者终端时刻资产负债比率的期望效用最大化问题。假设金融市场由1个无风险资产与多个风险资产构成，其中风险资产的价格过程由几何布朗运动刻画；投资者在整个投资时间水平内面临一个由几何布朗运动刻画的外生负债。利用随机动态规划方法，给出了相应的HJB方程与验证定理，并得到了最优投资策略与最优值函数的解析表达式。进一步，通过敏感性分析与数值算例发现：（1）外生负债的预期增长率与当前时刻的资产负债比率对最优投资策略没有影响；（2）在不考虑外生负债时，在最优策略下，投资到风险资产上的资金比例随着风险资产波动率或相对风险厌恶系数的增大而减小，而在考虑外生负债时，并非如此，只有满足一定条件时最优投资策略才是风险资产波动率或相对风险厌恶系数的减函数；（3）不考虑外生负债时，最优值函数是投资时间水平与风险资产预期收益率的增函数，风险资产波动率的减函数，但在考虑外生负债时该结论只在各参数满足一定关系时才成立，否则结论相反。

Asset-liability Management based on CRRA Utility Criterion

An optimization problem of maximizing the expected utility from an investor's terminal ratio of asset to liability is studied in a continuous-time incomplete financial market. Suppose that the financial market consists of one risk-free asset and multiple risky assets, whose price processes are modeled by geometric Brownian motions, and that during the investment process, the investor is faced with an uncontrollable exogenous liability, which is also described by a geometric Brownian motion. Corresponding HJB equation and verification theorem are provided, and the closed-form expressions for the optimal investment strategy and optimal value function are derived by adopting the stochastic dynamic programming approach. Furthermore, by employing sensitivity analysis and numerical examples, it can be found that: (1) The optimal investment strategy is independent of the appreciate rate of the exogenous liability and the current ratio of asset and liability;(2) in the case without exogenous liability, the optimal proportions of investing on the risky assets decrease as the volatility of the risky assets or the relative risk aversion coefficient increases, but in the case with exogenous liability, this result only holds when the parameters satisfy some conditions;(3) in the case without exogenous liability, the optimal value function increases as the investment time horizon or the appreciate rate of the risky assets becomes larger, and decreases as the volatility of the risky assets becomes larger, but this result also only holds when the parameters satisfy some conditions.