考虑通胀风险下的最优投资组合–消费问题

Optimal Portfolio-Consumption Problem Considering Inflation Risk

摘要: 本文研究通胀风险下带有随机收入的最优投资组合与消费问题。在考虑投资者随机寿命、劳动收入以及通货膨胀影响的框架下，构建包含无风险资产、股票和通胀指数债券的金融市场模型。投资者目标是在生命期内最大化其消费、遗产及最终财富的期望效用，同时可通过购买人寿保险对冲死亡风险。通过引入等价鞅测度与状态价格密度，将动态优化问题转化为静态优化问题，并利用CRRA (常相对风险厌恶)效用函数的性质，推导出最优消费、人寿保险保费及投资策略的显式解。结果表明，最优投资策略由对冲通胀的资产配置、收入风险的调整以及投资者的风险偏好共同决定。本研究为具有随机收入和寿命不确定的投资者在通胀环境下的资产配置与消费规划提供了理论依据与决策参考。

Abstract: This paper studies the optimal portfolio and consumption problem with stochastic income under inflation risk. Within a framework that accounts for investors’ stochastic lifetime, labor income, and the impact of inflation, a financial market model is constructed, consisting of a risk-free asset, stocks, and inflation-indexed bonds. The investor’s objective is to maximize the expected utility of consumption, bequest, and terminal wealth over their lifetime, while also hedging mortality risk through the purchase of life insurance. By introducing an equivalent martingale measure and a state price density, the dynamic optimization problem is transformed into a static optimization problem. Leveraging the properties of the CRRA (Constant Relative Risk Aversion) utility function, explicit solutions for the optimal consumption, life insurance premium, and investment strategies are derived. The results indicate that the optimal investment strategy is jointly determined by inflation-hedging asset allocation, adjustments for income risk, and the investor’s risk preference. This study provides a theoretical foundation and decision-making reference for asset allocation and consumption planning for investors with stochastic income and uncertain lifetime in an inflationary environment.