模型不确定下带通胀的最优消费和投资组合问题研究

Optimal consumption and portfolio under inflation and model uncertainty

本文探讨了带有递归偏好的投资者在考虑通胀情形下的最优消费和投资组合。由于投资者担心模型的误定,因此寻求稳健的决策规则。通过考虑模型误定和通胀的随机波动对消费和投资组合带来的影响,建立投资者决策的值函数所满足的HJB方程,根据特定的效用函数,推导了最优消费和投资决策的显示解。通过对数值模拟结果的分析可知,对模型不确定的担忧导致了短视需求的大幅减少,从而引起最优股权分配比率的下滑。当考虑低通胀波动率且股票不确定与通胀不确定为正相关时,相对于无通胀情形,通胀对冲需求增加了最优股权分配比率;当考虑高通胀波动率且股票不确定与通胀不确定为负相关时,相对于无通胀情形,通胀对冲需求则加剧了最优股权分配比率的下滑。

An investor is often faced with two types of uncertainties. One type is probability uncertainty or risk when variables related to environment have aknown probability distribution. The other type is Knightian uncertainty or model uncertainty when information on the concerned phenomenon is not muchavailable. When an investor makes a decision through a model, he/she is often concerned with the misspecification of a model and seeks robust decision rules. For an investor participating in a financial market, he/she is confxonted with an optimal consumption and portfolio decision under a continuous timeframework, which stems from Merton＇s initiating work in 1969 and 1971. Since then, many researchers further investigate this problem under different markethypotheses and obtain many significant results. Recently, the inflation factor has been included in the problem of the optimal consumption and portfolio, where an investor believes completely in theselected model. However, since the availability of market information is often ineffective, the suggested model could be misspecified. Thus, it is necessary toconsider model uncertainty in modelling a financial market. Especially, it is necessary that both inflation and model uncertainty are incorporated in the model ofan investor＇s optimal consumption and portfolio decisions. This paper studies the optimal consumption and portfolio in the case of inflation with the recursive preference of investors who seek robust decision rules.An investor participates in a financial market with a riskless asset （bond） and several risky assets （stocks）. The investor＇s objective is to maximize the expectedutility of consumption discounted by the inflation while he/she is considering the impact of the model misspecification and inflation on the consumption andportfolio. Through the method of stochastic conlxol, the corresponding HJB equation of an investor＇s value function is derived. For a special utility, the explicitsolution to the problem of the optimal consumption and portfolio is obtained. Using a numerical simulation, we find that the concern about model uncertaintyleads to a substantial reduction of myopic demand, causing the decline of the optimal equity allocation. However, when stock uncertainty and inflationuncertainty in the case of the low inflation volatility are considered a positive correlation relative to the non-inflationary situation, the inflation hedge demandincreases the optimal equity allocation. When stock uncertainty and inflation uncertainty in the case of the high inflation volatility are considered, a negativecorrelation relative to the non-inflationary situation, the inflation hedge demand exacerbates the decline of the optimal equity allocation. Moreover, economic interpretations of our results are as follows. First, the ratio of wealth invested in risky stocks is affected by inflation fluctuation. In thecase of low inflation, as real interest rates on deposits are negative interest rates, investors are more willing to increase the ratio of wealth invested in stocks inorder to hedge currency devaluation losses. On the other hand, in the case of the high inflation, investors are less willing to put money in a bank and invest instocks, but buy precious metals （gold, silver） with the hedging function. Second, as investors worry about the accuracy of the model, they usually consider theeffect of the model misspecification on the portfolio. Thus, they will lower the ratio of wealth invested in risky assets accordingly. In summary, this paper characterizes the investor＇s decision with both probability and model uncertainty which are reflected in the problem of investor＇soptimal consumption and portfolio decisions. By the modem stochastic control theory, we have obtained some results which have both theoretic values andactual economic significance.