M-V最优投资组合选择与最优投资消费决策
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摘要
投资组合理论的产生使得数理化方法真正进入到投资领域，使得数理金融学作为金融学的一个独立的分支迅速发展起来。但围绕投资组合理论，过去的一系列研究存在许多不足，如：均值-方差投资组合理论单纯地考虑一个确定的投资时域，并且考虑的市场环境比较简单；投资消费理论考虑的是一类单一的消费品，投资对象仅限于无风险证券和风险证券。而目前市场上消费品与投资对象日益丰富，原来的投资理论的一些结论不能满足实际的需求。为此，本文围绕均值-方差投资组合理论与投资消费理论开展了如下几个方面内容的研究。
(1) 确定时域的M-V最优投资组合选择。分别建立了股票价格服从跳跃扩散过程、考虑固定消费、市场系数为随机过程这三种情形下的均值-方差模型。运用动态规划原理与鞅方法求解模型，得到了这三种情形下的最优投资策略与有效前沿的解析解。与经典连续时间均值-方差模型进行了比较并通过实例分析了消费对投资的影响。结果表明：(ⅰ)本文的模型拓广了Zhou与Li的经典模型，与实际更加符合；(ⅱ)消费的存在影响投资者对投资策略的选择。在期望收益固定的情况下，消费越多，投资也越多。消费的增加(减少)会引起有效前沿向下(上)平移。从而揭示了固定消费与投资的内在联系。
(2) 随机时域的M-V最优投资组合选择。建立了离散时间、连续时间与跳跃扩散过程三种市场状态下随机时域的均值-方差模型，定义了相应的有效前沿。对前两种情形考虑退出时间是个随机变量，对最后一种情形考虑退出时间是个随机过程。分别得到了这三种情形下的最优投资策略与有效前沿的解析表达式。通过算例以及与确定时域对应情形的比较，发现：最优投资策略与随机退出时间的分布有关，确定时域的结论只是本文的一种特殊情形。
(3) 特殊消费的最优投资消费决策。与经典的投资消费问题考虑的消费不同，这里研究的是两类特殊的消费：固定的消费模式、消费对象为可存与非可存消费品的组合。建立了这两类特殊消费情形下的投资消费模型。分别得到了HARA效用函数与可分离、等弹性效用函数情形下的最优投资消费策略的显式解。分析了固定消费、可存消费品对投资的影响。得到了如下结论：(ⅰ)固定消费不会影响投资选择这一直觉并不正确，事实上，消费量越多，投资量会越少。这种影响程度决定于市场风险价格与无风险利率；(ⅱ)最优策略中，对可存品的消费与非可存品的消费决策不一样。因此，在进行投资消费决策时，有必要将消费品中的可存品与非可存品分开来考虑。
(4) 含期权的最优投资消费决策。随着期权等一系列衍生证券进入金融市场，期权已经日益成为投资者注目的投资对象。为适应实际需要，把一个欧式看涨期权作为一个投资对象，结合期权定价理论，建立了投资消费模型。本文考虑了三种情形：第一种是期权的买卖价格相同、市场系数为常值且风险证券是期权的标的物；第二种是期权的买卖价格相同、风险证券服从跳跃扩散过程而且风险证券是期权的标的物；第三种情形是期权买卖价格不同、市场系数为常值而且风险证券不一定是期权的标的物。分别得到了以上各种情形对应的最优投资消费策略的解析表达式。对第一种情形，还得到了对冲投资消费策略。通过对最优投资消费策略与对冲投资消费策略的分析，得到了如下结论：(ⅰ)当风险证券为期权
的标的物时，最优策略不唯一;当风险证券不是期权的标的物时，最优策略才‘可
能唯一;(ii)对冲策略一般不是最优策略。
The emergence of portfolio theory really makes mathematical methods enter the investment field. Thus, mathematical finance as an independent branch of the finance theory develops quickly. But there still exist shortcomings in previous study of portfolio theory, such as the mean-variance portfolio theory is only involved with the deterministic time horizon and its market is very simple; The investment consumption theory is only involved with a single consumption good and the investment objects are only a bond and some risky stocks. However, the consumption good and investment object are becoming more various. So the previous some conclusions of investment theory can not satisfy the requirement of real situations. The aim of this paper is to study mean variance portfolio theory and portfolio consumption theory further to fit better real situations. The main results are listed as follows.
(1) M-V portfolio selection of deterministic time horizon. The mean variance models are formulated respectively in three market cases: (i) The stock prices follow jump diffusion process; (ii) Fixed consumption is considered; (iii) The market coefficients are stochastic processes. By using stochastic dynamic programming principle and martingale approach to solve these models, the optimal investment strategies and the efficient frontier are presented explicitly. By comparing them with conclusions of the classical continuous time model and analyzing the influence of the fixed consumption on investment, the main results are derived as follows: (i) The models discussed here extend the classical model discussed by Zhou and Li[208] and can be better applied to the real situations; (ii) The selection of optimal investment strategies is affected by the fixed consumption. When the final expected return is fixed, investment on stocks grows with the consumption. The efficient frontier moves downward (upward) if consumpt
ion increase (decrease) . Thus, we characterize the internal relations between investment and consumption.
(2) M-V portfolio selection of random time horizon. The random time horizon mean variance models corresponding to three market situations :discrete time, continuous time and jump diffusion process are formulated respectively and the relevant efficient frontier is defined. In the former two situations, the exit time is assumed to be a random variable and in the last situation the exit time is assumed to be a stochastic process. By solving the three models respectively, the explicit expressions of the investment strategies and the efficient frontiers are presented. By number example and by comparing the constant time horizon with the random time horizon we find that the portfolio decisions are affected by distribution of the exit time and the random time horizon model extends the constant horizon model as a special case.
(3) portfolio consumption decision with special consumption. Being different from classical portfolio consumption problem, here two special portfolio consumption problems whose consumption are fixed consumption style and a combination of a perishable with a durable consumption good respectively are considered. The models
maximizing utility of consumption or final wealth corresponding to the above two cases are formulated. For the HARA utility function and separable isoelastic utility function, the optimal portfolio and consumption rules are derived explicitly. By analyzing the influences of fixed consumption style and durable consumption good on investment decisions, the following results are presented, (i) The intuition that the portfolio selection decision is not affected by the fixed consumption style is not true. As a fact, investment varies with consumption contrarily. The degree of such influence is decided by market risk price and riskless rate, (ii) The influence of durable consumption good on investment consumption decisions is different from that of perishable consumption good. So it is necessary to consider them separately when one invests and consumes.
(4) portfolio consumption decision concl
(1) 确定时域的M-V最优投资组合选择。分别建立了股票价格服从跳跃扩散过程、考虑固定消费、市场系数为随机过程这三种情形下的均值-方差模型。运用动态规划原理与鞅方法求解模型，得到了这三种情形下的最优投资策略与有效前沿的解析解。与经典连续时间均值-方差模型进行了比较并通过实例分析了消费对投资的影响。结果表明：(ⅰ)本文的模型拓广了Zhou与Li的经典模型，与实际更加符合；(ⅱ)消费的存在影响投资者对投资策略的选择。在期望收益固定的情况下，消费越多，投资也越多。消费的增加(减少)会引起有效前沿向下(上)平移。从而揭示了固定消费与投资的内在联系。
(2) 随机时域的M-V最优投资组合选择。建立了离散时间、连续时间与跳跃扩散过程三种市场状态下随机时域的均值-方差模型，定义了相应的有效前沿。对前两种情形考虑退出时间是个随机变量，对最后一种情形考虑退出时间是个随机过程。分别得到了这三种情形下的最优投资策略与有效前沿的解析表达式。通过算例以及与确定时域对应情形的比较，发现：最优投资策略与随机退出时间的分布有关，确定时域的结论只是本文的一种特殊情形。
(3) 特殊消费的最优投资消费决策。与经典的投资消费问题考虑的消费不同，这里研究的是两类特殊的消费：固定的消费模式、消费对象为可存与非可存消费品的组合。建立了这两类特殊消费情形下的投资消费模型。分别得到了HARA效用函数与可分离、等弹性效用函数情形下的最优投资消费策略的显式解。分析了固定消费、可存消费品对投资的影响。得到了如下结论：(ⅰ)固定消费不会影响投资选择这一直觉并不正确，事实上，消费量越多，投资量会越少。这种影响程度决定于市场风险价格与无风险利率；(ⅱ)最优策略中，对可存品的消费与非可存品的消费决策不一样。因此，在进行投资消费决策时，有必要将消费品中的可存品与非可存品分开来考虑。
(4) 含期权的最优投资消费决策。随着期权等一系列衍生证券进入金融市场，期权已经日益成为投资者注目的投资对象。为适应实际需要，把一个欧式看涨期权作为一个投资对象，结合期权定价理论，建立了投资消费模型。本文考虑了三种情形：第一种是期权的买卖价格相同、市场系数为常值且风险证券是期权的标的物；第二种是期权的买卖价格相同、风险证券服从跳跃扩散过程而且风险证券是期权的标的物；第三种情形是期权买卖价格不同、市场系数为常值而且风险证券不一定是期权的标的物。分别得到了以上各种情形对应的最优投资消费策略的解析表达式。对第一种情形，还得到了对冲投资消费策略。通过对最优投资消费策略与对冲投资消费策略的分析，得到了如下结论：(ⅰ)当风险证券为期权
的标的物时，最优策略不唯一;当风险证券不是期权的标的物时，最优策略才‘可
能唯一;(ii)对冲策略一般不是最优策略。
The emergence of portfolio theory really makes mathematical methods enter the investment field. Thus, mathematical finance as an independent branch of the finance theory develops quickly. But there still exist shortcomings in previous study of portfolio theory, such as the mean-variance portfolio theory is only involved with the deterministic time horizon and its market is very simple; The investment consumption theory is only involved with a single consumption good and the investment objects are only a bond and some risky stocks. However, the consumption good and investment object are becoming more various. So the previous some conclusions of investment theory can not satisfy the requirement of real situations. The aim of this paper is to study mean variance portfolio theory and portfolio consumption theory further to fit better real situations. The main results are listed as follows.
(1) M-V portfolio selection of deterministic time horizon. The mean variance models are formulated respectively in three market cases: (i) The stock prices follow jump diffusion process; (ii) Fixed consumption is considered; (iii) The market coefficients are stochastic processes. By using stochastic dynamic programming principle and martingale approach to solve these models, the optimal investment strategies and the efficient frontier are presented explicitly. By comparing them with conclusions of the classical continuous time model and analyzing the influence of the fixed consumption on investment, the main results are derived as follows: (i) The models discussed here extend the classical model discussed by Zhou and Li[208] and can be better applied to the real situations; (ii) The selection of optimal investment strategies is affected by the fixed consumption. When the final expected return is fixed, investment on stocks grows with the consumption. The efficient frontier moves downward (upward) if consumpt
ion increase (decrease) . Thus, we characterize the internal relations between investment and consumption.
(2) M-V portfolio selection of random time horizon. The random time horizon mean variance models corresponding to three market situations :discrete time, continuous time and jump diffusion process are formulated respectively and the relevant efficient frontier is defined. In the former two situations, the exit time is assumed to be a random variable and in the last situation the exit time is assumed to be a stochastic process. By solving the three models respectively, the explicit expressions of the investment strategies and the efficient frontiers are presented. By number example and by comparing the constant time horizon with the random time horizon we find that the portfolio decisions are affected by distribution of the exit time and the random time horizon model extends the constant horizon model as a special case.
(3) portfolio consumption decision with special consumption. Being different from classical portfolio consumption problem, here two special portfolio consumption problems whose consumption are fixed consumption style and a combination of a perishable with a durable consumption good respectively are considered. The models
maximizing utility of consumption or final wealth corresponding to the above two cases are formulated. For the HARA utility function and separable isoelastic utility function, the optimal portfolio and consumption rules are derived explicitly. By analyzing the influences of fixed consumption style and durable consumption good on investment decisions, the following results are presented, (i) The intuition that the portfolio selection decision is not affected by the fixed consumption style is not true. As a fact, investment varies with consumption contrarily. The degree of such influence is decided by market risk price and riskless rate, (ii) The influence of durable consumption good on investment consumption decisions is different from that of perishable consumption good. So it is necessary to consider them separately when one invests and consumes.
(4) portfolio consumption decision concl
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