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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Merton's portfolio problem ： ウィキペディア英語版 Merton's portfolio problem Merton's portfolio problem is a well known problem in continuous-time finance and in particular intertemporal portfolio choice. An investor must choose how much to consume and must allocate his wealth between stocks and a risk-free asset so as to maximize expected utility. The problem was formulated and solved by Robert C. Merton in 1969 both for finite lifetimes and for the infinite case. Research has continued to extend and generalize the model to include factors like transaction costs and bankruptcy. ==Problem statement== The investor lives from time 0 to time ''T''; his wealth at time ''t'' is denoted ''W''''t''. He starts with a known initial wealth ''W''0 (which may include the present value of wage income). At time ''t'' he must choose what amount of his wealth to consume, ''c''''t'', and what fraction of wealth to invest in a stock portfolio, ''π''''t'' (the remaining fraction 1 − ''π''''t'' being invested in the risk-free asset). The objective is : $\backslash max\; E\; \backslash left(\backslash int\_0^T\; e^u(c\_s)\; \backslash ,\; ds\; +\; e^u(W\_T)\; \backslash right)$ where ''E'' is the expectation operator, ''u'' is a known utility function (which applies both to consumption and to the terminal wealth, or bequest, ''W''''T'') and ''ρ'' is the subjective discount rate. The wealth evolves according to the stochastic differential equation ::$d\; W\_t\; =\; (+\; \backslash pi\_t(\backslash mu-r))W\_t\; -\; c\_t)\; \backslash ,\; dt\; +W\_t\; \backslash pi\_t\; \backslash sigma\; \backslash ,\; dB\_t$ where ''r'' is the risk-free rate, (''μ'', ''σ'') are the expected return and volatility of the stock market and ''dB''''t'' is the increment of the Wiener process, i.e. the stochastic term of the SDE. The utility function is of the constant relative risk aversion (CRRA) form: : $u(x)\; =\; \backslash frac,$ where $\backslash gamma$ is a constant which expresses the investor's risk aversion: the higher the gamma, the more reluctance to own stocks. Consumption cannot be negative: ''c''''t'' ≥ 0, while ''π''''t'' is unrestricted (that is borrowing or shorting stocks is allowed). Investment opportunities are assumed constant, that is ''r'', ''μ'', ''σ'' are known and constant, in this (1969) version of the model, although Merton allowed them to change in his intertemporal CAPM (1973). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Merton's portfolio problem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.