1、英文原文： 10 The Markowitz Investment Portfolio Selection Model The first nine chapters of this book presented the basic probability theory with which any student of insurance and investments should be familiar. In this final chapter, we discuss an important application of the basic theory: the Nobel Pr
2、ize winning investment portfolio selection model due to Harry Markowitz. This material is not discussed in other probability texts of this level； however, it is a nice application of the basic theory and it is very accessible. The Markowitz portfolio selection model has a profound effect on the inve
3、stment industry. Indeed, the popularity of index funds (mutual funds that track the performance of an index such as the S&P 500 and do not attempt to “beat the market”) can be traced to a surprising consequence of the Markowitz model: that every investor, regardless of risk tolerance, should hold th
4、e same portfolio of risky securities. This result has called into question the conventional wisdom that it is possible to beat the market with the “right” investment manager and in so doing has revolutionized the investment industry. Our presentation of the Markowitz model is organized in the follow
5、ing way. We begin by considering portfolios of two securities. An important example of a portfolio of this type is one consisting of a stock mutual fund and a bond mutual fund. Seen from this perspective, the portfolio selection problem with two securities is equivalent to the problem of asset alloc
6、ation between stocks and bonds. We then consider portfolios of two risky securities and a risk-free asset, the prototype being a portfolio of a stock mutual fund, a bond mutual fund, and a money-market fund. Finally, we consider portfolio selection when an unlimited number of securities is available
7、 for inclusion in the portfolio. We conclude this chapter by briefly discussing an important consequence of the Markowitz model, namely, the Nobel Prize winning capital asset pricing model due to William Sharpe. The CAPM, as it is referred to, gives a formula for the fair return on a risky security
8、when the overall market is in equilibrium. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice. 10.1 Portfolios of Two Securities In this section, we consider portfolios consisting of only two securities, 1S and 2S . These two securities could be a stock
9、mutual fund and a bond mutual fund, in which case the portfolio selection problem amounts to asset allocation, or they could be something else. Our objective is to determine the “best mix” of 1S and 2S in the portfolio. Portfolio Opportunity Set Lets begin by describing the set of possible portfolio
10、s that can be constructed from 1S and 2S . Suppose that the current value of our portfolio is d dollars and let 1d and 2d be the dollar amounts invested in 1S and 2S , respectively. Let 1R and 2R be the simple returns on 1S and 2S over a future time period that begins now and ends at a fixed future
11、point in time and let R be the corresponding simple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideration, 2211 111 RdRdRd . Hence, the return on the portfolio over the given time period is 21 1 RxxRR , where ddx 1 is the fraction of t
12、he portfolio currently invested in 1S . Consequently, by varying x , we can change the return characteristics of the portfolio. Now if 1S and 2S are risky securities, as we will assume throughout this section, then 1R , 2R , and R are all random variables. Suppose that 1R and 2R are both normally di
13、stributed and their joint distribution has a bivariate normal distribution. This may appear to be a strong assumption. However, data on stock price returns suggest that, as a first approximation, it is not unreasonable. Then, from the properties of the normal distribution, it follows that R is norma
14、lly distributed and that the distributions of 1R , 2R , and R are completely characterized by their respective means and standard deviations. Hence, since R is a linear combination of 1R and 2R , the set of possible investment portfolios consisting of 1S and 2S can be described by a curve in the pla
15、ne. To see this more clearly, note that from the identity 21 1 RxxRR and the properties of means and variances, we have 21 1 RRR xx , 22222 2211 112 RRRRR xxxx , where is the correlation between 1R and 2R , Eliminating x from these two equations by substituting 212 RRRRx , which we obtain from the e
16、quation for R , into the equation for 2R , we obtain 22222222221121212121212 2 RRRRRRRRRRRRRRRRRRR , which describes a curve in the RR plane as claimed. Notice that R and R change with x , while 1R, 1R, 2R, 2R, remain fixed. To emphasize the fact that R and R are variables, lets drop the subscript R from now on. Then, the preceding equation for 2R can be written as 20202 A , where A , 0 , 20 are parameters depending only on 1S and 2S with 0A and 020 . Indeed, 222 221121 21 RRRRRRA 212121 121 22 RRRRRR 0 (the inequality holding since 11 ), and
