【正文】 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 allocation between stocks and bonds. We then consider portfolios of two risky securities and a riskfree asset, the prototype being a portfolio of a stock mutual fund, a bond mutual fund, and a moneymarket fund. Finally, we consider portfolio selection when an unlimited number of securities is available 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 when the overall market is in equilibrium. Like the Markowitz model, the CAPM has had a profound influence on portfolio management practice. Portfolios of Two SecuritiesIn this section, we consider portfolios consisting of only two securities, and . These two securities could be a stock 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 and in the portfolio.Portfolio Opportunity SetLet39。英文原文：10　The Markowitz Investment Portfolio Selection ModelThe 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 Prize 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 investment industry. Indeed, the popularity of index funds (mutual funds that track the performance of an index such as the Samp。s begin by describing the set of possible portfolios that can be constructed from and . Suppose that the current value of our portfolio is dollars and let and be the dollar amounts invested in and , respectively. Let and be the simple returns on and over a future time period that begins now and ends at a fixed future point in time and let be the corresponding simple return for the portfolio. Then, if no changes are made to the portfolio mix during the time period under consideration,.Hence, the return on the portfolio over the given time period is,where is the fraction of the portfolio currently invested in . Consequently, by varying , we can change the return characteristics of the portfolio.Now if and are risky securities, as we will assume throughout this section, then , , and are all random variables. Suppose that and are both normally distributed 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 is normally distributed and that the distributions of , , and are pletely characterized by their respective means and standard deviations. Hence, since is a linear bination of and , the set of possible investment portfolios consisting of and can be described by a curve in the plane.To see this more clearly, note that from the identity and the properties of means and variances, we have,where is the correlation between and , Eliminating from these two equations by substituting , which we obtain from the equation for , into the equation for , we obtain,which describes a curve in the plane as claimed.Notice that and change with , while , , , , remain fixed. To emphasize the fact that and are variables, let’s drop the subscript from now on. Then, the preceding equation for can be written as,where , , are parameters depending only on and with and . Indeed, (the inequality holding since ), and(again since ). Further,.Consequently, the possible portfolios lie on the curve,，which we recognize as being the right half of a hyperbola with vertex at . (Figure ). Notice tha