investmentsandportfoliomanagementindexmodels(编辑修改稿)

investmentsandportfoliomanagementindexmodels(编辑修改稿)内容摘要：

o the risk premium of the index. The market risk premium is multiplied by the relative sensitivity, or beta, of the individual security. We call this thesystematicrisk premium because it derives from the risk premium that characterizes the entire market, which proxies for the condition of the full economy or economic system.The remainder of the risk premium is given by the first term in the equation, α. Alpha is anonmarketpremium. For example, α may be large if you think a security is underpriced and therefore offers an attractive expected return. Later on, we will see that when security prices are in equilibrium, such attractive opportunities ought to be peted away, in which case α will be driven to zero. But for now, let39。 s assume that each security analyst es up with his or her own estimates of alpha. If managers believe that they can do a superior job of security analysis, then they will be confident in their ability to find stocks with nonzero values of alpha.We will see shortly that the index model deposition of an individual security39。 s risk premium to market and nonmarket ponents greatly clarifies and simplifies the operation of macroeconomic and security analysis within an investment pany.Risk and Covariance in the SingleIndex ModelRemember that one of the problems with the Markowitz model is the overwhelming number of parameter estimates required to implement it. Now we will see that the index model simplification vastly reduces the number of parameters that must be estimated.Equation the systematic and firmspecific ponents of the overall risk of each security, and the covariance between any pair of securities. Both variances and covariances are determined by the security betas and the properties of the market index:Equations andimply that the set of parameter estimates needed for the singleindex model consists of only α, β, and σ(e) for the individual securities, plus the risk premium and variance of the market index.p. 251CONCEPTCHECK1The data below describe a threestock financial market that satisfies the singleindex model.The standard deviation of the market index portfolio is 25%.a.What is the mean excess return of the index portfolio?b.What is the covariance between stockAand stockB?c.What is the covariance between stockBand the index?d.Break down the variance of stockBinto its systematic and firmspecific ponents.The Set of Estimates Needed for the SingleIndex ModelWe summarize the results for the singleindex model in the table below.These calculations show that if we have:•nestimates of the extramarket expected excess returns, αi•nestimates of the sensitivity coefficients, βi•nestimates of the firmspecific variances, σ2(ei)•1 estimate for the market risk premium,E(RM)•1 estimate for the variance of the (mon) macroeconomic factor,then these (3n+ 2) estimates will enable us to prepare the entire input list for this singleindexsecurity universe. Thus for a 50security portfolio we will need 152 estimates rather than 1,325。 for the entire New York Stock Exchange, about 3,000 securities, we will need 9,002 estimates rather than approximately million!It is easy to see why the index model is such a useful abstraction. For large universes of securities, the number of estimates required for the Markowitz procedure using the index model is only a small fraction of what otherwise would be needed.Another advantage is less obvious but equally important. The index model abstraction is crucial for specialization of effort in security analysis. If a covariance term had to be calculated directly for each security pair, then security analysts could not specialize by industry. For example, if one group were to specialize in the puter industry and another in the auto industry, who would have the mon background to estimate the covariancebetweenIBM and GM? Neither group would have the deep understanding of other industries necessary to make an informed judgment of comovements among industries. In contrast, the index model suggests a simple way to pute covariances. Covariances among securities are due to the influence of the single mon factor, represented by the market index return, and can be easily estimated using the regressionEquation onp. 249.p. 252The simplification derived from the index model assumption is, however, not without cost. The “cost” of the model lies in the restrictions it places on the structure of asset return uncertainty. The classification of uncertainty into a simple dichotomy—macro versus micro risk—oversimplifies sources of realworld uncertainty and misses some important sources of dependence in stock returns. For example, this dichotomy rules out industry events, events that may affect many firms within an industry without substantially affecting the broad macroeconomy.This last point is potentially important. Imagine that the singleindex model is perfectly accurate, except that the residuals of two stocks, say, British Petroleum (BP) and Royal Dutch Shell, are correlated. The index model will ignore this correlation (it will assume it is zero), while the Markowitz algorithm (which accounts for the full covariance between every pair of stocks) will automatically take the residual correlation into account when minimizing portfolio variance. If the universe of securities from which we must construct the optimal portfolio is small, the two models will yield substantively different optimal portfolios. The portfolio of the Markowitz algorithm will place a smaller weight on both BP and Shell (because their mutual covariance reduces their diversification value), resulting in a portfolio with lower variance. Conversely, when correlation among residuals is negative, the index model will ignore the potential diversification value of these securities. The。