## Summary

Special analytical procedures and modeling techniques can make mean-variance portfolio theory more practical than it would be if the theory were used in its barest form The procedures and techniques discussed in this chapter include: (1) factor models to reduce the number of par ameters required to specify a mean-variance structure, (2) use of APT to add factors to the CAPM and also to avoid the equilibrium assumption that underlies the CAPM, (3) recognition of the errors inherent in computing parameter estimates from historical records of returns, and (4) blending of different types of parameter estimates to obtain informed and reasonable numerical results.

A factor model expresses the rate of return of each asset as a linear combination of certain specified (random) factor variables The same factors are used for each asset, but the coefficients of the linear combination of these factors are different for different assets In addition to the factor terms, there are a constant term a, and an error term et The coefficients of the factors are called factor loadings. In making calculations with the model, it is usually assumed that the error terms are uncorrected with each other and with the factors

A great advantage of a factor model is that it has far fewer parameters than a standard mean-variance representation, In practice, between three and fifteen factors can provide a good representation of the covariance properties of the returns of thousands of U S stocks

There are several choices for factors. The most common choice is the return on the market portfolio A factor model using this single factor is closely related to the CAPM. Other choices include various economic indicators published by the U S Government or factors extracted as combinations of certain asset returns It is also helpful to supplement a factor model by including combinations of company-specific financial characteristics,

When the excess market return is used as the single factor, the resulting factor model can be interpreted as defining a straight line on a graph with i ^ — if being the horizontal axis and r — iy the vertical axis. This line is called the characteristic line of the asset, Its vertical intercept is called alpha, and its slope is the beta of the CAPM. The CAPM predicts that alpha is zero (but in practice it may be nonzero).

Arbitrage pricing theory (APT) is built directly on a factor model. For the theory to be useful, it is important that the underlying factor model be a good representation in the sense that the error terms are uncorrected with each other and with the factors In that case, the error terms can be diversified away by forming combinations of a large number of assets.

The result of APT is that the coefficients of the underlying factor model must satisfy a linear relation In the special case where the underlying factor model has the single factor equal to the excess return on the market portfolio, the CAPM theory states that a = 0. This is a special case of APT, which states that the constant ci in the expression for the return of an asset is a linear combination of the factor loadings of that asset Again, the difficult part of applying APT is the determination of appropriate factors.

It is tempting to assume that the parameter values necessary to implement mean-variance theory—the expected returns, variances, and covariances for a Markowitz formulation, or the a,'s and /?,/s for a factor model representation—can be estimated from historical returns data Although some parameter values can be estimated this way, others cannot. In particular, for stocks the variances and covariances can be estimated to within reasonable accuracy by using about 1 year of weekly or daily returns data However, the expected rates of return (the means) are subject to a blurring phenomenon and therefore cannot be estimated to within workable accuracy, even if a record of 10 years of returns is employed This blurring phenomenon applies to the estimation of the a coefficients in factor models as well

The statistical analysis of estimates based on historical data tells us that we must supplement such estimates of expected returns with estimates obtained by other methods This conclusion is not altogether surprising It asserts that active portfolio management (as opposed to a passive strategy of investing only in the market portfolio and the risk-free asset) cannot be relegated to a pure computer analysis of historical data. Some additional intelligence is required If this intelligence can be cast into the form of estimates, with associated variances, these estimates can be logically combined with the estimates based on historical data to produce refined estimates with smaller errors An additional estimate of this type is provided by the CAPM formula itself.

The Markowitz mean-variance formulation of portfolio theory and the subsequent theories of CAPM, factor models, and APT provide an elegant foundation for single-period investment analysis These developments have elaborated the benefits of diversification and deepened our understanding of risk in a market environment These theories have also provided approaches that can be implemented, Indeed, this whole area has had a profound influence on the practice of portfolio management: index funds now abound, betas are computed and widely discussed in the financial community, large quadratic programming programs have been written to solve the Markowitz problem, numerous factor models have been constructed and tested, and trillions of dollars have been managed with at least some guidance from these ideas and methods

But mean-variance theory is not a universal investment panacea. The assumption that all investors focus exclusively on mean and variance is questionable, it is hard to estimate the required parameter values, it seems unlikely (as required of the equilibrium argument) that everyone has the same estimates of the parameter values, and the approach must be modified in a multiperiod framework, Each of these difficulties can be overcome to some extent by extending the model, living with approximations, or looking deeper into the properties of the assets under consideration A great deal of innovative effort has been so devoted. But ultimately, to make significant progress, we must expand the fundamental tools of analysis beyond mean-variance We must formulate a theory that, built on the insights of the mean-variance approach, treats uncertainty more explicitly and is directed at multiperiod situations.

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