Almost certainly, the discussion has been at such a high level ofÂ generality that it provides little concrete guidance for real investors. After some more similar, general, and abstract discussion of relatedÂ topics, such as capital asset pricing and risk, we hope to provide someÂ help in translating these general concepts into usable investmentÂ procedures.Â In order to define Markowitz’s efficient set of portfolios, it is necessary to know for each security its expected return, its variance, andÂ its covariance with each other security. If the efficient set were toÂ be selected from a list of only 1,000 securities, the volume of necessaryÂ inputs and the computational costs would be intolerably large. ItÂ would be necessary to have 1,000 statistics for expected return, 1,000Â variances, and 499,500 covariances.* It is not realistic to expect security analysts to provide this volume of inputs. If 20 analysts wereÂ responsible for the 1,000 stocks, each analyst would be responsibleÂ for providing almost 25,000 covariances. The volume of work wouldÂ be intolerable and, furthermore, it seems to be quite difficult to haveÂ an intuitive feeling about the significance of a covariance.
Because of this practical difficulty, the Markowitz portfolio modelÂ was exclusively of academic interest until William Sharpe suggestedÂ a simplification which made it usable.1 Since almost all securities areÂ significantly correlated with the market as a whole, Sharpe suggested that a satisfactory simplification would be to abandon the covariances of each security with each other security and to substitute information on the relationship of each security to the market. In his terms, it isÂ possible to consider the return for each security to be represented by the following equation:Â whereÂ Rtis the return on securityÂ i, atandÂ b,Lare parameters,Â ciis aÂ random variable with an expected value of zero, and / is the level ofÂ some index, typically a common stock price index. In words, the returnÂ on any stock depends on some constant (a) plus some coefficientÂ (b) times the value of a comprehensive stock index (say, the S & P “500”)Â plus a random component. Sharpe’s simplication reduces the number of estimates that the analyst must produce from 501,500 to 3,002 forÂ a list of 1,000 securities.*
There have been other efforts at simplification derived fromÂ Sharpe’s ideas. Cohen and PoagueÂ suggested that several indexesÂ rather than a single index be used, with the return for each securityÂ being related to the index most appropriate for itâperhaps some indexÂ of production which is a component of the aggregate Index of Industrial Production of the Federal Reserve Board. Their empirical resultsÂ suggest that the cost of using simplificationsâeither Sharpe’s orÂ theirsâis small. That is, the portfolios which are efficient as a resultÂ of their simplified processes are very similar to the efficient portfoliosÂ that result from Markowitz’s more complex process. Furthermore, if results are evaluated in terms of the two criteria, expected return andÂ risk, the efficient portfolios from the simple process are insignificantlyÂ worse than the efficient portfolios from the complex process.