The construction of an optimal portfolio is simplified if there is a single number that measures the desirability of including a stock in the optimal portfolio. If we accept the single-index model (Sharpe), such a number exists. In this case, the desirability of any stock is directly related to its excess return-to-beta ratio.
(Ri – RF)/ßi
Ri = expected return on stock i
RF = return on a riskless asset
ßi = expected change in the rate of return on stock i associated with a 1 percent change in the market return
If stocks are ranked by an excess return to beta (from highest to lowest), the ranking represents the desirability of any stock’s inclusion in a portfolio. The number of stocks selected depends on a unique cutoff rate such that all stocks with higher ratios of (Ri – RF)/ ßi will be included and all stocks with lower ratios excluded.
To determine which stocks are included in the optimum portfolio, the following steps are necessary:
1. Calculate the excess return-to-beta ratio for each stock under review and the rank from highest to lowest.
2. The optimum portfolio consists of investing in all stocks for which (Ri – RF)/ ßi is greater than a particular cutoff point C.
Sharpe notes that proper diversification and the holding of a sufficient number of securities can reduce the unsystematic component of portfolio risk to zero by averaging out the unsystematic risk of individual stocks. What is left is a systematic risk which, is determined by the market (index), cannot be eliminated through portfolio balancing. Thus, the Sharpe model attaches considerable significance to systematic risk and its most important measure, the beta coefficient (ß).
According to the model, the risk contribution to a portfolio of an individual stock can be measured by the stock’s beta coefficient. The market index will have a beta coefficient of +1.0. A stock with a beta of, for example, +2.0 indicates that it contributes far more risk to a portfolio than a stock with, say, a beta of + .05. Stocks with negative betas are to be coveted since they help reduce risk beyond the unsystematic level.
market movements. Risk in an efficient portfolio is measured by the portfolio beta. The beta for the portfolio is simply the weighted average of the betas of the component securities. For example, an optimal portfolio that has a beta of 1.35, suggests that it has a sensitivity above the + 1.0 attributed to the market. If this portfolio is properly diversified (proper number of stocks and elimination of unsystematic risk), it should move up or down about one-third more than the market. Such a high beta suggests an aggressive portfolio. Should the market move up over the holding period, this portfolio will be expected to advance substantially. However, a market decline should find this portfolio falling considerably in value.
In this way, establishing efficient portfolios (minimum risk for a given expected return) comprising broad classes of assets (e.g., stocks, bonds, real estate) lends itself to the mean-variance methodology suggested by Markowitz. Determining efficient portfolios within an asset class (e.g., stocks) can be achieved with the single index (beta) the model proposed by Sharpe.