Sharpe’s Optimal Portfolio

Sharpe’s Optimal Portfolio

Sharpe had provided a model for the selection of appropriate securities in a portfolio.

The selection of any stock is directly related to its excess return-beta ratio.

The excess return is the difference between the expected return on the stock and the riskless rate of interest such as the rate offered on the government security or treasury bill. The excess return to beta ratio measures the additional return on a security (excess of the riskless asset return) per unit of systematic risk or no diversifiable risk this ratio provides a relationship between potential risk and reward Ranking of the stocks are done on the basis of their excess return to beta. Portfolio managers would like to include stocks with higher ratios. The selection of the stocks depends on a unique cut-off rate such that all stocks with higher ratios of R.-R / B are included and the stocks with lower ratios are left off. The cut-off point is denoted by C*.

The steps for finding out the stocks to be included in the optimal portfolio are given below

Find out the “excess return to beta” ratio for each stock under consideration.

Rank them from the highest tothe lowest.

Proceed to calculate C for all the stocks according to the ranked order using the following formula.

The cumulated values of C start declining after a particular C and that point is taken as the cut-off point and that stock ratio is the dut-off ratio C.

This is explained with the help of an example.

Data for finding out the optimal portfolio are given below:

The riskless rate of interest is 5 per cent and the market variance is 10. Determine the cut-off point.

C calculations are given below

For Security 1

Here 0.7 is got from column 4 and 0.05 from column 6. Since the preliminary calculations are over, it is easy to calculate the C

The highest Ci.value is taken as the cutoff point i.e. C*. The stocks ranked above C* have high excess returns to beta than the cut-off C. and all the stocks ranked below C* have low excess returns to beta. Here, the cut-off rate is 8.29. Hence, the first four securities are selected. If the number of stocks is larger there is no need to calculate C_i values for all the stocks after the ranking has been done. It can be calculated until the C* value is found and after calculating for one or two stocks below it, the calculations can be terminated.

The C_i can be stated with mathematically equivalent way.

β_ip- the expected change in the rate of return on stock i associated with 1 per cent change in the return on the optimal portfolio.

R_p - the expected return on the optimal portfolio

β_ipand R_p cannot be determined until the optimal portfolio is found. lb find out the optimal portfolio, the formula given previously should be used. Securities are added to the portfolio as long as

The above equation can be rearranged with the substitution of equation:

Now we have,

R_i – R_f>β_ip(R_p – R_f)

The right hand side is the expected excess return on a particular stock based on the expected performance of the optimum portfolio. The term on the left hand side is the expected excess return on the individual stock. Thus, if the portfolio manager believes that a particular stock will perform better than the expected return based on its relationship to optimal portfolio, he would add the stock to the portfolio.