際際滷

際際滷Share a Scribd company logo

Sharpe index model

Download as PPT, PDF
A
Ashwini Das

The Sharpe model provides a simpler approach to portfolio optimization compared to the Markowitz model. It assumes the return of individual securities is linearly related to a single market index. This allows estimation of systematic and unsystematic risk for individual stocks based on their beta coefficient. An optimal portfolio is constructed by selecting stocks with the highest excess returns over the risk-free rate relative to their beta, up to the cutoff point where this ratio begins declining. The percentage invested in each stock is based on its beta and unsystematic risk. This results in a portfolio with the highest expected return for a given level of risk.

1 of 29
Downloaded 1,387 times
The Sharpe Index Model
Need for Sharpe Model In Markowitz model a number of co-variances have to be estimated. If a financial institution buys 150 stocks, it has to estimate 11,175 i.e., (N2 N)/2 correlation co-efficients. Sharpe assumed that the return of a security is linearly related to a single index like the market index.
Single Index Model Casual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index. When the Sensex increases, stock prices also tend to increase and vice versa. This indicates that some underlying factors affect the market index as well as the stock prices.
Stock prices are related to the market index and this relationship could be used to estimate the return of stock. Ri = 留i + 硫i Rm + ei where Ri expected return on security i 留i intercept of the straight line or alpha co-efficient 硫i slope of straight line or beta co-efficient Rm the rate of return on market index ei error term
Risk Systematic risk = 硫i2 variance of market index = 硫 i2 m 2 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk Thus the total risk= Systematic risk + Unsystematic risk = 硫 i2 m 2 + e i2
Portfolio Variance 2 錚錚 N 錚 2 錚 錚 N 2 2錚 2 = 錚錚 x i 硫 i 歎 m 錚 + 錚 x i e i 錚 p 錚 錚錚 i =1 錚 錚 i =1 錚 錚 錚
where 2p = variance of portfolio 2m = expected variance of market index e2i= Unsystematic risk xi = the portion of stock i in the portfolio
Example The following details are given for x and y companies stocks and the Sensex for a period of one year. Calculate the systematic and unsystematic risk for the companies stock. If equal amount of money is allocated for the stocks , then what would be the portfolio risk ? X stock Y stock Sensex Average return 0.15 0.25 0.06 Variance of return 6.30 5.86 2.25 eta 0.71 0.27
Company X Systematic risk = 硫i2 variance of market index = 硫i2 m2 = ( 0.71)2 x 2.25 = 1.134 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk = 6.3 1.134 =5.166 Total risk= Systematic risk + Unsystematic risk = 硫i2 m2 + ei2 = 1.134 + 5.166 = 6.3
Company Y Systematic risk = 硫i2 variance of market index = 硫i2 m2 = ( 0.27)2 x 2.25 = 0.1640 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk = 5.86 1.134 =5.166
2p = [ ( .5 x .71 + .5 x .27)2 2.25 ] + [ ( .5)2 (5.166) + (.5 )2 ( 5.696) ] = [ ( .355 + .135 )2 2.25 ] + [ ( 1.292 + 1.424 ) ] = 0.540 + 2.716 = 3.256
Corner portfolio The entry or exit of a new stock in the portfolio generates a series of corner portfolio. In an one stock portfolio, it itself is the corner portfolio . In a two stock portfolio, the minimum risk and the lowest return would be the corner portfolio. As the number of stocks increases in a portfolio, the corner portfolio would be the one with lowest return and risk combination.
Expected Return of Portfolio N R p = x i (留i + 硫i R m ) i =1
For each security 留i and 硫i should be estimated Portfolio return is the weighted average of the estimated return for each security in the portfolio. The weights are the respective stocks proportions in the portfolio.
Portfolio Beta N 硫 p = x i硫 i i =1
A portfolios beta value is the weighted average of the beta values of its component stocks using relative share of them in the portfolio as weights. 硫p is the portfolio beta.
Sharpes optimal portfolio Ri Rf 硫i
The selection of any stock is directly related to its excess return to beta ratio. where Ri = the expected return on stock i Rf = the return on a risk less asset 硫i = Systematic risk
Optimal Portfolio The steps for finding out the stocks to be included in the optimal portfolio are as: Find out the excess return to beta ratio for each stock under consideration Rank them from the highest to the lowest Proceed to calculate Ci for all the stocks according to the ranked order using the following formula
(R i R f )硫i 2 ei i =1 N 硫2 2 i 1 + m 2 i =1 ei 2 m N Ci
m2 = variance of the market index ei2 = stocks unsystematic risk
Cut-off point The cumulated values of Ci start declining after a particular Ci and that point is taken as the cut-off point and that stock ratio is the cut-off ratio C. ei2 - Unsystematic risk (Ri Rf) / 硫i Excess return to Beta
Example Data for finding out the optimal portfolio are given below Security number Mean return Excess return Beta ei2 Ri Ri Rf 1 19 14 1.0 20 23 18 1.5 30 12 3 11 6 0.5 10 12 4 25 20 2.0 5 13 8 1.0 6 9 硫 14 2 (Ri Rf) / 硫i 7 4 0.5 40 20 50 14 10 8 8 9 1.5 30 6 The riskless rate of intrest is 5% and the market variance is 10. Determine the cut off point .
C1 = (10 x .7)/ [ 1 + ( 10 x .05)] =4.67 C2 = (10 x 1.6)/ [ 1 + ( 10 x .125)] =7.11 C3 = (10 x 1.9)/ [ 1 + ( 10 x .15)] =7.6 C4 = (10 x 2.9)/ [ 1 + ( 10 x .25)] = 8.29* Cut-off point C5 = (10 x 3.3)/ [ 1 + ( 10 x .3)] = 8.25 C6 = (10 x 3.34)/ [ 1 +( 10 x .305)] = 8.25 C7= (10 x 3.79)/ [ 1 + ( 10 x .38)] = 7.90
The highest Ci value is taken as the cut-off point i.e C*. The stocks ranked above C* have high excess returns to beta than the cut off Ci and all the stocks ranked below C* have low excess return 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 ci 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.
Construction of the optimal portfolio After determining the securities to be selected, the portfolio manager should find out how much should be invested in each security.The percentage of funds to be invested in each security can be estimated as follows . Zi = (硫i / 2ei ) x [ (Ri Rf / 硫i) C ]
Zi = (硫i / 2ei ) x [ (Ri Rf / 硫i) C ] Z1 = (1/20) x ( 14 8.29) Z2 = (1.5 / 30) x ( 12 8.29) Z3 = (0.5 / 10) x ( 12 8.29) Z4 = (2 / 40) x ( 10 8.29) Total X1 = 0.285/ .743 = 0.38 X2 = 0.186 / .743 = 0.25 X3 = 0.186 / .743 = 0.25 X4 = 0.086 / .743 = 0.12 = 0.285 = 0.186 = 0.186 = 0.086 = .743
So the largest investment should be made in security 1 ( 0.38%) and the smallest in security 4 ( 0.12%). The characteristics of a stock that make it desirable can be determined before the calculations of an optimal portfolio is begun. The desirability of any stock is solely a function of its excess return to beta ratio.

More Related Content

Sharpe index model

  • 2. Need for Sharpe Model In Markowitz model a number of co-variances have to be estimated. If a financial institution buys 150 stocks, it has to estimate 11,175 i.e., (N2 N)/2 correlation co-efficients. Sharpe assumed that the return of a security is linearly related to a single index like the market index.
  • 3. Single Index Model Casual observation of the stock prices over a period of time reveals that most of the stock prices move with the market index. When the Sensex increases, stock prices also tend to increase and vice versa. This indicates that some underlying factors affect the market index as well as the stock prices.
  • 4. Stock prices are related to the market index and this relationship could be used to estimate the return of stock. Ri = 留i + 硫i Rm + ei where Ri expected return on security i 留i intercept of the straight line or alpha co-efficient 硫i slope of straight line or beta co-efficient Rm the rate of return on market index ei error term
  • 5. Risk Systematic risk = 硫i2 variance of market index = 硫 i2 m 2 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk Thus the total risk= Systematic risk + Unsystematic risk = 硫 i2 m 2 + e i2
  • 6. Portfolio Variance 2 錚錚 N 錚 2 錚 錚 N 2 2錚 2 = 錚錚 x i 硫 i 歎 m 錚 + 錚 x i e i 錚 p 錚 錚錚 i =1 錚 錚 i =1 錚 錚 錚
  • 7. where 2p = variance of portfolio 2m = expected variance of market index e2i= Unsystematic risk xi = the portion of stock i in the portfolio
  • 8. Example The following details are given for x and y companies stocks and the Sensex for a period of one year. Calculate the systematic and unsystematic risk for the companies stock. If equal amount of money is allocated for the stocks , then what would be the portfolio risk ? X stock Y stock Sensex Average return 0.15 0.25 0.06 Variance of return 6.30 5.86 2.25 eta 0.71 0.27
  • 9. Company X Systematic risk = 硫i2 variance of market index = 硫i2 m2 = ( 0.71)2 x 2.25 = 1.134 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk = 6.3 1.134 =5.166 Total risk= Systematic risk + Unsystematic risk = 硫i2 m2 + ei2 = 1.134 + 5.166 = 6.3
  • 10. Company Y Systematic risk = 硫i2 variance of market index = 硫i2 m2 = ( 0.27)2 x 2.25 = 0.1640 Unsystematic risk= Total variance Systematic risk ei2 = i2 Systematic risk = 5.86 1.134 =5.166
  • 11. 2p = [ ( .5 x .71 + .5 x .27)2 2.25 ] + [ ( .5)2 (5.166) + (.5 )2 ( 5.696) ] = [ ( .355 + .135 )2 2.25 ] + [ ( 1.292 + 1.424 ) ] = 0.540 + 2.716 = 3.256
  • 12. Corner portfolio The entry or exit of a new stock in the portfolio generates a series of corner portfolio. In an one stock portfolio, it itself is the corner portfolio . In a two stock portfolio, the minimum risk and the lowest return would be the corner portfolio. As the number of stocks increases in a portfolio, the corner portfolio would be the one with lowest return and risk combination.
  • 13. Expected Return of Portfolio N R p = x i (留i + 硫i R m ) i =1
  • 14. For each security 留i and 硫i should be estimated Portfolio return is the weighted average of the estimated return for each security in the portfolio. The weights are the respective stocks proportions in the portfolio.
  • 15. Portfolio Beta N 硫 p = x i硫 i i =1
  • 16. A portfolios beta value is the weighted average of the beta values of its component stocks using relative share of them in the portfolio as weights. 硫p is the portfolio beta.
  • 18. The selection of any stock is directly related to its excess return to beta ratio. where Ri = the expected return on stock i Rf = the return on a risk less asset 硫i = Systematic risk
  • 19. Optimal Portfolio The steps for finding out the stocks to be included in the optimal portfolio are as: Find out the excess return to beta ratio for each stock under consideration Rank them from the highest to the lowest Proceed to calculate Ci for all the stocks according to the ranked order using the following formula
  • 20. (R i R f )硫i 2 ei i =1 N 硫2 2 i 1 + m 2 i =1 ei 2 m N Ci
  • 21. m2 = variance of the market index ei2 = stocks unsystematic risk
  • 22. Cut-off point The cumulated values of Ci start declining after a particular Ci and that point is taken as the cut-off point and that stock ratio is the cut-off ratio C. ei2 - Unsystematic risk (Ri Rf) / 硫i Excess return to Beta
  • 23. Example Data for finding out the optimal portfolio are given below Security number Mean return Excess return Beta ei2 Ri Ri Rf 1 19 14 1.0 20 23 18 1.5 30 12 3 11 6 0.5 10 12 4 25 20 2.0 5 13 8 1.0 6 9 硫 14 2 (Ri Rf) / 硫i 7 4 0.5 40 20 50 14 10 8 8 9 1.5 30 6 The riskless rate of intrest is 5% and the market variance is 10. Determine the cut off point .
  • 24. C1 = (10 x .7)/ [ 1 + ( 10 x .05)] =4.67 C2 = (10 x 1.6)/ [ 1 + ( 10 x .125)] =7.11 C3 = (10 x 1.9)/ [ 1 + ( 10 x .15)] =7.6 C4 = (10 x 2.9)/ [ 1 + ( 10 x .25)] = 8.29* Cut-off point C5 = (10 x 3.3)/ [ 1 + ( 10 x .3)] = 8.25 C6 = (10 x 3.34)/ [ 1 +( 10 x .305)] = 8.25 C7= (10 x 3.79)/ [ 1 + ( 10 x .38)] = 7.90
  • 25. The highest Ci value is taken as the cut-off point i.e C*. The stocks ranked above C* have high excess returns to beta than the cut off Ci and all the stocks ranked below C* have low excess return to beta. Here the cut off rate is 8.29. Hence the first four securities are selected.
  • 26. If the number of stocks is larger there is no need to calculate ci 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.
  • 27. Construction of the optimal portfolio After determining the securities to be selected, the portfolio manager should find out how much should be invested in each security.The percentage of funds to be invested in each security can be estimated as follows . Zi = (硫i / 2ei ) x [ (Ri Rf / 硫i) C ]
  • 28. Zi = (硫i / 2ei ) x [ (Ri Rf / 硫i) C ] Z1 = (1/20) x ( 14 8.29) Z2 = (1.5 / 30) x ( 12 8.29) Z3 = (0.5 / 10) x ( 12 8.29) Z4 = (2 / 40) x ( 10 8.29) Total X1 = 0.285/ .743 = 0.38 X2 = 0.186 / .743 = 0.25 X3 = 0.186 / .743 = 0.25 X4 = 0.086 / .743 = 0.12 = 0.285 = 0.186 = 0.186 = 0.086 = .743
  • 29. So the largest investment should be made in security 1 ( 0.38%) and the smallest in security 4 ( 0.12%). The characteristics of a stock that make it desirable can be determined before the calculations of an optimal portfolio is begun. The desirability of any stock is solely a function of its excess return to beta ratio.