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# 3. sharpe index model

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### 3. sharpe index model

1. 1. PORTFOLIO MANAGEMENT
2. 2. INTRODUCTION The investor like to purchase securities with low risk and high return. Now for that purpose Markowitz model is good but we have to make lots of calculations in order to get the result. In Markowitz model we need N(N+2)/2 bits of information whereas in Sharpe 3N+2 bits of information is needed.
3. 3. Sharpe index model Casual observation of stock prices over a period of time reveals that most of stock prices move with the market index. When the Sensex increases the price increases and vice versa. Stock prices are related to the market index and this relationship could be used to estimate return on stock. Towards this purpose following equation can be used.
4. 4.  Ri= αi + βiRm + ejWhere Ri= expected return of security Iαi= alpha coefficient βi= beta coefficient Rm = the rate of return of market index ej = error term
5. 5.  According to the equation, the return of stock can be divided into two components, the return due to the market and the return independent of the market. βi indicates the sensitivity of stock return to the changes In market return. For example βi of 1.5 means the stock return is expected to increase by 1.5% if market increases by 1% and vice versa. The estimate of βi and αi can be obtained using regression analysis.
6. 6.  The single index model is based on the assumption that stocks vary together because of common movement in the stock market and there are no effects beyond the market (i.e. any fundamental factor effects) that accounts the stock co-movement. The expected return, standard deviation, and co-variance of single index model represents the joint movement of securities. The mean return isRi= αi + βiRm + ej
7. 7.  The variance of security’s return is σ²=βi²σ²m + σei² The covariance of returns between securities i and j isσij = βiβjσ²m
8. 8.  The variance of security has two components namely, systematic risk or market risk and unsystematic risk or unique risk. The variance explained by index is called systematic risk and the unexplained variance is called unsystematic risk.Systematic risk= βi² x variance of market index = βi²σ²mUnsystematic risk= total variance – systematic risk ei² = σ²i – systematic riskThus total risk= βi² σ² + ei²
9. 9.  From this the portfolio variance can be derived σ²p = [ (Σxiβi)²σ²m] + [ Σxi² ei²]σ²p = variance of portfolioσ²m = expected variance of indexei²= variation in security return not related to the market indexxi = the portion of stock i in the portfolio
10. 10. X stock Y stock SensexAverage return 0.15 0.25 0.06Variance of return 6.30 5.86 2.25β .71 .27Correlation coefficient .424Coefficient of determination .18
11. 11.  The coefficient of determination gives the percentage of the variation in the security’s return that is explained by variation of the market index return. In the X company stock return, 18 percent of variation is explained by index whereas 82% is not. Explained by index = variance x coefficient of determination = 6.3 x .18 = 1.134 Not Explained by index = variance x (1 – r²)= 6.3 x .82 = 5.166
12. 12. Company X Systematic risk= β² x variance of market index= (.71)2 x 2.25 = 1.134Unsystematic risk= total variance of security return - systematic risk= e2 = 6.3 – 1.134 = 5.166Total risk = 1.134 + 5.166= 6.3
13. 13. Company y Systematic risk = βi² x σ²m (.27)2 x 2.25 = 0.1640 Unsystematic risk = total variance of security return – systematic risk = 5.86 - .1640 = 5.696 σ²p = [ (Σxiβi)²σ²m] + [ Σxi² ei²] = {(.5 x .71 + .5 x .27)2 x 2.25} + {(.5) 2 (5.166) + (.5) 2 (5.696)} = (.540 + 2.716) = 3.256