Briefly describe the regression equation, the expected return-beta relationship, and the risk and covariance in the context of the Single Index Model. With the Single Index Model, please explain the “costs” of the simplification. Finally, briefly explain the diversification argument that is neatly highlighted by the Single Index Model (e.g. equation 8.16), employing the elements from pages 263-64.
Answer
Regression Equation =>y=a+b
Where
Where
N = Number of observations
X = a year index (decade)
Y = a + bx is said is the formation of a linear equation. X is said to be an independent variable while y is known as the dependent variable. B is the slope of line, and A is the intercept (value of y where x = 0).
Single Index Model:
Both the risk and return of a stock is measured by the SIM model. In 1963, William Sharpe came up with this model: Mathematical expression for Single Index model is:
The risk of a specific portfolio compares systematic risk, also called un-diversification risk and unsystematic risk which is also called a diversification risk. Common Risks to all types of security are known as systematic risk. A risk associated with individual assets is known as is known as unsystematic risk. Diversification of unsystematic risk can be done by involving a variety of assets in the portfolio. However, the same is not possible with systematic risk which is dependent upon the market. Market portfolio depends upon a list of 30-40 securities in a developed market. The UK and the US markets are considered to be fully developed today. Risk portfolio is sufficiently diversified in such markets, and risk exposure is limited to systematic risk only.
In figure 8.16, β_(p^2 ) α_(M^2 ) is considered to be the systematic risk as it is dependent upon the movements in the market.) being the non-systematic risk of portfolio variance, which is dependent upon firm-specific components (Teall, 1999).
What is the information ratio? In the context of 8.22, what does the information ration suggest about maximizing the overall Sharp ratio? Why is this important with respect to security analysis? How does the alpha factor into the discussion (using equation 8.20 and 8.23 to explain your answer)? On page 275, why does the author suggest that unlike alpha, the systematic component tradeoff is neither “virtue nor vice”?
Answer
The information ratio is a ratio used for an appraisal that is used for the measurement of the performance of an investment as compared to an Index, which is considered to be a benchmark. The additional amount of return that is made by the investor taking extra risk. The Sharpe ratio is a similar ratio, like information ratio, which indicates risk-adjusted returns. The difference between asset return and risk of the asset is divided by the standard deviation. A benchmark in Sharpe ratio is a risk-free return, while the information ratio uses a risky index as a benchmark. Information Ratio shows us the active portfolio contributed to the Sharpe ratio of a risky portfolio. Alpha and residual standard deviation determine the Sharpe ratio of a risky portfolio.
Figure 8.22 depicts that the information ratio must be maximized to maximize the Sharpe ratio of the overall active portfolio. The extra return is measured by this ratio, which is why it is an important aspect of the security analysis.
In figure 8.23, the alpha is negative, so a short position is assumed for the security in the optimal risky portfolio. If we prohibit a short position, then a negative value of alpha would have to be taken out of the optimized portfolio, and a zero will be assigned to the portfolio weight. A better diversification of the portfolio is observed with an increase in non-zero alpha values. Then at the expense of passive index portfolio, the overall weight of risky portfolios will rise (Teall, 1999).
Reference
Teall, J. L. (1999). Financial Market Analytics (2 ed.). Greenwood Publishing Group.
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