Single Index Model

Lokanandha Reddy Irala 2 Single Index Model MPT Revisited  Take all the assets in the world  Create as many portfolios possible  Compute the Expected Return & Risk (Variance or SD) for each of the portfolios  Plot all the portfolios in the return-risk space  Trace the Efficient Frontier  Super impose the indifference curves  Choose the optimal portfolio

Lokanandha Reddy Irala 3 Single Index Model Limitation-01 of MPT  Large Input requirements  In order to estimate the return and risk on a portfolio  expected return on each security  the variance of each security  the correlation coefficient between each possible pair of securities  Should we consider a portfolio of size n, then we need  n estimates of expected returns (one each for every security)  n estimates of the variances (one each for every security)  n(n-1)/2 estimates of correlation coefficients

Lokanandha Reddy Irala 4 Single Index Model Limitation-01 of MPT  If we consider a 150 stock portfolio, we would need 11, 175 correlation coefficients. This would be 31, 125 for a 250 stock portfolio.  This size of inputs severely limits s the application of Markowitz model.  This led to the development of models that used simplified correlation structures (reduced number of inputs) between stocks.  One such model is Single Index Model

Lokanandha Reddy Irala 5 Single Index Model Single Index Model  The basic premise The co-movement between stocks is due to a single common influence or index  Casual observation of stock prices reveals when the market goes up (as measured by any of the widely available stock market indexes), most stocks tend to increase in price and vice versa  One reason security returns might be correlated is because of a common response to market changes  A useful measure of this correlation might be obtained by relating the return on a stock to the return on a stock market index.

Lokanandha Reddy Irala 6 Single Index Model SIM Construction  When we regress the return (R) on a stock (i) with that of market over a period T, we shall get is the average return on the stock i is the component of security i’s return that is independent of the market's performance is the average return on the on the market index is a constant that measures the expected change in stock’s return given a change in market return

Lokanandha Reddy Irala 7 Single Index Model Regression-An Average relationship  The regression equation is an average relationship.  It will not hold good for each of the observation t (t=1, 2, ….T)  Consider an example. returns on a stock i and the market over a 10 year period.

Lokanandha Reddy Irala 8 Single Index Model Regression-The Residual t  Regression Eq.  The above equation doesn’t fully explain the relationship  When = 11 (t=1), we expect the equation to give us a R i of 10  The equation presents 11.10

Lokanandha Reddy Irala 9 Single Index Model Regression-The Residuals t

Lokanandha Reddy Irala 10 Single Index Model The Expected Value of the residual t  e i is a random variable and by construction, it has an expected value of zero

Lokanandha Reddy Irala 11 Single Index Model SIM  By Construction e i has an expected value of zero e i and R m will be uncorrelated, at least over the period to which the equation has been fit.  By Assumption e i is independent e j of for all values of i and j (but i≠ j)

Lokanandha Reddy Irala 12 Single Index Model SIM  The only reason stocks vary together, systematically, is because of a common co-movement with the market. There are no effects beyond the market (e.g., industry effects) that account for co-movement between securities.

Lokanandha Reddy Irala 13 Single Index Model SIM- Return & Risk  The Mean Return on a Security  The Variance of a Security  Covariance between the returns on two securities

Lokanandha Reddy Irala 14 Single Index Model SIM- Return & Risk  The return on the portfolio  The variance of a portfolio

Lokanandha Reddy Irala 15 Single Index Model SIM and Diversification  Consider a portfolio of size N  The Unique risk of the portfolio is given by  Let this portfolio is formed by investing equal amount in each of the stocks constituting this portfolio

Lokanandha Reddy Irala 16 Single Index Model SIM and Diversification  As we increase the size of the portfolio (N), the size of the portfolio unique risk decreases causing reduction in the total risk of the portfolio.  Increasing the portfolio size will not have any impact on the amount of the portfolio market risk

Lokanandha Reddy Irala 17 Single Index Model Single Index Model Thank You Questions?