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Topic 3 (Ch. 8) Index Models. A single-factor security market The single-index model Estimating the single-index model. A Single-Factor Security Market .
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Index Models
A Single-Factor Security Market
e.g. To analyze 50 stocks, the input list includes:
n = 50 estimates of expected returns
n = 50 estimates of variances
(n2 - n)/2 = 1,225 estimates of covariances
1,325 estimates
If n = 3,000 (roughly the number of NYSE stocks), we need more than 4.5 million estimates.
e.g.
Construct a portfolio with weights: -1.00; 1.00; 1.00, for assets A; B; C, respectively, and calculate the portfolio variance.
All these (interrelated) factors affect almost all firms. Thus, unexpected changes in these variables cause, simultaneously, unexpected changes in the rates of return on the entire stock market.
Suppose that we summarize all relevant economic factors by one macroeconomic indicator and assume that it moves the security market as a whole.
We can summarize the distinction between macroeconomic and firm-specific factors by writing the holding-period return on security i as:
Some securities will be more sensitive than others to macroeconomic shocks.
This approach leads to an equation similar to the single-factor model, which is called the single-index model, because it uses the market index to proxy for the common factor.
Intercept:
αi: the security i’s expected excess return when the
market excess return is zero.
βi: the security i’s sensitivity to the market index.
For every + (or -) 1% change in the market excess return, the excess return on the security will change by + (or -)βi%.
Residual:
ei is the zero-mean, firm specific surprise in the security return in time t.
nonmarket premium
part of a security’s risk premium is due to the risk premium of the market index
→ systematic risk premium
: variance attributable to the uncertainty of the
market index
: variance attributable to firm-specific uncertainty.
(total risk = systematic risk + firm-specific risk)
Note:
The covariance between RMand ei is zero because ei is defined as firm specific (i.e. independent of movements in the market).
The covariance between the rates of return on 2 securities:
The covariance between the return on stock i and the market index:
The correlation coefficient between the rates of return on 2 securities:
(product of correlations with the market index)
The set of estimates needed for the single-index model
For n = 50: need 152 estimates (not 1,325 estimates).
n = 3,000: need 9,002 estimates (not 4.5 million).
The index model and diversification
The excess rate of return on each security is:
The excess return on the portfolio of securities:
Note:
The portfolio has a sensitivity to the market given by:
(the average of the individual is)
It has a nonmarket return component of a constant
(intercept):
(the average of the individual alphas)
It has a zero mean variable:
(the average of the firm-specific components)
The portfolio’s variance is:
The systematic risk component of the portfolio variance (the component that depends on marketwide movements) is and depends on the sensitivity coefficients of the individual securities.
This part of the risk depends on portfolio beta and , and will persist regardless of the extent of portfolio diversification.
No matter how many stocks are held, their common exposure to the market will be reflected in portfolio systematic risk.
In contrast, the nonsystematic component of the portfolio variance is 2(eP)and is attributable to firm-specific components ei.
Because the eis are uncorrelated, we have:
where : the average of the firm-specific variances.
Because this average is independent of n, when n gets large, 2(eP) becomes negligible.
Thus, as more and more securities are added to the portfolio, the firm-specific components tend to cancel out, resulting in ever-smaller nonmarket risk.
As more and more securities are combined into a portfolio, the portfolio variance decreases because of the diversification of firm-specific risk.
However, the power of diversification is limited.
Even for very large n, part of the risk remains because of the exposure of virtually all assets to the common, or market, factor.
Therefore, this systematic risk is said to be nondiversifiable.
Estimating the Single-Index Model
suggests how we might go about actually measuring market and firm-specific risk.
Suppose that we observe the excess return on the market index and a specific asset over a number of holding periods.
We use as an example monthly excess returns on the S&P 500 index and GM stock for a one-year period.
We can summarize the results for a sample period in a scatter diagram:
The single-index model states that the relationship between the excess returns on GM and the S&P 500 is given by the following regression equation:
In this single-variable regression equation, the dependent variable plots around a straight line with an intercept and a slope .
The deviations from the line (e) are assumed to be mutually uncorrelated and uncorrelated with the independent variable.
The sensitivity of GM to the market, measured by GM, is the slope of the regression line.
The intercept of the regression line is GM, representing the average firm-specific return when the market’s excess return is zero.
Deviations of particular observations from the regression line in any period are denoted eGM, and called residuals (i.e. each of these residuals is the difference between the actual security return and the return that would be predicted from the regression equation describing the usual relationship between the security and the market).
Thus, residuals measure the impact of firm-specific events.
Estimating the regression equation of the single-index model gives us the security characteristic line (SCL).
The SCL is a plot of the typical excess return on a security as a function of the excess return on the market.
The estimate of beta coefficient (i.e. the slope of the regression line SCL):
The intercept of the regression line:
Hence, we can estimate the firm-specific variance:
which is equal to the standard deviation of the regression residual.
The Industry Version of the Index Model
instead of
If rf is constant over the sample period, both equations have the same independent variable rM and residual e.
Thus, the slope coefficient will be the same in the two equations.
The apparent justification for this procedure is that, on a monthly basis, rf(1 - ) is small.
But, note that for β≠1, the regression intercept will not equal the index model alpha.
This suggests that we might want a forecasting model for beta.
Current beta = a + b (Past beta)
Given estimates of a and b, we would then forecast future betas using the rule:
Forecast beta = a + b (Current beta)
Why not also investigate the predictive power of other financial variables in forecasting beta?
Rosenberg and Guy find the following variables help predict betas:
Variance of earnings.
Variance of cash flow.
Growth in earnings per share.
Market capitalization (firm size).
Dividend yield.
Debt-to-asset ratio.