Index Model

Question 1

Single-Index Model Input List

Answer 1

• Risk premium on the S&P 500 portfolio
• Estimate of the SD of the S&P 500 portfolio
• n sets of estimates of
– Beta coefficient
– Stock residual variances
– Alpha values

Question 2

Single factor model

Answer 2

Ri=E(Ri) + βi m +ei
βi = response of an individual security’s return to the common factor, m. Beta measures systematic risk.
m = a common macroeconomic factor that affects all security returns. The S&P 500 is often used as a
proxy for m.
ei = firm-specific surprises

• Reduces the number of inputs for
diversification
• Easier for security analysts to specialize

Question 3

How would (index model)scatter diagram of stock and S&P security characteristic line findings interpreted? (it is just a finding a beta)

Answer 3

• The model explains about 52% of the variation in HP.
• HP’s alpha is 0.86% per month(10.32% annually)
but it is not statistically significant.
• HP’s beta is 2.0348, but the 95% confidence interval is 1.43 to 2.53.

Question 4

Alpha and Security Analysis

Answer 4

1. Use macroeconomic analysis to estimate the risk premium and risk of the market index.
2. Use statistical analysis to estimate the beta coefficients of all securities and their residual variances, σ2 (ei).
3. Establish the expected return of each security absent any contribution from security analysis.
4. Use security analysis to develop private forecasts of the expected returns for each security.

Question 5

Is the Index Model Inferior to the Full-Covariance Model (min variance way)?

Answer 5

• Full Markowitz model may be better in principle, but
– Using the full-covariance matrix invokes estimation risk of thousands of terms.
– Cumulative errors may result in a portfolio that is actually inferior to that derived from the single-index model.
– The single-index model is practical and decentralizes macro and security analysis.

Question 6

Beta Book: Industry Version of the Index Model

Answer 6

• Use 60 most recent months of price data
• Use S&P 500 as proxy for M
• Compute total returns that ignore dividends
• Estimate index model without excess returns:
r = a + b Rm e*

Then adjust beta
• The average beta over all securities is 1. Thus, our best forecast of the beta would be that it is 1.
• Also, firms may become more “typical” as they age, causing their betas to approach 1.