BKM Chapter 8 - Index Models

Question 1

Single Factor Model

Answer 1

• Assumes that the return for a security is the sum of both expected and unexpected components
• Further, we assume that the return deviates from its expected
  return for two reasons —
  
  • unanticipated macroeconomic (economy-wide) events that affect all assets and
  • unanticipated firm-specific events

If all of the macro events can be summarized using a single factor, m, and security i’s sensitivity to that factor can be measured as ß_i, then
the return for security i can be written as:

r_i = E(r_i) + ß_im + e_i

Here, we are depicting the actual realized return for a specific security and suggesting that it reflects the security’s expected return, a return driven by a macroeconomic factor common to all securities and a return due to some firm-specific sources of variability.

Question 2

Single Factor Model - Variance of Returns

Answer 2

Using this equation for the single-factor model we can also depict the variance of returns as:

σ_i² = ß_i²σ_m² + σ²(e_i)

The first term reflects systematic risk that impacts all securities (perhaps to different degrees due to differences in the betas) and the second term reflects firm-specific risk.

Question 3

Single Factor Model - Covariance of Returns

Answer 3

Because a common factor, m, impacts the variance of returns for each security, the covariance of returns for any two securities is:

Cov (r_i, r_j) = ß_iß_jσ²_m

Question 4

Single Index Model

Answer 4

One limitation of the single factor model that we just presented is that there is no insight into how to quantify the single factor, m. But suppose we made the assumption that the S&P 500 index could be used as a proxy for the single factor and then used this more practical model to make estimates of the betas, variances and covariances?

Here, we’ll use M to denote the S&P 500 index value and for convenience reflect the excess returns over the risk free rate using R ƒ= r - r_f instead of r . Then, we can depict the excess returns for any specific stock, as:

R_i = Þ_i + ß_iR_M + e_i

Notice that this formula indicates that the return in excess of the risk free rate is equal to a constant, denoted by alpha, plus a return due to movements in the overall market and beta plus an unanticipated firm-specific component, e_i.

Question 5

Formular for estimator for ß using linear regression

Answer 5

Question 6

Single Index Model - Variance of Return

Answer 6

σ_i² = Variance(Þ_i) + Variance(ß_iR_M) + Variance(e_i)

=ß_i²σ_M²+σ²(e_i)

Question 7

Single Index Model - Covariances

Answer 7

The benefit of assuming an index model exists is to be able to estimate all of the covariances between risky assets for use in the portfolio optimization procedures we covered earlier. Here, it can be shown that the covariance between the returns of any two assets is:

Cov (R_i, R_j) = ß_iß_jσ_M².

This allows us to estimate the N„(N-1)/2 covariances needed for the Markowitz model by simply estimating N betas and the variance of the market return.In addition, by not having to estimate covariances among wildly disparate types of firms, such as IBM and GM, market professionals are able to specialize in specific industries and know that covariances are driven by correlations with the overall market.

Question 8

Single Index Model - Correlation

Answer 8

Corr(r_i, r_j) = Product of correlations with the market index = Corr(r_m, r_i) * Corr(r_m, r_j)

             = ß<sub>i</sub>ß<sub>j</sub>σ<sub>m</sub><sup>2</sup>/σ<sub>i</sub>σ<sub>j</sub>

Question 9

The Single Index Model - Diversification

Answer 9

It turns out that if we have N stocks in a portfolio then the portfolio is just the weighted average of the individual ’s, the portfolio is just the weighted average of the individual ’s and the unique risk is just the weighted average of each of the firm-specific components „e_i….

The variance of returns for the portfolio can be written as:

σ² = ß_p²σ_m² + Σw_i²σ²(e_i)

This simply says that the portfolio risk is equal to a systematic component and the weighted sum of all the unique risks.

As N gets large and if all w_i ƒ= 1/N, then the second term approaches zero and all that matters is the portfolio beta. The unique risk is diversified away and only the systematic risk matters.

Question 10

Portfolio Construction and the Index Model

• A key advantage of the index model
• Focus of the security analysis

Answer 10

A key advantage of the index model is the framework it creates for separating macroeconomic analysis (the analysis of economy-wide issues affecting the overall market) and security analysis (the analysis of the specific issues affecting individual stocks — the alphas).

Using the estimates of the risk and risk premiums for the market index, along with firm-specific beta estimates, we can identify the market-driven expected return and use it as a benchmark. Then security analysis can focus on trying to identify firms with a positive alpha — a return over and above that which can be expected based solely on its sensitivity to the market index and the expected risk premium for the market.

Question 11

The Index Portfolio

Answer 11

the market index (S&P 500 in this case) can be thought of as a passive portfolio

Question 12

The Optimal Risky Portfolio: Treynor-Black procedure

Answer 12

The idea is that the optimal portfolio will have a combination of w*_A invested in an active portfolio and the rest invested in the index portfolio, a passive portfolio. To determine the weights of the stocks in the active portfolio and the weights between the active and passive portfolio, the following steps are followed:

1. Calculate an initial weight for each stock based on its ratio of alpha to residual variance. This gives more weight to the stocks whose non-market returns (the alpha) are largest relative to their non-market risk (the residual variance). Stocks with negative alphas will be given a negative weight (shorted).
2. Scale the weights described above so that they add to one.
3. Compute the alpha for the active portfolio using the scaled weights from (2) above.
4. Compute the residual variance of the active portfolio, denoted σ²„(e_A), again using the weights from (2) above.
5. Calculate an initial weight for the active portfolio (which is accurate only if the beta for the active portfolio is 1.0) using the following ratio:
   w⁰_Aƒ = {σ_A/σ²(„e_A)} / {E„(R_M)/…σ²_M}
   This weight for the active portfolio reflects the non-market returns adjusted to reflect the non-market risk relative to the market returns adjusted to reflect the market risk.
6. Calculate the beta of the active portfolio using the weights from (2) above.
7. Adjust the weight in (5) above to account for the actual beta of the active portfolio, using the formula:
   w*_A= ƒw⁰_A/{1+ (1-ß_A)…w⁰_A}
8. The expected return and variance of the optimal risky portfolio can now be found. It will have 1 - w*_A invested in the market portfolio and the balance invested in the active portfolio, in proportion to the weights in (2) above.

Question 13

Information Ratio

Answer 13

When the index model is used as described above to identify the optimal portfolio, the Sharpe ratio of the optimal portfolio (S_P ) will exceed the Sharpe ratio of the passive index portfolio (S_M) by an amount that depends on the ratio of the active portfolio’s ratio of alpha to residual standard deviation, as shown below:

S_p²=S_M² + [Þ_A/σ(e_A)]²

The ratio of the alpha to the residual standard deviation is referred to as the Information Ratio.

Question 14

Comparing Index Model and Full-Covariance Model (Important)

Answer 14

The Markowitz approach

Identifies the portfolio of risky assets with the highest possible Sharpe ratio.

Advantage: allows you to take into account the full covariance matrix

(covariance is used to calculate the variance of the portfolio)

Disadvantage:

1. Requires a huge numbers of estimates to fill the covariance matrix
2. Introduces potentially significant parameter estimation errors.

The Single Index Model:

also achieves the goal of identifying the portfolio with the highest Sharpe ratio, but it does so under the assumption that the covariances are driven entirely by a common factor and therefore only has to estimate the expected returns and betas for each asset.

Advantage:

1.For large universes of securities, the number of estimates (3n+2) is only a fraction of what is needed for the Markowitz procedure.

• n estimates of expected excess return Þ
• n estimates of sensitivity coeffients ß
• n estimates of firm specific variance σ²(e_i)
• 1 estimate for market risk premium
• 1 estimate for the variance of the macro-economic factor, σ²_m

​2. Separates macroeconomic analysis and security analysis (search for positive alpha).

Disadvantage:

It ignores sources of covariance unrelated to the common index and may therefore miss opportunities for additional diversification.

Question 15

Adjusted Betas

• Why do betas tend to regress toward 1 over time

Answer 15

First, firms are often formed to offer something fairly unique in terms of products or management, but as they grow they will tend to diversify and begin to look more like other firms, causing their betas to become more like the average firm’s beta.

Second, there are statistical issues associated with estimating beta (substantial estimation error) and it would make sense to use a Bayesian
approach with a prior estimate of 1.0 (essentially a credibility adjustment).

There are various ways to reflect this tendency to regress towards 1.0, but Merrill uses the following simple formula:
Adjusted Beta ƒ= (2/3) * Estimated Beta… ‚+ (1/3) * 1

Question 16

Tracking Portfolios

Answer 16

Suppose you identify a portfolio that you believe earns a positive alpha, which represents a positive expected return over and above the risk free rate and the return based on the systematic risk (the beta) and the market return.

You could create a tracking portfolio that consists of T-bills and the S&P 500 Index so that the tracking portfolio has the same systematic risk as the portfolio that you think has a positive alpha. By buying the portfolio and at the same time shorting the tracking portfolio, the systematic exposure to the market will be eliminated. Your returns will consist solely of the positive alpha and the risk from the non-systematic residual risk. With a well diversified portfolio, this residual risk should be small, allowing you to earn the positive alpha without any exposure to the overall market.