Estimating Inputs for Mean-Variance Optimization

To apply mean-variance analysis in predicting expected returns, variances and correlation, an analyst first needs to determine the method of estimating inputs. The feasibility and accuracy of three methods are compared in the textbook for mean-variance optimization.

Historical Estimates

The method is to calculate means, variances and correlations directly from historical data. There are two problems with the method:

• Quantity of estimates: A very large number of parameters need to be estimated.
• Quality of estimates: There is a large amount of estimation error, especially for estimates of expected returns and covariances.

The method is feasible for asset allocation instead of portfolio formation.

Market Model Estimates: Unadjusted Beta

The market model uses the following familiar equation to estimate the return on asset i (R_i):

R_i = α_i + β_i R_M + ε_i

where:
α_i = average return on asset i unrelated to the market return. It is the intercept when R_M is 0.
β_i = the sensitivity of the return on asset i to the return on the market portfolio. It is the slope.
ε_i = an error term.

Assumptions of the market model:

• The expected value of the error term is 0. E(ε_i) = 0.
• The market return is uncorrelated with the error term, Cov (R_M, ε_i) = 0.
• The firm-specific error terms are uncorrelated among different assets.

Based on these assumptions, the market model makes the following predictions:

• Expected return for asset i: E(R_i) = α_i + β_i E(R_M).
• Variance of asset i: σ_i² = β_i² σ_M² + σ_εi².
  The first term, β_i² σ_M² , is the systematic risk of asset i, and the second term, σ_εi², is the non-systematic risk of asset i.
• Covariance between asset i and j: Cov_ij = β_i β_j σ_M²

We can use the market model to simplify the estimation process. For each of the n assets, we need to know α_i, β_i and σ_εi², and the expected return and variance for the market. That is 3n + 2 parameters.

To estimate the parameters of the market model, historical returns for an asset are regressed against returns for a market index. The outputs are unadjusted estimates (e.g. unadjusted β). These estimates can be used to compute the expected returns, the variances and covariances of those returns for mean-variance optimization.

The main problem is that the true market portfolio is unobservable.

Market Model Estimates: Adjusted Beta

Adjusted beta is a historical beta adjusted to reflect the tendency of beta to be mean-reverting. The motivation for adjusting beta estimates is that, on average, the beta coefficients of stocks seem to move toward 1 over time. The explanation is statistical. We know that the average beta over all securities is 1. Thus, before estimating the beta of a security, our best forecast of the beta would be that it is 1. When we estimate the beta over a particular sample period, we sustain some unknown sampling error of the estimated beta. The greater the difference between our beta estimate and 1, the greater is the chance that we incurred a large estimation error and that beta in a subsequent sample period will be closer to 1.

Merrill Lynch adjusts beta estimates in a simple way. It takes the sample estimate of beta and averages with 1, using weights of two-thirds and one-third:

Adjusted beta = 2/3 sample beta + 1/3 (1)

An adjusted beta tends to predict future beta better than historical beta.

Instability in the Minimum-Variance Frontier

The shape of the minimum-variance frontier changes as the risk and return attributes of individual assets change over time. A problem with standard mean-variance optimization is that small changes in inputs frequently lead to large changes in the weights of portfolios that appear on the minimum-variance frontier. This is the problem of instability in the minimum-variance frontier.

The problem of instability is practically important because the inputs to mean-variance optimization are often based on sample statistics, which are subject to random variation. Relatedly, the minimum-variance frontier is not stable over time. Besides the estimation error in means, variances, and covariance, shifts in the distribution of asset returns between sample time periods can give rise to this time instability of the minimum-variance frontier. This is called time instability.

To address the instability problem, analysts may consider the following responses:

• Add constraints against short sales.
• Improve the statistical quality of inputs to optimization.
• Use a statistical concept of the efficient frontier.

Practice Question 1

Which statement(s) is/are true regarding adjusted Beta?

I. The mean-reverting level of the beta is 0.
II. If the historical beta is greater than 1, the adjusted beta (as calculated by the Merrill Lynch's equation) will be more than the historical beta.
III. For the general form of the adjusted beta (forecast Beta_i,t = a₀ + a₁ Beta_{I, t-1}), a₀ + a₁ = 1.

Correct Answer: III only

I. It is 1. II. It will be less than the historical beta.

Practice Question 2

The beta of Microsoft stock has been estimated as 1.5 by Merrill Lynch using regression analysis on a sample of historical returns. The Merrill Lynch adjusted beta of Microsoft stock would be

A. 1.5.
B. 1.0.
C. 1.33.

Correct Answer: C

(2/3) * sample beta + 1/3 = 1.33.

Practice Question 3

Betas show a tendency to evolve toward ______ over time.

A. 1.
B. 0.
C. -1.

Correct Answer: A

Practice Question 4

Suppose that the market model for the excess returns of stocks A and B is estimated with the following results:

R_A = 1.0% + 0.9R_M + ε_A
R_B = -2.0% + 1.1R_M + ε_B
σ_M = 20%
σ(ε_A) = 30%
σ(ε_B) = 10%.

What is the standard deviation of each stock and covariance between them?

Correct Answer: σ_i² = β_i² σ_M² + σ²(ε_i)

σ_A = (0.9² x 0.2² + 0.3²)^1/2 = 35%

σ_B = (1.1² x 0.2² + 0.1²)^1/2 = 24%

Cov_i,j = β_i β_j σ_M² = 0.9 x 1.1 x 0.2² = 0.04.

Practice Question 5

Suppose you held a well-diversified portfolio with a very large number of securities, and that the market model holds. If the σ of your portfolio was 0.25 and σ_M was 0.20, the β of the portfolio would be approximately:

Correct Answer: 1.25

σ_i² = β_i² σ_M² + σ²(ε_i)

σ²(ε_i) = 0, so β_i = 0.25/0.2 = 1.25.

Practice Question 6

An investment fund manager analyzes 100 stocks using historical estimates in order to construct a mean-variance efficient portfolio constrained by these 100 securities. The manager will need to calculate ________ covariances.

A. 100.
B. 4,950
C. 10,000.

Correct Answer: B

The covariance between any two stocks needs to be calculated (100 + 99 + 98 + ...)

Practice Question 7

The market model makes the following assumptions. Which assumptions are correct?

I. The error term is normally distributed.
II. The variance of the error term is identical across assets.
III. The errors are uncorrelated with the market return.
IV. The error terms are uncorrelated among different assets.
V. The expected value of the error term is zero.

Correct Answer: III, IV and V

Practice Question 8

Responses to instability include:

I. Prohibiting short sales so that all portfolio weights are positive.
II. Employing forecasting models that provide better forecasts than historical estimates.
III. Avoiding rebalancing until significant changes occur in the efficient frontier.

Correct Answer: I, II and III

Practice Question 9

Consider the market model. The alpha of a stock is 3%. The return on the market index is 12%. The risk-free rate of return is 4%. The stock earns a return that exceeds the risk-free rate by 12.6% and there are no firm-specific events affecting the stock performance. The β of the stock is

A. 1.2
B. 0.8
C. 1.13

Correct Answer: C

12.6% + 4% = 3% + β 12% => β = 1.13. Notice the 3% is the α.

Practice Question 10

Suppose you held a well-diversified portfolio with a very large number of securities, and that the market model holds. If the σ of your portfolio was 0.20 and σ_M was 0.16, the β of the portfolio would be approximately

A. 1.25
B. 0.8
C. 1.0

Correct Answer: A

σ_i² = β_i² σ_M² + σ²(ε_i)

σ²(ε_i) = 0, so β_i = (0.2/0.16) = 1.25.