Cross-Section of Returns Flashcards
What is the Sharpe Ratio
The Sharpe ratio measures the asset’s excesses return over total risk and is defined as
SR=(μ_i-r_f)/σ_i
It describes how much excess return you receive for the volatility of holding a riskier asset.
The tangency portfolio has the maximum possible Sharpe ratio. The Sharpe ratio can be illustrated as the slope of the line through the risk-free return to the portfolio.
Testing the mean-variance efficiency of a portfolio is equivalent to testing whether the Sharpe ratio of the portfolio is the maximum of the set of Sharpe ratios of all possible portfolios, i.e. alpha is statistically indistinguishable from zero for all portfolios.
CAPM Time Series Test
There are different hypothesis we can test:
A. Test whether all assets lie on the security market line by (a) testing alphas individually using t-tests (H_0: α_i=0), or by (b) testing for joint significance of the alphas (a Wald-test)
B. Test whether the market portfolio has the highest Sharpe Ratio, i.e., testing if the ex post tangency portfolio and the market portfolio are identical (Gibbons-Ross-Shanken test).
Asymptotic Joint Test
t-tests of individual alphas (=0) may be too strict. Assets may have non-zero alphas (over limited time) even if the CAPM is true. Instead, we consider an economy-wide test of the CAPM by testing whether all α_i’s are jointly zero at time t. This can be considered a test of whether the average α is zero. It requires the covariance structure of the α_i’s. Since the population covariance structure is unknown, we have to rely on sample estimates. This results in an asymptotic Wald-test for joint significance of the alphas:
Type I Error
Reject H_0, although H_0 is true (i.e., a false positive, size) with probability is α
Type II Error
Failure to reject H_0, when H_0 is true (i.e., a false negative) with probability is β
Correct Inference
Accept H_0 and H_0 is true (i.e., true negative) with probability 1-α.
Reject H_0 when H_a is true (i.e., true positive, power) with probability 1-β.
Size of the Test
Size is the probability of making a Type I error (probability of falsely rejecting H_0 when H_0 is true). For a finite sample test, size is equal to the chosen significance level (e.g., 5%). For an asymptotic test, there may be serious deviations between the true size and the chosen significance level α, especially if the sample is small.
Power of Test
Power is the probability of making a Type II error (Failure to reject H_0, when H_0 is true (i.e., a false negative) with probability β). A reasonable power is typically set at 80%, i.e., Type II error rate is β=20%.
Trade of Power and Size
There is a trade-off between error rates α and β. For any given sample size, the effort to reduce one type of error generally results in increasing the other type of error. For a given test, the only way to reduce both error rates is to increase the sample size, and this may not be feasible
Testing the CAPM - Implications for Empirical Design
Number of securities N should be kept small relative to T. We have thousands of securities trading at any point in time. But we cannot just throw stocks away to decrease N. Instead, form portfolios (usually based on firm characteristics). N then refers to the number of portfolios NOT to the number of stocks. However, then the tangency portfoliio will consist of fewer assets (less diversification, bigger standard deviation σ_q, smaller δ, lower power). So, there is a trade-off between having too many or too few portfolios. Especially if your T is small, you should be careful. N should be big enough to form a well-diversified tangency portfolio.
Rule of thumb N=10.
When do we need cross sectional returns?
Time-series regressions only work for portfolios and when factors are returns, so if we wish to test whether investing in a single asset yields alpha, or if a factor that is not a return yields re-turns beyond risk compensation, we need cross-sectional tests.
Testing the CAPM in Cross Sectional Tests
A zero intercept (α) is not the only testable implication of the CAPM. Alternatively, the CAPM also implies a linear positive relationship between beta and the expected excess return i.e. the SML line.
- Beta Estimation: Estimate betas for all assets (firms, portfolios) making N time-series regressions
- Formulate the Regression: Perform a cross-sectional regression at a point in time or averaged over time, where the dependent variable is the excess return (return over the risk-free rate) of each asset, and the independent variable is the beta of each asset –> Regress N average returns on N betas
Interpretation; In CAPM, 𝛼 (the intercept) should statistically not differ from zero, and the slope 𝛽 should equal the market risk premium. Significant deviations would imply CAPM does not hold.
Testing the CAPM in Cross Sectional Tests - Empirical Challenges
Problem I: The market portfolio is not observable (Roll’s critique). A common proxy for the market return is the value-weighted portfolio of all traded equities.
Problem II: Which risk-free rate to use. Is there a risk-free rate, and is it reasonable to assume it is non-stochastic?
Problem III: Betas are not observable. We ca estimate using OLS, but which sample period (and data frequency) should be use to estimate OLS?
Why do we test Portfolios?
The economic issue when testing individual firms is that (a) CAPM is a one-period model where (b) beta do not change through time. In real life, firms change (products, size, market)
The statistical issue when testing individual firms is
(1) Some firm returns are highly correlated (cross-sectional correlation). OLS SE are too small
(2) Betas are estimated, and if the standard error of the betas is too large, the point estimate will be biased so γ ̂_0 is too large and γ ̂_1 is too small. Standard errors are also too small.
Testing the CAPM in CS Tests: Approach to Problem I fixed firm beta
(1) Time-varying betas. Compute rolling betas, so each year compute beta based on the past data.
(2) Portfolio sort. Build portfolios based on estimated betas, e.g., each year take the estimated beta and sort stocks into portfolios according to their beta. Hold the portfolio for one year and rebalance.
Interim conclusion: Economically sensible to compute time-varying betas for firms. For portfo-lios, constant betas are easier to justify (if CAPM holds, stocks can switch portfolios when sys-tematic risk changes). Portfolio betas have lower standard errors as they are more diversified.
