2.8 Alpha, Beta, and Hypothesis Testing

Question 1

Ex-Ante vs Ex-Post Alpha

Answer 1

Ex-Ante: EXPECTED consistent superior return generated through skill

Ex-Post: BACKWARD-LOOKING superior performance based on realised return. skill + luck

Question 2

Simple Linear Regression

Answer 2

Y = a +bX + e

Question 3

Ordinary Least Squares OLS Regression

Answer 3

Best fit of data points that minimises sum of squared error terms

Question 4

Explain issues of OLS:
1. Outliers
2. Autocorrelation
3. Heteroskedasticity

Answer 4

1. Outlier: Data sets with fat-tailed (Leptokurtic) distributions = higher error
2. Autocorrelation: illiquid assets get smoothed or appraised praises
3. Heteroskedasticity: variance varies over time (<,> shapes)

Question 5

OLS Goodness of fit = r-squared

Answer 5

r-squared close to one means that the model is a good fit (i.e., variable X explains most of the variance in variable Y). Thus, all else equal, larger values of r-squared are preferred.

e.g 0.8 means that 80% of the returns/variation can be explained by systematic risk, therefore 20% alpha

Question 6

T-test Statistical significance

Answer 6

t test = estimated value / standard error

at 5% level, t-statistic must exceed critical value of 1.96 for it to be statistically significant

Question 7

In the context of hypothesis testing: Test Statistic =

Answer 7

(Estimated Value - Hypothesized Value) / Standard Error of Statistic

Question 8

p value

Answer 8

Result generated by the statistical test that indicates the probability of obtaining a test statistic by chance that is equal to or more extreme than the one that was actually observed (under the condition that the null hypothesis is true).

The p-value that the test generated is then compared to the level of significance that the researcher chose.

Question 9

Type I vs Type II errors in hypothesis testing

Answer 9

Type I: REJECT null hypothesis when it is true
Type II: FAIL TO REJECT when it is false

Question 10

List the three major types of model misspecification in the context of estimating systematic risk.

Answer 10

Omitted (or misidentified) systematic return factors
Misestimated betas
Nonlinear risk-return relationships

Question 11

Does ex ante alpha lead to ex post alpha?

Answer 11

Not necessarily. While ex ante alpha may be viewed as expected idiosyncratic return, ex post alpha is realized idiosyncratic return. Simply put, ex post alpha is the extent to which an asset outperformed or underperformed its benchmark in a specified time period. Ex post alpha can be the result of luck and/or skill. To the extent that an investor suffers bad luck, ex ante will not guarantee ex post alpha.

Question 12

Does an analyst select a p-value or a significance level in preparation for a test?

Answer 12

Significance level. P Value is the output.

Question 13

What are the two steps to an analysis of ex ante alpha using historical data?

Answer 13

An asset pricing model or benchmark must be used to divide the historical returns into the portions attributable to systematic risks (and the risk-free rate) and those attributable to idiosyncratic effects.

The remaining returns, meaning the idiosyncratic returns (i.e., ex post alpha), should be statistically analyzed to estimate the extent, if any, to which the superior returns may be attributable to skill rather than luck.

Question 14

What is the goal of an empirical investigation of abnormal return persistence?

Answer 14

To identify ex ante alpha

Question 15

Provide two common interpretations of the investment term alpha.

Answer 15

• Alpha refers to any excess or deficient investment return after the return has been adjusted for the time value of money (the risk-free rate) and for the effects of bearing systematic risk (beta).
• Alpha can also refer to the extent to which the skill, information, and knowledge of an investment manager generates superior risk-adjusted returns (or inferior risk-adjusted return in the case of negative alpha).
• Note that the first interpretation can include high returns from luck.

Question 16

Provide two common interpretations of the investment term beta.

Answer 16

• Beta is the proportion by which an asset’s excess return moves in response to the market portfolio’s excess return (the return of the asset minus the return of the riskless asset).
• Beta refers to any of a number of measures of risk or the bearing of risk, wherein the underlying risk is systematic (shared by at least some other investments and usually unable to be diversified or fully hedged without cost) and is potentially rewarded with expected return without necessarily specifying that the systematic risk is the risk of the market portfolio.

Question 17

What are two methods of detecting outliers in a statistical analysis?

Answer 17

Detection through visual inspection of plots
Ordered listings of the variables and regression residuals

Question 18

What is the term for investment products trying to deliver systematic risk exposure with an emphasis on doing so in a highly cost-effective manner?

Answer 18

Beta drivers (or passive indexers)

Question 19

What is the relationship between selection bias and self-selection bias in hedge fund datasets?

Answer 19

Selection bias is a distortion in relevant sample characteristics from the characteristics of the population, caused by the sampling method of selection or inclusion used by the data manager.

If the selection bias originates from the decision of fund managers to report or not to report their returns, then the bias is referred to as a self-selection bias.