8. Alpha, Beta, and Hypothesis Testing

Question 1

1. Provide two common interpretations of the investment term alpha.

Answer 1

• 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 2

2. Provide two common interpretations of the investment term beta.

Answer 2

• 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 3

3. Does ex ante alpha lead to ex post alpha?

Answer 3

• 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 4

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

Answer 4

• 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 5

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

Answer 5

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

Question 6

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

Answer 6

• To identify ex ante alpha

Question 7

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

Answer 7

• Beta drivers (or passive indexers)

Question 8

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

Answer 8

• The significance level. The p-value is the output of the statistical computations.

Question 9

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

Answer 9

• 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.

Question 10

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

Answer 10

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

Question 11

alpha driver

Answer 11

An investment that seeks high returns independent of the market is this.

Question 12

alternative hypothesis

Answer 12

is the behavior that the analyst assumes would be true if the null hypothesis were rejected.

Question 13

asset gatherers

Answer 13

are managers striving to deliver beta as cheaply and efficiently as possible, and include the largescale index trackers that produce passive products tied to well-recognized financial market benchmarks.

Question 14

backfill bias fund

Answer 14

or instant history bias, is when the funds, returns, and strategies being added to a data set are not representative of the universe of fund managers, fund returns, and fund strategies.

Question 15

backfilling

Answer 15

typically refers to the insertion of an actual trading record of an investment into a database when that trading record predates the entry of the investment into the database.

Question 16

backtesting

Answer 16

is the use of historical data to test a strategy that was developed subsequent to the observation of the data.

Question 17

beta creep

Answer 17

is when hedge fund strategies pick up more systematic market risk over time.

Question 18

beta driver

Answer 18

An investment that moves in tandem with the overall market or a particular risk factor is this.

Question 19

beta expansion

Answer 19

is the perceived tendency of the systematic risk exposures of a fund or asset to increase due to changes in general economic conditions.

Question 20

beta nonstationarity

Answer 20

is a general term that refers to the tendency of the systematic risk of a security, strategy, or fund to shift through time.

Question 21

causality

Answer 21

The difference between true correlation and causality is that causality reflects when one variable’s correlation with another variable is determined by or due to the value or change in value of the other variable.

Question 22

cherry-picking

Answer 22

is the concept of extracting or publicizing only those results that support a particular viewpoint.

Question 23

chumming

Answer 23

is a fishing term used to describe scattering pieces of cheap fish into the water as bait to attract larger fish to catch.

Question 24

confidence interval

Answer 24

is a range of values within which a parameter estimate is expected to lie with a given probability.

Question 25

data dredging

Answer 25

or data snooping, refers to the overuse and misuse of statistical tests to identify historical patterns.

Question 26

data mining

Answer 26

typically refers to the vigorous use of data to uncover valid relationships.

Question 27

economic significance

Answer 27

describes the extent to which a variable in an economic model has a meaningful impact on another variable in a practical sense.

Question 28

ERP

Answer 28

equity risk premium is the expected return of the equity market in excess of the risk-free rate.

Question 29

equity risk premium puzzle

Answer 29

is the enigma that equities have historically performed much better than can be explained purely by risk aversion, yet many investors continue to invest heavily in low-risk assets.

Question 30

ex ante alpha

Answer 30

is the expected superior return if positive (or inferior return if negative) offered by an investment on a forward-looking basis after adjusting for the riskless rate and for the effects of systematic risks (beta) on expected returns.

Question 31

ex post alpha

Answer 31

is the return, observed or estimated in retrospect, of an investment above or below the risk-free rate and after adjusting for the effects of beta (systematic risks).

Question 32

full market cycle

Answer 32

is a period of time containing a large representation of market conditions, especially up (bull) markets and down (bear) markets.

Question 33

Model misspecification

Answer 33

is any error in the identification of the variables in a model or any error in identification of the relationships between the variables.

Question 34

null hypothesis

Answer 34

is usually a statement that the analyst is attempting to reject, typically that a particular variable has no effect or that a parameter’s true value is equal to zero.

Question 35

overfitting

Answer 35

is using too many parameters to fit a model very closely to data over some past time frame.

Question 36

passive beta driver

Answer 36

strategy generates returns that follow the up-and-down movement of the market on a one-to-one basis.

Question 37

process drivers

Answer 37

are beta drivers that focus on providing beta that is fine-tuned or differentiated. They compete against the asset gathers with the low fees.

Question 38

product innovators

Answer 38

At one end of the spectrum are these, which are alpha drivers that seek new investment strategies offering superior rates of risk-adjusted return.

Question 39

p-value

Answer 39

is a 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).

Question 40

return driver

Answer 40

represents the investments, the investment products, the investment strategies, or the underlying factors that generate the risk and return of a portfolio.

Question 41

significance level

Answer 41

is used in hypothesis testing to denote a small number, such as 1%, 5%, or 10%, that reflects the probability that a researcher will tolerate of the null hypothesis being rejected when in fact it is true.

Question 42

spurious correlation

Answer 42

The difference between true correlation and this term is that this idiosyncratic in nature, coincidental, and limited to a specific set of observations.

Question 43

Survivorship bias

Answer 43

is a common problem in investment databases in which the sample is limited to those observations that continue to exist through the end of the period of study.

Question 44

test statistic

Answer 44

is the variable that is analyzed to make an inference with regard to rejecting or failing to reject a null hypothesis.

Question 45

type I error

Answer 45

also known as a false positive, is when an analyst makes the mistake of falsely rejecting a true null hypothesis.

Question 46

type II error

Answer 46

also known as a false negative, is failing to reject the null hypothesis when it is false.

Question 47

Consider Sludge Fund, a fictitious fund run by unskilled managers that generally approximates the S&P 500 Index but does so with an annual expense ratio of 100 basis points (1%) more than other investment opportunities that mimic the S&P 500. Using Equation 8.1 and assuming that the S&P 500 is a proxy for the market portfolio, the ex ante alpha of Sludge Fund would be approximately –100 basis points per year. This can be deduced from assuming that βi = 1 and that [E(Ri,t) – E(Rm,t)] = –1% due to the expense ratio.

Answer 47

E ( R(i,t) − R(f) ) = alpha i + Beta i (E (R(m,t) ) − R(f)) manipulate for alpha: Alpha = Beta (E(R(m,t) - R(f)) - E(R(i,t) - Expense Ratio - R(f)) 1 (0.1 - 0) - (0.1 + 0.01 - 0) = -0.01

Question 48

Consider Trim Fund, a fund that tries to mimic the S&P 500 Index and has managers who are unskilled. Unlike Sludge Fund from the previous section, Trim Fund has virtually no expenses. Although Trim Fund generally mimics the S&P 500, it does so with substantial error due to the random incompetence of its managers. However, the fund is able to maintain a steady systematic risk exposure of βi = 1. Last year, Trim Fund outperformed the S&P 500 by 125 basis points.

Answer 48

R(it) - R(f) = Beta (R(mt) - R(f) + E(it) rearrange to = E(it) Beta = 1 R(it) - R(mt) = 1.25% R(it) = 10% R(mt) = 11.25% R(f) = 0 (no risk) Step One: Press 0.1125 → - → 0 Step Two: Press ÷ → 1 Step Three: Press - → 0.10 + → 0 Step Four: Press = Answer: 0.0125