Lecture 5

Empirical Testing of the asset pricing models; Multi-factor Models; and Market Anomalies

Outline of the Lecture

Review: Regression Analysis Application of Regression to the

Testing of the Asset-pricing models Beta, B/M ratio, and size effect:

Fama-French three factor model Macro variables and asset pricing:

Chen-Roll-Ross three factor model Market Anomalies

Regression Analysis

Regression analysis: a statistical tool for the investigation of relationships between variables. In order to see the association between

them; and For the purpose of predicting the future

value of the dependent variable. In a single factor CAPM, we wish to test

whether the market index is a common factor which affects assets' returns.

Market Model Regression

For each asset i, one can estimate

We often use a firm’s monthly stock return data for the previous 5 years to estimate the beta. Why do not use daily data? Why uses 5 year long data?

)1(itftmtiiftit

rrrr

First-Pass Regression

First-pass regression: estimate alphas, betas on individual stocks using the market model regression (1).

Then calculate the average (excess) return of each stock and the market index during the sample period.

Record the estimates of the variance of the residuals for individual stocks.

Example: Problem 1, Chpt. 13

To do the first –pass regression on asset A: The excess return on the market index is

the independent variable; The stock A’s excess return is the

dependent variable; Click “Tools” and find “Data Analysis” tool

pack (if you cannot find it, you need to click “add-in” and add the “Analysis toolpak” on your Excel), and then click “Regressions”.

Regression Procedure

In the Input Y Range: fill “c3 to c14”; In the Input X Range: fill “b3 to b14”; In the Output options: click “New

Worksheet Ply”, insert the name “Stock A”; Then click “OK.” The regression result of equation (1) for

Stock A will appear on the separate worksheet named “Stock A.”

How to Read the Results

The R2 coefficient is given in Cell B5; For multiple variables regression, you should report the adjusted R2 – in Cell B6.

is in B17 (with the t-stat in D17); is in B18 (with the t-stat in D18); the estimated

ei is equal to D13. Both estimated for stock A are not

statistically different from zero. Check the t-stat. (Or check P-value. For any significance, P-value should be lower than 0.10 or 0.05).

Interpretations

R-Squared is the coefficient of “goodness-of-fit.”

The higher the R2, the better the line fits the observations. For the multiple regressions, we need to use the

“adjusted R-Squared.” In the index model, R2 = systematic

risk/total risk. T-statistics tests whether an estimated

parameter is statistically significantly different from zero. The significance of the test can be checked with the p-value from the table.

Second-Pass Regression

Second-pass regression:

where betas, excess returns, and residual variances (unique risks) are obtained from the first-pass regressions.

)2(2

210 iei

ifivrr

Hypothesis Testing

According to the CAPM,

However, Lintner, Miller and Scholes found that Sample average, using annual data:

The empirical SML is too flat -- it rejects the CAPM.

fm rr 120 ,0

.310.,042.,127.210 .165.

fmrr

Roll’s Critique

Roll’s major point is that “the CAPM is not testable unless the exact composition of the true market portfolio is known and used in the test.”

Joint null hypothesis underlying the test: security markets are efficient and returns behave according to a pre-specified model (such as the CAPM).

Another problem: The CAPM is concerned with expected returns, whereas we can only observe actual returns.

Measurement Error in Beta

Tests using individual stocks may suffer from the error-in-variable problem.

Beta cannot be measured in the first-pass regression without error. When this happens, the slope coefficient in the regression equation (2) will be biased downward and the intercept biased upward.

Testing the CAPM with Portfolios:Black,

Jensen, Scholes (1972)

To overcome the error-in-variable problem, Black, Jensen and Scholes formed 10 portfolios based on the magnitude of estimated individual betas, then estimated, using monthly data:

They found more supportive evidence,

ippvr

10

%42.1%,08.1%,359.0 10 fm rr

Testing the CAPM with Portfolios: Fama-MacBeth (1973)

Fama and MacBeth (1973) formed 20 portfolios based on the magnitude of estimated individual betas, then estimated:

They found that , and are not significant. The slope, is less than the market risk premium, but not significantly so.

iepppp vr 23

2

210

Fama-French 3 Factor Model

Fama and French (1993) run the regression

where a1>0, a4>0 and insignificant; a2<0, a3<0 and significant. “Beta is dead!”

FF interpret size and P/B as proxies for unobservable risk factors that have been omitted from the beta-only asset pricing relation.

iiiiiPEaBPaSizeaaaR )/()/()(

43210

An Alternative View on B/M

Lakonishok, Shleifer and Vishny (1994): show that the high B/M firms generally have

earnings declines over the preceding 3-5 years.

Claim that the market “over-reacts” to these firms poor performance, and the price of these firms gets pushed too low.

The price “recovers” when the firms do not do as badly as expected, and the firms on average experience high returns.

Inconsistency with the FF Interpretation

The premia related to size and P/B are significant primarily due to the large premia observed in January. Outside January the premia are insignificant, see Daniel and Titman (1997).

If the premia represent compensation for risk, it is reasonable to expect that compensation to be earned uniformly throughout the year, it is an unusual kind of risk that manifests itself only in one month.

Human Capital and the CAPM: JW study (1996)

Two more factors should be considered: the most important non-traded asset is

human capital; Betas are cyclical with business cycles.

Jaganathan and Wang (1996) used a proxy for changes in the value of human capital (based on the rate of change in aggregate labor income), default spread (a proxy for business conditions), as well as size and beta, and they found that the improvements of these tests are quite dramatic, see Figure 13.2 and Table 13.2.

Stocks and Bonds in Business Cycles

In general, expected returns on stocks and long term bonds move together.

Default spread, term spread, and dividend yields are measures of business conditions. Default spread: the difference between the yield for

corporate bonds and the long term Treasury bonds. The larger spread indicates a worsening business condition.

Term Spread: the difference between long term Treasury bonds vs. short term Treasury bills. A negative term spread indicates a higher chance of a recession in the near future.

Possible Explanations

Expected returns on stocks and bonds are lower when economic conditions are strong and higher when conditions are weak.

When business conditions are poor, income is low and expected returns on stocks and bonds must be high to induce substitution from consumption to investment.

Variations in expected returns with business conditions is due to variation in the risks of bonds and stocks.

Gains through Timing the Cycle

Since stocks fall prior to a recession, investors want to switch out of stocks and into Treasury bills, returning to stocks when prospects for economic recovery look good.

Based on research done by Jeremy Siegel, the excess returns from timing is 1.8% (4.8%) per year if you can predict the peak and trough one month (4 months) before it occurs.

Predict the Business Cycle?

Wall Street economists desperately try to predict the next recession or upturn. That is, they have to watch and analyse the leading economic indicators. Economic forecast data for the U.S. can be

obtained from the Website of Fed’s Philadelphia’s office.

Beating the stock market by analysing real economic activity ahead of any other agents requires the skill that forecasters do not yet have.

Chen, Roll and Ross (1986)

CRR (an example of APT multi-factor model) examines the following macro variables: YP: Yearly growth rate in industrial production MP: monthly growth rate in industrial production DEI: change in expected inflation UI: unanticipated inflation UPR: unanticipated change in default spread (Baa

and under - Aaa) UTS: unanticipated change in the term structure

(long term gov’t bond - T-bill rate)

CRR (1986) 3 Factor Results

They use the traditional 2-pass method to estimate factor risk premiums (’s)

Note: items with (*) are not statistically significant.

Years YP MP DEI UI UPR UTS Const.

58-84 4.431 13.98 -0.111* -0.672* 7.941 -5.87 4.112

58-67 0.417 15.76 0.014* -0.133* 5.584 0.535 4.868

68-77 1.819 15.65 -0.264* -1.42* 14.35 -14.33 -2.544

78-84 13.55 8.937 -0.07* -0.373* 2.15 941 12.54

Pricing Anomalies:a Summary

January effect. Possible explanation: tax loss selling?

Turn-of-the-month effect. Size. M/B ratios. Reversal and momentum.

Momentum and Reversal Effects

Many studies have documented that: Short-term momentum: there are positive

short term auto-correlation of stock returns, for individual stocks and the market as a whole. (“short” here refers to periods on the order of three to twelve months).

Long-term reversal: stocks that have had the lowest returns over any given five-year period tend to have high returns over the subsequent five years.