Applied Macro Finance

Transcription

1 Master in Money and Finance Goethe University Frankfurt Week 2: Factor models and the cross-section of stock returns Fall 2012/2013 Please note the disclaimer on the last page

2 Announcements Next week (30 October, 5pm to 8pm) double session with Marco Dion, global head of quant equity research at JP Morgan, on factor models in practice 2

3 Ideas of the last class 1. Structure of global financial markets Introducing financial assets and asset classes The structure varies significantly across countries and regions Stocks, bonds and cash are the main asset classes of institutional investors and therefore this course treats them in detail Available investment set to investors 2. Stylized facts Historical Behavior of Asset Returns Equity premium in detail 3. Brief introduction to CAPM Motivation for this course: global (macro) factors/state variables drive asset prices 3

4 Today Structure of this course General Stocks Bonds Evidence Observed returns and risk premia Stylized facts on the yield curve Foundations Forecastability & predictability Financial crisis etc. Valuation techniques Yields, duration, estimation of the yield curve Theory Asset pricing (factor model, APT) CAPM APT Expectations hypothesis Factor model Applications Technical analysis and trading rules Value factor Momentum factor Portfolio optimization and risk models Forecasting the yield curve Finance industry Overview asset management industry Guest lecture on industry factors Investment process 4

5 Topics of class 2 1. Brief look at the Arbitrage Pricing Model 2. Regression-based tests of linear factor models Time-series regressions Cross-sectional regressions Fama-MacBeth Procedure 3. Empirical evidence Fama and French (1996), part 1 5

6 Brief introduction to the Arbitrage Pricing Theory (APT) Reference: review article by Huberman, G., and Z. Wand (2005) Arbitrage Pricing Theory, Federal Reserve Bank of New York Staff Reports, no APT is one approach to motivate the analysis of FF96 First developed by Ross (1976) Assumptions: one-period model, every investor believes that the stochastic properties of the security returns are consistent with a factor structure Main point: If equilibrium prices offer no arbitrage opportunities (over the static assets), then the expected returns are linear function of the risk premia. Intuition using Arrow-Debreu security pricing: Arrow-Debreu model: k fundamental securities span all possible states of nature Each asset s payoff can be described as the payoff on a portfolio of the fundamental assets Arbitrage opportunity: a zero-cost portfolio of the assets of the economy with a positive expected return If the market clearing rules out arbitrage opportunities then the price of each asset is the weighted average of the prices of fundamental assets. 6

7 Brief introduction to the APT cont. Formal statement: Agents believe that the random one-period returns satisfy the linear factor model (n assets, k factors) Normalization: E(f)=0, E(e)=0 then E(r)=μ The APT asserts that there exists a constant a (for each n) such that where (with ι being a vector of ones) (*) Exact arbitrage pricing (a=0): If some portfolio of the assets is risk free, then λ 0 is the return on this portfolio. The vector λ 1 is referred to as the risk premium and the matrix β as the beta or loading on the factor risk. The matrix Z is often the covariance matrix E(ee ). Note that (*) intuitively states that the adjusted pricing error should be bounded by a. 7

8 Brief introduction to the APT cont. Give the logic of the Arrow-Debreu framework, it is quite natural that excess returns are explained by returns of factor portfolios. However, we can also use macroeconomic factors in the model. In this case, we need to adjust the testing procedure and the interpretation of the results accordingly. 8

9 Topics of class 2 1. Brief look at the Arbitrage Pricing Model 2. Regression-based tests of linear factor models Time-series regressions Cross-sectional regressions Fama-MacBeth Procedure 3. Empirical evidence Fama and French (1996) 9

10 Regression-based tests of linear factor models Reference: Cochrane (2005). Asset Pricing, chapter 12. Time-series regression: Start with the CAPM: single-factor model, factor is market excess return Excess returns on asset i: CAPM relationship: expected return is related to the expected factor outcome If we use the market excess return itself for asset i: where λ is referred to as the expected return premium. It is the compensation that the investors require to be willing to take the market risk. The CAPM relationship implies that in regressions the α-term should be zero which can be tested using a time-series regression. 10

11 Time-series regression: Testing strategy: Run time-series regression for each asset: Estimate the factor risk premium by computing the mean: Use t-statistic to test whether the α-term is zero for each regression. OR: Test whether the α-term of all regression are jointly zero. The resulting χ 2 - test statistic is more complex and takes the covariance structure of the regression errors (Σ) into account. The F-test statistic is given by Gibbons, Ross and Shanken (1989), GRS: 11

12 Time-series regression cont.: K factor model: Regression equation (using vector notation): Asset pricing model: Again, we test whether the α-terms are zero (i.e. zero pricing error). Test statistic: where 12

13 Cross-sectional regression: K factor model: We now turn to the question why average returns vary across assets. The model states that expected return of asset i should be higher if it exhibits a larger exposure to a factor and its corresponding risk premium. Two-pass regression: (general setup) Stage 1: Find estimates of the betas using a time-series regression: Stage 2: Estimate factor risk premia λ in a cross sectional regression (see scatterplot) Again, the pricing errors (α) should be zero. Please note that the model does not impose any restrictions on the intercept a. 13

14 Cross-sectional regression cont. (OLS regression): The estimates of the second stage: Further: The test, whether all pricing errors are zero, is given by: The test statistic is refers to a single factor. The Chi-squared statistic would imply N-K degrees of freedom in the case of a K factor model. Issue: We have not accounted for the fact that the beta terms are estimated. 14

15 Cross-sectional regression cont. (OLS regression): We have to take into account that the beta terms are estimated and that the residuals in the cross-sectional regression are correlated, which calls for a GLS regression Under the assumption that the errors are i.i.d over time and independent of the factors the results are then given by Then the test statistic is given by 15

16 Time-series vs. cross-sectional regressions: Recall the time-series regression Cross-sectional regression: Recall that we want to test whether the model prices the returns. The pricing model gives us testable restrictions. Asset pricing (AP) model: AP models that are based on a linear factor structure imply that the average premium of an asset is described by a weighted average of risk premia on global factors and by nothing else (strict interpretation). and by nothing else marks the difference between the factor model and the asset pricing implications. It can be tested whenever we have the risk premium of the asset on the LHS and the risk premia of the factors on the RHS. To test the AP model in the case of the time-series regression, we need to have returns on the RHS. In the case of the cross-sectional regression, we can use any kind of factor (e.g. macroeconomic factors) in the 1st equation because the AP implications are tested in the 2nd equation. 16

17 Approach by Fama and MacBeth (1973): The authors propose an alternative to running cross-sectional regressions. Step1: Find beta estimates using a time series regression They use a rolling window of 5 years (alternative specifications like using the full sample possible) Step 2: Run a cross-sectional regression for each period Estimates for α i and λ are given by with the sampling errors 17

18 Approach by Fama and MacBeth (1973) cont.: Estimates in matrix notation (K factor model) Statistic for testing whether the pricing errors are jointly zero Recall: Using individual stocks leads to unreliable results due to the immense amount of dispersion. Betas are measured with an significant measurement error. Therefore it is common nowadays to use portfolios rather single stocks to test the model. This way idiosyncratic risks are reduced. Portfolio betas are more stable over time and therefore its estimates are more accurate. The next question is how to compose the portfolios. Early studies grouped stocks by their beta in an initial period and then examined the behavior in the subsequent period. Nowadays portfolio are grouped by characteristics like size, book-tomarket, industries etc. In the view of Cochrane, these grouping based on these characteristics stems also from the fact that this grouping is a more natural behavior of investors than sorting stocks by, say, their name. 18

19 Standard framework for testing AP models: (Cochrane p. 437) Reference: Fama, E. F., and MacBeth, J. D. (1973) Risk, Return and Equilibrium: Empirical Tests, Journal of Political Economy, 71,

20 Topics of class 2 1. Brief look at the Arbitrage Pricing Model 2. Regression-based tests of linear factor models Time-series regressions Cross-sectional regressions Fama-MacBeth Procedure 3. Empirical evidence Fama and French (1996) 20

21 Fama and French (1996): Fama French three-factor model (FF3FM) Summary of the 6 preceding papers on the FF3FM Expected returns: Factors on the RHS are returns: 1. excess return on the broad market portfolios 2. SMB (small minus big) difference between the return of small stocks minus the return on a portfolio of large stocks 3. HML (high minus low) difference between the return on a portfolio of high-book-to-market stocks and the return on a portfolio of low-book-to-market stocks Time-series regression: Model is tested using the F-test statistic by Gibbons, Ross and Shanken (1989) 21

22 Fama and French (1996) cont. Data: Monthly data on stocks of the NYSE, AMEX and Nasdaq Rf: one-month Treasury bill rate observed at the beginning of the month Construction of the factors: At the end of June of each year: stocks are sorted by its market equity/capitalization (ME) and split into two halves: small (S) and big (B) Stocks are independently sorted by its book-to-market equity (BE/ME) into 3 quantiles: low (L=bottom 30%), middle (M=middle 40%) and high (H=top 30%) Within each group of portfolios, factor returns are given by value-weighted monthly returns on the portfolios (rebalancing in every June) Construction of the 25 portfolios of interest: Sorting analogous to the factors using quintiles as the breaking point, twodimensional sort results into 25 portfolios 22

23 Recall: Regression-based tests of linear factor models cont. Fama and French (1996) cont. Average premium tends to be higher for stocks with high book-to-market value (BE/ME) and small stocks Low B/M ( expensive ) High B/M (value, cheap ) Small cap Large cap Most simple implementation in a portfolio: overweight stocks with high value and small size 23

24 Fama and French (1996) cont. Result: intercept near zero (on average 0.093), GRS89 rejects the null hypothesis that the models explains the data at a a level of

25 Fama and French (1996) cont. Average R-squared of

26 Fama and French (1996) cont. Re-do analysis with portfolios built on momentum (ranging from 1 to 5 years), we observe short-term momentum and long-term reversal in the portfolios 26

27 Fama and French (1996) cont. 27

28 Fama and French (1996) cont. The p-value of the GRS test-statistic increases to for the return of the past 13 to 60 months which documents the failure of the FF3FM to capture the long-term reversal effect 28

29 Fama and French (1996) cont. The FF3FM does captures the cross-sectional variation of portfolio sorted by size and by the book-to-market ratio. This effect is not captured by the CAPM. However, the FF3FM fails to capture the momentum effects. Fama and French (1995) show that book-to-market ratio and the slope on the HML are proxies for relative distress i.e. weak companies with low earning tend to have high BE/ME and large loading on the HML factor Fama and French (1993) show that the model also explains well portfolio sorted by BE/ME alone Fama and French (1997) show that the model also explains well industry returns, i.e. portfolios sorted by earnings yield (E/P), cash flow yield (C/P) and sales growth Stocks with low earnings yield, low cash flow yield and high sales growth are typically strong firms which load negatively on HML 29

30 References: Fama, E. F., and K. R. French (1993) Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, 33, Fama, E. F., and K. R. French (1995) Size and Book-to-Market Factors in Earnings and Returns, Journal of Finance, 50, Fama, E. F., and K. R. French (1996) Multifactor Explanations of Asset Pricing Anomalies, Journal of Finance, 51, Fama, E. F., and K. R. French (1997) Industry Cost of Equity, Journal of Financial Economics, 43, Gibbons, M., Ross, S. A., and J. Shanken (1989) A Test of the Efficiency of a Given Portfolio, Econometrica, 57,

31 Sneak preview: Information Ratio Next week, Marco may use the term information ratio, we is briefly described here. A portfolio is commonly defined by the vector the relative weights of all assets in the portfolio. In practice, a portfolio manager is often asked to outperform a given benchmark portfolio. For example, the MSCI World in US-dollars is one commonly used commercial benchmark among equity managers. The composition (i.e. the portfolio weights) is determined by the criteria including the market capitalization. So-called active portfolio managers are given the freedom to chose portfolio weights that may differ significantly from the weights of the benchmark. The chosen weights minus the benchmark weights are sometimes called the active weights. The residual return is the portfolio return, which is not described by the benchmark. The expected value of the residual return is called alpha. In the mean-variance framework, risk is defined in terms of the standard deviation. Analogously, the active risk is defined as the standard deviation of the residual return. The information ratio (IR) of any portfolio P is defined as the annualized residual return divided by the annualized residual risk (assuming ω>0): Rules of thumb: IR=0.5 is good, IR=0.75 is very good, IR=1 is exceptional Please note the similarity to the Sharpe ratio SR. SR relates absolute risk to absolute returns and IR relates residual risk to residual return. 31

32 Conclusion 1. Review: Factor models 2. Regression-based tests of linear factor models Time-series regressions Cross-sectional regressions Fama-MacBeth Procedure 3. Empirical evidence: Fama and French (1996) Fama and French 3 factor model (FF3FM) FF3FM explains variations in BE/ME and the size effect, but fails to capture the momentum effect. 32

33 Disclaimer Disclaimer: This document is distributed for educational purposes only and should not be considered investment advice or an offer of any security for sale. The views expressed in this document are those of the author and do not necessarily represent those of current and past employers as well as affiliated companies. The views do not represent a recommendation of any particular security, strategy or investment product. 33