Extra Slides on Asset Pricing for Day 2

8/3/2019 Extra Slides on Asset Pricing for Day 2

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Predicting Betas: Correcting for Mean Reversion

In order to correct for mean reversion Because the average beta is 1

We calculate the following:

2008 Patrick J. Kelly 97

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2 += taHistoricBeBetaAdjusted

Estimating beta for illiquid stocks

Assume that the stock is not traded in dates t 1, t2, What would be the prices in the non-traded days?

Same as in the last trading day, thus zero return Large return in the first trading day after the break

Predicted by the market model: Rt =*RM,t In the first trading day after the break, the stocks return will accumulate all

idiosyncratic noise over the non-traded period

Betas will be biased! Dimson (1979): regression including lead and lag values of

the market index

Rj,t =j +l=-l1:l2j,lRM,t+l +j,tTrue beta is a sum of all lead-lag betas: l=-l1:l2j,l

Estimating beta for illiquid stocks

The trade-to-trade approach: Compute stock returns from the last traded day to the next traded

day (if necessary - over 2, 3 or nt days)

Compute market returns for the same periods Run a regression with matched multi-period returns:

Rj,nt =j,nt +jRM,nt +t=0:nt-1j,t How to control for heteroscedasticity?

WLS: the variances are proportional to nt

OLS regression with data divided bynt(Rj,nt / nt) =j*nt +j(RM,nt /nt) + (t=0:nt-1j,t /nt)

What if the stock has a large weight in the index?

Endogeneity problem In the extreme, when the index is dominated by one stock, this

stock will have beta of 1 by construction

In Russia, this is a problem for Gazprom, Lukoil, How to solve it?

Usual way: exclude this stock from the index

Theoretical solution: use IV approach with other blue chips (orindustry indices) as instruments

The evidence

CAPM


CAPM predictions

should be zero. If not, there may be missing factors.

is the only relevant factor Relation between and returns is linear Over long periods the return on the market is greater than

the risk free return

In general, riskier stocks should earn higher return on average Market portfolio is mean-variance efficient If you cross-sectionally regress on risk premia, estimates

should equal the average market risk premium ()



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Related findings- Fama French 1992 - Size and Beta


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Testing CAPM

Approach 1: time series regressionWe observe the series of returns of the asset and of the market

index

We estimate beta (in-sample) and check whether other variables(const, D/P, P/E, B/M, ) explain remaining time variation in the

assets return

Approach 2: cross-sectional regressionWe check whether observed betas are (linearly) related to stock

returns (as in the Security Market Line)

Time series test

Run time series regression for N assets: Ri,t-RF =i +i(RM,t-RF) +i,t (+iXi,t-1)

Usually, N portfolios Grouped by industry / size / To reduce idiosyncratic variation present in individual stocks

H0:i=0 (i=0) for any i=1,,NThere are no systematic deviations from the return predicted by the

CAPM Assuming that Ri,t ~ IID Normal, estimate by OLS

F-test for the joint significance of and coefficients

Results

Early tests, up to 80s Did not reject CAPM But: betas are unstable over time

Gibbons, Ross, and Shanken (1989) Data: US, 1926-1982, monthly returns of 11 industry portfolios and

VW-CRSP market index For each individual portfolio, standard CAPM is not rejected But the joint test rejects CAPM

CLM (1997, Table 5.3) Data: US, 1965-1994, monthly returns of 10 size portfolios, VW-

CRSP market indexJoint test rejects CAPM, esp. in the earlier part of the sample

period

Fama-French (1993)

Run CAPM regressions using 342 months of data for 25 sizeand book-to-market portfolios

Strongly reject CAPM. F=1.91 with p-value 0.004 2010 Patrick J. Kelly 107

Intuition: Tests of Mean Variance Efficiency

Cochrane (2001, Chapters 2 and 5) shows that any expectedreturn can be related to any mean-variance efficient portfoliolying on the efficient frontier:

This structure suggests a natural test for efficiency:

Test whether is non-zero.This is the basic intuition of most tests of portfolio efficiency

Differences across tests are mostly econometric refinements 2011 Patrick J. Kelly 108

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Test of Mean Variance Efficiency

Gibbons, Ross, Shenken (1989)

Wherefis a return based factor or portfolio return on the mean-variance efficient frontier, ET(f) is the sample mean of the factor,

and(f) the sample standard deviation, Tis the number of

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Cross-sectional tests

Main idea: Ri,t =0 +1i +i,t (+2Xi,t)

H0: asset returns lie on the SMLThe intercept is the risk-free rate: 0 = RFThe slope is the market risk premium:1 = mean(RM-RF) > 0There are no additional effects: 2 = 0

We dont know true betas!We first need to estimate them measurement error Usually, use predicted betas (estimated over thepreviousperiod)

The realized returns may be very different from the expectedreturns (in SML)We need to estimate this cross-sectional regression over many

periods (to measure averageeffect)

Two-stage procedure

Recursive procedure: for each t(e.g., every month), repeat Time-seriesregression over the previous K periods to estimate beta

Ri,t-k= i + iRM,t-k+ i,t-k, for k=1:K Cross-sectional(CS) regression

Ri,t-RF = 0 + 1 i + i,t (+ 2 2i + 3 i)

Fama-MacBeth approach (first in 1973) Running CS regression for each month t, we get the time series of coefficients0,t,1,t,

Compute mean and st. deviation ofs from these time series: No need for s.e. of coefficients in the cross-sectional regressions! Shankens correction for the estimation error in betas

Assuming normal IID returns, t-statistic = Tavg/ ()

Results

Fama-MacBeth (1973): estimate CS regression Ri,t-RF =0 +1i +22i +32,i +i,t

1 > 0: positive effect of beta2 = 0: no non-linear effects of beta3 = 0: no impact of idiosyncratic risk

Cochrane (2001) Sort all NYSE stocks into 10 size deciles, add government and

corporate bonds

Run separate time-series regressions to estimate 12 portfolio betas Regress sample average portfolio returns against estimated betas Compare fitted SML with the one predicted by the CAPM

CAPM does well in explaining stock vs. bond returns, but poorly inexplaining large vs. small-cap stocks Cross-Sectional Approach: Fama-McBeth (1973)

Tests if high beta is associated with high returns and viceversa.

General format:Time-series regressions to get betas for test portfolios

Usually beta sorted porttflios Monthly cross-sectional regressions to test if high beta is associated

with high return

Cochrane (1999) shows that GMM panel regressions areidentical under some assumptions



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General FM73 Algorithm

Use monthly data from years 1 through 7 to estimate CAPM Beta1-7 for eachcompany.

Use to rank stocks (rank1-7) into 20 equally sized groups (call portfolios formed onrank1-7 portfolio1-7)

Use monthly data from years 8 through 12 to estimate CAPM Beta8-12 (= ) for eachcompany.

Find the average Beta8-12 and average (equally weighted) return for each Portfolio1-7in each month of year 13 (i.e. letting delisted companies to drop out each month).

Roll one year forward and repeat steps 1-4 till done. Example in step 1 use data from years 2 through 8

For each month run a cross-sectional regression using the average Betas and returnscalculated in step 4.

Collect the time series of the coefficients (s)


Example from FF92

They look at whether CAPM works. It doesnt.

Here is what they do:The sort stocks by size and by the stocks pre-ranking beta.

Pre-ranking betas: betas are calculated for each stock over the 5 years ofmonthly data preceding (not including) the current year (from t-5 to t-1).

Stocks are sorted by size and then each size portfolio is sorted by the pre-ranking beta.

For each month in year t, calculate equally weighted portfolio returns For each size/pre-ranking-beta portfolio calculate Beta for the entire sample,

including one lag of the value weighted market to correct for non-synchronous

trading.

Assign the betas to each stock i n the portfolio and run monthly cross-sectionalregressions.


Average Slopes (t-Statistics) from Month-by-Month Regressions of Stock Returns on ,B,Size,Book-to-Market Equity, Leverage, and E/P: July 1963 to December 1990


A word of caution about tests: Rolls Critique (1977)

If a portfolio against which returns are measured is ex-postefficient, then no security will have abnormal returns

If the portfolio is inefficient then any abnormal return ispossible

If alpha is non-zero is it because the market portfolio is inefficientor is it because there is mispricing?

If returns are linear in Beta, all that proves is that the marketis ex-post efficient

If they are not linear in beta you do not know if it is because themarket is

ex-post inefficient or because there is a missing factor


Rolls Critique (1977) continued

Only true test if the true market portfolio is ex-post efficient True market portfolios is unobservable. So, good luck.

This critique implies Time-series tests of efficiency are the better tests to show a model

might work

Cross-sectional tests are better to show that the model might not work


The evidence

ICAPM and APT



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What are the factors

Neither APT nor ICAPM are too clear about possiblefactors

ICAPM: factors forecast changes in the investment opportunity setAPT: factors are tradable portfolios that covary with undiversifiable

sources of risk.


Chen, Roll, Ross (1986): Testing APT or is it ICAPM?

In the US:

Unexpected Inflation Term structure (Long government bonds short) Default premium (High risk corporate bonds long

government bonds)

Growth in Industrial production Consumption Growth Oil prices


APT only test: Conner and Korajczyk (1993)

Factor Analysis of stock returns Find evidence of 1 to as many as 6 factors

More recent work provides evidence of both market-wideand exchange-specific factors [Goyal, Perignon, and Villa(2008)]

In US 2 factors common to all stock, plus unique (separate) factorsof NASDAQ and NYSE.


Data Driven Risk Factors

Research has identified portfolios of stock that appear to getextra high returns:

Small firms out perform large firms at least they did 1926 through 1980

Value stocks out perform Glamour/Growth stocks Fama and French developed a three factor model that

includes:

Market Size factor (SMB)Value/Growth factor (HML)


Size and Book to Market


446 The Journal of FinanceTable V

Average Monthly Returns on Portfolios Formed on Size andBook-to-Market Equity; Stocks Sorted by ME (Down) and thenBE/ME (Across): July 1963 to December 1990In June of each year t, the NYSE, AMEX, and NASDAQ stocks that meet the CRSP-COMPUSTAT data requirements are allocated to 10 size portfolios using the NYSE size (ME)breakpoints. The NYSE, AMEX, and NASDAQ stocks in each size decile are then sortedinto 10 BE/ME portfolios using the book-to-market ratios for year t - 1. BE/ME is the bookvalue of common equity plus balance-sheet deferred taxes for fiscal year t - 1, over marketequity for December of year t - 1. The equal-weighted monthly portfolio returns are thencalculated for July of year t to June of year t + 1.

Average monthly return is the time-series average of the monthly equal-weighted portfolioreturns (in percent).

The All column shows average returns for equal-weighted size decile portfolios. The All rowshows average returns for equal-weighted portfolios of the stocks in each BE/ME group.

Book-to-Market PortfoliosAll Low 2 3 4 5 6 7 8 9 High

All 1.23 0.64 0.98 1.06 1.17 1.24 1.26 1.39 1.40 1.50 1.63Small-ME 1.47 0.70 1.14 1.20 1.43 1.56 1.51 1.70 1.71 1.82 1.92ME-2 1.22 0.43 1.05 0.96 1.19 1.33 1.19 1.58 1.28 1.43 1.79ME-3 1.22 0.56 0.88 1.23 0.95 1.36 1.30 1.30 1.40 1.54 1.60ME-4 1.19 0.39 0.72 1.06 1.36 1.13 1.21 1.34 1.59 1.51 1.47ME-5 1.24 0.88 0.65 1.08 1.47 1.13 1.43 1.44 1.26 1.52 1.49ME-6 1.15 0.70 0.98 1.14 1.23 0.94 1.27 1.19 1.19 1.24 1.50ME-7 1.07 0.95 1.00 0.99 0.83 0.99 1.13 0.99 1.16 1.10 1.47ME-8 1.08 0.66 1.13 0.91 0.95 0.99 1.01 1.15 1.05 1.29 1.55ME-9 0.95 0.44 0.89 0.92 1.00 1.05 0.93 0.82 1.11 1.04 1.22Large-ME 0.89 0.93 0.88 0.84 0.71 0.79 0.83 0.81 0.96 0.97 1.18

controlling for size, book-to-market equity captures strong variation in aver-age returns, and controlling for book-to-market equity leaves a size effect inaverage returns.B. The Interaction between Size and Book-to-Market Equity

The average of the monthly correlations between the cross-sections ofln(ME) and ln(BE/ME) for individual stocks is - 0.26. The negative correla-tion is also apparent in the average values of ln(ME) and ln(BE/ME) for theportfolios sorted on ME or BE/ME in Tables II and IV. Thus, firms with lowmarket equity are more likely to have poor prospects, resulting in low stockprices and high book-to-market equity. Conversely, large stocks are morelikely to be firms with stronger prospects, higher stock prices, lower book-to-market equity, and lower average stock returns.

The correlation between size and book-to-market equity affects the regres-sions in Table III. Including ln(BE/ME) moves the average slope on ln(ME)from -0.15 (t = -2.58) in the univariate regressions to -0.11 (t = -1.99)in the bivariate regressions. Similarly, including ln(ME) in the regressions

Fama-French Three Factor Model

Market Risk Premium SMB is a portfolio long Small stocks and short large or Big

stocks and is called SMB for Small Minus Big

HML is a portfolio long Value stocks and short Growthstocks

(value stocks are called High book to market stocks, by financeprofessor types and Growth stocks are called Low book tomarket stocks. So, HML is High Minus Low.


( ) HMLSMBrrErrEiHMLiSMBtMiMti ,,,

][][ ++=


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Do the Fama-French Factors Work in the US?Table Ya

lntrrceprs from excess stuck return regressions for 25 stuck purtfulios formed on size and book-to-market equity: July 1963 IO December IYYI,342 months.

Book-to-market equity (HE/ME) quintiles

Sizequinlile

Smal234Uig

Low 2

0.31 0.620.35 0.630.34 0.580.4 1 0.270.34 0.30

t, I(43 4 High Low 2 3 4 High~~

(i) K(r) - W(r) = (I + OI~ERM(I) + dDEF(f) + t$r)0.71 0.80 0.92 0.75 1.73 2.20 2.61 2.870.77 0.75 0.93 0.93 I.91 2.60 2.85 3.030.60 0.73 0.89 I.00 I.9Y 2.28 3.01 3.1 I0.4) 0.69 0.96 1.34 1.01 1.96 2.X8 3.350.25 0.50 0.53 1.35 1.27 1.17 2.36 2.14

(ii) R(r) - RF(r) = u + ~[RM(I) - RF(r)] + ($0sm; - 0.22 0.15 0.30 0.42 0.54 - O.YO 0.73 1.54 2.19 2.532 - 0. IX 0.17 0.36 0.3) 0.53 - 1.00 I.05 2.35 2.7Y 3.013 - 0. I6 0.1s 0.23 0.39 0.50 - 1.12 1.25 1.n2 3.20 3.194 - 0.05 - 0.14 0.12 0.35 0.57 - 0.50 - 1.50 1.20 2.91 3.71Big - 0.04 - 0.07 - 0.07 0.20 0.21 - 0.4Y - 0.95 - 0.70 1.89 1.41


S I l l ; L l l 0.242 0.523 0.524 0.69Big 0.76

Smnll - 0.342 -0.113 - 0.1 I4 0.09Big 0.21

(iii) X(r) - RF(I) = u + sSMtqr) + hlfML&) + t(r)0.46 0.49 0.53 0.55 0.97 I.92 2.240.58 0.64 0.58 0.64 2.00 2.40 2.760.61 0.52 0.60 0.66 2.00 2.58 2.250.39 0.50 0.62 0.79 2.78 1.55 2.070.52 0.43 0.51 0.44 3.41 2.23 1.84

(iv) K(r) - W(r) = 0 + h[RM(r) - RF(r)] + sSM&I) + h/IML(f) + V(I)- 0.12 - 0.05 0.01 0.00 - 3.16 - 1.47 - 0.73- 0.01 0.08 0.03 0.02 - 1.24 - 0.20 1.04

0.04 - 0.04 0.05 0.05 - 1.42 0.47 - 0.47- 0.22 - 0.08 0.03 0.13 I 07 - 2.65 - 0.99- 0.05 -0.13 - 0.05 - 0.16 3.27 - 0.67 - 1.46

Sm;ill - 0.352 - 0.1 I3 - 0.124 0.0XBig 0.21~~_.___

See footnote under table 9c.

(v) H(l) - RF(f) = u + h[RM(r) - RF(r)] + sSMB(r) + hHML(r) + ntTERM(r) + dDEF(r) + &)- 0. I3 - 0.05 0.0 I 0.00 - 3.24 - 1.58 - 0.79- 0.02 0.0x 0.04 0.02 ~ 1.29 - 0.24 1.10

0.04 - 0.03 0.06 0.05 - 1.45 0.48 - 0.42- 0.22 - 0.08 0.04 0. I 1.04 - 2.67 - 0.94- 0.05 - 0.13 - 0.06 - 0.17 3.29 - 0.72 - 1.46_ ~~~ ~~~~__.~~~.~.

2.52 2.492.61 2.562.66 2.612.51 2.852.20 1.70

0.22 0.140.51 0.340.7 I 0.560.33 I 240.69 - I.41

0.20 0.090.67 0.290.79 0.560.47 1.230.73 - I.51

Though, the GRS Test Marginally rejects.Ooops!

Example with the Fama-French 3 Factor Model


Factor data are from: http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Extra Slides on Asset Pricing for Day 2

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