IM3 Asset Pricing Models

Warwick Business School

INVESTMENT MANAGEMENT(IB357)Week 3: Asset Pricing Models

Vikas Raman


Outline Reviewing the CAPM Using the CAPM

estimating beta and the market risk premium Testing the CAPM

Roll’s critique Results

Factor models the market factor model APT Fama and French Three-Factor Model


CAPM If everyone should be holding the market (plus cash, or borrowing), then

no one holds or wants to hold idiosyncratic risk it follows that you only get rewarded for holding market risk but not for

holding idiosyncratic risk: you get rf interest for lending money without risk you get E[rM] for holding the market which means you get a premium of E[rM] - rf for taking the risk of the market which means you should get a premium of b(E[rM] - rf) for taking the market risk

of a share with a beta of b which means you should expect a return of

E[rY] = rf + b(E[rM] - rf) for holding share Y


The CAPM Suppose you were a CAPM-style investor holding the world wealth

portfolio, and someone offered you another stock to invest in.

What rate of return would you demand to hold this stock?

The answer before the CAPM might have depended upon the standard deviation of a stock's returns. After the CAPM, it is clear that you care about the effect of this stock on the TANGENCY portfolio.

The lower the average correlation a stock has with the rest of the assets in the portfolio, the lower its expected return!


A Revision Question Suppose that you are holding a portfolio M and somebody

offers you a share i, under what conditions would i help improve your portfolio? we are looking for a precise answer: ALPHA

Precise Answer: computeαi = E[Ri] – {RF + βi(E[RM] - RF)}

if αi > 0, you can improve your portfolio by buying Y if αi < 0, you can improve your portfolio by short-selling Y if αi = 0, you should neither buy nor short Y


Jensen’s Alpha On the SML:

βi

E[Ri]

RF

M

0

E[RM]

1

E[RM]-RF

E[RB]

E[RA]

βA βB

A

B

C

A is below the SML. It has a negative alpha.

Which one would you buy and which one would you short?

αA < 0

αB > 0 B is above the SML. It has a positive alpha.


Jensen’s Alpha Definition: αi = E[Ri] – {RF + βi(E[RM] - RF)}

According to CAPM, equilibrium asset prices should make Jensen’s alpha equal to zero for any asset.

An asset with αi > 0 offers expected return higher than the equilibrium. This asset is an attractive investment and therefore it must be underpriced.

An asset with αi < 0 offers expected return lower than the equilibrium. This asset is not an attractive investment and therefore it must be overpriced.

Hence, Jensen’s alpha is an important performance measure.


Implications If the portfolio M is mean-variance efficient, then by definition it cannot be improved by

adding or subtracting any share to it so for any share Y, aY = 0 , so:

Note 1: this is not some prediction or some theory that depends on efficient markets, rational investors, absence of taxes etc. It is a simple mathematical consequence of M being mean-variance efficient

Note 2: it holds whether the means, correlations and standard deviations are: market consensus about the future, my own subjective beliefs about the future based on my prejudices and private

information the historic (realised) behaviour of pricesthough of course the portfolio M will differ in each case as will the interpretation of mY


Usage of CAPM ( common sense) Valuation

the value of a company is the value of the cash flow it generates forecast the cash flows of the company estimate the beta of the cash flows use the CAPM to get the discount rate: ( Peter Corvi)

Portfolio Construction identify securities which are mispriced – off the SML use portfolio optimiser to construct portfolio

Performance Measurement see next week

Need to estimate beta and market risk premium

b Market Risk PremiumFDR r


Market Risk Premium – estimates from fundamentals Dividend growth models

value of share is present value of future dividends if dividend share of GDP bounded, long run dividend growth limited by growth

in economyMRP = return on equities – interest rate

= dividend yield + dividend growth – interest rate○ dividend yield say 4%○ dividend growth = GDP growth say 2%/year real○ real interest rate say 2%

gives MRP of say 4% Assumes that return to holders of shares comes solely and entirely from

dividends, but if return comes from capital repayments (eg cash takeovers, share buybacks) or if substantial new share issues, argument does not work


Market Risk Premium – estimates from fundamentals Implied MRP

Value of equity = Expected Earnings Next Period/Required Return on Equity MRP = Required Return on Equity – Interest Rate

Example:If S&P 500 Index is at 900 and the expected earnings next period (value weighted average of all companies that constitute the index) is 9000, then required rate of return on the US market (proxied by S&P 500) is 10%.If the risk-free rate is 4%, then MRP = 10% - 6% = 4%


Market Risk Premium – estimates from fundamentals From economics of risk aversion The market risk premium exists to compensate people for the risk of the

market so the more risk there is in the economy, and the more people dislike risk, the

higher the equity risk premium should be per unit of equity risk risk aversion is measured by the risk aversion coefficient; 0 is indifference to risk,

and the higher the number the more risk averse people are experiments suggest that typical risk aversion is about 3 – implies that someone

with wealth of 100 would be indifferent if offered a gambel which could equally well send them to 90 or 114 (see next slide)

looking at consumption per head suggests that people do not face much risk in aggregate – consumption growth varies by about 1% per annum

Mehra and Prescott (1985) show in a simple model that the equity risk premium should equal market volatility (say 16%) x risk aversion coefficient (3) x volatility of consumption (1%) = 0.5% per annum – the Equity Premium Puzzle


Risk Aversion (NFE) Economists assume people

maximise the expected utility of wealth, where utility increases with wealth, but at a decreasing rate

A standard utility function is power utility, U(W) = W1-g /(1-g) where g is the risk aversion coefficient

With g = 3, U(90) = -90-2/2= -6.2E-5; U(100) = -5.0E-5; U(114) = -3.8E-5

so 50% 90 + 50% 114 has same utility as 100

60 80 100 120 140

Utilit

y

Wealth

Power Utility


Source: Barclays Equity Gilt Study 2004(figures for UK).

Some Data


The Market Risk Premium - Empirical Issues with historical premium.:

need very long run (standard error is M /T; with market volatility of say 16%, need 250 years to be 95% confident of being 2%/yr);

quality of data, tax issues time-varying? (development of much more sophisticated financial

markets, changing levels of risk); survivorship bias

○ tend to focus on US○ what about markets that disappeared?○ what about disappeared asset classes?○ is current state of world as good as might have been expected at the

outset?


Estimating Beta Estimate historical beta using regression:

regress ri,t-rF,t on rM,t-rF,t (but in practice can do with returns rather than excess returns, unless interest rates are very volatile)

use shorter return intervals to get more data, but problems with thin trading, bid-ask bounce, stale prices if too frequent

use long data run to get more data, but problems with stability if too long For individual share, may use monthly returns over 60 months, or weekly

returns over 1 year but assumes reasonably liquid market

For portfolio, can use much more frequent data, shorter runs because idiosyncratic risk of portfolio much smaller than of individual share


Estimating Beta in Practice Take BP shares, and

estimate beta regressing returns against FTSE estimates depend on whether

using daily or weekly returns is it really varying? estimate 5% confidence

intervals using 2 stand errors

2004 2005 2006 2007 2008 2009 20100.4

0.6

0.8

1

1.2

1.4

1.6

1.8

BP's beta against the FTSE-100based on daily returns over

250 days

Beta

(+/-

2 sd

evs)

2004 2005 2006 2007 2008 2009 20100.40.60.8

11.21.41.61.8

BP's beta against the FTSE-100based on weekly returns over

250 days

Beta

(+/-

2 sd

evs)


What the CAPM says The only empirical prediction of the CAPM is that the market

portfolio is ex ante efficient cannot construct a portfolio today that can be expected to have a higher

return, lower volatility in future than the market portfolio


Ex ante and ex post Given returns on securities over a period, can compute actual means and

standard deviations can identify portfolios that would have given maximum return for minimum risk these are the ex post efficient portfolios (ex post because the portfolio can only

be identified afterwards) heavily weighted towards stocks that happen to have done well

CAPM is about constructing portfolios that are expected to have maximum return for given risk CAPM asserts only that the market portfolio is efficient ex ante (that is, using

information you can use to construct a portfolio in advance) market portfolio is unlikely to be efficient ex post CAPM says that no method of constructing a portfolio that does not involve

foresight on average produces higher return for less risk than the market portfolio


The Roll Critique (Roll 1977) The only economic content to the CAPM is the assertion that the market

portfolio is ex ante efficient But the market portfolio is unobservable in practice – it consists of all

investor wealth To test the CAPM need a proxy – say some index

tests may support the CAPM because the proxy is efficient even if the market portfolio is not

tests may reject the CAPM because the proxy is not efficient even if the market portfolio is

The critique does not say the CAPM is wrong or meaningless but to use and test the CAPM need to commit to some proxy for the market generally use the market-weighted domestic equity index (though could

consider other possibilities)


Source: Fama and French, 2004

Early tests (Black, Jensen, Scholes, 1972, Fama and MacBeth, 1973) showed beta was priced, that idiosyncratic risk was not priced, but that the expected return on a zero beta portfolio is significantly greater that rF.

But recent evidence finds a very flat relationship between Beta and returns: relationship is not as strong as the theory suggests.

How Does the CAPM Perform Empirically?


How Does the CAPM Perform Empirically? Also, a great deal of anomalous evidence has

come out suggesting that things other than beta are important in determining expected returns:The Small Firm Effect (Keim, 1981)The Book-to-Market Effect (Stattman, 1980; Rosenberg, et

al., 1985) The Momentum Effect (Jegadeesh and Titman, 1993)



Source: Mertens (2002) (http://www.elmarmertens.com/lecturenotes/financenotes)

http://www.elmarmertens.com/lecturenotes/financenotes

http://www.elmarmertens.com/lecturenotes/financenotes



B/M Quintile

Monthly excess returns sorted by Size and B/M quintiles Source: Fama and French (1993)


Why does the CAPM not work well? Have actually made some very strong assumptions

Investors only care about mean and variance: but may worry about skew – would you rather: 200 (Pr 10%) 120 (Pr 90%)

100 or 100 ? 100 (Pr 90%) 20 (Pr 10%)(Both the gambles have the same mean and standard deviation, but differ in skewness) all investors care only about risk to wealth next period: but if I am a long term investor I

may prefer an investment that pays more if say interest rates are low next period and less if they are high

all investors are indifferent to tax: but some investors may prefer to receive return as capital gain rather than dividends for tax reasons

the stock market is the sum of wealth: but there is a lot of other wealth (real estate, bonds, human capital) and theory says that beta should be estimated on entire wealth portfolio

no transaction costs and unlimited shortselling: but stock market participation is limited, and it is not possible to short sell one’s human capital


Other Issues Investors have a common information set and interpret it

rationally: but not clear how rational agents should form expectations about future events, and significant evidence that investors have biases, bounded rationality

Tests tend to assume stability of parameters: evidence of time varying market price of risk, beta

Many models that try to relax one or other assumption: models tend to introduce other priced factors (such as dividend yield,

sensitivity to other economic variables apart from the market return) no consensus on a better model CAPM remains as a flawed standard


Equilibrium Models So far looked at equilibrium models

characterise demands of investors and find prices for securities that brings supply/demand equilibrium

but empirical results are disappointing maybe we are being too ambitious given that demands are affected by

complex factors○ beliefs, fears, superstitions○ different endowments, horizons and objectives○ existence of substitutes (housing, human capital)○ institutional frictions (herding, league tables)

Try a more limited approach: start with a simple description and then explore the impact of arbitrageurs (prices may not be right, but they should not allow easy certain profits)


Arbitrage Pricing Theory An arbitrage opportunity arises when an investor can earn

riskless profits without making a net investment. The law of one price states that if two assets are equivalent in

all economically relevant respects, then they should have the same market price.

During the arbitrage activity, investors will bid up the price where it is low and force it down where it is expensive. As a result they eliminate the arbitrage opportunities.

Security prices should satisfy a “no-arbitrage condition”.


Arbitrage Pricing Theory APT (Ross, 1976) predicts a security market line as

CAPM and shows a linear relation with expected return and risk.

According to APT:Security returns are described by a factor modelThere are sufficient securities to diversify away idiosyncratic

riskWell-functioning security markets do not allow for the

persistence of arbitrage opportunitiesThere are no taxes. There are no transaction costs.

○ Notice that there are considerably fewer than with the CAPM.


Single Factor Model A first attempt at characterising behaviour of returns

use excess returns (ie actual – risk free rate) for ease In SFM, assume return is a linear function of some factor common to all stocks,

and idiosyncratic noise: so Ri,t = ai + biFt + ei,t

bi is a constant, measures i’s sensitivity to the factor F (will assume +ve); E[ei,t] = 0, so e is noise; Cov[Ft , ei,t] = 0, so e is idiosyncratic;

With F equal to the market return, this model is called the Market Model Suppose we have enough history and the a’s and b’s are stable enough that we

(and everyone else) can estimate them accurately: how would we make up a good portfolio? (assuming we want maximum return for minimum risk)


Optimum Portfolio

Obvious strategy:the attractive shares are those that have a high a per unit of

b – buy themsell the shares that have a low a per unit of bensure that you end up with zero beta netend up with high alpha portfolio, no factor exposure, and

just idiosyncratic riskif you avoid putting too much in one share, you will have

diversified away almost all the idiosyncratic riskso you will have a staggeringly good deal!


Simple Example 1000 G shares with a = 6%, b = 1, and idio = 20% 1000 B shares with a = 5%, b = 1, and idio = 20% I go short £1 of each of the B shares, and go long £1 each of

the G shares the portfolio’s beta = 0 the expected return is 6% on £1000 – 5% on £1000 = £10 this looks a fantastic deal

APT: securities must be priced such that these arbitrage strategies are not profitable.


Arbitrage Pricing Theory The risk-premiums of well-

diversified portfolios with different betas should be proportional to their betas.

The expected return on all well-diversified portfolios must lie on the straight line from the risk-free asset.

The equation of the line will also show the expected return on all well-diversified portfolios.


Arbitrage Pricing Theory Take M, market index portfolio

as a well-diversified portfolio. Since M is well-diversified,

should be on the line and its beta is 1.

Thus, the equation of the line is:

E[rY] = rf + b(E[rM] - rf)

The CAPM is an example of a one-factor APT model


Identification of factors APT fails to offer any guidance on either the number of factors or their

identity – they have to be estimated empirically Can deduce the factors by doing a factor analysis on security returns

factor analysis finds the best groupings of securities that minimises the variance of residual returns

Roll and Ross (1980) identified at least three significant factors (using 42 portfolios of 30 US stocks, daily returns, 1962-72)

Alternatively, can pre-specify factors likely to be important. Chen, Roll and Ross (1986) suggest that four most important factors are the unanticipated changes in inflation slope of term structure (difference between long and short rates of interest) risk premium on corporate debt (difference in yield between low and high grade

debt) industrial output.


Fama and French Three-Factor Model Three – factor empirical model

(Fama and French ,1993) The factors:

Return on the broad market index (M)

Return on the portfolio long in small stocks, short in big stocks (small minus big, or SMB)

Return on the portfolio long in high BM-ratio stocks, short in low BM-ratio stocks (high minus low, or HML)

][][][][ HMLHMLSMBSMBMMI RERERERE bbb


What do these factors measures? SMB: measure of “size risk”

small companies logically should be expected to be more sensitive to many risk factors as a result of their relatively undiversified nature and their reduced ability to absorb negative financial events.

HML: measure of “distress risk”high B/M is usually an indication that their public market

value has plummeted because of hard times. would be exposed to greater risk of bankruptcy than their

more highly valued counterparts. Other interpretations?

Fama and French Three-Factor Model


APT vs. CAPM APT looks like a multi-beta CAPM, with the expected excess return being

the product of a beta showing the exposure to a risk source, and a k being the price of risk also assumes frictionless markets, with shorting, borrowing etc

But the assumptions are different: APT places no restrictions on what the factors are nor on the price of factor risk

exposure multi-beta CAPM requires that factors enter into utility functions and economics

may well suggest plausible ranges or signs for factor prices CAPM is an equilibrium theory so “uneconomic” agents will change the result;

APT makes no assumptions about other players However, tendency to give economic interpretations to factors blurs

distinction …


APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio.

APT can be extended to multifactor models. There is exactly one well-diversified portfolio in the

CAPM: the market. There can be more than one in the APT.

APT vs. CAPM


Summary Two approaches to risk-return:

equilibrium model of CAPM and derivatives factor models such as Market model and APT

Equilibrium models identify the “factors” and justify why they should attract a premium – gives more confidence that premium will persist

Factor models unconstrained in choice of factors, so can fit data more readily, but do not explain why they should attract risk premia

Both types of model imply a distinction between systematic and idiosyncratic risk, with the latter not being rewarded

No consensus about the “right” model Key issues in applying models (beta estimation, estimation of risk premia) common

to all Reading: BKM Chs 9-11

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IM3 Asset Pricing Models