Lecture 10: The Capital Assets Pricing Model: Theoretical and ...

Managerial Economics

Lecture Ten:Standard theory of finance

Recap

• Minsky’s “Financial Instability Hypothesis”– Finance-driven cyclical behavior

“Modern” Finance?

• “Modern Finance Theory” has many components:– Sharpe’s “Capital Asset Pricing Model” (CAPM)– Modigliani-Miller’s “Dividend Irrelevance Theorem”– Markowitz’s risk-averse portfolio optimisation

model– Arbitrage Pricing Theory (APT)

• “Modern” as compared to pre-1950 theories that emphasised behaviour of investors as explanation of stock prices, value investing, etc.– Initially appeared to explain what “old finance”

could not– But 50 years on…

• Foundation is Sharpe’s CAPM:

The Capital Assets Pricing Model

• How to “predict the behaviour of capital markets”?• Extend theories of investment under certainty...

– to investment under conditions of risk• Based on neoclassical utility theory• Investor maximises utility subject to constraints• Utility:

– Positive function of expected return ER

– Negative function of risk (standard deviation) R

• Constraints are available spectrum of investment opportunities


Increasing utility:Higher expected

returns & lower risk

Indifferencecurves

Investment opportunities

“Efficient”opportunitieson the edgeZ inferior

to C (lower ER) and B (higher sR)

Optimalcombination

for thisinvestor

Border (AFBDCX) is Investment Opportunity Curve (IOC)


• IOC reflects correlation of separate investments. Consider 3 investments A, B, C:– A contains investment A only

• Expected return is ERa,

• Risk is sRa

– B contains investment B only• Expected return is ERb,

• Risk is sRb

– C some combination of a of A & (1-a) of B• ERc=aERa + (1-a)ERb

Rc Ra Rb ab Ra Rbr 2 2 2 21 2 1. ( ) . . . . .

Variance due to A

Perceived correlation of A

with B(varies between -

1 & +1)


• If rab=1, C lies on straight line between A & B:

Rc Ra Rb ab Ra Rbr 2 2 2 21 2 1. ( ) . . . . .

This is 1



Rc Ra Rb ab Ra Rb

Ra Rb Ra Rb

r

2 2 2 2

2 2 2 2

1 2 1

1 2 1

. ( ) . . . . .

. ( ) . . . .

This can be factored



Rc Ra Rb ab Ra Rb

Ra Rb Ra Rb

Ra Rb Ra Rb

r

2 2 2 2

2 2 2 2

2

1 2 1

1 2 1

1 1

. ( ) . . . . .

. ( ) . . . .

. ( ). . ( )

Identical to straight line relation for expected return:


EERR

RR

AA

BB

CC

WWhheenn rraabb== 11

IInnvveessttmmeennttOOppppoorrttuunniittyy

CCuurrvvee


• If rab=0, C lies on curved path between A & B:

Rc Ra Rb ab Ra Rbr 2 2 2 21 2 1. ( ) . . . . .

This is zero

Hence this is zero



Rc Ra Rb ab Ra Rb

Ra Rb

r

2 2 2 2

2 2 2 2

1 2 1

1

. ( ) . . . . .

. ( )



Rc Ra Rb ab Ra Rb

Ra Rb

Ra Rb

r

2 2 2 2

2 2 2 2

2

1 2 1

1

1

. ( ) . . . . .

. ( )

. ( )Straight line relation

Hence lower risk for diversified portfolio(if assets not perfectly correlated)


EERR

RR

AA

BB

CC

IInnvveessttmmeennttOOppppoorrttuunniittyy

CCuurrvvee

WWhheenn rraabb<< 11

Fall in sR due todiversification

when investmentsare not perfectly

correlated


• Sharpe assumes riskless asset P with ERP=pure interest rate, RP=0.

• Investor can form portfolio of P with any other combination of assets

• One asset combination will initially dominate all others:


Only asset combinationwhich can efficiently

be combined withriskless asset P in

a portfolio

Efficiency: maximise expected return& minimise risk given constraints


• Assume limitless borrowing/lending at riskless interest rate = return on asset P

• Investor can move to anywhere along PZ line by borrowing/lending

• Problem:– P the same for all investors (“simplifying

assumption”)– But investor perceptions of expected return, risk,

investment correlation will differ• Solution:

– assume “homogeneity of investor expectations”– utterly unrealistic assumption, as is assumption of

limitless borrowing by all borrowers at riskless interest rate. So...


• Defended by appeal to Friedman’s “Instrumentalism”:– “the proper test of a theory is not the realism of its

assumptions but the acceptability of its implications”

• Consequence of assumptions:– spectrum of available investments/IOC identical

for all investors– P same for all investors– PZ line same for all investors– Investors distribute along line by

borrowing/lending according to own risk preferences:


Highlyrisk-averse

Thrill seeker...


• Next, the (perfect) market mechanism– Price of assets in will rise– Price of assets not in will fall– Price changes shift expected returns– Causes new pattern of efficient investments

aligned with PZ line:


Capital market line

Range of efficient assetcombinationsafter market

priceadjustments:

more than just one efficient portfolio


• Theory so far applies to combinations of assets• Individual assets normally lie above capital market

line (no diversification)

• Can’t relate between ERi & i

• Can relate ERi to “systematic risk”:

• Investment i can be part of efficient combination g:– Can invest (additional) in i and (1-) in g

• =1 means invest solely in i;• =0 means some investment in i (since part of

portfolio g);• Some <0 means no investment in i;• Only =0 is “efficient”


Efficientcombination g

Single investmenti which is part of

portfolio g

Additional investmentin i is zero (a=0) here


• Slope of IOC and igg’ curve at tangency can be used to derive relation for expected return of single asset

E P r E PR ig

R

RRi

i

g

g

• This allows correlation of variation in ERi to variation in ERg (undiversifiable, or systematic, or “trade cycle” risk)

• Remaining variation is due to risk inherent in i:


Risk peculiarto asset i

Higher return for assets more stronglyaffected by trade cycle (systematic risk)


• Efficient portfolio enables investor to minimise asset specific risk

• Systematic risk (risk inherent in efficient portfolio) can’t be diversified against

• Hence market prices adjust to degree of responsiveness of investments to trade cycle:– “Assets which are unaffected by changes in

economic activity will return the pure interest rate; those which move with economic activity will promise appropriately higher expected rates of return.” [OREF II]


• Crux/basis of model: markets efficiently value investments on basis of expected returns/risk tradeoff

• Modigliani-Miller extend model to argue valuation of firms independent of debt structure (see OREF II)

• Combination: the “efficient markets hypothesis”• Focus on portfolio allocation across investments at a

point in time, rather than trend of value over time• Argues investors focus on “fundamentals”:

– Expected return– Risk

• So long as assumptions are defensible…


• “In order to derive conditions for equilibrium in the capital market we invoke two assumptions. First, we assume a common pure rate of interest, with all investors able to borrow or lend funds on equal terms. Second, we assume homogeneity of investor expectations: investors are assumed to agree on the prospects of various investments–the expected values, standard deviations and correlation coefficients described in Part II. Needless to say, these are highly restrictive and undoubtedly unrealistic assumptions.

• However, since the proper test of a theory is not the realism of its assumptions but the acceptability of its implications, and since these assumptions imply equilibrium conditions which form a major part of classical financial doctrine, it is far from clear that this formulation should be rejected–especially in view of the dearth of alternative models leading to similar results.”

• Sharpe later admits to qualms…

The CAPM: Reservations

• “People often hold passionately to beliefs that are far from universal. The seller of a share of IBM stock may be convinced that it is worth considerably less than the sales price. The buyer may be convinced that it is worth considerably more.” (Sharpe 1970)

• However, if we try to be more realistic:– “The consequence of accommodating such aspects of reality

are likely to be disastrous in terms of the usefulness of the resulting theory... The capital market line no longer exists. Instead, there is a capital market curve–linear over some ranges, perhaps, but becoming flatter as [risk] increases over other ranges. Moreover, there is no single optimal combination of risky securities; the preferred combination depends upon the investors’ preferences... The demise of the capital market line is followed immediately by that of the security market line. The theory is in a shambles.” (Sharpe 1970 emphasis added)

The CAPM: Evidence

• Sharpe’s qualms ignored & CAPM takes over economic theory of finance

• Initial evidence seemed to favour CAPM– Essential ideas:

• Price of shares accurately reflects future earnings– With some error/volatility

• Shares with higher returns more strongly correlated to economic cycle

– Higher return necessarily paired with higher volatility

• Investors simply chose risk/return trade-off that suited their preferences

– Initial research found expected (positive) relation between return and degree of volatility

– But were these results a fluke?

The CAPM: Evidence

• Volatile but superficially exponential trend– As it should be if economy growing smoothly

DJIA 1920-2001

-2000

0

2000

4000

6000

8000

10000

12000

11/25/1921 8/4/1935 4/12/1949 12/20/1962 8/28/1976 5/7/1990 1/14/2004

The CAPM: Evidence

• Sharpe’s CAPM paper published 1964• Initial CAPM empirical research on period 1950-

1960’s– Period of “financial tranquility” by Minsky’s theory

• Low debt to equity ratios, low levels of speculation– But rising as memory of Depression recedes…

• Steady growth, high employment, low inflation…• Dow Jones advance steadily from 1949-1965

– July 19 1949 DJIA cracks 175– Feb 9 1966 DJIA sits on verge of 1000 (995.15)

• 467% increase over 17 years

– Continued for 2 years after Sharpe’s paper• Then period of near stagnant stock prices

The CAPM: Evidence

• Dow Jones “treads water” from 1965-1982– Jan 27 1965: Dow Jones cracks 900 for 1st time– Jan 27 1972: DJIA still below 900! (close 899.83)

• Seven years for zero appreciation in nominal terms• Falling stock values in real terms

– Nov. 17 1972: DJIA cracks 1000 for 1st time• Then “all hell breaks loose”

– Index peaks at 1052 in Jan. ‘73– falls 45% in 23 months to low of 578 in Dec. ’74– Another 7 years of stagnation– And then “liftoff”…

The CAPM: Evidence• Fit shows average exponential growth 1915-1999:• index well above or below except for 1955-1973

Log of Dow Jones Industrial Average 1915-1999

y = 6E-05x + 1.4228

R2 = 0.9031

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

7/5/1914 4/13/1926 1/20/1938 10/29/1949 8/7/1961 5/16/1973 2/22/1985 12/1/1996 9/9/2008 6/18/2020

Date

Lo

g C

losi

ng

Val

ue

Steady above trend growth 1949-Steady above trend growth 1949-1966: Minsky’s “financial 1966: Minsky’s “financial tranquility”tranquility”

Sharpe’s paper Sharpe’s paper publishedpublished

Dec 12 1974: bottoms at Dec 12 1974: bottoms at 578578

Crash of ’73: 45% Crash of ’73: 45% fall in 23 months…fall in 23 months…

Bubble takes off in Bubble takes off in ‘82…‘82…

CAPM fit to this data looks pretty good!CAPM fit to this data looks pretty good!

Jan 11 ’73: Peaks at Jan 11 ’73: Peaks at 10521052

CAPM fit doesn’t CAPM fit doesn’t look so hot any look so hot any more…more…

An aside: what caused ’73 crash?• Many blame OPEC Oil Embargo for economic downturn

• October 17, 1973: OPEC bans export of oil to countries that supported Israel in 1973 Arab-Israeli War• Oil price rises 4-fold from US$2.50 to $10 a barrel

• BUT embargo occurred 9 months after market peak• Quarterly GDP growth fell from 10.1% in Q1 1973 to

-1.6% in Q3 ‘73 before embargo• Market already down 8% by October 17 to 963

• Rose in immediate aftermath to embargo• Real acceleration of crash occurred with Franklin

National Bank of New York crisis from May-October 1974

• May 1 854; Dec 12 578: 32% fall in 7 months• Minsky’s “financial fragility” more important• OPEC-inspired inflation may have made crisis less

severe: reduction of real debt burden

The CAPM: Evidence

• By mid-1990s, clear CAPM and reality weren’t related– “The Dow ... stood at around 3,600 in early 1994.– By 1999, it had passed 11,000, more than tripling in

five years...– However, over the same period, basic economic

indicators did not come close to tripling.– U.S. … gross domestic product rose less than 30%,

and almost half of this increase was due to inflation.– Corporate profits rose less than 60%, and that from

a temporary recession-depressed base.– Viewed in the light of these figures, the stock price

increase appears unwarranted and, certainly by historical standards, unlikely to persist…” (Schiller: 4)

The CAPM: Evidence• Schiller’s inflation adjusted S&P P:E ratio

• Very cyclical• Notice peak in

1966: Minsky’s nominated date for transition from robust to fragile financial system

• 2000 valuations highest in history

• Not “rational calculation” of CAPM but “irrational exuberance”

The CAPM: Evidence

• Price far more volatile than earnings;• Volatility should be similar if CAPM true:

• Growth in earnings “slow & steady” compared to highly volatile prices

The CAPM: Evidence

• CAPM predicts no cyclical behavior to prices (apart from that caused by earnings): “you can’t time the market”…

• Schiller finds clear cyclicality: when P:E ratio high, next ten year stock market returns low & vice versa

• 1990-2000 valuation “off the chart”!

• Negative returns for next decade highly likely…

The CAPM: Evidence

• “The relation between price-earnings ratios and subsequent returns appears to be moderately strong...

• longterm investors who can commit their money to an investment for ten full years do well when prices were low relative to earnings at the beginning of the ten years and do poorly when prices were high at the beginning of the ten years.

• Long-term investors would be well advised, individually, to stay mostly out of the market when it is high, as it is today, and get into the market when it is low.” (12)

• Critics of CAPM thus dismiss it—but even its one-time champions do too– Fama & French

The CAPM: Evidence According to Fama 1969

• Evidence supports the CAPM– “This paper reviews the theoretical and empirical

literature on the efficient markets model… We shall conclude that, with but a few exceptions, the efficient markets model stands up well.” (383)

• Assumptions unrealistic but that doesn’t matter:– “the results of tests based on this assumption

depend to some extent on its validity as well as on the efficiency of the market. But some such assumption is the unavoidable price one must pay to give the theory of efficient markets empirical content.” (384)


• CAPM good guide to market behaviour– “For the purposes of most investors the efficient

markets model seems a good first (and second) approximation to reality.” (416)

• Results conclusive– “In short, the evidence in support of the efficient

markets model is extensive, and (somewhat uniquely in economics) contradictory evidence is sparse.” (416)

• Just one anomaly admitted to– Large movements one day often followed by large

movements the next—“volatility clustering”…


• “one departure from the pure independence assumption of the random walk model has been noted … large daily price changes tend to be followed by large daily changes. The signs of the successor changes are apparently random, however, which indicates that the phenomenon represents a denial of the random walk model but not of the market efficiency hypothesis… But since the evidence indicates that the price changes on days follow ing the initial large change are random in sign, the initial large change at least represents an unbiased adjustment to the ultimate price effects of the information, and this is sufficient for the expected return efficient markets model.” (396)

• 35 years later, picture somewhat different…


• “The attraction of the CAPM is that it offers powerful and intuitively pleasing predictions about how to measure risk and the relation between expected return and risk.

• Unfortunately, the empirical record of the model is poor—poor enough to invalidate the way it is used in applications.

• The CAPM's empirical problems may reflect theoretical failings, the result of many simplifying assumptions…

• In the end, we argue that whether the model's problems reflect weaknesses in the theory or in its empirical implementation, the failure of the CAPM in empirical tests implies that most applications of the model are invalid.” (Fama & French 2004: 25)

The CAPM: Evidence According to F&F 2004

• Clearly admits assumptions dangerously unrealistic:– “The first assumption is complete agreement given

market clearing asset prices at t-1, investors agree on the joint distribution of asset returns from t-1 to t. And this distribution is the true one—that is, it is the distribution from which the returns we use to test the model are drawn. The second assumption is that there is borrowing and lending at a risk free rate, which is the same for all investors and does not depend on the amount borrowed or lent.” (26)• Bold emphasis: model assumes all investors

effectively know the future

• Assumptions, which once “didn’t matter” (see Sharpe earlier) are now crucial…


• “The assumption that short selling is unrestricted is as unrealistic as unrestricted risk-free borrowing and lending… But when there is no short selling of risky assets and no risk-free asset, the algebra of portfolio efficiency says that portfolios made up of efficient portfolios are not typically efficient. This means that the market portfolio, which is a portfolio of the efficient portfolios chosen by investors, is not typically efficient. And the CAPM relation between expected return and market beta is lost.” (32)

• Still some hope that, despite lack of realism, data might save the model…


– “The efficiency of the market portfolio is based on many unrealistic assumptions, including complete agreement and either unrestricted risk-free borrowing and lending or unrestricted short selling of risky assets. But all interesting models involve unrealistic simplifications, which is why they must be tested against data.” (32)

• Unfortunately, no such luck…– 40 years of data strongly contradict all versions of

CAPM• Returns not related to betas• Other variables (book to market ratios etc.) matter• Linear regressions on data differ strongly from risk

free rate (intercept) & beta (slope) calculations from CAPM


• Tests of the CAPM are based on three implications… – “First, expected returns on all assets are linearly

related to their betas, and no other variable has marginal explanatory power.

– Second, the beta premium is positive, meaning that the expected return on the market portfolio exceeds the expected return on assets whose returns are uncorrelated with the market return.

– Third, … assets uncorrelated with the market have expected returns equal to the risk-free interest rate, and the beta premium is the expected market return minus the risk-free rate.” (32)


• “There is a positive relation between beta and average return, but it is too "flat." … the Sharpe-Lintner model predicts that– the intercept is the risk free rate and– the coefficient on beta is the expected market

return in excess of the risk-free rate, E(RM) - R.– The regressions consistently find that the intercept

is greater than the average risk-free rate…, and the coefficient on beta is less than the average excess market return” (32)


• Average Annualized Monthly Return versus Beta for Value Weight Portfolios Formed on Prior Beta, 1928-2003

• “the predicted return on the portfolio with the lowest beta is 8.3 percent per year; the actual return is 11.1 percent. The predicted return on the portfolio with the highest beta is 16.8 percent per year; the actual is 13.7 percent.” (33)


• The hypothesis that market betas completely explain expected returns …– Starting in the late 1970s… evidence mounts that

much of the variation in expected return is unrelated to market beta…” (34)

– Fama and French (1992) update and synthesize the evidence on the empirical failures of the CAPM… they confirm that size, earnings-price, debt equity and book-to-market ratios add to the explanation of expected stock returns provided by market beta.” (36)

– Best example of failure of CAPM as guide to building investment portfolios: Book to Market (B/M) ratios provide far better guide than Beta…


• “Average returns on the B/M portfolios increase almost monotonically, from 10.1 percent per year for the lowest B/M group to an impressive 16.7 percent for the highest.

• But the positive relation between beta and average return predicted by the CAPM is notably absent… the portfolio with the lowest book-to-market ratio has the highest beta but the lowest average return.

• The estimated beta for the portfolio with the highest book-to market ratio and the highest average return is only 0.98. With an average annualized value of the riskfree interest rate, Rf, of 5.8 percent and an average annualized market premium, Rm - Rf, of 11.3 percent, the Sharpe-Lintner CAPM predicts an average return of 11.8 percent for the lowest B/M portfolio and 11.2 percent for the highest, far from the observed values, 10.1 and 16.7 percent.”


• Average Annualized Monthly Return versus Beta for Value Weight Portfolios Formed on B/M, 1963-2003

• Simple regression of data gives Simple regression of data gives oppositeopposite relationship to CAPM: relationship to CAPM: return rises as beta fallsreturn rises as beta falls!!

“Beta is dead, long live…”?

• CAPM clearly a failure– As predictor of stock market behaviour– As guide to formation of return-maximising, risk-

minimising portfolio• Early “success” possibly due to unusual “financial

tranquility” nature of post-WWII to 1966 stock market• New model of finance needed

– To explain behaviour of asset markets in general– To guide portfolio selection

• Next week: the developing Really New Finance:– The Inefficient Market Hypothesis– The Fractal Market Hypothesis– Chaos, heterogeneity and “minority games”

Lecture 10: The Capital Assets Pricing Model: Theoretical and ...

Economy & Finance

Transcript of Lecture 10: The Capital Assets Pricing Model: Theoretical and ...