CAPM Asset Pricing, William Sharpe, Fischer Black

1. Prepared By
Anuj Vijay Bhatia
F1401 (FPRM 14)
Theory of Finance
Institute of Rural Management Anand (IRMA)

2. Capital Market Equilibrium: Introduction
• CAPM is centerpiece of modern financial economics.
• CAPM gives a model of risk-return relationship.
• Capital market theory attempts to explain how investors place dollar/rupee value on
securities traded in the capital market.
• This theory extends the concept of market equilibrium in order to determine the
market price for risk and the appropriate measure of risk for a single asset.
• Based upon Markowitz (1952, 1959) Mean-Variance Portfolio, Sharpe (1964), Lintner
(1965), and Mossin (1966) have derived and developed the static Capital Asset Pricing
Model.
• During the four past decades, the CAPM has been the benchmark of asset pricing
models, and most empirical apply it to calculate asset returns and the cost of capital.
• The Capital Asset Pricing Model (CAPM) is a model to explain why capital assets are
priced the way they are.
• The CAPM was based on the supposition that all investors employ Markowitz
Portfolio Theory to find the portfolios in the efficient set. Then, based on individual
risk aversion, each of them invests in one of the portfolios in the efficient set.

3. Assumptions of CAPM
1. Investors are risk-averse individuals who maximize the expected utility of their end-of-period wealth.
2. Investors are price takers and have homogeneous expectations about asset returns that have a joint normal
distribution.
3. There exists a risk-free asset such that investors may borrow or lend unlimited amounts at the risk-free rate.
4. The quantities of assets are fixed. Also, all assets are marketable and perfectly divisible.
5. Asset markets are frictionless and information is costless and simultaneously available to all investors.
6. There are no market imperfections such as taxes, regulations, or restrictions on short selling.
Some Implications of Assumptions
• If markets are frictionless, the borrowing rate equals the lending rate, and we are able to develop a linear efficient
set called the Capital Market Line.
• If all assets are divisible and marketable, we exclude the possibility of human capital as we usually think of it. In
other words, slavery is allowed in the model. We are all able to sell (not rent for wages) various portions of our
human capital (e.g., typing ability or reading ability) to other investors at market prices.
• Another important assumption is that investors have homogeneous beliefs. They all make decisions based on an
identical opportunity set. In other words, no one can be fooled because everyone has the same information at the
same time.
• Also, since all investors maximize the expected utility of their end-of- period wealth, the model is implicitly a one-
period model.

4. Capital Asset Prices:
A Theory of Market Equilibrium Under Conditions of Risk
William F. Sharpe
The Journal of Finance
September 1964

5. CAPM: Sharpe’s Version
Sharpe (1964) in his paper ‘Capital Asset Prices: A Theory of Market
Equilibrium Under Conditions of Risk’ built upon the Modern Portfolio
Theory of Markowitz.
Objective: Pricing of assets under conditions of risk. In other words, to
construct a market equilibrium theory of asset prices under conditions of
risk.
Research Problem:
• No theory was present which could determine the price of risk
resulting from investors preferences.
• Difficult to determine price-risk relationship.
• Which risk can be eliminated by diversification and which risk
component cannot? Which is the relevant risk component for
diversification?

6. Capital Market Equilibrium: Introduction
• In equilibrium, capital asset prices have
adjusted so that the investor, if he follows
rational procedures (primarily
diversification), is able to attain any
desired point along a capital market line.
• He may obtain a higher expected rate of
return on his holdings only by incurring
additional risk.
• In effect, the market presents him with two
prices: the price of time, or the pure
interest rate (shown by the intersection of
the line with the horizontal axis) and the
price of risk, the additional expected return
per unit of risk borne (the reciprocal of the
slope of the line).

7. Optimal Investment Policy for Individual
The Investors Preference Function
• Sharpe (1964) provides the model of individual investor behavior under conditions of
risk.
• An investor, who thinks in probabilistic terms, takes action based only on two
parameters – its expected value and standard deviation.
• Total Utility Function:
• where Ew indicates expected future wealth and σw the predicted standard deviation of
the possible divergence of actual future wealth from Ew.
• Investors prefer higher wealth to lower and they are risk averse. Thus, their
Indifference Curves (ICs) are upward-sloping.
• To simplify, we take Rate of Return instead of Wealth:
• W1 is the present level of wealth and Wt is the terminal wealth.
• R is the Rate of Return.
• Expressing U in terms of R:

8. The Investors Preference Function (Cont..)
• Figure 2 summarizes the model of investor
preferences in a family of indifference curves;
successive curves indicate higher levels of
utility as one moves down and/or to the right.
Investment Opportunity Curve
• Investor maximizes utility – Point F
• Efficient Plans: (1) the same ER and a lower
σR, (2) the same σR and a higher ER or (3) a
higher ER and a lower σR.
• Thus investment Z is inefficient since
investments B, C, and D (among others)
dominate it. The only plans which would be
chosen must lie along the lower right-hand
boundary (AFBDCX)- the investment
opportunity curve.

9. Investment Opportunity Curve
• Consider two investment plans -A and B, each including one or
more assets.
• If the proportion a of the individual's wealth is placed in plan A
and the remainder (1-a) in B, the expected rate of return of the
combination will lie between the expected returns of the two
plans:
• The predicted standard deviation of return of the combination is:
• Correlation Coefficient can range from -1 to +1.
• If the two investments are perfectly correlated, the combinations
will lie along a straight line between the two points.
• If they are less than perfectly positively correlated, the standard
deviation of any combination must be less than that obtained with
perfect correlation.
• Thus the combinations must lie along a curve below the line AB.
AZB shows such a curve for the case of complete independence
(rab = 0); with negative correlation the locus is even more U-
shaped.

10. Pure Rate of Interest (Dealing with Risk-Less Assets)
• Let P be the risk-less asset, risk = σRp = 0.
• ERp = Pure Rate of Interest
• If an investor places a of his wealth in P and the remainder in some
risky asset rate of return:
• The standard deviation of such a combination would be:
• but since σRp = 0, this reduces to:
• Straight lines shows combinations of Erp and σRp
• Best Point: Where Line is Tangent to the curve.
• Investor can lend as well as borrow at pure interest rate.
• Even if borrowing is impossible, the investor will choose ∅ (and lending if
his risk-aversion leads him to a point below ∅ on the line P ∅). Since a large
number of investors choose to place some of their funds in relatively risk-
free investments, this is not an unlikely possibility. Alternatively, if
borrowing is possible but only up to some limit, the choice of ∅ would be
made by all but those investors willing to undertake considerable risk. These
alternative paths lead to the main conclusion, thus making the assumption of
borrowing or lending at the pure interest rate less onerous than it might
initially appear to be.

11. Equilibrium in the Capital Market
• Two Assumptions: (1) All investors can borrow and lend at pure
rates (2) homogeneous expectations about risk and return.
• Each investor will view his alternatives in the same manner.
• Figure 5: Set of price alternatives for individuals A, B and C.
• An investor with the preferences indicated by indifference
curves A1 through A4 would seek to lend some of his funds at
the pure interest rate and to invest the remainder in the
combination of assets shown by point ∅ since this would give
him the preferred over-all position A*.
• An investor with the preferences indicated by curves B1
through B4 would seek to invest all his funds in combination ∅,
while an investor with indifference curves C1 through C4 would
invest all his funds plus additional (borrowed) funds in
combination ∅ in order to reach his preferred position (C*).
• All would attempt to purchase only those risky assets which
enter combination ∅.
Capital asset prices must, of course, continue to change until a set of prices is attained for which every asset enters
at least one combination lying on the capital market line.

12. • All possibilities in the shaded area can be attained with
combinations of risky assets, while points lying along the
line PZ can be attained by borrowing or lending at the pure
rate plus an investment in some combination of risky
assets. Certain possibilities (those lying along PZ from
point A to point B) can be obtained in either manner. For
example, the ER, σR values shown by point A can be
obtained solely by some combination of risky assets;
alternatively, the point can be reached by a combination of
lending and investing in combination C of risky assets.
• It is important to recognize that in the situation shown in
Figure 6 many alternative combinations of risky assets are
efficient (i.e., lie along line PZ), and thus the theory does
not imply that all investors will hold the same combination.
• On the other hand, all such combinations must be perfectly
(positively) correlated, since they lie along a linear border
the ER, σR region.
• This provides a key prices of capital assets and different
types of risk.
Equilibrium in the Capital Market (Cont..)

13. The Prices of Capital Assets
• Linear relation between risk and return for combinations of
risky assets. What about individual assets?
• Moreover, such points may be scattered throughout the
feasible region, with no consistent relationship between
their expected return and total risk (σR). However, there
will be a consistent relationship between their expected
returns and what might best be called systematic risk.
• Consider Asset i and efficient combination of assets g.
Three points: i (a = 1), g (a = 0) and g’ (a < 0).
• Curve igg’ is tangent to CML (PZ) at g. All curves must be
tangent to CML in equilibrium as (1) It represents efficient
combination (2) they are continuous at that point.
• Tangency imply that the curve intersects PZ. But then assets
would lie to the right of the CML, an obvious impossibility
since the capital market line represents feasible values of ER
and σR

14. The Prices of Capital Assets (Cont..)
• If the relationship between the return of asset i and that of
combination g is viewed in a manner similar to that used in
regression analysis.
• Imagine that we were given a number of (ex post) observations
of the return of the two investments. The points might plot as
shown in Fig. 8.
• The scatter of the Ri observations around their mean (which will
approximate ERi) is, of course, evidence of the total risk of the
asset - σRi. But part of the scatter is due to an underlying
relationship with the return on combination g, shown by βig, the
slope of the regression line.
• The response of Ri, to changes in Rg (and variations in Rg itself)
account for much of the variation in Ri. It is this component of
the asset's total risk which we term the systematic risk.
• The remainder, being uncorrelated with Rg, is the unsystematic
component. This formulation of the relationship between Ri, and
Rg can be employed ex ante as a predictive model. βig becomes
the predicted response of Ri to changes in Rg.
• Then, given σRg (the predicted risk of Rg), the systematic portion
of the predicted risk of each asset can be determined.

15. The Prices of Capital Assets (Cont..)
• All assets entering efficient combination g must have
(predicted) βig and ER, values lying on the line PQ.
• Prices will adjust so that assets which are more responsive to
changes in Rg will have higher expected returns than those
which are less responsive.
• Obviously the part of an asset's risk which is due to its
correlation with the return on a combination cannot be
diversified away when the asset is added to the combination.
• Since βig indicates the magnitude of this type of risk it should
be directly related to expected return. The relationship
illustrated in Figure 9 provides a partial answer to the question
posed earlier concerning the relationship between an asset's risk
and its expected return.
• The fact that rates of return from all efficient combinations will
be perfectly correlated provides the justification for arbitrarily
selecting any one of them.
• Alternatively we may choose instead any variable perfectly
correlated with the rate of return of such combinations. The
vertical axis in Figure 9 would then indicate alternative levels
of a coefficient measuring the sensitivity of the rate of return of
a capital asset to changes in the variable chosen.

16. Concluding Sharpe
• All efficient combinations will be perfectly correlated and a useful
interpretation of the relationship between an individual asset's expected return
and its risk.
• Diversification enables the investor to escape all but the risk resulting from
swings in economic activity - this type of risk remains even in efficient
combinations.
• And, since all other types can be avoided by diversification, only the
responsiveness of an asset's rate of return to the level of economic activity is
relevant in assessing its risk. Prices will adjust until there is a linear relationship
between the magnitude of such responsiveness and expected return.
• 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.

17. Capital Market Equilibrium
with Restricted Borrowings
Fischer Black
The Journal of Business
July 1972

18. CAPM
• Several authors have contributed to the development of a model describing
the pricing of capital assets under conditions of market equilibrium.
• The model states that under certain assumptions the expected return on any
capital asset for a single period will satisfy –
• Where,
• Assumptions:
a) Homogeneous expectations
b) Returns are normally distributed
c) Investors maximize their end-period wealth and they are risk averse
d) An investor may take a long or short position of any size in any asset,
including the riskless asset. Any investor may borrow or lend any amount
he wants at the riskless rate of interest.

19. • Of these assumptions, the one that has been felt to be the most restrictive is
(d). Assumption (d), however, is not a very good approximation for many
investors, and one feels that the model would be changed substantially if
this assumption were dropped.
• In addition, several studies have suggested that the returns on securities do
not behave as the simple capital asset pricing model described above
predicts they should.
• Pratt analyzes the relation between risk and return in common stocks in the
1926-60 period and concludes that high-risk stocks do not give the extra
returns that the theory predicts they should give.
• Friend and Blume use a cross-sectional regression between risk-adjusted
performance and risk for the 1960-68 period and observe that high-risk
portfolios seem to have poor performance, while low-risk portfolios have
good performance.
CAPM

20. • Black, Jensen, and Scholes analyze the returns on portfolios of stocks at different levels of βi in the
1926-66 period. They find that the average returns on these portfolios are not consistent with
equation of CAPM.
• Their estimates of the expected returns on portfolios of stocks at low levels of βi are consistently
higher than predicted by equation, and their estimates of the expected returns on portfolios of
stocks at high levels of βi are consistently lower than predicted by equation of CAPM.
• Black, Jensen, and Scholes also find that the behavior of well- diversified portfolios at different
levels of βi is explained to a much greater extent by a "two-factor model" than by a single-factor
market model.
• They show that a model of the following form provides a good fit for the behavior of these
portfolios -
• This model suggests that in periods when Rz is positive, the low βi portfolios all do better
than predicted by equation of CAPM, and the high βi portfolios all do worse than predicted
by equation of CAPM. In periods when Rz is negative, the reverse is true: low βi portfolios
do worse than expected, and high βi portfolios do better than expected.
• In the post-war period, the estimates obtained by Black, Jensen, and Scholes for the mean of
Rz were significantly greater than zero. One possible explanation for these empirical results is
that assumption (d) of the capital asset pricing model does not hold.
Black, Jensen and Scholes (1972)

21. • Efficient frontier portfolios have a number of interesting characteristics,
independent derived by Merton and Roll. Two of these are:
1. Any portfolio that is a combination of two frontier portfolios is itself on the
efficient frontier.
2. Every portfolio on the efficient frontier, except the global minimum variance
portfolio, has a “companion” portfolio on the bottom (inefficient) half of the
frontier with which it is uncorrelated. Because it is uncorrelated, the companion
portfolio is referred to as Zero-Beta Portfolio of the efficient portfolio. If we
choose the market portfolio M and its zero-beta companion portfolio Z, then we
will obtain a CAPM like equation:
Black’s Zero Beta Portfolio Z

22. • Portfolio M is identified by the investors as the market
portfolio that lies on the efficient set.
• Now, suppose that we can identify all portfolios that are
uncorrelated with the true market portfolio.
• This means that their returns have zero covariance with the
market portfolio, and they have the same systematic risk (i.e.,
they have zero beta).
• Because they have the same systematic risk, each must have
the same expected return.
• Portfolios A and B are both uncorrelated with the market
portfolio M and have the same expected return, E(Rz).
• However, only one of them, portfolio B, lies on the
opportunity set. It is the minimum variance zero-beta portfolio
and it is unique.
• Portfolio A also has zero beta, but it has a higher variance and
therefore does not lie on the minimum variance opportunity
set.

23. Fischer Black (1972): Capital Market Equilibrium with
Restricted Borrowing
• The expected return on every asset, even when there is no riskless asset and
riskless borrowing is not allowed, is a linear function of its β.
• If we compare this equation with CAPM, we see that the introduction of riskless
asset simply replaces E(Rz) with Rf.
• This equation holds for any asset and thus for any portfolio.
• Setting βi = 0, we see that every portfolio with β equal to zero must have the same
expected return as portfolio z.
• Since the return on portfolio z is independent of the return on portfolio m, and
since weighted combinations of portfolios m and z must be efficient, portfolio z
must be the minimum-variance zero- β portfolio.
• The upshot of this proof is that the major results of the CAPM do not require the
existence of a pure riskless asset.
• Beta is still the appropriate measure of systematic risk for an asset, and the
linearity of the model still obtains. The version of the model given by equation
above is usually called the two-factor model.

24. Black (1972)
• Black (1972) uses this equation to show the results when investors
face restrictions on borrowings.
• In this case, at least some investors will choose portfolios on the High
risk-premium portion of the efficient frontier.
• Put differently, investors who otherwise wish to borrow and leverage
their portfolios but who find it impossible or costly will instead tilt
their portfolios towards high-beta stocks away from low beta stocks.
• As a result, prices of high beta stocks will rise and their risk premiums
will fall.
• The SML will be flatter than the simple CAPM.
• The risk premium on the market is smaller (because the expected
return on the zero-beta portfolio is greater than the risk-free rate) and
therefore the reward to bearing beta risk is smaller.

25. CAPM and The Real World
Concluding Remarks

26. CAPM & The Real World
• In limited ways, portfolio theory and the CAPM have become accepted tools in
the practitioner community.
• Many investment professionals think about the distinction between firm specific
and systematic risk and are comfortable with the use of beta to measure systematic
risk.
• Still, the nuances of the CAPM are not nearly as well established in the
community.
• New ways of thinking about the world (that is, new models or theories) displace
old ones when the old models become either intolerably inconsistent with data or
when the new model is demonstrably more consistent with available data.
• To some extent, the slowness with which the CAPM has permeated daily practice
in the money management industry also has to do with its precision in fitting data,
that is, in precisely explaining variation in rates of return across assets.

27. CAPM & The Real World: Some Empirical Evidence
• The CAPM was first published by Sharpe in the Journal of Finance (the journal of the
American Finance Association) in 1964 and took the world of finance by storm.
• Early tests by Black, Jensen, and Scholes (1972) and Fama and MacBeth (1973) were
only partially supportive of the CAPM: average returns were higher for higher-beta
portfolios, but the reward for beta risk was less than the predictions of the simple theory.
• While this sort of evidence against the CAPM remained largely within the ivory towers of
academia, Roll’s (1977) paper “A Critique of Capital Asset Pricing Tests” shook the
practitioner world as well. Roll argued that since the true market portfolio can never be
observed, the CAPM is necessarily untestable.
• The publicity given the now classic “Roll’s critique” resulted in popular articles such as
“Is Beta Dead?” (Wallace, 1980) that effectively slowed the permeation of portfolio
theory through the world of finance.
• Although Roll is absolutely correct on theoretical grounds, some tests suggest that the
error introduced by using a broad market index as proxy for the true, unobserved market
portfolio is perhaps not the greatest of the problems involved in testing the CAPM.

28. CAPM & The Real World: Some Empirical Evidence
• Fama and French (1992) published a study that dealt the CAPM an even harsher blow.
They claimed that once you control for a set of widely followed characteristics of the
firm, such as the size of the firm and its ratio of market value to book value, the firm’s
beta (that is, its systematic risk) does not contribute anything to the prediction of future
returns.
• It seems clear from these studies that beta does not tell the whole story of risk. There
seem to be risk factors that affect security returns beyond beta’s one-dimensional
measurement of market sensitivity.
• Despite all these issues, beta is not dead.
• Jagannathan and Wang (1996) shows that when we use a more inclusive proxy for the
market portfolio than the S&P 500 (specifically, an index that includes human capital) and
allow for the fact that beta changes over time, the performance of beta in explaining
security returns is considerably enhanced.
• We know that the CAPM is not a perfect model and that ultimately it will be far from the
last word on security pricing.
• However, CAPM provides a useful framework for thinking rigorously about the
relationship between security risk and return.