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Ch. 8 Risk and Return
━ the purpose of the chapter:
1) Develop some theories linking risk and return in a competitive economy.
2) Show how to use these theories to estimate return that investors require for taking risk in
different investments.
8-1 Harry Markowitz and the Birth of Portfolio Theory
━ Markowitz‟s (1952) work:
1) Diversification can reduce the standard deviation of portfolio returns.
2) Work out the basic principles of portfolio construction, the foundation for the theories about
the relationship between risk and return.
━ distributions of returns:
1) When measured over a short interval (say, one day) the past rates of return on any stock
conform closely to a normal distribution.
2) The distribution of returns over long periods (say, one year) would be better approximated
by a lognormal distribution.
Both the normal and the lognormal distributions can be completely defined by two numbers:
the expected return and the standard deviation of returns.
Investors need consider only these two measures.
notes: 1) One can show that any distribution function is characterized by its statistical moments.
The normal distribution is the only stable distribution which is fully characterized by its
first two moments.
2) Usually we refer to distributions that are stable under addition. This means that the
addition of two distributions will result in the same type of distribution.
3) Obviously, returns on assets cannot be normally distributed because the largest negative
return possible, given limited liability of the investor, is minus 100%. Unfortunately,
the assumption of normally distributed returns implies that there is a finite possibility
that returns will be less than minus 100%.
4) Fama (1965a) has investigated the empirical distribution of daily returns on NYSE
securities and discovered that they are distributed symmetrically but that the empirical
distribution has „fat tails‟ and no finite variance. In fact, he shows that the distribution
of daily return is the stable Paretian distribution. You may be referred to the paper:
“The Behavior of Stock Market Prices,” Journal of Business, January 1965, pp. 34-105.
5) Fama (1065b) has shown that as long as the distribution is symmetric and stable,
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investors can use measures of dispersion other than the variance (e.g., the
semi-interquartile range) and the theory of portfolio choice is still valid. Pleas refer to
the paper: “Portfolio analysis in a Stable Paretian Market,” Management Science,
January 1965, pp. 404-419.
6) The equation for a lognormal distribution is given by
(ln x )
2
1
f ( x) e 2
2
,x 0.
x 2
For our application of return distribution, we define x as pt/pt-1. Hence, there is no limit
to the positive returns that may be realized on a successful investment, but the
maximum negative return is minus 100%.
━ return and risk for individual stocks:
Figure 8-2: For the same level of expected return, investors prefer lower risk.
Figure 8-3: For the same level of risk, investors prefer higher expected return.
implied assumption: risk aversion
━ return and risk for a portfolio
I. two stocks: 12 = 0.4
Stock xi ri i
----------------------------------------------------------------
1 (Coca-Cola) x1 10% 18.2%
2 (Reebok) x2 15% 27.3%
~x~x ~
r
=> 1r1 2 r2
r 10 x1 15 x2
(18 .2) 2 x1 ( 27 .3) 2 x2 2(0.4)(18 .2)( 27 .3) x1 x2
2 2
We could change r and by different combinations of the two stocks (Figure 8.4).
Q: Which of these combinations is best? That depends on your tastes for expected return and
risk.
1) maximizing return => put all money in stock 2 ( x 2 = 1).
2) minimizing risk => keep a small investment in stock 2 ( x1 = 0.805). (See footnote 4.)
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3) general analysis: need utility function over the space of r and , U( r , )
the shape of the indifference curve, assuming risk aversion:
i) To maintain the same level of utility, ↑ => r ↑
ii) The MRS between and r is increasing.
iii) Indifference curve shifting upwards means increasing utility.
a helpful diagram for making the best choice:
notes: 1) Figure out the shapes of indifference curve, assuming risk neutral or risk loving
investors.
2) Here, we deal with economic agents whose preferences over probability
distribution can be represented by a function of the mean and the variance of these
probability distributions alone. This is known as mean-variance analysis.
3) Recall that any distribution function can be characterized by its statistical
moments. We write P(W|M) to indicate that the distribution function P is
determined by M, where W is wealth and M is the set of moments of the
distribution. For the expected utility of a probability distribution, we have
therefore in general: U(M) = V(P(W|M)) = v(W )dP(W | M ) , i.e., expected
utility is a function of all moments of the distribution P, where v (W ) is some
expected utility index. Thus, one way to use expected utility theory to obtain a
preference function on mean and variance is to restrict the set of possible
probability distributions to those which depend upon their first two moments
only. Since the normal distribution is the only stable distribution which is fully
characterized by its first two moments, we have U(M) = U( r , ).
4) As discussed earlier, the support of the normal distribution is the entire real line
and this makes it unsuitable for many economic applications. A second possibility,
which places no constraints on the distribution function P, is to require expected
utility index v (W ) to be quadratic such as v (W ) W 2 W , . For a
quadratic expected utility index, the expected utility function depends exclusively
upon the mean and variance of arbitrary distributions. However, some properties
of the quadratic expected utility index are either inconvenient or implausible.
5) Recall also that stable under addition means that the addition of two distributions
will result in the same type of distribution. Hence, ~ x1~ x2 ~ is normally
r r1 r2
distributed too.
II. many stocks
N
~ x ~ x ~ ... x ~ ,
r 1r1 2 r2 N rN x
i 1
i 1 , xi 0 .
Figure 8.5:
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the feasible set (the opportunity set): the set of all possible portfolios that can be formed from
the N stocks
the efficient set (the efficient portfolios): the set of portfolios that (1) offer maximum expected
return for any level of risk and (2) offer minimum risk for any level of expected return
point A: maximum-return portfolio
point B: minimum-risk portfolio
point C: maximum-utility portfolio
a note: To find the efficient set, we can employ quadratic programming to solve the problem:
N N
Min 2 x i x j ij
x i 1 j 1
N N
s.t. ri xi k ,
i 1
x
i 1
i 1 , xi 0 , i 1,2,...,N
━ extending the range of investment opportunities by borrowing and lending
● assumption: You can lend and borrow money at some risk-free rate of interest r f .
● lending: Invest some of money in T-bills and place the remainder in portfolio S.
~ x r (1 x ) ~
r f rS => r x rf (1 x ) rS , (1 x ) S , 0 x 1
The line joining rf and S represents different combinations of ( r , ). See Figure 8-6.
● borrowing: Borrow money at rate of rf and invest them as well as your own money in stock S.
~ x r (1 x ) ~ , where x = the percentage of the money borrowed for each dollar of
r rS
f
your own money
=> r x rf (1 x ) rS , (1 x ) S , x > 0
The line to the right of S represents additional possibilities from this investment strategy.
● examples: r f = 5%, S: ( rS , S ) = (15%,16%)
1) lending money, x = 1/2 => r = 10%, = 8%
2) borrowing an amount equal to your initial wealth and investing everything in S
=> x = 1, r rf 2 rS 25%, 2 S 32%
● the extended opportunities:
For any level of risk, you can get the highest expected return by a mixture of portfolio S and
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borrowing or lending. Investors are thus better off with the extended opportunities.
the diagram for investing in T-bills and S (preferring lower risk):
the diagram for borrowing and investing all money in S (preferring higher risk):
● portfolio S: the point where the line of the extended opportunities tangent to the efficient set
r rf
It offer the highest expected risk premium (r – rf) per unit of standard deviation (i.e., ).
It is called the best portfolio.
● the separation theorem:
We can separate the investor‟s job into two stages:
1) Select the best portfolio S.
2) S must be mixed with borrowing or lending to obtain a maximum-utility portfolio.
implied assumption: All investors have homogeneous expectations on the expected returns R
and the variance-covariance matrix of return .
● What does portfolio S look like?
In a competitive market, you have the same information and face the same opportunities as
everyone else. => Each investor should hold the same portfolio as everyone else.
Since the market portfolio is simply the sum of all individual holdings, each should hold the
market portfolio.
a note: The line representing the extended opportunities is called the capital market line
r rf
(CML), indicating the relation ri rf ( m ) i .
m
8-2 The Relationship between Risk and Return
━ the capital asset pricing model (CAPM) and the security market line (SML)
● two benchmark investments:
1) T-bills: f = 0, risk premium = 0
2) market portfolio: m = 1, risk premium = rm rf
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The sloping line passing through the two points in the space of ( r , ) is called the security
market line (SML). See Figure 8.7.
● What is the expected risk premium when beta 0 or 1?
William Sharpe‟s (1964) CAPM: In a competitive market, the expected risk premium varies
in direct proportion to beta. For any security i, the relationship is given by
ri rf i ( rm rf ) or ri rf i ( rm rf ) .
i 0.5 => The expected risk premium is half rm rf .
i 2 .0 => The expected risk premium is twuce rm rf .
notes on CML and SML:
1) The portfolio selection theory gives the CML as examined earlier.
2) The SML can be derived from the CML by assuming that market equilibrium exists (i.e.,
the prices of all assets must adjust until the supply of all assets equals the demand
holding them).
3) Only efficient portfolios or securities plot on the CML. But, efficient and inefficient ones
plot on the SML.
a helpful diagram:
4) CAPM: ri rf i ( rm rf ) = risk free rate of return + risk premium
where risk premium = the quantity of risk the market price of risk.
Notice that the market price of risk is given for all investors.
━ some estimates of expected returns in March 2004 (Table 8.2)
Assuming r f = 1.0% and rm rf = 7.0%.
━ proof by reasoning:
1) Risk premium always reflect the contribution to portfolio risk.
Hence, contribution of stock i ↑ => ( ri rf )↑.
2) If a portfolio is efficient, then the expected risk premium of a stock is proportional to its
contribution to the portfolio risk, i.e., ri rf = c contribution. This means that there must
be a straight-line relationship between each stock‟s expected risk premium and its marginal
contribution to the risk of the portfolio.
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a diagram for an efficient portfolio:
a diagram for an inefficient portfolio:
3) The marginal contribution of a stock to the risk of the portfolio is measured by beta.
4) If the market portfolio is efficient, there must be a straight-line relationship between each
stock‟s risk premium and its beta.
5) The CAPM thus boils down to the statement that the market portfolio is efficient.
━ If a stock does not lie on the SML, its price is not in equilibrium.
● cases for the stocks lying below the SML (Figure 8.8):
1) stock A, A = 0.6
Construct a portfolio p with the same risk p = 0.6:
weight r
market portfolio 0.6 (x) rm 1
T-bills 0.4 (1-x) rf 0
=> p = x m = 0.6, rp xrm (1 x ) rf rf x ( rm rf ) rf 0.6( rm rf )
=> We can get a higher expected return by this investment strategy.
=> price of stock A ↓ => rA ↑
2) stock B, B = 1.6
Construct a portfolio q with the same risk q = 1.6:
weight r
market portfolio 1.6 (1+x) rm 1
T-bills -0.6 (-x) rf 0
=> q = (1+x) m = 1.6
rq (1 x ) rm ( x ) rf rf (1 x ) rm (1 x ) rf rf (1 x )( rm rf )
= rf 1.6( rm rf )
=> Similarly, we can get a higher expected return by this investment strategy.
=> price of stock B ↓ => rB ↑
● Are there stocks lying above the SML?
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1) Market portfolio lies on the line. => Stocks on average lie on the line.
2) None lies below the line. => None lies above the line.
● An investor can always obtain any expected risk premium r-rm by mixing the market portfolio
and a risk-free loan. First, find = (r-rf)/(rm-rf). Then, (1) if 1 , invest dollars in M for
every dollar owned; (2) if 1 , invest dollars in M by borrowing 1 for every dollar
owned.
8-3 Validity and Role of the Capital Asset Pricing Model
━ the introductive words:
The CAMP captures the ideas of risk and return in a simple way. But that doesn‟t mean that the
CAPM is ultimate truth. It has several unsatisfactory features.
We will look at some alternative theories. But the development of theories is still going on.
━ a test for the CAPM (Black, 1993)
● Test if ( rp , p ) of 10 portfolios plot along the SML.
a note: The CAMP is better tested with returns ad betas of portfolios because estimating betas
of portfolios is more reliable.
● results (Figure 8.9): 1) Returns did increase with beta.
2) The fitted line (the actual SML) is flatter than the market line.
● findings on two fronts against the CAPM from critics:
1) Return has not risen with beta in recent years (e.g., 1966-1991 shown in Figure 8.10).
2) The CAPM predicts that beta is the only reason that expected returns differ. However, returns
have been related to other factors such as size and book-to-market ratio (Figure 8.11).
notes: a) Fama and French (1992) found that size, leverage, earnings-price ratios, and book
value to market value of equity all have a significant impact on univariate tests on
average return. In multivariate tests, size and book to market equity value are the
major explanatory factors. (footnote 18.)
b) Fama and French (1993) thus develop a three-factor model. Please refer to their
paper: "Common Risk Factors in the Returns on Stocks and Bonds," Journal of
Financial Economics 33, 3-56.
● defenses:
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1) two problems with testing the model:
a) The CAPM is concerned with expected returns, whereas we can observe only actual
returns. Actual stock returns reflect expectations, but they also embody lots of noise.
=> Focus on the longest period for which there is reasonable data.
b) The market portfolio should contain all risky investments, whereas most market indexes
contain only a sample of common stocks (footnote 16). According to Roll (1977), the
slope of the SML will be underestimated because these proxy market portfolios employed
are probably not as efficient as the true market portfolio.
2) The size and book-to-market effects are simply chance results that stem from data snooping.
━ assumptions behind the CAPM:
1) Investment in T-bills is risk-free. But, their real return is uncertain because of uncertainty
about inflation.
2) Investors can borrow and lend at the same rate of interest. Generally, borrowing rates are
higher than lending rates.
3) Investors are content to invest their money in few benchmark portfolios (i.e., T-bills and the
market portfolio).
a note: See Jensen (1972) (footnote21) and Copeland and Weston (1992) for the summary of
modified CAPMs.
8-4 Some Alternative Theories
━ Douglas Breeden‟s (1979) work:
consumption beta: the sensitivity of security‟s return to changes in investors‟ consumption
consumption CAPM: linking security‟s return directly to uncertainty about consumption (Figure
8.12)
━ Stephen Ross‟s (1976) arbitrage pricing theory (APT)
● the theory:
It assumes that each stock‟s return depends on (1) pervasive macroeconomic factors and on (2)
events that are unique to that company (noise):
~ a b (~ ~ ~
r to t )
1 rf a c 1 ) b2 ( rf a c 2 ) r . . bk ( rf a c k o r e
tor .
where ~factor i = return associated with factor i, bi = the stock‟s sensitivity to factor i, e = noise.
r
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notes: 1) The underlying assumption is that actual returns on an individual security follow a
generating process which is linear in a finite number of factors with an additional
non-factor random element.
The theory does not require that there be some distribution of security returns and
that there be a market portfolio which contains all risky assets that are
mean-variance efficient.
2) The theory doesn‟t say what factors are.
3) Some stocks will be more sensitive to a particular factor than other stocks.
● diversification and expected risk premium:
For any individual stock, there are two sources of risk. The risk that stems from the pervasive
macroeconomic factors can not be eliminated by diversification. The expected risk premium
on a stock is affected by macroeconomic risk, not by unique risk. Thus, the APT formula is
t
r rf b1 (rf a c 1 o r rf ) b2 (rf a c 2 o rf ) . . bk (rf a c k o rf )
t r . t r
where rfactori rf = expected risk premium associated with factor i.
notes: 1) The factors in arbitrage pricing represent special portfolios of stocks that tend to be
subject to a common influence.
2) the two statements made by the formula and the risk-free arbitrage
3) Securities must be priced according to this formula to prevent risk-free arbitrage,
hence the name of the model.
● a comparison of the CAPM and APT
1) similarities:
a) Both models stress that expected return depends on the risk stemming from economywide
influences and is not affected by unique risk.
b) The CAPM is a special case of the APT if the expected risk premium on each of other
factor portfolios is proportional to the portfolio‟s market beta.
2) attractive features of the APT:
a) The market portfolio does not necessarily play any role in the APT. Therefore, the
portfolio does not have to be identified and we don‟t have to worry about the problem of
measuring it.
b) We can test the theory using the data on a sample of risky assets.
3) disadvantage of the APT:
The theory does not indicate what the underlying factors are and these factors can change
over time.
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● Elton, Gruber, and Mei‟s (1994) empirical study
a note: Step 2 is to measure the expected risk premium on the theoretical portfolios which are
affected by only one single factor.
● empirical tests of the APT:
1) Several studies test the model by examining the number of factors in the return generating
process that were priced. Their evidence supports the multi-factor model. When the number
of securities is increased, both the number of factors that enter the model and the number of
significant priced factors increased. In general, the findings indicate extreme instability in the
relationships between returns and factors
2) testing on the hypothesis that the APT would be able to account for the differences in
average returns between small and large firms:
Reinganum(1981): Even after taking account of all risk as specified by the APT, the small
firms have superior risk-adjusted returns.
Chen (1983): Evidence related to the small firm effect is contrary to Reinganum.
━ the three-factor model by Fama and French (1993)
● size and book-to-market factors:
Fama and French (1995) provide evidence that these factors are related to company
profitability and therefore may be picking up risk factors that are left out of the simple CAPM.
● the model: r rf bmarket (rmarket factor ) bsize (rsize factor ) bbook tomarket (rbook tomarket factor )
bmarket ( rm rf ) bsize ( rS rB ) bbook to m a r k rHt rL )
(e
where rS = the expected return on the portfolio representing (mimicking) small-firm stocks
rB = the expected return on the portfolio representing big-firm stocks
rH = the expected return on the portfolio representing value (high book-to-market ratio)
stocks
rL = the expected return on the portfolio representing growth (low book-to-market
ratio) stocks
● Fama and French‟s (1997) empirical study
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