- 1. The Capital Asset Pricing Model
- This chapter has one of the most important models in investment
modeling.It addresses the question of what is a reasonable price
for an asset.The same model also gives some very good investment
advice.The results of the chapter build upon the Markowitz
mean-variance portfolio theory of Chapter 6.
2.
- Everyone is a mean-variance optimizer, as in Chapter 6.
- Everyone assigns to returns ofall availablemarket assets the
same mean values, variances, and covariances (available from
Morning Star, for example).
- There is a unique risk-free rate of borrowing and lending
available to all, with no transaction costs.
3.
- Under the above assumptions, everyone would rely on theone-fund
theorem .They would buy a single (efficient) fund of risky assets,
and then also borrow or lend at the risk-free rate.The mix of the
fund and the risk-free asset would depend on a investors attitude
towards risk.
- Basic Question: What is the one fund (shared by all investors)
?
4.
- Example of theMarket Fund , after Table 7.1.
- Suppose there are only three risky assets in the market, as
follows:
5.
- The table illustrates the definitions of capitalization
(sharesprice) and capitalization weights.
- The weights are NOT the same as the relative shares in the
market.
- Each weight is the ratio of the capital value of the asset to
the total market capital value.
6.
- If you invested in this market fund, you would use the weights
in the last column; a $1,000 investment would give you
- [(3/20)1000]/6 = 25 shares of Mahler Inc
- [(3/10)1000]/4 = 75 shares of Mozart Inc.
- [(11/20)1000]/5.5 = 100 shares of Verdi, Inc.
7.
- Theactual market fundwould be comprised ofeveryinvestment asset
available.Just as in the example, each investment weight would be
the ratio of the capital value of the asset to the total market
capital value.
- Funds similar to the market fund actually exist, and are
calledindex funds .Of course, they do not include every available
asset, but they may well include 500 or more.Index funds typically
out-perform most actively managed funds.Vanguard Securities offers
many such funds (e.g. Index Trust 500)
8.
- Index funds have been somewhat resisted by the financial world,
including many private investors.The first reaction is usually that
surely an actively managed fund will give better performance.The
facts, however, are otherwise.About 90% of actively managed funds
perform worse than the S&P 500.You can also check performance
with the Morning Star services.
9.
- If everyone buys just one fund, and their purchases add up to
the market, then that one fund must be the market as well.
- The fund must contain shares of every stock in proportion to
that stocks representation in the entire market.
- If everyone else solved the one-fund problem, we would not need
to.
- Suppose everyone elsesolves the mean-variance portfolio problem
with their common estimates, and places orders in the market to
acquire their portfolios.This solution isefficient , because it
minimizes the total variance of the return.
10.
- Equilibrium Arguments (Contd)
- If the orders placed do not match what is available, the prices
must change.Prices of assets under heavy demand will go up, prices
of assets under light demand will go down.The price changes will
affect the estimates of asset returns directly.Therefore, investors
will recalculate their optimal portfolios.The process continues
until demand exactly matches supply; that is, until there is an
equilibrium.
11. Flow Chart Visualization Solve mean-variance portfolio
problem for efficient portfolio. Place orders. Supply=
Demand?YesEquilibrium No Prices adjustments occur. Asset return
changes cause portfolio data changes. 12.
- Everyone would buy just one portfolio in this idealized world,
and it would be the market portfolio.
- Prices adjust to drive the market to efficiency.
- After the adjustments, the portfolio will be efficient, so we
would not need to make the calculations.
- This argument is most plausible when applied to assets traded
repeatedly over time, which is certainly the case with the stock
market.
- The fact that index funds perform well provides some
verification that the equilibrium argument is plausible.
13.
- Final Word: If you have better information than your rivals,
you will want your portfolio to include relatively large
investments in the stocks you think are undervalued.In a
competitive market you are unlikely to have a monopoly of good
ideas.In that case, there is no reason to hold a different
portfolio of common stocks from anybody else.In other words, you
might just as well hold the market portfolio.That is why many
professional investors invest in a market-index portfolio, and why
most others hold well-diversified portfolios.
14.
- M is the market portfolio
- E{r} = r f+--------------
15.
- The line shows the efficient set , starting at the risk-free
point, and passing through the market portfolio.
- The CML shows the relation between the expected rate of return
and the risk of return (as measured by the standard deviation), for
efficient assets or portfolios of assets.
- The CML is also called thepricing line .Prices should adjust so
that efficient assets fall on this line. equationsw.r.t.
16. 17.
- r f= 6%, E{r M } = 12%, M= 15%.
- John Eager wants to retire in 10 years.For this he needs
$1,000,000.He currently has $1,000.At the market rate, it would
take about 60 years for $1,000 to grow into $1,000,000.If he can
get 100% return each year he concludes he will grow $1,000 into
$1,000,000 in 10 years.
18.
19. Example 7.2. The Capital Market Line E{r} M r f r f = 0.10,M
= ( M , E{r M }) = (0.12, 0.17),OD = (0.4,0.14). Oil Drilling 20.
21. The Pricing Model
- The CML relates the expected rr ofany efficient portfolioto its
standard deviation.Another step beyond the CML is to show how the
expected rate of return ofany individual assetrelates to its
individual risk.That is what the capital asset pricing model
does.
22. 23.
- Example(We use t instead of ).
- The equation for the standard deviation of the rate of the
return combining the market portfolio with any asset i is
- x = f(t) = [t 2 i 2+ 2 t (1-t) iM+ (1-t) 2 M 2 ]
- The expected return for this combination becomes
- y = g(t) = t E{r i } + (1-t) E{r M }
- Note f(0) = M , g(0) = E{r M } .
24.
- dy/dt = g (t) = E{r i } - E{r M }
- dx/dt = f (t) = [t i 2+ (1-2 t) iM+ (t-1) M 2 ]/f(t)
- f (t)| t=0= ( iM- M 2 )/f(0) = ( iM- M 2 )/ M
25.
26.
- If the market portfolio M is efficient, then the expected
return E{r i } ofanyasset i satisfies
- E{r i } r f= i(E{r M } r f )
27.
- For any t, consider the portfolio consisting of a portion t
invested in asset i and a portion 1-t invested in the Market
Portfolio M.(t < 0 corresponds to selling short the asset.)
- The expected rate of return of this portfolio is
- y = g(t) = t E{r i } + (1-t) E{r M }
- The standard deviation of the rate of return is
- x = f(t) = [t 2 i 2+ 2 t (1-t) iM+ (1-t) 2 M 2 ]
28.
- As t varies, the values (f(t), g(t)) trace out a curve in the
expected return-sd diagram, as shown below.
- In particular, the point on the curve (f(0),g(0)) for
- t = 0 corresponds to the market portfolio M.
29.
- This curve cannot cross the capital market line.If it did, the
portfolio corresponding to a point above the capital market line
would violate the definition of the capital market line as being
the efficient boundary of the feasible set.Hence as t passes
through zero,the curve must be tangent to the capital market line
at M .This tangency is the condition that we exploit to derive the
formula.
- The tangency condition can be translated into the condition
that the slope of the curve (f(t), g(t)) is equal to the slope of
the capital market line at the point M, where t = 0.
30. 31.
- If we solve this equation for E{r i } we get
- E{r i } = r f+ [(E{r M } r f )/ M 2 ] iM
- Equivalently, we have E{r i } - r f= i(E{r M } r f ).
- iis called thebetaof asset i.Sometimes the subscript is
omitted.
32.
- The Morning Star service estimates betas.
- Since E{r i } - r f= i(E{r M } r f ), E{r i } - r fis called
theexpected excess rate of return of asset i .It is the amount by
which the rate of return is expected to exceed the risk-free
rate.
- (E{r M } r f ) is called theexpected excess rate of return of
the market portfolio .
- THE CAPM says the expected excess rate of return of an asset is
proportional to the expected excess rate of return of the market
portfolio.The constant of proportionality factor is .
33.
- Because i= iM / M 2 , it is anormalized version of the
covariance of an asset with the market portfolio . The excess rate
of return for the asset is directly proportional to its covariance
with the market.
- Generally speaking, we expect aggressive assets/companies or
highly leveraged companies to have high betas.Conservative
companies whose performance is unrelated to the general market
behavior are expected to have low betas.We expect that companies in
the same business will have similar beta values.
34. 35. 36.
- The CAPM changes our concept of the risk of an asset fromto .It
is still true that, overall, we measure the risk of aportfolioin
terms of .But this does not translate into a concern for the s
ofindividual assets .For those, the proper measure is their s.
37.
- After Table 7.2.Betas and Sigmas for Some U.S. Companies
(1979)
38.
- The concept of beta is well-accepted.
- Various financial service organizations (e.g., Morning Star)
provide beta and other estimates.
- Estimates may be based on 6 to 18 months of weekly values.
- Companies in the same business should have similar betas:
compare, for instance, JC Penny with Sears Roebuck, and Exxon with
Standard Oil of California.
39.
- We never know the beta and sigma values we only haveestimatesof
them.
- Generally speaking, we expect aggressive companies or highly
leveraged companies to have high betas, whereas conservative
companies whose performance is unrelated to the general market
behavior are expected to have low betas.Also we expect that
companies in the same business will have similar, but not
identical, beta values.
40.
- cov{U + V, Z} = cov{U,Z} + cov{V,Z}
- cov{a U + b V, Z} = cov{a U,Z} + cov{b V,Z}
- = a cov{U,Z} + b cov{V,Z}
41.
- Suppose a portfolio P has 2 assets with returns r 1 , r 2and
weights w 1 , w 2 .Let r Mdenote the market return.We know the
portfolio return is r = w 1r 1+ w 2r 2 .Let Pdenote the beta of the
portfolio (ratio of the portfolio covariance with the market and M
2).
- P= cov{r ,r M } / M 2= cov{ w 1r 1+ w 2r 2 , r M } / M 2
- = cov{ w 1r 1 , r M } / M 2+cov{w 2r 2 , r M }/ M 2
- = w 1cov{r 1 , r M } / M 2+ w 2cov{r 2 , r M }/ M 2
42.
- Beta of a Portfolio (Contd)
- The two dimension formulageneralizes to n assets:
- P= cov{r ,r M } / M 2= w 1 1+ w 2 2++ w n n .
- The portfolio beta is just the weighted average of the betas of
the individual assets in the portfolio .The weights are those that
define the portfolio.
- Risk neutral portfolio: P= 0 .
43. The Security Market Line (SML) 44. 45. 46. 47.
- Under the equilibrium conditions assumed by the CAPM, any asset
should fall on the SML.
- The SML expresses the reward-risk structure of assets according
to the CAPM, and emphasizes that the risk of an asset is a function
of its covariance with the market or, equivalently, a function of
its beta.
48.
- There are several types of risk with an investment:
- - nonsystematic ,idiosyncratic , orspecificrisk.
- The systematic risk is risk associated with the market as a
whole.The second type of risk is uncorrelated with the market, and
can be reduced by diversification.
- We can use the CAPM to quantify these two types of risks.
49.
- r i= r f+ i(r M- r f ) + i (***)
- To begin with, we view this equation simply as a definition of
the random variable i .Namely,
- i = r i [ r f+ i(r M- r f )]
- The CAPM provides some information on i .Note firstthat
50.
- i 2 var(r i ) = i 2 M 2+ var( i )
51. 52. 53.
- Implications. For asset i, its risk is the sum of
- (1) i 2 M 2 , thesystematicrisk, and
- (2) var( i ), thenonsystematicorspecificrisk.
- The systematic risk is the risk associated with the market as a
whole.It cannot be reduced by diversification, because every asset
with nonzero beta contains this risk.
54.
- The specific risk is uncorrelated with the market andcan be
reduced by diversification .
- It is the systematic risk, measured by beta,that is most
important.It directly combines with the systematic risk of other
assets.
- There is a limit to how much diversification can achieve in
reducing risk.
55.
- M is the market portfolio
- E{r} = r f+--------------
56.
- For asset i, its risk is the sum of
- (1) i 2 M 2 , thesystematicrisk, and
- (2)var( i ), thenonsystematicorspecificrisk.
- If it hasonly systematic risk , then its standard deviation
is
57. 58.
- Now considerother fundswith the samevalue, i , as asset i.The
CAPM implies all these funds have an expected return of
- But this is the expected return of asset i, E{r i }.Suppose
these other assets have nonsystematic risk.Then each will have
avariance, for some , of
- i 2 M 2+ var( ) > i 2 M 2= i 2 .
59. 60.
- Assets with the same beta as asset i, but which also have
systematic risk, have the same expected return as asset i but do
not fall on the CML.
- Bottom Line. The horizontal distance of an asset point from the
CML is a measure of the nonsystematic risk of the asset.
61.
- CAPM & Investment Implications
- A CAPM purist is one who completely believes the CAPM theory as
applied to publicly traded securities.A purist would just purchase
the market fund and some risk-free securities (e.g., U.S. Treasury
bills), adjusting the relative investment in the two according to
her/his tolerance for risk.
- Individual investors cannot easily purchase the market
fund.They can, however, purchase an index fund.These funds allocate
their investments in order to duplicate the portfolio of a major
stock market index, such as the S&P 500 or the Wiltshire
2,000.
62.
- The CAPM requires the assumption that everyone has identical
information about the expected returns and variance of returns of
all assets.The assumption is certainly open to criticism.
- Therefore, akey questionfor an investor is the following:
- Do I possess superior information to that required by the
Markowitz model and the CAPM?
- With superior information, it is likely that one can do
better.
63.
- Few people quarrel with the idea that investors require some
extra return for taking on risk ....
- Investors do appear to be concerned principally with those
risks that they cannot eliminate by diversification.
- The CAPM captures these ideas in a simple way.That is why many
financial managers find it the most convenient tool for coming to
grips with the slippery notion of risk.
64.
- There are two problems with the CAPM.
- - First, it is concerned withexpected returns , whereas we can
observe onlyactual returns .Stock returns reflect expectations, but
they also embody lots of noise the steady flow of surprises that
gives many stocks standard deviations of 30 or 40 percent per
year.
- - Second, the market portfolio should compriseall risky
investments.... Most market indexes contain only a sample of common
stocks.
65.
- A classic paper by Fama and MacBeth avoids the main pitfalls
that come from having to work with actual rather than expected
returns.Fama and MacBeth (Risk, Return and Equilibrium: Empirical
TestsJ. of Political Economy , 81, 607-636 ,May, 1973) grouped all
New York Stock Exchange stocks into 20 portfolios.They then plotted
the estimated beta of each portfolio in one 5-year period against
the portfolios average return over asubsequent5-year period.1
Figures 8-10 show what they found.You can see that the estimated
beta of each portfolio told investors quite a lot about its future
return.
66. 67.
- If the CAPM is correct, investors would not have expected any
of these portfolios to perform better or worse than a comparable
package of Treasury bills and the market portfolio.Therefore, the
expected return on each portfolio, given the market return, should
plot along the sloping lines in Figures 8 10.Notice that the actual
returns do plot roughly along those lines.
68.
- 1 Fama and MacBeth first estimated the beta of each stock
during one period and then formed (the 20) portfolios on the basis
of these estimated betas.Next they reestimated the beta of each
portfolio by using the returns in the subsequent period.This
ensured that the estimated betas for each portfolio were largely
unbiased and free from error.Finally, these portfolio betas were
plotted against returns in an even later period.
69. Performance Evaluation
- Many institutional portfolios (pension funds, mutual funds) now
have their performance evaluated using the CAPM framework.
- The following example illustrates the evaluation ideas and the
use the CAPM.
70.
Std. Dev. 12.39 9.43 0.47 Geom. Mn 12.34 11.63 7.60
Cov(ABC,S&P) 107 Beta 1.20375 1 Jensen 0.104 0 Sharpe
0.43577368 0.46669 RR Percentages Year ABC S&P T - bills 1 14
12 7 2 10 7 7.5 3 19 20 7.7 4 - 8 - 2 7.5 5 23 12 8.5 6 28 23 8 7
20 17 7.3 8 14 20 7 9 - 9 - 5 7.5 10 19 16 8 Avg. 13.00 12.00 7.60
71. 72. 73. 74. 75.
- According to the CAPM, the value of J should be zero when true
expected returns are used.Hence J measures, approximately, how much
the performance of ABC has deviated from the theoretical value of
zero.A positive value of J presumably implies that the fund did
better than the CAPM prediction (but of course we recognize that
approximations are introduced by the use of a finite amount of data
to estimate the important quantities.
76.
- The Jensen index can be indicated on the security market line
(Figure 7.5 a).
77.
- Sharpe Index (Note. Figure 7.5 in the text has mistakes)
- The slope of the heavy dashed line is the
78. We compare S for ABC (0.43577) with S for the market
(0.46669).The conclusion is that ABC is not efficient. 79. 80.
- CAPM as a Pricing Formula
81. 82. 83.
84. 85. 86.
- Certainty Equivalent Form of the CAPM
87.
- Certainty Equivalent Form of the CAPM (Contd)
88. 89. 90.
- Practical Implication.If we want the CAPM for two assets, and
have the CAPM for each, we can get the CAPM for the two in total by
adding the certainty equivalent forms for each, or equivalently,
using the latter equation.
- The reason for linearity can be traced back to the principle of
no arbitrage .... This linearity of pricing is therefore a
fundamental tenet of financial theory (in the context of perfect
markets) ....
91.
- Example 7.7(Certainty Equivalent version of Example 7.5)
- invests 10% of its money at r f= 0.07
- invests 90% of its money at the market rate,
- Its expected return in a year is 0.10.07 + 0.90.15 = 0.142
- The epected value of a $100 share in a year will be 100 1.142 =
$114.20.
- Thevalue is 0.100 + 0.91.0 = 0.9and P=90
- Certainty Equivalent Version
- P = (1 + r f ) -1 [E{Q} - (cov(Q,r M )/ M 2)(E{r M } r f
)]
- = (1 + r f ) -1 [E{Q} - P)(E{r M } r f )]
- = (1.07)1 [114.20- 900.08] = $100