1 Chapter 13: The Capital Asset Pricing Model Copyright Prentice
Hall Inc. 1999. Author: Nick Bagley Objective The Theory of the
CAPM Use of CAPM in benchmarking Using CAPM to determine correct
rate for discounting Slide 2 2 Chapter 13 Contents 13.1 The Capital
Asset Pricing Model in Brief 13.2 Determining the Risk Premium on
the Market Portfolio 13.3 Beta and Risk Premiums on Individual
Securities 13.4 Using the CAPM in Portfolio Selection 13.5
Valuation & Regulating Rates of Return Slide 3 3 Introduction
CAPM is a theory about equilibrium prices in the markets for risky
assetsCAPM is a theory about equilibrium prices in the markets for
risky assets It is important because it providesIt is important
because it provides a justification for the widespread practice of
passive investing called indexing a way to estimate expected rates
of return for use in evaluating stocks and projects Slide 4 4 13.1
The Capital Asset Pricing Model in Brief Developed in the 1960s by
Sharp, and independently by Lintner, and MossinDeveloped in the
1960s by Sharp, and independently by Lintner, and Mossin It answers
the questionIt answers the question What would equilibrium risk
premiums be if people had the same set of forecasts of expected
returns, risk, and correlationspeople had the same set of forecasts
of expected returns, risk, and correlations all chose their
portfolios according the principles of efficient diversificationall
chose their portfolios according the principles of efficient
diversification Slide 5 5 So whats wrong with - analysis The
assumptions of the last chapter appeared fully acceptableThe
assumptions of the last chapter appeared fully acceptable In fact
it may appear to be pedantic to mention them at all Why develop a
new model for risk-return if the present model aint broke?Why
develop a new model for risk-return if the present model aint
broke? Slide 6 6 -analysis: Estimation We did not spell it out, but
if you recall the mnemonic for obtaining the portfolio volatility
in the -model, (given n- shares in the portfolio,) we neededWe did
not spell it out, but if you recall the mnemonic for obtaining the
portfolio volatility in the -model, (given n- shares in the
portfolio,) we needed n-means (no problem) n-standard deviations
(no problem) n*(n-1)/2 correlations (? problem) Slide 7 7
-analysis: Estimation All parameters need estimation, and there are
n*(n+1)/2 + n parametersAll parameters need estimation, and there
are n*(n+1)/2 + n parameters Assume a portfolio of, say, 2,000
shares represent the market, then we need to estimate more than
2,000,000 parameters, most of which are correlationsAssume a
portfolio of, say, 2,000 shares represent the market, then we need
to estimate more than 2,000,000 parameters, most of which are
correlations Slide 8 8 -analysis: Estimation Recall that when you
estimate parameters, it is done with only a given level of
confidenceRecall that when you estimate parameters, it is done with
only a given level of confidence Confidence improves with the
number of observationsConfidence improves with the number of
observations In practice the parameters have time dependence, so
old data introduces errorIn practice the parameters have time
dependence, so old data introduces error For 2,000 shares, and a
99% confidence, about 20,000 parameters will be in errorFor 2,000
shares, and a 99% confidence, about 20,000 parameters will be in
error Slide 9 9 -analysis: Estimation The errors may, or may not,
be significant to your investment decision, but their existence
calls for further analysis In any case, the data collection,
verification, and processing, is a significant use of analytical
resources Slide 10 10 -analysis: Wishes After we have the estimated
parameters, finding the optimal portfolio requires quadratic
programming, and this again requires heavy use of computational
resourcesAfter we have the estimated parameters, finding the
optimal portfolio requires quadratic programming, and this again
requires heavy use of computational resources The problem is
similar to knowing the position and velocity of every star in the
Milky Way, and attempting to predict their futures by computing
individual interactions Slide 11 11 -analysis: Guidance Principles
for Simplification An important principle of financial modeling is
to create equations that capture the key factors parsimoniouslyAn
important principle of financial modeling is to create equations
that capture the key factors parsimoniously Another important
principle is to attempt to develop simple modelsAnother important
principle is to attempt to develop simple models Linear models are
then preferred to quadratic models Slide 12 12 The Astrophysics of
Finance In the Milky Way problem, an astronomer should specify
exactly what needs to be predicted, and give attention to the
variables that most affect it So, if he wants to know when the next
star will come close enough to Sol to disturb the Oort cloud then
close stars need individual analysis distant stars may be treated
homogeneously Slide 13 13 Specifying the Model In the last chapter
we examined diversifying a homogenous portfolio, and we observed
that there were two kinds of riskIn the last chapter we examined
diversifying a homogenous portfolio, and we observed that there
were two kinds of risk diversifiable or individual risk
Nondiversifiable or market risk Slide 14 14 Specifying the Model We
also observed that in the limit as the number of securities becomes
large, we obtained the formulaWe also observed that in the limit as
the number of securities becomes large, we obtained the formula
This formula tells us that the correlations are of crucial
importance in the relationship between a portfolio risk and the
stock risk Slide 15 15 Specifying the Model In the homogenous
model, we saw that there was individual- and market-riskIn the
homogenous model, we saw that there was individual- and market-risk
Assume that each equitys return is the composition of two random
variables:Assume that each equitys return is the composition of two
random variables: one associated with the markets return one
associated with the company-specific return Slide 16 16 Specifying
the Model: Assumptions Company-specific return on any stock
xCompany-specific return on any stock x is not correlated to the
company-specific return on any other stock y is correlated with the
market return The risk-free rate is constant during the investment
the periodThe risk-free rate is constant during the investment the
period Slide 17 17 Assumptions Investors forecasts agree with
respect to expectations, standard deviations, and correlations of
the returns of risky securities Therefore all investors hold risky
assets in the same relative proportions Investors behave optimally
In equilibrium, prices adjust so that aggregate demand for each
security is equal to its supplyIn equilibrium, prices adjust so
that aggregate demand for each security is equal to its supply
Slide 18 18 Market Portfolio Since every investors relative
holdings of the risky security is the same, the only way the asset
market can clear is if those optimal relative proportions are the
proportions in which they are valued in the market placeSince every
investors relative holdings of the risky security is the same, the
only way the asset market can clear is if those optimal relative
proportions are the proportions in which they are valued in the
market place Market PortfolioMarket Portfolio Slide 19 19 CML and
the CAPM CAPM says that in equilibrium, any investors relative
holding of risky assets will be the same as in the market
portfolioCAPM says that in equilibrium, any investors relative
holding of risky assets will be the same as in the market portfolio
Depending on their risk aversions, different investors hold
portfolios with different mixes of riskless asset and the market
portfolioDepending on their risk aversions, different investors
hold portfolios with different mixes of riskless asset and the
market portfolio Slide 20 20 CAPM Formula Slide 21 21 Active v.
Passive Management CAPM implies that, on average, the performances
of active portfolio managers is equal to that of passive managers
employing just the market portfolio and the risk-free securityCAPM
implies that, on average, the performances of active portfolio
managers is equal to that of passive managers employing just the
market portfolio and the risk-free security Diligent managers do
outperform passive managers, but only to the degree that their
diligence is rewardedDiligent managers do outperform passive
managers, but only to the degree that their diligence is rewarded
Slide 22 22 Reward Only for Market Risk The risk premium on any
individual security is proportional only to its contribution to the
risk of the market portfolio, and does not depend on its
stand-alone riskThe risk premium on any individual security is
proportional only to its contribution to the risk of the market
portfolio, and does not depend on its stand-alone risk Investors
are rewarded only for bearing market riskInvestors are rewarded
only for bearing market risk Slide 23 23 13.2 Determining the Risk
Premium on the Market Portfolio CAPM states thatCAPM states that
the equilibrium risk premium on the market portfolio is the product
of variance of the market, 2 Mvariance of the market, 2 M weighted
average of the degree of risk aversion of holders of risk,
Aweighted average of the degree of risk aversion of holders of
risk, A Slide 24 24 Comment CAPM explains the difference between
the riskless interest rate and the expected rate of return on the
market portfolio, but not their absolute levels The absolute level
of the equilibrium expected rate of return on the market portfolio
is determined by such factors as expected productivity household
inter-temporal preferences for consumption Slide 25 25 Example: To
Determine A Slide 26 26 13.3 Beta and Risk Premiums on Individual
Securities If risk is defined as that measure such that as it
increases, a risk-averse investor would have to be compensated by a
larger expected return in order that she would continue to hold it
in her optimal portfolio, then the measure of a securitys risk is
its beta, tells you how much the securitys rate of return changes
when the return on the market portfolio changes Slide 27 27
Comment: = 1 A security with a = 1 on average rises and falls with
the marketA security with a = 1 on average rises and falls with the
market a 10% (say) unexpected rise (fall) in the market return
premium will, on average, result in a 10% rise (fall) in the
securitys return premium Slide 28 28 Comment: 1 A security with a 1
on average rises and falls more than the marketA security with a 1
on average rises and falls more than the market With a = 1.3, a 10%
(say) unexpected rise (fall) in the market return premium will, on
average, result in a 13% rise (fall) in the securitys return
premium Such a security is said to be aggressiveSuch a security is
said to be aggressive Slide 29 29 Comment: 1 A security with a 1 on
average rises and falls less than the marketA security with a 1 on
average rises and falls less than the market With a = 0.7, a 10%
(say) unexpected rise (fall) in the market return premium will, on
average, result in a 7% rise (fall) in the securitys return premium
Such a security is said to be defensiveSuch a security is said to
be defensive Slide 30 30 CAPM Risk Premium on any Asset According
the the CAPM, in equilibrium, the risk premium on any asset is
equal the product ofAccording the the CAPM, in equilibrium, the
risk premium on any asset is equal the product of (or Beta) the
risk premium on the market portfolio Slide 31 31 Security Market
Line The plot of a securitys risk premium (or sometimes security
returns) against security beta Note that the slope of the security
market line is the market premiumNote that the slope of the
security market line is the market premium By CAPM theory, all
securities must fall precisely on the SML (hence its name)By CAPM
theory, all securities must fall precisely on the SML (hence its
name) Slide 32 32 Practical Example Some simulated data was
generated under the assumptions that: the market portfolio return
has an expected value of 0.15, a volatility of 0.20, and index 0 =
50 the share z has a return of 0.12, a volatility of 0.25, and
price 0 = 30 (no dividends)the market portfolio return has an
expected value of 0.15, a volatility of 0.20, and index 0 = 50 the
share z has a return of 0.12, a volatility of 0.25, and price 0 =
30 (no dividends) the correlation between the returns is 0.90; and
the risk-free rate is 0.05the correlation between the returns is
0.90; and the risk-free rate is 0.05 Slide 33 33 Slide 34 34 Data
Set Used In order to display the material clearly, only one year of
data is generated, and is collected monthly, resulting in 13 sets
of pricesIn order to display the material clearly, only one year of
data is generated, and is collected monthly, resulting in 13 sets
of prices In a real simulation, much more data must be collected in
order to provide an adequate confidence interval for parameter
estimatesIn a real simulation, much more data must be collected in
order to provide an adequate confidence interval for parameter
estimates Slide 35 35 Transformation of Prices into Returns The
prices are transformed into monthly holding period returns
(mhpr_Ind, and mhpr_Z) The mhprs are transformed into annual rates,
compounded annually The annual rates compounded annually are
transformed to annual rates compounded continuously Slide 36 36
Table of Prices Slide 37 37 Slide 38 38 Financial Calculators
Everything could have been done using a modern standard-issue
financial or scientific calculatorEverything could have been done
using a modern standard-issue financial or scientific calculator
Remember, the correct rate to use is the annual rate compounded
continuously, and that month-to-year conversions of standard
deviation involve a square root of 12 Take care to enter the market
rate as the independent variable, x Slide 39 39 Accuracy Issue We
assumed that the s and s are constants, but they are random
variables too In order to achieve adequate confidence, a large
sample is needed Small movements in price are masked by transaction
prices The result is a compromise between currency and
confidenceThe result is a compromise between currency and
confidence Slide 40 40 Model and Measured Values of Statistical
Parameters Slide 41 41 Comment The illustrated trajectory is
typical for monthly data collected over a yearThe illustrated
trajectory is typical for monthly data collected over a year
Caution: avoid using small data sets to estimate CAPM
parametersCaution: avoid using small data sets to estimate CAPM
parameters Slide 42 42 Regression Line The slope of the regression
line of dependent stock against independent market returns is
betaThe slope of the regression line of dependent stock against
independent market returns is beta Slide 43 43 Slide 44 44
Observation All securities, (not just efficient portfolios) plot
onto the SML, if they are correctly priced according to the CAPMAll
securities, (not just efficient portfolios) plot onto the SML, if
they are correctly priced according to the CAPM Slide 45 45 The
Beta of a Portfolio When determining the risk of a portfolioWhen
determining the risk of a portfolio using standard deviation
results in a formula thats quite complex using beta, the formula is
linear Slide 46 46 Computing Beta Here are some useful formulae for
computing betaHere are some useful formulae for computing beta
Slide 47 47 13.4 Using the CAPM in Portfolio Selection Whether or
not CAPM is a valid theory, indexing is attractive to investors
becauseWhether or not CAPM is a valid theory, indexing is
attractive to investors because historically it has performed
better than most actively managed portfolios it costs less to
implement that active management Slide 48 48 A Paradox Resolved The
last chapter posed a paradox with two securities co-existing, one
having a lower standard deviation and higher return than the
otherThe last chapter posed a paradox with two securities
co-existing, one having a lower standard deviation and higher
return than the other If we accept the CAPM as a valid theory, we
have a resolutionIf we accept the CAPM as a valid theory, we have a
resolution Both securities lie on the SML, and both securities lie
below the CMLBoth securities lie on the SML, and both securities
lie below the CML Slide 49 49 risk and risk A security has two
kinds of risk: risk that may be diversified away, and risk that is
associated with the marketA security has two kinds of risk: risk
that may be diversified away, and risk that is associated with the
market The CAPM theory states that the lower return on the riskier
security implies that it has a lower level of market risk, and this
is the only relevant risk The riskier security contains relatively
more (irrelevant) security-specific risk Slide 50 50 A Brand
Manager Most investors have the opportunity to eliminate most
individual risk from their portfolio; but consider a product
managers exposure to riskMost investors have the opportunity to
eliminate most individual risk from their portfolio; but consider a
product managers exposure to risk If a brand managers productsIf a
brand managers products perform well, promotion, higher salary, and
greater autonomy followperform well, promotion, higher salary, and
greater autonomy follow perform badly, humiliation, unemployment
and poverty followperform badly, humiliation, unemployment and
poverty follow Slide 51 51 A Brand Manager Now assume that a new
product is available for inclusion in the brand, but given its
-risk and expected return, it falls below the sml, and hence is not
in the investors interestsNow assume that a new product is
available for inclusion in the brand, but given its -risk and
expected return, it falls below the sml, and hence is not in the
investors interests The manager discovers that the new product
reduces his total risk, and acts in his own interests (rather than
the investors), and accepts the product (agency problem) Slide 52
52 The Portfolio Manager Remember (last chapter) we had not
resolved the issue how to evaluate the performance of a portfolio
manager, but given the CAPM a resolution is at handRemember (last
chapter) we had not resolved the issue how to evaluate the
performance of a portfolio manager, but given the CAPM a resolution
is at hand If your portfolio is producing actual returns with a
lower beta than the sml specifies (with statistical significance),
then you should certainly not be firedIf your portfolio is
producing actual returns with a lower beta than the sml specifies
(with statistical significance), then you should certainly not be
fired Slide 53 53 The Portfolio Manager The further a well
diversified portfolio consistently lies above (below) the sml, the
better (worse) the fund managers performanceThe further a well
diversified portfolio consistently lies above (below) the sml, the
better (worse) the fund managers performance There are several
measures of this distance, but this topic is better left for
another day Slide 54 54 How to Win Investment Games You may have
been asked to take part in an investment game where you given
$100,000 to manage for a semester; winner takes all The
overwhelming chances are that the winning student uses poor
financial practices Slide 55 55 How to Win Investment Games
(Continued) The criteria of success for the game differs
significantly from real-life investing, so your strategy for
winning is likely to be different If you diversify away
unsystematic risk--even if you have some kind of informational
advantage over your competition--you are very unlikely to win the
game To win, you need individual risk to separate you from the
crowd Unlike a real investor you dont have real downside-risk your
upside-potential materializes only by being first Slide 56 56 13.5
Valuation and Regulating Rates of Return Beta may be used to obtain
the discount factor for a projectBeta may be used to obtain the
discount factor for a project Assume a project is similar to the
projects undertaken by another firm, BetafulAssume a project is
similar to the projects undertaken by another firm, Betaful Betaful
is financed by 20% short-term debt, and 80% equity, and its is 1.3
(assume debt is risk-free)Betaful is financed by 20% short-term
debt, and 80% equity, and its is 1.3 (assume debt is risk-free)
Your optimal capital structure is 40% (risk- free) debt, and 60%
equityYour optimal capital structure is 40% (risk- free) debt, and
60% equity Slide 57 57 Valuation and Regulating Rates of Return
Assume the market rate is 15%, and the risk- free rate is 5%Assume
the market rate is 15%, and the risk- free rate is 5% Compute the
beta of Betafuls operationsCompute the beta of Betafuls operations
Slide 58 58 Valuation and Regulating Rates of Return Beta of
Betafuls operations is equal to the beta of our new operationBeta
of Betafuls operations is equal to the beta of our new operation To
find the required return on the new project, apply the CAPMTo find
the required return on the new project, apply the CAPM Slide 59 59
Valuation and Regulating Rates of Return Assume that your company
is just a vehicle for the new project, then the beta of your
unquoted equity isAssume that your company is just a vehicle for
the new project, then the beta of your unquoted equity is Slide 60
60 Valuation and Regulating Rates of Return Assume that your
company has an expected dividend of $6 next year, and that it will
grow annually at a rate of 4% forever, the value of a share
isAssume that your company has an expected dividend of $6 next
year, and that it will grow annually at a rate of 4% forever, the
value of a share is Slide 61 61 Valuation and Regulating Rates of
Return Regulators use the CAPM to establish a fair rate of return
on invested capital in public utilities, given the level of
riskRegulators use the CAPM to establish a fair rate of return on
invested capital in public utilities, given the level of risk
