Capital Asset Pricing Model
Introduction
• The CAPM was developed in the mid 1960’s by
three researchers William Sharp, John Linter
and Jan Mossin independently.
• CAPM is really the extension of Markowitz
Portfolio theory.
• The PT is a description how a rational investor
should build efficient portfolio and select the
optimal portfolio.
• The CAPM derives the relationship btw the
expected return and risk of individual securities
and portfolios in the capital market if every one
behaved in the way the PT suggest.
Fundamental notions of PT.
•
•
•
•
•
•
•
•
•
Risk and return are two important characteristics of every investment.
Risk is measured by the variability in returns and investors attempt to
reduce the variability of returns through diversification of investments.
With a given set of securities, any number of portfolios may be created by
altering the weight of investment in each security.
Diversification help to reduce risk, but a well diversified portfolio does not
become risk free.
Portfolio M is the most diversified portfolio.
Some variabilities are undiversified e.g systematic risk.
The real risk of a portfolio is the market risk which can’t be eliminated
through diversification.
A rational investor would expect the return on a security to be
commensurate with the risk.
The CAPM gives the nature of the relationship btw the expected return and
the systematic risk of a security.
Assumptions of CAPM
• Investors make their investment decisions on the basis
of risk- return analysis.
• The purchase and sale of a security can be undertaken
in infinitely divisible units.
• Purchase and sale by a single investor can not effect the
prices.
• There is no transaction cost.
• There is no personal income taxes.
• The investor can lend and borrow any amount of funds
at risk free rate.
• The investor can sell short any amount of any share.
• The investors share the homogeneity of expectations
from the market.
Efficient frontier with riskless
lending and borrowing.
• The PT deals with the portfolio of risky assets.
• According to the theory, an investor faces an efficient
frontier containing the set of efficient portfolios of risky
assts.
• Now it is assumed that there is a riskless asset available
in the market.
• A riskless asset is one whose return is certain e.g. Tbills.
• The investor can invest a certain portion of his
investment in the riskless assets which would be
equivalent to lending at the risk free asset’s rate of return
namely Rf.
Efficient frontier with riskless
lending and borrowing……
• Similarly it may be assumed that an investor
may borrow at the risk free rate for the purpose
of investing in a portfolio of risky assets.
• The EF arises from a feasible set of portfolios of
risky assets is concave in shape.
• When an investor is assumed to use riskless
lending and borrowing in his investment activity
the shape of the EF transforms into a straight
line.
• Let see the example of portfolio A, B and C.
Efficient frontier with riskless
lending and borrowing……
• Portfolio B is and optimal portfolio with Rp=15%, σp= 8%
Rf= 7% ώ = 40% in risk free asset and 60% is risky asset.
• Rc = ώRm + (1-ώ)Rf
• σc= ώσm + (1-ώ) σf
• σc= ώσm
•
Now for borrowing funds by the investor
• ώ = 1 the investor’s funds is fully committed to the risky
portfolio.
• ώ < 1, only a portion of the funds is invested in the risky
portfolio.
• ώ > 1, it means the investor is borrowing at the risk free
rate and investing an amount larger than his own
investments in the risky portfolio.
Efficient frontier with riskless
lending and borrowing……
• Rc = ώRm - (ώ-1)Rf
• The first term of the equation represents the
gross return earned by investing the borrowed
funds as well as investor’s own funds in the risky
portfolio.
• The second term of the equation represents the
const of borrowing funds which is deducted from
the gross return to obtain the net return of
levered portfolio.
• σl= ώσm
Efficient frontier with riskless
lending and borrowing……
• The return and risk of the levered portfolio are larger than those of
risky portfolio.
• The risk return of all levered portfolios would lie in a straight line to
the right of the risky portfolio B.
• So the introduction of borrowing and lending gives us an efficient
frontier that is a straight line through line.
• The line segment from Rf to B includes all the combinations of the
risky portfolios and the risk free assets.
• The line segment beyond the point B represents all the levered
portfolios ( that is combinations of the risky portfolio with borrowing)
• Borrowing increase risk and return and lending reduces the risk and
return of the portfolio.
• Thus the investor can use the borrowing and lending the attain the
desired level of risk.
• More risk aversor well prefer lending and less risk aversor well
prefer borrowing.
Capital Market Line
• All investors are assumed to have identical expectations.
• Every investor will seek to combine the same risky
portfolio B with different levels of lending or borrowing
according to his desired level of risk.
• Because all investors hold the same risky portfolio, then
it will include all risky securities in the market.
• This known as market portfolio M.
• Each security will be held in the proportion which the
market value of the security bears to the total market
value of all risky securities in the market.
• All investors will hold combination of only two assets,
Market portfolio and a riskless security.
CML
• All the combination will lie along the straight line
representing the EF.
• This line is formed by the action all investors
mixing the market portfolio with the risk free
asset is known as the CML.
• Re = Rf + [(Rm – Rf)/σm] σe
• Where the subscript e denotes an efficient portfolio.
• The risk free return Rf represents the reward for waiting. In the other
words the price of time.
• The term (Rm – Rf)/σm represents the price of risk or risk
premium.
CML
• When the risk of the Efficient Portfolio σe
• is multiplied with this term, we get the risk
premium available for the particular
efficient portfolio under consideration.
• (Expected return) = (Price of time) + (Price
of risk) ( Amount of risk)
• So the CML provides the risk return
relationship and a measure of risk for
efficient portfolio.
Security market line
• The CML shows the risk and return relationship and would all lie
along the capital market line.
• All portfolios other than the efficient ones will lie below the capital
market line.
• The CAPM specifies the relationship between expected return and
risk for all securities and all portfolios, whether efficient or inefficient.
• Only relevant risk is systematic risk measure by beta β.
• It follows that the expected return of a security or of a portfolio
should be related to the risk of that security or portfolio measured by
β.
• β is the measure of sensitivity of the security to changes in the
market.
• β of the market is one.
• The relationship between the expected return and β can be
determined graphically.
• The straight line joining these two points is as SML.
Security market line
• Mathematically this relationship can be
written as:
• Ri = Rf + βi(Rm – Rf)
• The risk premium is directly proportional
the β.
• Expected return on a security = risk free rate + ( beta x
risk premium of market)
CAPM
•
A model that describes the relationship between risk and expected
return and that is used in the pricing of risky securities.
•
The general idea behind CAPM is that investors need to be compensated in
two ways: time value of money and risk. The time value of money is
represented by the risk-free (rf) rate in the formula and compensates the
investors for placing money in any investment over a period of time. The
other half of the formula represents risk and calculates the amount of
compensation the investor needs for taking on additional risk. This is
calculated by taking a risk measure (beta) that compares the returns of the
asset to the market over a period of time and to the market premium (Rmrf).
Short Comings of CAPM
•
•
•
•
•
The model assumes that either asset returns are (jointly) normally distributed random variables or
that investors employ a quadratic form of utility. It is however frequently observed that returns in
equity and other markets are not normally distributed. As a result, large swings (3 to 6 standard
deviations from the mean) occur in the market more frequently than the normal distribution
assumption would expect.[3]
The model assumes that the variance of returns is an adequate measurement of risk. This might
be justified under the assumption of normally distributed returns, but for general return
distributions other risk measures (like coherent risk measures) will likely reflect the investors'
preferences more adequately. Indeed risk in financial investments is not variance in itself, rather it
is the probability of losing: it is asymmetric in nature.
The model assumes that all investors have access to the same information and agree about the
risk and expected return of all assets (homogeneous expectations assumption).
The model assumes that the probability beliefs of investors match the true distribution of returns. A
different possibility is that investors' expectations are biased, causing market prices to be
informationally inefficient. This possibility is studied in the field of behavioral finance, which uses
psychological assumptions to provide alternatives to the CAPM such as the overconfidence-based
asset pricing model of Kent Daniel, David Hirshleifer, and Avanidhar Subrahmanyam (2001)[4].
The model does not appear to adequately explain the variation in stock returns. Empirical studies
show that low beta stocks may offer higher returns than the model would predict. Some data to
this effect was presented as early as a 1969 conference in Buffalo, New York in a paper by
Fischer Black, Michael Jensen, and Myron Scholes. Either that fact is itself rational (which saves
the efficient-market hypothesis but makes CAPM wrong), or it is irrational (which saves CAPM,
but makes the EMH wrong – indeed, this possibility makes volatility arbitrage a strategy for
reliably beating the market).
Short Comings of CAPM
•
•
•
•
The model assumes that given a certain expected return investors will prefer lower
risk (lower variance) to higher risk and conversely given a certain level of risk will
prefer higher returns to lower ones. It does not allow for investors who will accept
lower returns for higher risk. Casino gamblers clearly pay for risk, and it is possible
that some stock traders will pay for risk as well
The model assumes that there are no taxes or transaction costs, although this
assumption may be relaxed with more complicated versions of the model
The market portfolio consists of all assets in all markets, where each asset is
weighted by its market capitalization. This assumes no preference between markets
and assets for individual investors, and that investors choose assets solely as a
function of their risk-return profile. It also assumes that all assets are infinitely
divisible as to the amount which may be held or transacted
The market portfolio should in theory include all types of assets that are held by
anyone as an investment (including works of art, real estate, human capital...) In
practice, such a market portfolio is unobservable and people usually substitute a
stock index as a proxy for the true market portfolio. Unfortunately, it has been shown
that this substitution is not innocuous and can lead to false inferences as to the
validity of the CAPM, and it has been said that due to the inobservability of the true
market portfolio, the CAPM might not be empirically testable. This was presented in
greater depth in a paper by Richard Roll in 1977, and is generally referred to as Roll's
critique
Short Comings of CAPM
• The model assumes just two dates, so that there is no
opportunity to consume and rebalance portfolios
repeatedly over time. The basic insights of the model are
extended and generalized in the intertemporal CAPM
(ICAPM) of Robert Merton, and the consumption CAPM
(CCAPM) of Douglas Breeden and Mark Rubinstein.
• CAPM assumes that all investors will consider all of their
assets and optimize one portfolio. This is in sharp
contradiction with portfolios that are held by individual
investors: humans tend to have fragmented portfolios or,
rather, multiple portfolios: for each goal one portfolio
SML Vs CML
• 1. The CML is a line that is used to show the rates of return, which
depends on risk-free rates of return and levels of risk for a specific
portfolio. SML, which is also called a Characteristic Line, is a
graphical representation of the market’s risk and return at a given
time.
• 2. While standard deviation is the measure of risk in CML, Beta
coefficient determines the risk factors of the SML.
• 3. While the Capital Market Line graphs define efficient portfolios,
the Security Market Line graphs define both efficient and nonefficient portfolios.
• 4. The Capital Market Line is considered to be superior when
measuring the risk factors.
• 5. Where the market portfolio and risk free assets are determined by
the CML, all security factors are determined by the SML.
Pricing of Securities with CAPM
• The CAPM provides a framework for
assessing whether the a security is under
priced, overpriced or correctly priced.
• Under priced when the estimated return is
more than the expected return.
• Over priced when the estimated return is
less than the expected return.
• Correctly price when the estimated return
is equal to expected retun.