The Capital Asset Pricing Model CAPM

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Topic 4 The Capital Asset Pricing Model (CAPM)

• The CAPM
• ? The market portfolio
• ? The capital market line
• ? The risk premium on the market portfolio
• ? Expected returns on individual securities
• ? The security market line
• Some extensions of the CAPM

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The CAPM

• An equilibrium model specifying the relationship
  between risk and expected return on risky assets.
  
• Assumptions
• Investors are price-takers (i.e. their trades do
  not affect security prices).
• Investors have a single-period investment horizon.

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• Investments are limited to publicly traded
  financial assets (e.g. stocks bonds), and to
  risk-free borrowing or lending arrangements.
• Investors pay no taxes on returns and no
  transaction costs (commissions and
  servicecharges) on trades in securities.
• All investors are rational mean-variance
  optimizers.
• All investors have the same expectations (i.e.
  identical estimates of expected returns,
  variances, and covariances among all assets).

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Model implication 1 The market portfolio

• All investors will choose to hold a portfolio of
  risky assets in proportions that duplicate
  representation of the assets in the market
  portfolio (M), which includes all traded assets.
• The proportion of each asset in the market
  portfolio equals the market value of the asset
  divided by the total market value of all assets.

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• Why?
• All investors arrive at the same
  determination of the optimal risky portfolio, the
  portfolio on the efficient frontier identified by
  the tangency line from T-bills to that frontier.
• As a result, the optimal risky portfolio of
  all investors is simply the market portfolio.

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Model implication 2 The capital market line (CML)

• Not only will the market portfolio be on the
  efficient frontier, but it also will be
  thetangency portfolio to the optimal capital
  allocation line (CAL) derived by every investor.
• As a result, the capital market line (CML),
  the line from the risk-free rate through the
  market portfolio, M, is also the best attainable
  capital allocation line.
• All investors hold M as their optimal risky
  portfolio, differing only in the amount invested
  in it versus in the risk-free asset.

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• Mutual fund theorem
• Since all investors choose to hold a market
  index mutual fund, we can separate portfolio
  selection into 2 components
• a technological problem creation of market index
  mutual funds by professional managers
• a personal problem depends on an investors risk
  aversion, allocation of the complete portfolio
  between the mutual fund and risk-free assets.
• Note In reality, different investment
  managers do create risky portfolios that differ
  from the market index (in part due to the use of
  different input lists in the formation of the
  optimal risky portfolio).

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Model implication 3 Risk premium on M

• The risk premium on the market portfolio will be
  proportional to its risk and the degree of risk
  aversion of the representative investor
• where
• E(rM) expected return on M
• variance of M
• average degree of risk aversion
  across
• investors

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• Recall Each individual investor chooses a
  proportion y, allocated to the optimal portfolio
  M, such that
• Risk-free investments involve borrowing and
  lending among investors. Any borrowing position
  must be offset by the lending position of the
  creditor. This means that net borrowing and
  lending across all investors must be zero, and
  thus the average position in the risky portfolio
  is 100, or y 1.
• Setting y 1 and rearranging, we obtain the
  risk premium on the market portfolio.

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Model implication 4 Expected returns on
individual securities

• The contribution of an asset to the risk of the
  market portfolio
• Risk-averse investors measure the risk of
  the optimal risky portfolio (i.e. the market
  portfolio) by its variance.
• We would expect the reward, or the risk
  premium on individual assets, to depend on the
  contribution of the individual asset to the
  variance of the market portfolio.

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To calculate the variance of the market
portfolio, we use the following bordered
covariance matrix with the market portfolio
weights
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The contribution of GEs stock to the
variance of the market portfolio
The covariance of GE with the market portfolio is
proportional to the contribution of GE to the
variance of the market portfolio. In other
words, we can measure an assets contribution to
the risk of the market portfolio by its
covariance with the market portfolio.
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• The reward-to-risk ratio for investments in GE

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• The market portfolio has a risk premium of E(rM)
  - rf and a variance of , for a
  reward-to-risk ratio of
• This ratio is called the market price of
  risk, because it quantifies the extra return that
  investors demand to bear portfolio risk (i.e.
  tells us how much extra return must be earned per
  unit of portfolio risk).

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• In equilibrium, all investments should offer the
  same reward-to-risk ratio.
• If the ratio were better for one investment
  than another, investors would rearrange their
  portfolios, tilting toward the alternative with
  the better trade-off and shying from the other.
• Such activity would impact on security
  prices until the ratios were equalized.

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?In equilibrium, the reward-to-risk ratio of GE
stock must equal that of the market portfolio.
Otherwise, if the reward-to-risk ratio of GE gt
the markets, investors can increase their
portfolio reward for bearing risk by increasing
the weight of GE in their portfolio. Until
the price of GE stock rises relative to the
market, investors will keep buying GE stock.
The process will continue until stock prices
adjust so that reward-to-risk ratio of GE equals
that of the market.
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The same process, in reverse, will equalize
reward-to-risk ratios when GEs initial
reward-to-risk ratio lt that of the market
portfolio. ? ? ?
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The ratio measures
the contribution of GE stock to the variance of
the market portfolio as a fraction of the total
variance of the market portfolio. The
ratio is called beta and is denoted by ?. ?
The expected return-beta relationship
More generally, for any asset i
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Note Risk-averse investors measure the risk
of the optimal risky portfolio by its variance.
We would expect the reward (the risk premium on
individual assets) to depend on the contribution
of the individual asset to the risk of the
portfolio. The beta of an asset measures
the assets contribution to the variance of the
market portfolio as a fraction of the total
variance of the market portfolio. Hence we
expect, for any asset, the required risk premium
to be a function of beta. The CAPM
confirms this intuition the securitys risk
premium is directly proportional to both the beta
the risk premium of the market portfolio that
is, the risk premium equals .
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• If the expected return-beta relationship holds
  for any individual asset, it must hold for any
  combination of assets.
• Suppose that some portfolio P has weight wk
  for stock k, where k takes on values 1, , n.
• Then
• where E(rP)

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• This result has to be true for the market
  portfolio itself
• Note
• The market beta (i.e. the weighted average beta
  of all assets) is 1.
• Betas gt 1 are considered aggressive in that
  investment in high-beta assets entails
  above-average sensitivity to market swings.
• Betas lt 1 can be described as defensive.

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Model implication 5 The security market line
(SML)

• The expected return-beta relationship can be
  portrayed graphically as the security market line
  (SML).
• Because the market beta is 1, the slope is
  the risk premium of the market portfolio.

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M
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Compare SML to CML

• The CML graphs the risk premiums of efficient
  portfolios (i.e., portfolios composed of the
  market and the risk-free asset) as a function of
  portfolio standard deviation.
• This is appropriate because standard
  deviation is a valid measure of risk for
  efficiently diversified portfolios that are
  candidates for an investors overall portfolio.

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• The SML, in contrast, graphs individual asset
  risk premiums as a function of asset risk.
• The relevant measure of risk for individual
  assets held as parts of well-diversified
  portfolios is not the assets standard deviation
  or variance.
• It is, instead, the contribution of the
  asset to the portfolio variance, which we measure
  by the assets beta.
• The SML is valid for both efficient
  portfolios and individual assets.

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Uses of the SML

• The SML provides a benchmark for the evaluation
  of investment performance.
• Given the risk of an investment, as measured
  by its beta, the SML provides the required rate
  of return necessary to compensate investors for
  risk.
• Because the SML is the graphic
  representation of the expected return-beta
  relationship, fairly priced assets plot exactly
  on the SML that is, their expected returns are
  commensurate with their risk.

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The difference between the fair and actually
expected rates of return on an asset is called
the assets alpha, denoted ?. e.g. The
market return is expected to be 14, a stock has
a beta of 1.2, and the T-bill rate is 6.
The SML would predict an expected return on the
stock of 6 1.2(14 6) 15.6. If one
believed the stock would provide an expected
return of 17, the implied alpha would be 1.4 (
17 - 15.6).
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• The CAPM is also useful in capital budgeting
  decisions.
• For a firm considering a new project, the
  CAPM can provide the required rate of return that
  the project needs to yield, based on its beta, to
  be acceptable to investors.
• Managers can use the CAPM to obtain this
  cutoff internal rate of return (IRR), or hurdle
  rate for the project.

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Q The risk-free rate is 6 and the expected
return on the market portfolio is 14.
A firm considers a project that is expected
to have a beta of 0.6. a. What is
the required rate of return on the
project? b. If the expected internal rate of
return of the project is 19, should it
be accepted?
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The CAPM tells us that an acceptableexpected
rate of return for the project is which
becomes the projects hurdle rate. If the
internal rate of return of the project is 19,
then it is desirable. Any project with an
rate of return ? 10.8 should be rejected.
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The CAPM and the Index Model
Recall For any asset i and the
(theoretical) market portfolio, the CAPM
expected return-beta relationship is
Recall The index model (in excess return
form) or
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• From the index model, the covariance between the
  returns on stock i and the market index
• Notes
• We can drop ?i from the covariance terms because
  ?i is a constant and thus has zero covariance
  with all variables.
• The firm-specific or nonsystematic component is
  independent of the marketwide or systematic
  component (i.e. Cov(ei, RM) 0).

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• ?
• ? The index model beta coefficient turns out to
  be the same beta as that of the CAPM expected
  return-beta relationship, except that we replace
  the (theoretical) market portfolio of the CAPM
  with the well-specified and observable market
  index.

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• If the index M in the index model represents the
  true market portfolio, we can take the
  expectation of each side of the index model
• A comparison of the index model relationship
  to the CAPM expected return-beta relationship
  shows that the CAPM predicts that ?i should be
  zero for all assets.
• The alpha of a stock is its expected return
  in excess of (or below) the fair expected return
  as predicted by the CAPM.
• If the stock is fairly priced, its alpha
  must be zero.

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Thus, if we estimate the index model for
several firms, using the index model as a
regression equation, we should find that the ex
post or realized alphas (the regression
intercepts) for the firms in our sample center
around zero. The CAPM states that the
expected value of alpha is zero for all
securities, whereas the index model
representation of the CAPM holds that the
realized value of alpha should average out to
zero for a sample of historical observed returns.
Empirical example Michael Jensen
examined the alphas realized by U.S. mutual
funds.
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He shows these alphas indeed seem to be
distributed around zero.
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• The market model
• The market model states that the return
  surprise of any security is proportional to the
  return surprise of the market, plus a
  firm-specific surprise
• If the CAPM is valid
• and

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• The market model equation becomes identical
  to the index model.
• Thus, the terms index model and market
  model are used interchangeably.

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Some Extensions of the CAPM
Risk-free lending but no risk-free borrowing

• Zero-beta portfolio
• Every portfolio on the efficient frontier
  (except for the global minimum-variance
  portfolio) has a companion portfolio on the
  bottomhalf (the inefficient part) of the
  minimum-variance frontier with which it
  isuncorrelated.
• Because the portfolios are uncorrelated, the
  companion portfolio is referred to as the
  zero-beta portfolio of the efficient portfolio.

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Efficient portfolios and their zero-beta
companions
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From any efficient portfolio such as P draw
a tangency line to the vertical axis. The
intercept will be the expected return on
portfolio Ps zero-beta companion portfolio,
denoted Z(P). The horizontal line from the
intercept to the minimum-variance frontier
identifies the standard deviation of the
zero-beta portfolio. Notice that different
efficient portfolios such as P and Q have
different zero-beta companions.
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• The market portfolio M has a companion zero-beta
  portfolio on the minimum-variance frontier, Z,
  with an expected rate of return E(rZ).
• Note that since M and Z are uncorrelated, we
  have covrM, rZ 0.
• The zero-beta CAPM
• (i.e. rf has been replaced by E(rZ))
• Note The zero-beta CAPM is still valid when (i)
  there is no risk-free asset at all or (2) the
  borrowing rate is higher than the lending rate.
  

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Liquidity and the CAPM

• Liquidity refers to the cost and ease with which
  an asset can be converted into cash (i.e. sold).
• Recall one assumption of the CAPM that all
  trading is costless. In reality, no security is
  perfectly liquid, in that all trades involve some
  transaction cost.
• Investors prefer more liquid assets with
  lower transaction costs, so that all else equal,
  relatively illiquid assets trade at lower prices
  (i.e. the expected return on illiquid assets must
  be higher). Thus, an illiquidity premium must be
  impounded into the price of each asset.

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• Trading costs
• Commission.
• Bid-asked spread
• bid price the price at which a dealer is willing
  to purchase a security.
• asked price the price at which a dealer will
  sell a security.
• bid-asked spread the difference between a
  dealers bid and asked price.

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• A simple example
• Consider a world in which market risk premium is
  ignored. Thus, the expected rate of return on
  all securities will equal the risk-free rate
  (rf).
• Assume that there are only two classes of
  securities liquid (L) and illiquid (I).
• The liquidation cost cL of a class L stock
  to an investor with an investment horizon of h
  years will reduce the per-period rate of return
  by cL/h.
• Thus, if you intend to hold a class L
  security for h periods, your expected rate of
  return net of transaction costs is rf - cL/h.

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Class I assets have higher liquidation
costs cI that reduce the per-period return by
cI/h, where cI gt cL. Thus, if you intend
to hold a class I security for h periods, your
expected rate of return net of transaction costs
is rf - cI/h. These net rates of return
would be inconsistent with a market in
equilibrium, because with equal gross rates of
return (rf) all investors would prefer to invest
in zero-transaction-cost asset (the risk-free
asset). As a result, both class L and
class I stock prices must fall, causing their
expected returns to rise until investors are
willing to hold these shares.
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Suppose, therefore, that each gross return
is higher by some fraction of liquidation cost.
Specifically, assume that the gross
expected return on class L stocks is rf xcL and
that of class I stocks is rf ycI. The
net rate of return on class L stocks to an
investor with a horizon of h (rf xcL) -
cL/h rf cL(x - 1/h). The net rate of
return on class I stocks (rf ycI) - cI/h
rf cI(y - 1/h).
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Now we can determine equilibrium
illiquidity premiums. For the marginal
investor with horizon h, the net return from
class I and L stocks is the same rf
cL(x - 1/h) rf cI(y - 1/h) ? ?
The expected gross return on illiquid stocks
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Recall that the expected gross return on
class L stocks is rL rf cLx. Thus, the
illiquidity premium of class I versus class L
stocks is As expected, equilibrium
expected rates of return are bid up to compensate
for transaction costs.
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• If we allow for market risk premium, we would
  find that the illiquidity premium is simply
  additive to the risk premium of the usual CAPM.
• Thus, we can generalize the CAPM expected
  return-beta relationship to include a liquidity
  effect
• where f(ci) is a function of trading costs
  ci that measures the effect of the illiquidity
  premium given the trading costs of security i.
• The usual CAPM equation is modified because
  each investors optimal portfolio is now affected
  by liquidation cost as well as risk-return
  considerations.

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• Trading frequency may well vary inversely with
  trading costs.
• An investor who plans to hold a security for
  a given period will calculate the impact of
  illiquidity costs on expected rate of return
  illiquidity costs will be amortized over the
  anticipated holding period.
• Investors who trade less frequently thus
  will be less affected by high trading costs.
• The reduction in the rate of return due to
  trading costs is lower, the longer the security
  is held.

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Hence, in equilibrium, investors with long
holding periods will, on average, hold more of
the illiquid securities, while short-horizon
investors will more strongly prefer liquid
securities. This clientele effect
mitigates the effect of the bid-ask spread for
illiquid securities. The end result is
that the illiquidity premium should increase with
the bid-ask spread at a decreasing rate.
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The Relationship Between Illiquidity and Average
Returns