The Capital Asset Pricing Model

1
The Capital Asset Pricing Model

• The Risk Return Relation Formalized

2
Summary

• As we discussed, the market pays investors for
  two services they provide (1) surrendering their
  capital and so forgoing current consumption and
  (2) sharing in the total risk of the economy.
• The first gets you the time value of money.
• The second gets you a risk premium whose size
  depends on the share of total risk you take on.
• From this we wrote E(R) Rf ?
• We refine this to E(R) Rf Units Price

3
Summary

• We need a reference for measuring risk and choose
  the total risk the market has to divide or the
  market portfolio as that reference.
• Define the market portfolio as having one unit of
  risk (Var(Rm) 1 unit of risk). Other assets
  will be evaluated relative to this definition of
  one unit of risk.
• From E(R) Rf Units Price we can see that
  Price E(Rm) Rf. (Note Units 1 for Rm.)
• In other words we also identified the price per
  unit risk (the market risk premium).

4
Summary

• The hard part is to show that any assets
  contribution to the total risk of the economy or
  Var(Rm) is determined not by Var(Ri) but rather
  by Cov(Ri, Rm).
• Standardize Cov(Ri, Rm) so that we measure the
  risk of each asset relative to our definition of
  one unit and we get beta
• ßi Cov(Ri, Rm)/Var(Rm)
• The number of units of risk for asset i is ßi.
• So E(Ri)Rf ßi(E(Rm) Rf)Rf Units Price.

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Risk and Return

• When we are concerned with only one asset its
  risk and return can be measured, as discussed,
  using expected return and variance of return.
• If there is more that one asset (so portfolios
  can be formed) risk becomes more complex.
• We will show there are two types of risk for
  individual assets
• Diversifiable/nonsystematic/idiosyncratic risk
• Nondiversifiable/systematic/market risk
• Diversifiable risk can be eliminated without cost
  by combining assets into portfolios. (Big Wow.)

6
Diversification

• One of the most important lessons in all of
  finance concerns the power of diversification.
• Part of the total risk of any asset can be
  diversified away (its effect on portfolio risk
  is eliminated) without any loss in expected
  return (i.e. without cost).
• This also means that no compensation needs to be
  provided to investors for exposing their
  portfolios to this type of risk.
• Why should the economy pay you to hold risk that
  you can get rid of for free (or which is not part
  of the aggregate risk that all agents must some
  how share).
• This in turn implies that the risk/return
  relation is actually a systematic risk/return
  relation.

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Diversification Example

• Suppose a large green ogre has approached you and
  demanded that you enter into a bet with him.
• The terms are that you must wager 10,000 and it
  must be decided by the flip of a coin, where
  heads he wins and tails you win.
• What is your expected payoff and what is your
  risk?

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Example

• The expected payoff from such a bet is of course
  0 if the coin is fair.
• We can calculate the standard deviation of this
  position as 10,000, reflecting the wide swings
  in value across the two outcomes (winning and
  losing).
• Can you suggest another approach that stays
  within the rules?

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Example

• If instead of wagering the whole 10,000 on one
  coin flip think about wagering 1 on each of
  10,000 coin flips.
• The expected payoff on this version is still 0
  so you havent changed the expectation.
• The standard deviation of the payoff in this
  version, however, is 100.
• Why the change?
• If we bet a penny on each of 1,000,000 coin
  flips, the risk, measured by the standard
  deviation of the payoff, is 10. The expected
  payoff is of course still 0.

10
Example

• The example works so well at reducing risk
  because coin flips are independent.
• If the coins were somehow perfectly correlated we
  would be right back in the first situation.
• Suppose all flips after the first always landed
  the same way, what good is bothering with 10,000
  flips?
• With one dollar bets on 10,000 flips, for flip
  correlations between zero (independence) and one
  (perfect correlation) the measure of risk lies
  between 100 and 10,000.
• This is one way to see that the way an asset
  contributes to the risk of a large portfolio is
  determined by its correlation or covariance with
  the other assets in the portfolio.

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Covariances and Correlations The Keys to
Understanding Diversification

• When thinking in terms of probability
  distributions, the covariance between the returns
  of two assets (A B) equals Cov(A,B) ?AB
• When estimating covariances from historical data,
  the estimate is given by
• Note An assets variance is its covariance with
  itself.

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Correlation Coefficients

• Covariances are difficult to interpret. Only the
  sign is really informative. Is a covariance of
  20 big or small?
• The correlation coefficient, ?, is a normalized
  version of the covariance given by
• Correlation CORR(A,B)
• The correlation will always lie between 1 and -1.
• A correlation of 1.0 implies ...
• A correlation of -1.0 implies ...
• A correlation of 0.0 implies ...

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• Risk and Return in Portfolios Example
• Two Assets, A and B
• A portfolio, P, comprised of 50 of your total
  investment
• invested in asset A and 50 in B.
• There are five equally probable future outcomes,
  see below.

• In this case
• VAR(RA) 191.6, STD(RA) 13.84, and E(RA)
  16.
• VAR(RB) 106.0, STD(RB) 10.29, and E(RB)
  12.
• COV(RA,RB) 21
• CORR(RA,RB) 21/(13.8410.29) .1475.
• VAR(RP)84.9, STD(Rp)9.21, E(Rp)14½ E(RA) ½
  E(RB)
• Var(Rp) or STD(RP) is less than that of either
  component!

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What risk return combinations would be possible
with different weights?
Asset A

½ and ½ portfolio

Asset B

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What risk return combinations would be possible
with a different correlation between A and B?
Asset A

Asset B

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Symbols The Variance of aTwo-Asset Portfolio

• For a portfolio of two assets, A and B, the
  portfolio variance is

Or,
For the two-asset example considered
above Portfolio Variance .52(191.6)
.52(106.0)
2(.5)(.5)21
84.9 (check for yourself)
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For General Portfolios

• The expected return on a portfolio is the
  weighted average of the expected returns on each
  asset. If wi is the proportion of the investment
  invested in asset i, then

• Note that this is a linear relationship.

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For General Portfolios

• The variance of the portfolios return is given
  by
• Not simple and not linear but very powerful.

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In A Picture (N 2)
Var(RA) Cov(RA, RB)
Cov(RB, RA) Var(RB)
Portfolio variance is a weighted sum of these
terms.
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In A Picture (N 3)
Var(RA) Cov(RA,RB) Cov(RA,RC)
Cov(RB,RA) Var(RB) Cov(RB,RC)
Cov(RC,RA) Cov(RC,RB) Var(RC)
Portfolio variance is a weighted sum of these
terms.
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In A Picture (N 10)










Portfolio variance is a simple weighted sum of
the terms in the squares. The blue are
covariances and the white the variance terms.
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In A Picture (N 20)




















Which squares are becoming more important?
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It is important to note that the level of
correlation or equivalently the level of
systematic risk is a choice you make.
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Implications of Diversification

• Diversification reduces risk. If asset returns
  were uncorrelated on average, diversification
  could eliminate all risk. They are actually
  positively correlated on average.
• Diversification will reduce risk but will not
  remove all of the risk. So,
• There are effectively two kinds of risk
• Diversifiable/nonsystematic/idiosyncratic risk.
• Disappears in well diversified portfolios.
• It disappears without cost, i.e. you need not
  sacrifice expected return to reduce this type of
  risk.
• Nondiversifiable/systematic/market risk.
• Does not disappear in well diversified
  portfolios.
• Must trade expected return for systematic risk.
• Level of systematic risk in a portfolio is an
  important choice for an individual.

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DIVERSIFICATION ELIMINATES UNIQUE RISK
Portfolio Standard Deviation
Note this level is a choice
Diversification is costless!!
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Nonsystematic/diversifiable risks

• Examples
• Firm discovers a gold mine beneath its property
• Lawsuits
• Technological innovations
• Labor strikes
• The key is that these events are random and
  unrelated across firms. For the assets in a
  portfolio, some surprises are positive, some are
  negative. On average, across assets, the
  surprises offset each other if your portfolio is
  made up of a large number of assets.

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Systematic/Nondiversifiable risk

• We know that the returns on different assets are
  positively correlated with each other on average.
  This suggests that economy-wide influences
  affect all assets.
• Examples
• Business Cycle
• Inflation Shocks
• Productivity Shocks
• Interest Rate Changes
• Major Technological Change
• These are economic events that affect all assets.
  The risk associated with these events is
  systematic (system wide), and does not disappear
  in well diversified portfolios.

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Measuring Systematic Risk

• How can we estimate the amount or proportion of
  an asset's risk that is diversifiable or
  non-diversifiable?
• The Beta Coefficient is the slope coefficient in
  an OLS regression of stock returns on market
  returns
• Beta is a measure of sensitivity it describes
  how strongly the stock return moves with the
  market return.
• Example A Stock with ? 2 will on average go up
  20 when the market goes up 10, and vice versa.

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Betas and Portfolios

• The Beta of a portfolio is the weighted average
  of the component assets Betas.
• Example You have 30 of your money in Asset X,
  which has ?X 1.4 and 70 of your money in Asset
  Y, which has ?Y 0.8.
• Your portfolio Beta is
• ?P .30(1.4) .70(0.8) 0.98.
• Why do we care about this feature of betas?
• It shows directly that an assets beta measures
  the contribution that asset makes to the
  systematic risk of a portfolio!
• Also note that this is a linear relation.

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Recap What is Beta?

• (1) A measure of the sensitivity of a stocks
• return to the returns on the market
  portfolio.
• (2) A standardized measure of a stocks
  contribution to the risk of a well diversified
  portfolio.
• (3) A measure of a stocks systematic risk
  (again, per unit risk or relative to the risk of
  the market portfolio).

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The Capital Asset Pricing Model (CAPM)

• Given that
• some risk can be diversified,
• diversification is easy and costless,
• rational investors diversify,
• There should be no premium associated with
  diversifiable risk.
• The question becomes What is the equilibrium
  relation between systematic risk and expected
  return in the capital markets?
• The CAPM is the best-known and most-widely used
  equilibrium model of the risk/return (systematic
  risk/return) relation.

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CAPM Intuition Recap

• ERi RF (risk free rate) Risk Premium
• Appropriate Discount Rate
• Risk free assets earn the risk-free rate (think
  of this as a rental rate on capital).
• If the asset is risky, we need to add a risk
  premium.
• The size of the risk premium depends on the
  amount of systematic risk for the asset (stock,
  bond, or investment project) and the price per
  unit risk.
• Could a risk premium ever be negative?

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The CAPM Intuition Formalized
or,

• The expression above is referred to as the
  Security Market Line (SML).

34
Using the CAPM to Select a Discount Rate

• Three inputs are required
• (i) An estimate of the risk free interest rate.
• The current yield on short term treasury bills is
  one proxy.
• Practitioners tend to favor the current yield on
  longer-term treasury bonds but this may be a fix
  for a problem we dont fully understand.
• Remember to adjust the market risk premium
  accordingly.
• (ii) An estimate of the market risk premium,
  E(Rm) - Rf.
• Expectations are not observable.
• Generally use a historically estimated value.
• (iii) An estimate of beta. Is the project or a
  surrogate for it traded in financial markets? If
  so, gather data and run an OLS regression. If
  not, you enter a very fuzzy area.

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The Market Risk Premium

• The market is defined as a portfolio of all
  wealth including real estate, human capital, etc.
• In practice, a broad based stock index, such as
  the SP 500 or the portfolio of all NYSE stocks,
  is generally used.

The Market Risk Premium Is Defined As

• Historically, the market risk premium has been
  about 8.5 - 9 above the return on treasury
  bills.
• The market risk premium has been about 6.5 - 7
  above the return on treasury bonds.

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Problems

• The current risk free rate is 4 and the expected
  risk premium on the market portfolio is 7.
• An asset has a beta of 1.2. What is the expected
  return on this asset? Interpret the number 1.2.
• An asset has a beta of 0.6. What is the expected
  return on this asset?
• If we invest ½ of our money in the first asset
  and ½ of our money in the second, what is our
  portfolio beta and what is its expected return?
• Relative to the first asset our portfolio has a
  smaller expected return, why?
• Does this mean the first asset is better than the
  portfolio?

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Problem

• The current risk free rate is 4 and the expected
  risk premium on the market portfolio is 7.
• You work for a software company and have been
  asked to estimate the appropriate discount rate
  for a proposed investment project.
• Your companys stock has a beta of 1.3.
• The project is a proposal to begin cigarette
  production.
• RJR Reynolds has a beta of 0.22.
• What is the appropriate discount rate and why?