CAPM

Lecture 6

The CAPM

Learning outcomes

• By the end of this lecture you should: – Be able to interpret and apply the CAPM, both for individual assets and portfolios

– Know what the assumptions of the CAPM is why they are relevant

– Know how to partition the risk of an asset into a systematic and an unsystematic part, and why it is important

2

Quick recap

• Last lecture we saw how the separation theorem implies all investors hold the same optimal risky portfolio, P*

• In equilibrium, the demand for risky assets must equal the supply of risky assets

• Therefore the optimal risky portfolio must equal the market portfolio, M

• The properties of a risky asset in that portfolio will determine how attractive it is

3

Quick recap

• The relevant properties of our combined portfolio is its risk and expected return

• If we combine several assets, some risk may be diversified away

• How much of an asset’s risk can be diversified away and how much cannot depends on its covariance with the other assets in the portfolio

4

Quick recap

• Last lecture, we derived an expression for the relation between the covariance of an assets returns with the market and the required return of the asset:

• We call this expression the Capital Asset Pricing Model, or the CAPM

fMM

Mifi rrErrCovrrE 2

,

5

Interpretation of the CAPM

• On the LHS we have the expected return of the asset

• We typically refer to this as the required return • It is the return required by the market to compensate for the relevant risk of the asset

• In equilibrium no asset is “better” than some other asset. Each asset is exactly compensated for its risk.

fMM

Mifi rrErrCovrrE 2

,

6

Two special cases of the CAPM

• For the market portfolio M itself, the CAPM simplifies to:

• For the risk free asset the CAPM simplifies to:

MM

fMffMM

MfM

fMM

MMfM

rErE

rrErrrErrE

rrErrCovrrE

2

2

2

,

ff

fMM

ff

fMM

Mfff

rrE

rrErrE

rrErrCov

rrE

2

2

0

,

7

Two special cases of the CAPM

• For M and f the CAPM is simply an identity • Their expected returns are set by the specific preference of people that trade in the market

• These are very hard to figure out and we’ll take the properties of M and f as given (exogenous)

• Once we know these, the CAPM will tell us how any other asset are priced in relation to them

8

Interpretation of the CAPM

• In particular, we take the expected excess return of the market, E(rM)-‐rf, as given

• We often refer to this expected excess return as the market risk premium

• This is the reward we can expect to get for taking on the risk of the market portfolio

• Recall that the covariance of an asset with the market portfolio measures how much risk the asset contributes

fMM

Mifi rrErrCovrrE 2

,

9

The Security Market Line

• We can illustrate this in a graph similar to the CAL

• Each asset is compensated with excess return in relation to its risk in the market portfolio

• We often denote the risk ratio in the CAPM with i

• According to the CAPM, all assets plot on a straight line between f and M in -‐ E(r) space

• We call this line the Security Market Line, SML

rf

M

2

,

M

Mii

rrCov

irESML

10

• An asset’s measures how much risk the assets contributes in the market portfolio, relative to the average contribution

• i > 1 means that Cov(ri, rM) > Var(rM), or that the asset contributes more risk than the average asset

• i < 1 means that Cov(ri, rM) < Var(rM), or that the asset contributes less risk than the average asset

11

The Security Market Line

• Since nobody likes risk, investors will demand higher returns of assets that contribute more risk in the portfolio

• We can interpret the SML much like we interpreted the CAL: – An asset with = 0.5 contributes half the risk of the average asset and gets half the

reward in terms of excess returns

– An asset with = 2 contributes twice the risk of the average asset and gets twice the reward in terms of excess returns

rf

M

2

,

M

Mii

rrCov

irESML

12

A regression formulation

• Although the CAPM tells us the expected return of an asset, we know that realized returns may be higher or lower than that

• It is useful to phrase the CAPM as a regression model, in which we relate realized returns to each other and allow for some random error, :

• We’ll make the usual assumption on , e.g. that it is normally distributed around zero and uncorrelated with the dependent variables, i.e. rM

fMifi rrrr

13


• Recall from statistics, the univariate OLS regression model:

• The value of is

• In our case, X = rM – rf and Y = ri and = rf:

• This is the CAPM expression that we derived last lecture

XY

XVarYXCov ,

M

Mi

fM

fMi

rVarrrCov

rrVarrrrCov ,,

14

,i Mi f M f

M

Cov r rr r r r

Var r


• We see that by taking expectations on both sides we can return to our previous formula:

• Given this formulation, we can calculate the risk of an asset by

taking the variance on both sides:

• The second step follows as rf is a constant and the third step is applying our usual manipulations of variances

0fMifi

fMifi

fMifi

fMifi

rrErrEErErErErE

rrrErErrrr

,22fMifMii

fMii

fMifi

fMifi

rrCovVarrrVarrVar

rrVarrVarrrrVarrVar

rrrr

15


• Since rf is still a constant, we can simplify the expression further:

• Finally, let’s note that since is uncorrelated to rM, the last term equals zero:

• We see that variance of an asset can be partitioned into two parts

,2

,22

2

MMii

fMfMii

rCovVarrVarrVar

rrCovVarrrVarrVar

2222

2

Mii

Mii VarrVarrVar

16

Systematic risk

• The first term in the expression is called systematic risk or market risk

• Recall that is a measure of an assets covariance with the market

• The systematic risk is the part of an asset’s risk that is common with the market

• Since this risk is common with the market it cannot be diversified away in the market portfolio

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17

Example

• Suppose we run some industrial firm. Many things could affect our returns.

• For instance, there could be an oil embargo or an earth quake

• These events would affect all firms, so we say that these risks are systematic

• Since all firms are affected, their returns would move together in this situation

• Therefore we cannot diversify the risk away

18

Unsystematic risk

• The second term in the expression is called unsystematic risk or idiosyncratic risk

• The unsystematic risk is the part of an asset’s risk that is particular to the asset itself

• Since this risk comes from sources that do not affect the market it can be diversified away in the market portfolio

2222Mii

19

Example

• Suppose that we step on a nail and cannot work or that an accidental fire burns our plant down

• This would be bad for our returns, but other firms would not be affected

• We call these risks unsystematic • Since other firms are unaffected, we could diversify such risks away by combining many stocks in a portfolio

• It is unlikely that everyone steps on a nail the same day

20

Implications of the CAPM

• The CAPM basically states that only systematic risk is priced

• You are not compensated for taking on unsystematic risk (which can be diversified away anyway)

• Unsystematic risk hurts as much as systematic risk unless it’s diversified away

• The implication is that we should always well-‐diversified portfolios

• It is very hard to beat the market, but it is very easy to lose to it

21

The SML and the CAL

• The SML and the CAL are similar in that they relate expected returns to risk measures

• Individual assets will typically plot under the CAL, as it relates expected returns to total risk, which typically includes some (unpriced) unsystematic part

• All assets will plot on the SML, as it relates expected returns to (priced) systematic risk only

22

The SML and the CAL

• We can see the two types of risk in a graph of the CAL

• The risk of an asset left of the CAL is systematic

• The risk of an asset right of the CAL is unsystematic (and earns no extra expected return)

rf

irECAL

A

P*

A Systematic risk

Unsystematic risk

23

The market portfolio

• Recall that the in equilibrium the market portfolio is the optimal risky portfolio

• By construction, all risk in M is common with itself (the market)

• There is no unsystematic risk in M = P* as seen in the last graph

• There is no unsystematic risk in any portfolio on the CAL, i.e. when on the CAL we get paid for all the risk we take

24

The market portfolio

• If we interpret the as the systematic risk of an asset relative to that of the market, it is obvious that the market portfolio must be one

• Mathematically: 1,2

2

2M

M

M

MMM

rrCov

25

Portfolio

• The of a portfolio is the weighted average of the of the assets in that portfolio

• Consider the portfolio P:

• We manipulate both sides to get the :

• To calculate Cov(w1r1 + w2r2, rM), we set up the covariance matrix:

22211

2

,,

M

M

M

MP rrwrwCovrrCov

2211 rwrwrP

rM

w1r1 Cov(w1r1,rM)

w2r2 Cov(w2r2,rM)

26

Portfolio

• We see that

• Substituting:

• Since the market portfolio consists of all assets on the market and has = 1, the (value weighted) average beta on the market is one

MMMMM rrCovwrrCovwrrwCovrrwCovrrwrwCov ,,,,, 121122112211

2211212

211

2

,,, wwrrCovwrrCovwrrCov

M

M

M

M

M

MPP

27

Assumptions of CAPM

• Investors are price takers – No investor is large enough relative to the market to influence equilibrium prices by her trades

• Investors have identical investment horizons and agree on the statistical properties of all assets – All investors agree on expected returns and covariances of all assets

• Perfect capital markets – There are no financial frictions such as short selling constraints, transaction costs, taxes etc.

28

Assumptions of CAPM

• Investors are rational mean-‐variance optimizers – All investors maximize a utility function somewhat like ours and do so correctly

• There is a risk free asset available to all – All investors can borrow and invest in this at the same rate

• All investors can trade all assets – We disregard assets such as human capital

29

Using the CAPM

• The CAPM has implications for our portfolio choice, e.g. avoid unsystematic risk

• The CAPM allows us to evaluate investment performance by relating returns to the priced risk taken

• We can separate market effects from unsystematic effects

• We can use the CAPM to calculate required returns for non-‐traded assets or firm projects

30

CAPM

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