Money, Banking & FinanceLecture 5

The Capital Asset Pricing Model CAPM

Aims

• Analyse the determinants of the equilibrium expected return on an individual security.

• To show how the risk premium on an asset is determined.

• Explain the capital asset pricing model.

• Show that the riskiness of an individual asset is given by its ‘beta’.

The Capital Market Line

• Assume once again that the investor can borrow and lend at the same rate.

• The transformation line that is tangential to the efficient set is the capital market line CML

• This defines the single composition of risky assets the investor wants to hold. This is called the market portfolio M

• The CML is the transformation line that is tangential to the efficient frontier.

CML

• The market portfolio represents the point on the efficient frontier which maximises the slope of the CML.

• The optimal proportions of risky assets at M maximise expected return E(Rm)-Rf.

• If all investors are at M then they earn the same excess return per unit of risk.

• The slope of the CML represents the market price of risk.

Capital Market Line• The CML provided a linear relation between

expected return and risk that describes the proportion of a risk-free asset and an efficient portfolio of assets (market portfolio) that an investor can hold.

• The same basic function can be used to derive an expression for the expected return on an inefficient investment other than the market portfolio.

• Or indeed for a single stock

Individual stock return

• At the Market portfolio ‘M’ all the risky assets are held in the optimal proportions by all investors. This represents a market equilibrium.

• Since all ‘n’ assets are held at M, there is a set of expected returns E(Ri) corresponding to point M on the efficient frontier.

• The equation representing the equilibrium returns for asset ‘i’ recognises that when held as part of a wider portfolio it could reduce the risk of the total portfolio depending on the covariance.

• The riskiness of asset ‘i’ when considered as part of a diversified portfolio is not its own variance but the covariance between Ri and the market return Rm.

CML and the market portfolio

E(Rp)

M

CML

Rf

E(Rm)

σm

α

E(Rm)-Rf

Slope of CML

m

fm RRE

)(tan

So to recap - The Capital Market Line

• The transformation line that is tangential to the efficient set and has intercept at the risk-free rate Rf is the capital market line CML

• This defines the single composition of risky assets the investor wants to hold. This is called the market portfolio M

• The CML is the transformation line that is tangential to the efficient frontier.

CML and SML

• CML: E(Rp) = Rf + λσp, where λ is the slope.• Interpretation of the CML is that it represents the

return available to an investor with no risk or the additional return that can be expected as a reward for holding the investment’s risk – this is λσp and is known as the risk premium.

• The risk premium is the product of the market price of risk λ and the amount of risk taken given by σp.

• The amount of market risk at ‘M’ is σM

SML

• The SML is similar to the CML.• The individual expected return of a share consists

of 2 elements. The risk-free return and a risk premium.

• The risk premium for an individual share is not the product of the market price of risk λ and the risk of the share σi but the covariance relationship between the share and the market portfolio which is defined by - Beta

Beta

• Beta represents an asset’s systematic (market or non-diversifiable) risk

• The CAPM at the point ‘M’ on the efficient frontier gives the risk adjusted equilibrium return on asset ‘i’

• E(Ri) – Rf = βi[E(Rm) – Rf]• βi = Cov(Ri,Rm)/σ2

m

• The risk premium is βi[E(Rm) – Rf] and represents the reward for taking risk above that of the risk-free rate

Deriving the SML

• The covariance between the returns of an individual share return and the market return will tell us how much the inclusion of that asset in the portfolio will reduce the risk on the portfolio as a whole.

• To derive the SML let us look at a 2-asset portfolio an asset A and the market M

• E(Rp) = ωE(RA) + (1 – ω)E(Rm)

Two-asset portfolio

CML

M

A

E(Rp)

σpσm

E(Rm)

Rf

The expected return from a marginal investment in the

inefficient portfolio is

)()()(

mAp RERE

d

RdE

The marginal risk produced by a marginal investment in A is

),(4),(2)1(22

),()1(2)1(

),()1(2)1(

22

222221

2222

21

mAmAmA

mAmAp

mAmAp

RRCovRRCov

RRCovd

d

RRCov

At the point M ω=0

m

mmA

m

mmAp

MmAmp

RRCovRRCov

d

d

RRCovd

d

2

2

2

2221

),(),(

2),(2

21

21

Tangent at point M on the frontier AM is:

2

2

),(

)()(

),(

)()()()(

mmA

mmA

mmmA

mA

p

ppp

RRCov

RERE

RRCov

RERE

d

RdE

d

d

d

RdE

The CML is also tangent to frontier at M. So:

fmfA

fmfmmA

m

mA

m

mmAfmmA

m

fm

mmA

mmA

RRERRE

RRERRERERE

RRCov

RRCovRRERERE

RRE

RRCov

RERE

)()(

)()()()(

,

,)()()(

)(

,

)()(

2

2

2

2

Security Market Line

β

E(Rp)

SML

Rf

1

E(Rm)

Interpreting Beta

m

AmA

mA

mAmA

m

mA

RRCov

RRCov

,

,

2

,

,

Beta

• The numerator represents the systematic risk of asset A

• The denominator represents the total risk of the market portfolio

• Beta is an index of the amount of share A’s systematic risk relative to the market portfolio

• The beta value will tell us how much the expected return on a share should rise or fall relative to the market.

If the expected return on the market rises by 10%

• E(Ri) will rise > 10% if β > 1• E(Ri) will rise < 10% if β < 1• E(Ri) will rise = 10% if β = 1• Higher beta shares will outperform the

market in a bull run and lower beta shares will under-perform

• Conversely high beta shares will fall faster in a bear market.

Measurement of beta

• Plot the risk premium of the asset against the risk premium of the market – slope is beta

• Regress risk premium of the asset against the risk premium of the market – beta is the regression coefficient

• (Ri – Rf) = αi + βi(Rm – Rf) + ui; αi = 0• αi = 0 means that when the market risk premium is

zero so should the individual shares that make up the portfolio have zero risk premium.

• So what if αi ≠ 0?

Alpha

• Over a long period alpha values should be zero.• But if not it can used to provide investor advice.• αi < 0 shares should be sold and αi > 0 should be

bought. Meaning share price of {i} is mispriced.• Negative alpha means that returns are below the

equilibrium predicted by the CAPM, therefore share prices will fall until yields rise to the equilibrium. The act of selling will drive down the price.

• Vice versa for positive alpha

Summary

• We have examined the CAPM framework• The CAPM is used to measure the systematic risk

of an individual share.• The beta value measures the degree of

responsiveness of the expected return on the share relative to movements in the expected return of the market.

• Alpha is interpreted as an indication of mispricing and has been used as justification for investment strategy