The Capital Asset The Capital Asset Pricing ModelPricing Model

Review

Review of portfolio diversification

Capital Asset Pricing ModelCapital Market Line (CML)

Security Market Line (SML)

It is the equilibrium model that underlies all modern financial theory.

Derived using principles of diversification with simplified assumptions.

Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development.

Capital Asset Pricing Model (CAPM)

Single-period investment horizon.

Investors forecasts agree with respect to expectations, standard deviations, and correlations of the returns of risky securities- Therefore all investors hold risky assets in the same

relative proportions

Investors behave optimally - In equilibrium, prices adjust so that aggregate

demand for each security is equal to its supply

Assumptions

Since every investor’s relative holdings of the risky security is the same, the only way the asset market can clear is if those optimal relative proportions are the proportions in which they are valued in the market place

All investors will hold the same portfolio for risky assets – market portfolio.

Market Portfolio

Capital Market Line (CML) and the CAPM

CAPM says that in equilibrium, any investor’s relative holding of risky assets will be the same as in the market portfolio

Depending on their risk aversions, different investors hold portfolios with different mixes of riskless asset and the market portfolio

Example (online two stocks)

Capital Market Line

m

fm

frm

fmr

rErslope

rrEr

Er

Capital Market Line

E(r)

E(rM)

rf

MCML

m

What determines the market risk premium?

Risk premium on the the market depends on the average risk aversion (A) of all market participants.

Example:

2)( Mfm ArrE

0.220.0

06.014.0

,06.0,20.0,14.0

2

22

A

rErAArEr

rEr

m

fmmfm

fmm

Beta and Security Market Line

If risk is defined as the measure such that as it increases, a risk-averse investor would have to be compensated by a larger expected return in order for her to hold it in her optimal portfolio, then the measure of a security’s risk is its beta, not its standard deviation!

tells you how much the security’s rate of return changes when the return on the market portfolio changes

CAPM Risk Premium on any Asset

According to the CAPM, in equilibrium, the risk premium on any asset is equal to the product of

- (or ‘Beta’), and

- the risk premium on the market portfolio

ifmfiifmfi

m

imi

rErrErrErrEr

2

The Security Market Line

- The plot of a security’s risk premium Eri-rf (or sometimes security returns) against security beta is the security market lineNote that the slope of the security market line is the

market premium

By CAPM theory, all securities must fall precisely on the SML (hence its name)

Security Market Line (SML)

E(r)

E(rM)

rf

SML

= 1.0

SML: ri = rf + i[E(rm) - rf]

i= [COV(ri,rm)] / m2

Slope SML = E(rm) - rf

= market risk premium

Security Market Line (SML) Relationships

The Beta of a Portfolio in CAPM

When determining the risk of a portfolio - using standard deviation results in a formula

that’s quite complex

- using beta, the formula is linear

i

iinnrwrwrw wwwwnn

...2211...2211

2

1

,,1

2... 2

2211

jijijiji

niiirwrwrw www

nn

E(rm) - rf = .08 rf = .03

x = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

y = .6

E(ry) = .03 + .6(.08) = .078 or 7.8%

Examples for SML

Graph of Sample Calculations

E(r)

Rx=13%

SML

1.0

Rm=11%

Ry=7.8%

3%

1.25x

.6y

.08

»By

Disequilibrium Example

E(r)

15%

SML

1.0

Rm=11%

rf=3%

1.25

Suppose a security with a of 1.25 is offering expected return of 15%.

According to SML, it should be 13%.

Under-priced: offering too high of a rate of return for its level of risk.

Disequilibrium Example

Alpha

Underpriced stocks plot above the SMLOverpriced stocks plot below the SMLThe difference between the fair and actually

expected rates of return on a stock is called the stock’s alpha.- Example: an alpha fund with alpha=1%, beta=0.5,

standard deviation=0.15, market return=0.14, risk free rate is 0.06, market standard deviation=0.2