2. 2
Capital Asset Pricing Model (CAPM)
Arbitrage Pricing Theory (APT)
All assets can be organized on the Security market line (SML)
in the Risk - Return space.
Expected return of i-th asset (security) can be calculated as:
R R R Ri f M f i= + −( ) β
where: Ri … expected return on security i
Rf … risk-free return (interest rate)
RM … expected return on the market portfolio
RM - Rf … excess return of market portfolio
βi … security’s beta which measures the
sensitivity of the return on asset i to
the return in the market as a whole
File name: 1 CAPM - APT
Ready on Feb 8 - 9, 1999.
File name: 1 CAPM - APT
Ready on Feb 8 - 9, 1999.
3. 3
Assumptions of CAPM (1)
• No transaction costs
• All assets are infinitely divisible
• No taxation
• No single investor can affect the price
(perfect market)
• Investors make decisions solely in terms of
expected returns and standard deviations
• Unlimited short sales are allowed
4. 4
Assumptions of CAPM (2)
• Unlimited lending and borrowing of funds at
the (single) risk-less rate
• Homogeneous expectations concerning the
mean and variance of assets
• All investors have identical expectations
with respect to the portfolio decision inputs
(1.exp. returns, 2.variances, 3.correlations)
• All assets (eg, including human capital) are
marketable
5. 5
Characteristic line
( ) ( )R R R Ri f M f i− = − β
SML can be rewritten as:
(Ri - Rf ) … excess return of the security i
(RM - Rf ) … excess return of the market
6. 6
Beta estimates
( ) ( )R R R Ri f M f i i− = − +β ε
Estimating beta from historical returns using
regression
Beta is a slope of a characteristic line of i-th security.
The “single factor” CAPM was extended to describe
the optimal intertemporal consumption decisions of
investors who face multiple sources of risk, such as
uncertainty over future earnings, prices of consumption
goods, investment opportunities etc.
The “multifactor” CAPM (also referred to as multi-beta
CAPM) incorporates these extra market sources of risk
7. 7
Alpha
An investor can be convinced, that the security is wrongly priced
according to CAPM.
His estimate will differ by αI
( )[ ]α βi i
i n v e s t o r
f M f iR R R R= − − −
α i i
i n v e s t o r
i
C A P M
R R= −
If αi > 0 the investor believes that the security is undervalued
If αi < 0 the investor believes that the security is overvalued
8. 8
Impact on the characteristic line
( ) ( )R R R Ri
i n v e s t o r
f i M f i i− = + − +α β ε
Excess return of the security (Ri
investor
- Rf ) is composed of:
1) difference between investor’s estimate
and CAPM estimate (αi)
2) excess return of the market times beta (RM - Rf ) *βi
3) an error term (εi)
9. 9
Arbitrage pricing theory (APT)
• More recent and different approach to
determining the asset prices
• More general than CAPM which takes into
account mean and variance of asset returns
• The basic postulate of the APT is that the
market risk is itself made up of a number of
separate systematic factors
• Law of one price: two assets that are the
same can not sell at different prices
10. 10
Factor (index) models
• Return on any security is related to a set of
systematic factors, for example:
– growth of real GDP (unanticipated changes)
– unanticipated changes in interest rates
– unanticipated inflation
– impact of the market itself
– other unanticipated variables
• Not only to the market excess return
11. 11
Single factor model - return
R a b F ei i i i= + +*
Uncertain return on an i-th security is determined by:
F uncertain value of a factor
ai expected value of i-th security in case
the value of factor F = 0
bi sensitivity of i-th security to factor F
ei uncertain error term
12. 12
Single factor model - risk
σ σ σi i F e ib2 2 2 2
= +*
Risk of an i-th security is determined by:
σF
2
variance of factor F
σei variance of an error term
Covariance between assets i and j is:
σ σi j i j Fb b= 2
13. 13
Factor model - assumptions
• error term and factor are not correlated
• error terms of any two assets are not
correlated
• returns of assets are correlated since they
depend on the same factor
E 2093: Inv
Valu & Port
Mgmt
