ARBITRAGE PRICING THEORY
Less complicated than the CAPM Primary assumption
Each investor, when given the opportunity to increase the return of his or her portfolio without increasing its risk, will proceed to do so.
The mechanism for doing so involves the use of arbitrage portfolios.
Factor models Assumption
Security returns are related to an unknown number of unknown factors
iiii eFbar 1
An investor owns three stocks, and the current market value of his or her holdings in each one is 4000000. The investor’s current investable wealth W0 is equal to 12000000. Everyone believes that these three stocks have the following expected returns and sensitivities:
I ri bi stock 1 15% 0.9 stock 2 21% 3.0 stock 3 12% 1.8 Do these expected returns and factors sensitivities
represent an equilibrium situation? If not, what will happen to stock prices and expected returns to restore equilibrium?
Factor models Principle of arbitrage
Arbitrage The process of earning riskless profits by taking ad
vantage of differential pricing for the same physical asset or security.
The sale of a security at a relatively high price and the simultaneous purchase of the same security(or its functional equivalent) at a relatively low price
Factor models Principle of arbitrage
A factor model implies that securities or portfolios with equal factor sensitivities will behave in the same way except for nonfactor risk.
Securities or portfolios with the same factor sensitivities should offer the same expected returns. If they do not, then almost arbitrage opportunities exist. Investor will take advantage of these opportunities, causing their elimination.
Factor models Arbitrage portfolios
A portfolio that does not require any additional funds from the investor
Has no sensitivity to any factor Its expected return is positive
Xi denotes the change in the investor’s holdings of security i
0%12%21%150
08.10.39.00
00
321
321
321
xxxrx
xxxbx
xxxx
ii
ii
i
Factor models Arbitrage portfolios
There is an infinite number of combinations of values for X1,X2,X3 that satisfy these two equations.
Arbitrarily assigning a value of 0.1 to X1 X1=0.1 X2=0.075 X3=-0.175
The expected return= 15%*0.1+21%*0.075+12%*(-0.175) =0.975%
Factor models Arbitrage portfolios
Summary This arbitrage portfolio is attractive to any inv
estor who desires a higher return and is not concerned with nonfactor risk
It requires no additional dollar investment, it has no factor risk, and it has a positive expected return.
Pricing effects What are the consequences of buying
stocks 1 and 2 and selling stock 3? Buying a stock will push up its current price
P0 yet will have no impact on the stock’s expected end-of-period price P1. As a result, its expected return will decline.
Conversely, selling s stock such as stock 3 will push down its current price and result in a rise in its expected return.
The buying-and-selling activity will continue until all arbitrage possibilities are significantly reduced or eliminated.
Pricing effects What are the consequences of buying
stocks 1 and 2 and selling stock 3? At the end point there will exist an
approximately linear relationship between expected returns and sensitivities of the following sort:
ii br 10
The APT pricing equation The expected return on any security is, in eq
uilibrium, a linear function of the security’s sensitivity to the factor bi 。
are constants
Any security that has a factor sensitivity and expected return such that it lies off the line will be mispriced according to the APT and will present investors with the opportunity of forming arbitrage portfolios.
ii br 10 0 1
Interpreting the APT pricing equation For any asset with bi=0
In the case of riskfree asset
imply
0ir
fi rr
ifi
f
brr
r
1
0
Interpreting the APT pricing equation Pure factor portfolio
Has unit sensitivity to the factor If there were other factors, such a portfolio would
be constructed so as to have no sensitivity to them.
is the expected excess return on a portfolio that has unit sensitivity to the factor.(factor risk premium)
0.1* pb
fp
fp
rr
rr
*
*
1
1
1
Interpreting the APT pricing equation Second version of the APT pricing
equation
denote the expected return on a portfolio that has unit sensitivity to the factor.
ir
f
f
iffi
r
r
brrr
11
0
1 )(
*1 pr
Two-factor models The security returns are generated by
the following factor model:
Consider a situation in which there are four securities that have the following expected returns and sensitivities. Consider an investor who has $5000000 invested in each of the securities. Are these securities priced in equilibrium?
1211 FbFbar iiii
Arbitrage portfolios
X1=0.1 X2=0.088 X3=-0.108 X4=-0.08 Rp=0.1x15%+0.088x21%-0.108x12%-0.08x8%=1.
41%>0
02.37.05.12
028.139.0
0
4321
4321
4321
xxxx
xxxx
xxxx
The arbitrage portfolio involves the purchase of stock 1 and 2, funded by selling stocks 3 and 4.
The buying-and-selling pressures will drive the prices of stocks 1 and 2 up and of stock 3 and 4 down.
In return, the expected returns of stocks 1 and 2 will fall and of stocks 3 and 4 will rise.
Investors will continue to create such arbitrage portfolios until equilibrium is reached.
ir
21
22110
%2%4%8 ii
iii
bb
bbr
Pricing effects Consider a well-diversified portfolio that has
unit sensitivity to the first factor and zero sensitivity to the second factor
b
2211
2
1
)(
0
1
iiffi
i
i
bbrrr
b
b
Pricing effects Consider a portfolio that has zero sensitivity
to the first factor and unit sensitivity to the second factor
2211
22
2
1
)()(
1
0
ififfi
f
brbrrr
r
b
b
FACTOR MODELS
SECURITY PRICING FORMULA:
ri = 0 + 1 b1 + 2 b2 +. . .+ KbK
whereri = rRF +(1rRFbi12rRF)bi2+
rRFbiK
FACTOR MODELS
where r is the return on security iis the risk free rateb is the factore is the error term
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