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What is Arbitrage Pricing Theory?

Arbitrage Pricing Theory: 
It is developed by Stephen Ross in 1976. It is a linear model and used for estimating the asset price. It helps to determine the assets are overvalued or undervalued. With the help of arbitrage investors can make a profit through price differences of asset without any investment. There are many macro economic factors (like inflation) and market and security specific factors which affect the price of an assets and it is difficult  to determine which factor consider or which should not. There are some factors which affect the price of security like-
·         Inflation
·         Gross Domestic Product
·         Interest rate
·         Market index
·         Investors confidence
·         Shift of yield curve etc.
      To calculate the expected return on stock you have to consider these factors which affect the price of that stock. 
If alpha is greater than zero then the expected return is greater it means the security is overpriced. And if alpha is smaller than zero it means expected return is smaller and security is under priced.

Arbitrage: It is an activity in which any investor can buy a security from one market and simultaneously sell it to other market for making risk less profit through price discrimination of same security. The person who engages in this buying and selling activity is known as arbitrager.

Formula:
E(r)i = rf + βi1 RP1 + βi2 RP2 + βi3 RP3 +.... βin RPn + ei
Where,
E(r)i = expected rate of return
rf = risk free rate
βi = beta sensitive to the associated factor
RP = risk premium
Ei= error term (unsystematic risk)
Assumptions:
·         All investors have homogeneous interest.
·         There is no arbitrage and if it is exist then the investors take the advantages of it and turned into equilibrium state.
·         It has no transaction cost, no tax.

Advantages:
·         It includes more than one factor which affects the price of security.
·         With the help of it determine the Is the security overprice or under priced?
·         It helps to make arbitrage free market.

Disadvantages:
·         It does not specify the factors which affect the price of the security.
·         It is difficult to identify the factors which affect the price of security and expected rate of return.

Example: Find out the expected return of portfolio A if the beta of portfolio is 1.52 and risk free rate is 2%. Through research it has been analysed that there are 2 factors which affect the portfolio A most is Inflation and interest rate. The beta of inflation is 0.96 and risk premium is 5%. The beta and risk premium of interest rate is 2.12 and 3% respectively.

Solution:
E(r)i = rf + βi1 RP1 + βi2 RP2 + βi3 RP3 +.... βin RPn + ei
= 2 + 0.96 + 5 + 2.12 + 3
= 13.08%

Example: Using the arbitrage pricing theory find out the expected return if the risk free rate is 2% from the following information:

Security
Factor
Beta (β1)
factor
Beta (β2)
A
2.33
2.10
0.98
0.78
B
1.3
1.26
1.25
1.95
C
0.65
1.45
2.56
0.89

Solution:
E(r)i = rf + βi1 f1 + βi2 f2 + βi3 f3 +.... βin fn + ei
Security
A
B
C
Calculation
=2.1*2.33 + 0.78*0.98
= 4.90+0.76
= 5.66
= 1.26*1.3+ 1.95*1.25
=1.64+2.44
=4.08
= 1.45*0.65+ 0.89*2.56
=0.94+2.28
=3.22






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