Arbitrage Pricing Theory

Arbitrage refers to riskless profits that are generated not because of a net investment but on account of exploiting the difference that exists in the price of identical financial instruments due to market imperfections. Eventually, these arbitrage opportunities are eliminated and supply becomes equal to demand. Thus, the innate tendency for the price of a security to change is dispelled once the market is in equilibrium.

The aim of any investor is to maximize returns for a given level of risk or to minimize the risk for the given return. Risks can be broadly classified as systematic or unsystematic. Unsystematic risk is firm-specific and can be eliminated by holding a diversified portfolio while systematic risk refers to market risk that can at best be mitigated by hedging. Hence, the market rewards an investor for undertaking systematic risk. For more on ROI one may refer to the article titled, "calculating return on investment".

Arbitrage Pricing Theory
Arbitrage pricing theory (APT) was expounded by Stephen Ross in the year 1976. According to this theory, the expected return of a stock (or portfolio) is influenced by a number of independent macro-economic variables. These macro-economic variables are referred to as risk factors. Since, the arbitrage pricing theory (model) gives the expected price of an asset, arbitrageurs use APT to identify and profit from mispriced securities.

Arbitrage Pricing theory - Assumptions
The arbitrage pricing theory is based on a number of assumptions. These are as follows:

• Arbitrage pricing theory assumes that asset/portfolio returns can be described by a multi-factor model and proceeds to derive the expected returns relation that follows from that assumption.
• Since the intention is to maximize returns, the investor holds a number of securities so that unsystematic risk becomes negligible.
• In time, arbitrageurs will exhaust all potential opportunities for riskless profits and the market will be in equilibrium.

Arbitrage Pricing Theory - Factors
Although, arbitrage pricing theory does not explicitly state the relevant macro economic factors, it has been observed that the following factors tend to influence the price of the security under consideration.

• Change in industrial production or GDP.
• Unanticipated inflation or deflation.
• Shifts in the Yield Curve.
• Investor confidence measured by surprises in default risk premiums for bonds.

Arbitrage Pricing Theory - Equation
Arbitrage pricing theory specifies asset (stock or portfolio) returns as a linear function of the aforementioned factors. APT gives the expected return on asset i as:

E(R_i) = R_f + b₁*(E(R₁) - R_f) + b₂*(E(R₂) - R_f) + b₃*(E(R₃) - R_f) + …+ b_n*(E(R_n) - R_f)

Here
R_f = Risk free interest rate (i.e. interest on Treasury Bonds)
b_i = Sensitivity of the asset to factor_i
E(R_i) - R_f) = Risk premium associated with factor_i where i = 1, 2,...n

For more on Treasury Bonds one may refer to the following articles: how do treasury bonds work and bond equivalent yield formula.

Example of Portfolio Arbitrage
Portfolio arbitrage can be understood with the help of a simple example.
Let us assume that an investor has 2 portfolios: A and B

Where
Expected return on portfolio A or E(R_A) is 20%
b_A or the systematic risk of portfolio A is 1.5
Expected return on portfolio B or E(R_B) is 10%
b_B or the systematic risk of portfolio B is 1

Consider portfolio C where E(R_C) = 20% and b_C = 1.2. Since portfolio C yields the same return as A but is less risky as evidenced by its systematic risk, an arbitrage profit should be possible. If we construct a portfolio D by allocating weights to Portfolio A and B in proportion of 40% and 60% then,

E(R_D) = .4*E(R_A) +.6*E(R_B) = .4*.2 + .6*.1 = .14 or 14%
b_D = .4*b_A + .6*b_B = .4*1.5 +.6* 1 = 1.2

Both portfolio C and D have the same level of risk but generate different returns. Thus, an arbitrageur can profit by shorting D and by using the proceeds of the short sale to buy portfolio C. Since the arbitrage pricing model gives the expected price of an asset, arbitrageurs use arbitrage pricing theory to identify and profit from mispriced securities.