# Arbitrage Pricing Theory

Arbitrage Pricing Theory is one of the most popular finance theories of the world. The theory talks about the asset pricing principles and thereby helps and influences the pricing of shares. Arbitrage Pricing Theory is also popularly known as the APT model of finance theory.
The APT model says that the expected return from any financial asset can be represented in the form of a liner function. The linear function is modeled on the factors like various theoretical market indices and macro-economic factors. It is being assumed that the factors here are exposed to sensitive changes, which is represented by a constant factor-specific beta coefficient.

According to the Arbitrage Pricing Theory model, a risky asset needs to satisfy the following linear relation:

E(rj) = rf + bj1RP1 + bj2RP2 + … bjnRPn

Rj = E(rj) + bj1F1 + bj2F2 + …. bjnFn + εj
Where,
E(rj) carries the values for risky asset’s expected return
rf carries the value for risk-free rate
RPk carries the value for risk premium of the factor
Fk carries the value for macroeconomic factor
bjk carries the value for sensitivity of the asset to factor k and is also called factor loading
εj carries the value for risky asset’s idiosyncratic random shock with mean zero
This model represents the uncertain return of the asset ‘j’ in relationship among ‘n’ factors. Every factor considered here are assumed to be a random variable with mean zero.

The Arbitrage Pricing Theory primarily describes the mechanism where the arbitrage by the investors may bring the mis-priced asset back into its expected price. Here we need to give attention to that fact that under true arbitrage, the investor locks-in a guaranteed payoff while under APT arbitrage the investor locks-in a positive expected payoff. While seen in the light of Arbitrage Pricing Theory, arbitrage actually consists of trading in two different assets where at least one is mis-priced. Usually the arbitrageur sells the relatively costlier asset and then buys the relatively cheaper asset. 