# A Simple Explanation About the Arbitrage Pricing Theory

According to the arbitrage pricing theory, the return on a portfolio is influenced by a number of independent macro-economic variables.

Aparna Iyer

Last Updated: Oct 06, 2017

**Basics**

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 model gives the expected price of an asset, arbitrageurs use APT to identify and profit from mispriced securities.

**Assumptions**

This theory is based on a number of assumptions. These are as follows:

- The 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.

**Factors**

Although this 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.

**Equation**

The 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}+ b

_{1}*(E(R

_{1}) - R

_{f}) + b

_{2}*(E(R

_{2}) - R

_{f}) + b

_{3}*(E(R

_{3}) - 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

**Example**

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 this theory to identify and profit from mispriced securities.