# Explanation of Arbitrage Pricing theory Ι Principle of Arbitrage Theory

Modern portfolio theory helps an investor to identify his optimal portfolio from umpteen number of security portfolios that can be constructed. Elaborate on Arbitrage Pricing Theory and the principle of Arbitrage theory.

## Explanation of Arbitrage pricing theory

The Arbitrage Pricing Theory (APT) has been developed by Stephen Ross. It can calculate the expected return without taking recourse to the market portfolio.

It is a multi-factor model for determining the required rate of return which means that it takes into account a number of economy-wide factors that can affect the security prices. APT calculates relations among expected returns that will rule out arbitrage by investors.

APT starts with the assumption that security returns are related to an unknown number of unknown factors. These factors can be GDP (Gross domestic product), a market interest rate, the rate of inflation or any other random variable that impacts security prices. For simplicity, let us assume that there is only one factor (such as the GDP growth rate) that impacts the security price.

In this one factor APT model, the security return is: R_{i} = a_{i} + b_{i}F_{i}+ ɛ_{i}

Where F_{i} is the factor, a_{i} is the expected return on the security i if the factor has a value of zero, b_{i} is the sensitivity of security i to this factor (also known as factor loading for security i), and ɛ_{i} is the random error term.

#### Principle of Arbitrage theory

APT shows that for well-diversified portfolios, if the portfolio’s expected return (price) is not equal to the expected return predicted by the portfolio’s sensitivities (bi), then there will be an arbitrage opportunity.

According to APT, an investor will explore the possibility of forming an arbitrage portfolio to increase the expected return on his current portfolio without increasing its risk.

An arbitrage opportunity arises if an investor can construct a zero investment portfolio with no risk, but with a positive profit. Since no investment is required, an investor can create large positions, in long and short, to secure large levels of profits.

An arbitrage portfolio does not require any additional commitment of funds. Let Xi represents the change in the investor’s holding of security i (as a proportion of total wealth; it is, therefore, the proportion of security i in the arbitrage portfolio). Thus the requirement of no new investment can be expressed as: X1 + X2 + X3 = 0

An arbitrage portfolio has no sensitivity to any factor. The sensitivity of a portfolio is the weighted average of the sensitivities of the securities in the portfolio to that factor, this requirement can be expressed as b1X1 + b2X2 + b3X3 =0.