# Arbitrage Pricing Theory - APT

## What is the 'Arbitrage Pricing Theory - APT'

Arbitrage pricing theory is an asset pricing model based on the idea that an asset's returns can be predicted using the relationship between that asset and many common risk factors. Created in 1976 by Stephen Ross, this theory predicts a relationship between the returns of a portfolio and the returns of a single asset through a linear combination of many independent macroeconomic variables.

## BREAKING DOWN 'Arbitrage Pricing Theory - APT'

The arbitrage pricing theory (APT) describes the price where a mispriced asset is expected to be. It is often viewed as an alternative to the capital asset pricing model (CAPM), since the APT has more flexible assumption requirements. Whereas the CAPM formula requires the market's expected return, APT uses the risky asset's expected return and the risk premium of a number of macroeconomic factors. Arbitrageurs use the APT model to profit by taking advantage of mispriced securities, which have prices that differ from the theoretical price predicted by the model. By shorting an overpriced security, while concurrently going long in the portfolio the APT calculations were based on, the arbitrageur is in a position to make a theoretically risk-free profit.

## Arbitrage Pricing Theory Equation and Example

APT states that the expected return on a stock or other security must adhere to the following relationship:

Expected return = r(f) + b(1) x rp(1) + b(2) x rp(2) + ... + b(n) x rp(n)

Where,

r(f) = the risk-free interest rate

b = the sensitivity of the asset to the particular factor

rp = the risk premium associated with the particular factor

The number of factors will range depending on the analysis. There can be a few or dozens; it depends on which factors an analyst chooses for the analysis. In addition, the exact factors do not have to be the same across analyses. As an example calculation, assume a stock is being analyzed. The following four factors have been identified, along with the stocks sensitivity to each factor and the risk premium associated with each factor:

Gross domestic product growth: b = 0.6, rp = 4%

Inflation rate: b = 0.8, rp = 2%

Gold prices: b = -0.7, rp = 5%

Standard and Poor's 500 index return: b = 1.3, rp = 9%

The risk-free rate is 3%.

Using the above APT formula, the expected return is calculated as:

Expected return = 3% + (0.6 x 4%) + (0.8 x 2%) + (-0.7 x 5%) + (1.3 x 9%) = 15.2%