# UNDERSTANDING THE ARBITRAGE PRICING MODEL

__What is Arbitrage?__

Arbitraging is a technique which involves simultaneous buying and selling of an asset from varied exchanges, locations, or platforms to encash in on the price difference (which is usually minor in percentage terms). To an arbitrageur, temporarily mispriced securities depict a short-term shot to profit virtually, that too risk-free.

In the APT context, arbitrage consists of trading in two securities – with at least one being mispriced. The arbitrageur creates their portfolio by recognizing correctly priced assets (one per risk-factor, plus one) and then weighting the assets such that the portfolio beta per factor is the same as for the mispriced asset. While conducting an arbitrage trade, the quantity of the underlying asset bought and sold should be the same.

__Understanding Arbitrage Pricing Theory (ATP)__

The Arbitrage Pricing Theory (APT) is a concept in asset pricing. It states that returns on an asset can be anticipated by taking into account the linear relationship of the asset’s expected returns and the macroeconomic factors that influence its systematic risk. The theory was created in 1976 by American economist, Stephen Ross, as an advancement to CAPM (Capital Asset Pricing Model).

The APT is a more elastic and complicated alternative to the CAPM. It provides investors and analysts with the chance to customize their analysis and research. Still, it is tougher to apply, as it takes a substantial amount of time to determine all the factors that may influence the asset’s price.

__Assumptions in the Arbitrage Pricing Theory:__

Unlike CAPM, where only a single risk factor of the overall market is taken into consideration, the APT model looks at several macroeconomic or systematic factors that determine the risk and return of a specific security. Moreover, these factors are ones that cannot be diversified.

The APT not only recommends that investors need to diversify their portfolio, but also asks them to choose their risk and return profile based on the premiums and sensitivity towards macroeconomic risk factors. Risk-taking investors will exploit the differences in anticipated and actual returns on the asset by using arbitrage (how the theory got its name).

__The formula of APT : __

In the APT model, the returns of an asset follow a factor intensity structure if the returns could be expressed;

The following formula is used : *ri = ai + βi1 * F1 + βi2 * F2 + … + βkn * Fn + εi*, where

- ~ ai is a constant for the asset;
- ~ F is a macroeconomic or systematic factor;
- ~ β is the sensitivity of the asset or portfolio concerning the specified factor;
- ~ εi is the asset’s idiosyncratic random shock with an expected mean of zero, also recognized as the error term.

The APT formula is *E(ri) = rf + βi1 * RP1 + βi2 * RP2 + … + βkn * RPn,*

- ~ where rf is the risk-free rate of return,
- ~ β is the beta or sensitivity of the asset or portfolio concerning the specified factor
- ~ RP is the risk premium.

** For example**, the following four factors have been identified as explaining a stock’s return and its sensitivity to each factor and the risk premium associated with each factor have been calculated:

Gross domestic product (GDP) growth: ß = 0.8, RP = 4%

Inflation rate: ß = 0.9, RP = 3%

Gold prices: ß = -0.7, RP = 6%

Nifty index return: ß = 1.4, RP = 8%

The risk-free rate is 4%

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

*Expected return = 4% + (0.8 x 4%) + (0.9 x 3%) + (-0.7 x 6%) + (1.4 x 8%) = (calculate) *

__Issues with the APT__

An underlying issue with the APT is accurately determining the level of risk (which applies to any given asset). It may be feasible to find a ‘factor portfolio’ where the risks are identical. However, generally, the level of risk is determined by systematic factors.

__COMPARISON OF CAPM and the APT__

APT may be informational over the medium to long term but is not considered to be as precise in the short term. The CAPM, on the other hand, is a snapshot which appears to be more valid in the short term. The APT focuses on risk factors rather than assets, so it has the advantage of not having to create an equivalent portfolio to assess risk.

The CAPM presumes that there is a linear relationship between the assets, whereas the APT supposes that there is a linear relationship between risk factors.

This means that where there no linear relationship exists, the models are unable to adequately indicate outcomes.

However, both the CAPM and the APT make fairly unrealistic assumptions in that assets are freely attainable and desirable, there are no fees incurred in the acquisition of assets and that all investors tend to think alike and come to the similar judgments. This seems intuitively contradictory, as the most successful investors are likely to be those who can spot potential which has remained unnoticed by the market as a whole. Indeed, when all investors do think alike, a ‘bubble’ can be created which inflates the asset price and tones down the risks inherent in the asset. In this situation, assessing the risk of an asset-based on the climate of the market is likely to be far riskier than can be predicted by either the CAPM or the APT. Theoretically, it could be asserted that using a CAPM or APT analysis is likely to increase the propensity for ‘bubbles’ to emerge, as they are using static predictions of investor behaviour.

Therefore, although the CAPM and APT are useful as rule-of-thumb heuristics of the market as it currently operates, they are both static models which use a limited number of factors to predict risk in a highly complex market. Although they are established on mathematical principles, they are subjective in that the analyst performing the calculation has the freedom to decide which factors are applicable in each particular case.

__References __

1.) Chen, N. F.; Ingersoll, E. (1983) – “Exact Pricing in Linear Factor Models with Finitely Many Assets: A Note” *Journal of Finance*.

2.) The Arbitrage Pricing Theory Prof. William N. Goetzmann, Yale School of Management

3.) The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning, Richard Roll and Stephen A. Ross.