# Arbitrage Pricing Theory (APT)

As its name implies, the Arbitrage Pricing Theory, or APT, describes a mechanism used by investors to identify an asset, such as a share of common stock, which is incorrectly priced. Investors can subsequently bring the price of the security back into alignment with its actual value.

## The Arbitrage Pricing Theory Model

The APT model was first described by Steven Ross in an article entitled The Arbitrage Theory of Capital Asset Pricing, which appeared in the Journal of Economic Theory in December 1976.  The Arbitrage Pricing Theory assumes that each stock's (or asset's) return to the investor is influenced by several independent factors.

### The APT Formula

Furthermore, Ross stated the return on a stock must follow a very simple relationship that is described by the following formula:

Expected Return = rf + b1 x (factor 1) + b2 x (factor 2)... + bn x (factor n)

Where:

• rf = the risk free interest rate, which is the interest rate the investor would expect to receive from a risk-free investment.  Typically, U.S. Treasury Bills are used for U.S. dollar calculations, while German Government bills are used for the Euro
• b = the sensitivity of the stock or security to each factor
• factor = the risk premium associated with each entity

The APT model also states the risk premium of a stock depends on two factors:

• The risk premiums associated with each of the factors described above
• The stock's own sensitivity to each of the factors; like the beta concept

Risk Premium = r - rf = b(1) x (r factor(1) - rf) + b(2) x (r factor(2) - rf)... + b(n) x (r factor(n) - rf)

If the expected risk premium on a stock were lower than the calculated risk premium using the formula above, then investors would sell the stock.  If the risk premium were higher than the calculated value, then investors would buy the stock until both sides of the equation were in balance.  Arbitrage is the term used to describe how investors could go about getting this formula, or equation, back into balance.

### Factors Used in the Arbitrage Pricing Theory

It's one thing to describe the APT theory in terms of simple formulas, but it's another matter entirely to identify the factors used in this theory.  That's because the theory itself does not tell the investor what those factors are for a particular stock or asset, and for a very good reason.  In practice, and in theory, one stock might be more sensitive to one factor than another.  For example, the price of a share of ExxonMobil might be very sensitive to the price of crude oil, while a share of Colgate Palmolive might be relatively insensitive to the price of oil.

In fact, the Arbitrage Pricing Theory leaves it up to the investor, or analyst, to identify each of the factors for a particular stock.  Therefore, the real challenge for the investor is to identify three items:

• Each of the factors affecting a particular stock
• The expected returns for each of these factors
• The sensitivity of the stock to each of these factors

Identifying and quantifying each of these factors is no trivial matter, and is one of the reasons the Capital Asset Pricing Model remains the dominant theory to describe the relationship between a stock's risk and return.

Keeping in mind the number and sensitivities of a stock to each of these factors is likely to change over time, Ross and others identified the following macro-economic factors they felt played a significant role in explaining the return on a stock:

• Inflation
• GNP or Gross National Product
• Investor Confidence
• Shifts in the Yield Curve

With that as guidance, the rest of the work is left to the stock analyst.

### APT versus the Capital Asset Pricing Model

As mentioned, the Arbitrage Pricing Theory and the Capital Asset Pricing Model (CAPM) are the two most influential theories on stock and asset pricing today.  The APT model is different from the CAPM in that it is far less restrictive in its assumptions.  APT allows the individual investor more freedom to develop a model that explains the expected return for a particular asset.

Intuitively, the APT makes a lot of sense because it removes the CAPM restrictions, and basically states "The expected return on an asset is a function of many factors as well as the sensitivity of the stock to these factors."  As these factors move, so does the expected return on the stock, and therefore its value to the investor.

In the CAPM theory, the expected return on a stock can be described by the movement of that stock relative to the rest of the market.  The CAPM is just a simplified version of the APT, whereby the only factor considered is the risk of a particular stock relative to the rest of the market, as described by the stock's beta.

From a practical standpoint, CAPM remains the dominant pricing model used today.  When compared to the Arbitrage Pricing Theory, the Capital Asset Pricing Model is both elegant and relatively simple to calculate.

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