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# Arbitrage Pricing Theory – Definition

### Arbitrage Pricing Theory Definition

The arbitrage pricing theory is a model used to estimate the fair market value of a financial asset on the assumption that an asset’s expected returns can be forecasted based on its linear pattern or relationship to several macroeconomic factors that determine the risk of the specific asset. The theory deals with specifically financial assets such as bonds, stocks, derivatives, commodities and currencies.

Arbitrage generally refers to the act of exploiting the price differences in a financial asset in different markets to make profits by simultaneously purchasing at a low price  in one market and selling the same asset at a higher price in a different market. It is generally considered a risk free investment. The person who tries to profit from such arbitrage opportunity due to price imbalance is called an arbitrageur.

However, in the context of the arbitrage pricing theory, an arbitrage involves trading in the same market and it is not necessarily a risk-free operation – but it does offer a high probability of success. Also, it may involve trading in two different assets where an investor sells one asset that is overvalued as per the theory to buy another that is viewed undervalued. We shall discuss the arbitrage pricing model in detail to understand better how it works.

### A Little More on Arbitrage Pricing Theory

Stephen Alan “Steve” Ross, an economist and a sterling professor of economics and finance is credited for development of the arbitrage pricing theory in 1976. Professor Ross suggested the theory as an alternative to the Capital Asset Pricing Model (CAPM). Arbitrage pricing theory is more of a complex multiple macroeconomic factor alternative to CAPM which is a single factor pricing.

The theory works to point out assets that are estimated to be mispriced. The model calculates what should be the fair market value of a financial asset and if the existing market price differs with the theoretical estimates from the arbitrage pricing theory, the asset is considered mispriced. The theory suggests that the returns on assets follow a linear pattern. Consequently, an asset that deviates in returns from the linear pattern, presents an arbitrage opportunity that an investor can leverage.

For instance, the theoretical fair market value of stock “A” can be  determined using the Arbitrage pricing model to be \$20.However, the stock may be currently trading at a market price of \$11, which to an arbitrageur the stock can be considered mispriced. Therefore, an investor would buy the stock, based on the belief that further market price action will quickly “correct” the market price back to the \$20 per share price.

Alternatively, it may involve trading in two assets. For example, the theoretical fair market value of stock “B” can be determined using the Arbitrage pricing model to be \$50.However, the stock may be currently trading at a market price of \$65. Therefore, an investor would sell the stock, based on the belief that it is theoretically overpriced and if the market action corrects it, the price will drop. The proceeds may be used to purchase Stock “A” which offers better short term profitability.

### Mathematical Model of the Arbitrage Pricing Theory

The arbitrage pricing theory is a multifactor model for financial asset valuation. Multi-factor portfolios are a financial modeling strategy in which multiple factors which could be grouped into macroeconomic or company-specific statistical factors are used to analyze and explain asset prices. However, the choice of macroeconomic factors as well as how many are used in the calculation is a subjective choice.

The number and nature of these factors is likely to change over time and between economies. Therefore, using the arbitrage pricing theory means different investors will have varying results depending on their choice of macroeconomic factor. Nevertheless, there are four or five factors which have often proven to be reliable and are consistently used in an attempt to predict most financial assets returns. They include:

Other commonly used macroeconomic factors considered in calculation may include:

•         Gross Domestic Product (GDP),
•         Commodities prices,
•         Market indices; short-term interest rates, the difference in long-term and short-term interest rates diversified stock index such as the S&P 500 or NYSE Composite, oil prices, gold or other precious metal prices, and Currency exchange rates

It takes a great amount of research to determine the level of sensitivity to changes in each macroeconomic factor. Selecting a macroeconomic factor to use in the calculation of an asset’s returns depends on several characteristics that the factor should manifest including:

1)      Have unexpected movements which subsequently impacts on an asset price.

2)      Their influence cannot be easily mitigated as they carry systemic risk.

3)      The information available concerning a macroeconomic factor should be timely and accurate.

4)      Their influence and risks to a financial asset can theoretically be proven on economic grounds.

Formula:

E(rj) = rf + β1*RP1 + β2*RP2 + ……..  + βn*RPn

Representation:

E(rj)Expected return on the financial asset j

rf – Risk-free rate of return

ßnThe level of volatility or sensitivity of an asset’s price with respect to changes by macroeconomic factor n

RPnRisk premium of macroeconomic factor n

### Beta Coefficient (βn)

In the mathematical model each factor is represented by a factor-specific beta coefficient (β). The Beta coefficient represents how an asset is volatile to the overall market based on its previous performances. It can also be described as a measure of systematic risk, of an individual asset in comparison to the unsystematic risk of the entire market.

Systematic risk, also known as un-diversifiable risk is refers to the risk of the entire market declining. The 2008 financial crisis in the United States is a good example of a systematic-risk event.no amount of diversification by investors during that period could prevent them from losing value in their stock portfolios.

Unsystematic or diversifiable risks, on the other hand, are risks associated with an individual stock. For instance, when in 2015 Lumber Liquidators (LL) made a surprise announcement that it had been selling hardwood flooring with dangerous levels of formaldehyde a dangerous drying chemical. The stock to the company plummets due to panic. However, and investor can mitigate such a risk by having  a diversified portfolio.

Generally, if a stock has a beta of one it indicates that its price activity is strongly correlated with the market. It means that the asset is an investable asset with little risks involved. However, adding a stock to a portfolio with a beta of one doesn’t add any risk to the portfolio, but also it doesn’t mean that the same portfolio will guarantee an excess return on a financial asset.

Also, a beta coefficient that is greater than one usually means that the financial asset is highly volatile and tends to move up and down with the market. For example, if a portfolio has a beta of 1.25 in relation to the Standard & Poor’s 500 Index (S&P 500), it is theoretically 25 percent more volatile than the S&P 500 Index. Therefore, if the index rises by 10 percent, the portfolio rises by 12.5 percent. If the index falls by 10 percent, the portfolio falls by 12.5 percent.

Whereas, a beta that is less than one may indicate either an investment with lower volatility than the market like treasury bills, or a volatile investment whose price movements are not highly correlated with the market like precious metals such as gold. In addition, negative betas are possible for investments that tend to go down when the market goes up, and vice versa, for example, put options.

The beta coefficients of various macroeconomic factors in the Arbitrage pricing model are estimated by using linear regression. In the graph; beta is represented by the slope of the line through a regression of data points from an individual stock’s returns against those of the market.

While beta offers useful information in determining a security’s short-term risk and for analyzing volatility to arrive at an asset returns using arbitrage pricing model, it does have a few shortcomings. Since Beta is found by statistical analysis of historical data points, that is, individual, daily share price returns in comparison with the market’s daily returns over precisely the same period. Therefore, it becomes less meaningful for investors looking to predict a stock’s future movements.

Additionally, because beta relies on historical data, it doesn’t factor in any new information on the market, stock or portfolio for which it’s used. Also, it is less useful for long-term investments, since a stock’s volatility can change significantly from year to year depending upon the company’s growth stage and other factors.

Finally, the beta coefficient assumes that stock returns are normally distributed from a statistical perspective. However, financial markets are prone to volatility and in essence returns aren’t always normally distributed. Therefore, what beta might predict about a stock’s movement may not always hold.

### Risk-Free Rate of Return(rf )

Risk-free return is the theoretical return attributed to an investment that guarantees zero risk with no likelihood of default or losing one’s money in the process. The risk-free rate represents the interest on an investment that would be expected by an investor from an absolutely risk-free venture over a specified period.

However, in practice a risk-free rate does not  exist because even the most safest investments considered risk-free still carry a very small amount of risk. The benchmark for a risk free rate in the market is short-term government-issued securities like the three-month U.S. Treasury bill. The choice is due to the fact that the government never defaults on its debt payment. Even the government is facing a low cash flow they can simply print more money to cover its interest payment and principal repayment obligations.

Therefore, the   U.S. Treasury bill (T-bill) short-term risk-free rate is the benchmark rate against which other returns are measured. Investors that purchase a security with some measure of risk higher than a US Treasury bill will consequently demand a higher level of return than the risk-free return.

A risk premium is the return in excess of the risk-free rate of return an investment is expected to yield; the difference between the return earned and the risk-free return represents the risk premium of an asset.an asset’s risk premium is a form of compensation for investors who tolerate the extra risk, compared to that of a risk-free asset, in a given investment.

For example, high-quality corporate bonds issued by established corporations that are earning large profits typically have very little risk of default. Consequently, such bonds pay a lower interest rate, or yield. However, those bonds issued by less-established companies with uncertain profitability and relatively higher default risk would issue a higher interest rate to attract investors with an appetite for risk.

Think of a risk premium as a form of hazard payment for an investment. An Investor expects compensation for the amount of risk they undertake in the form of a risk premium, or additional returns above the rate of return on a risk-free investment like the U.S. treasury bills. Therefore, Just as employees who have relatively dangerous jobs receive hazard pay as compensation for the risks they undertake, risky investments must provide an investor with the potential for larger returns to compensate for the risks of the investment.

However, the prospect of earning a risk premium does not mean investors can actually get it because it is possible the borrower may default. Therefore, Investors risk losing their entire money because of the uncertainty of the undertaken investment especially since a successful investment outcome is not guaranteed. What encourages investors to take on risky investment is the risk premium incentive offering potentially bigger payoffs.

### Arbitrage Pricing Theory Example

The following four macroeconomic factors have been identified as explaining a stock A’s return .The stock’s sensitivity to each factor and the risk premium associated with each factor have been calculated as listed below:

• Gross domestic product (GDP) growth: ß = 0.6, RP = 4%
• Inflation rate: ß = 0.8, RP = 2%
• Gold prices: ß = -0.7, RP = 5%
• Standard and Poor’s 500 index return: ß = 1.3, RP = 9%
• The risk-free rate is 3%

One is required to calculate the expected return using the Arbitrage pricing theory formula.

Solution;

E(ri) = rf + β1 * RP1 + β2 * RP2 + … + βn * RPn

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

### Capital Asset Pricing Model (CAPM)

Before the development of the arbitrage pricing theory in 1970, another model, the Capital Asset Pricing Model (CAPM) had earlier been developed in the 1960s. Economists Jack Treynor, William F. Sharpe, John Lintner, and Jan Mossin had developed CAPM to determine the theoretical expected return for an asset given the relationship to the level of risk assumed. CAPM is used widely in pricing risky financial assets, particularly stocks.

CAPM formula:

E(ri) = rf + βi * (E(rM) – rf)

Where;

rf – is the risk-free rate of return

βi – is the asset’s or portfolio’s beta in relation to a benchmark index

E(rM) – is the expected benchmark index’s returns over a specified period

E(ri) – is the theoretical appropriate rate that an asset should return given the inputs

(E(rM) – rf) – The difference between the return earned and the risk-free return represents the risk premium of an asset.

### CAPM Relationship to Arbitrage Pricing Model

The capital asset pricing model (CAPM) determines the theoretical appropriate rate that an asset should return given the level of risk assumed. The Arbitrage pricing theory as an alternative introduced a framework that explains the expected theoretical rate of return of an asset, or portfolio, in equilibrium as a linear function of the risk of the asset, or portfolio, with respect to a set of factors capturing systematic risk.

The underlying philosophy of arbitrage pricing theory is to improve some aspects of CAPM. This seeks to give complete answers and to consider multi-causal factors. Most experts consider it a more complete model because it explains with too diverse variables that influence the return that can be expected from an asset.

At first glance, the CAPM and Arbitrage pricing theory formulas may seem almost identical. However, while the CAPM formula has only one factor and one beta and requires the input of the expected market return, the Arbitrage theory formula, on the other hand, uses an asset’s expected rate of return and the risk premium of multiple macroeconomic factors which requires the asset’s beta in relation to each separate factor.

Since the CAPM is a one-factor model and simpler to use, investors may want to use it to determine the expected theoretical appropriate rate of return rather than using arbitrage pricing theory, which requires users to quantify multiple factors to analytically determine relevant factors that might affect the asset’s returns.

The Arbitrage pricing theory uses fewer assumptions and may be harder to implement than CAPM. Nevertheless, like CAPM, the theory assumes that a macroeconomic factor model can effectively describe the correlation between risk and expected returns for an asset. However, Unlike the CAPM the theory does not assume that investors hold efficient portfolios.

The main advantage of the theory over CAPM lies in the fact that the arbitrage model is a variable with more utility since an individualized asset can be taken as a reference without taking into account the entire portfolio and other available assets. Therefore, it is a model that can be conveniently utilized, not only by financial companies but also by entities from other sectors as well by individual investors.

### Academic Research on Arbitrage Pricing Theory

An empirical investigation of the arbitrage pricing theory, Roll, R., & Ross, S. A. (1980). The Journal of Finance, 35(5), 1073-1103.

Performance measurement with the arbitrage pricing theory: A new framework for analysis, Connor, G., & Korajczyk, R. A. (1986). Journal of financial economics, 15(3), 373-394.

The arbitrage pricing theory: is it testable?, Shanken, J. (1982). The journal of FINANCE, 37(5), 1129-1140.

International arbitrage pricing theory, Solnik, B. (1983). The Journal of Finance, 38(2), 449-457.

The empirical foundations of the arbitrage pricing theory, Lehmann, B. N., & Modest, D. M. (1988). Journal of financial economics, 21(2), 213-254.

Liquidity risk and arbitrage pricing theory, Cetin, U., Jarrow, R. A., & Protter, P. (2004). Finance and stochastics, 8(3), 311-341.

A critical reexamination of the empirical evidence on the arbitrage pricing theory, Dhrymes, P. J., Friend, I., & Gultekin, N. B. (1984).

The arbitrage pricing theory: Some empirical results, Reinganum, M. R. (1981). The Journal of Finance, 36(2), 313-321.

Measuring the pricing error of the arbitrage pricing theory, Geweke, J., & Zhou, G. (1996). The review of financial studies, 9(2), 557-587.

Some empirical tests in the arbitrage pricing theory: Macro variables vs. derived factors, Chen, S. J., & Jordan, B. D. (1993). Journal of Banking & Finance, 17(1), 65-89.

The arbitrage pricing theory, macroeconomic and financial factors, and expectations generating processes, Priestley, R. (1996). Journal of Banking & Finance, 20(5), 869-890.