# Arbitrage Pricing Theory (APT)

## Arbitrage Pricing Theory (APT)

Arbitrage Pricing Theory (APT) is a model used for identifying an asset, which is priced incorrectly, i.e. either the asset is undervalued or overvalued. After identifying such assets, investors gets the actual price of the asset and he/she can bring the mispriced value of the asset to its actual price. As discussed earlier, the APT model assumes that the return on an asset is affected by several factors and the impact of these factors on the asset.

These factors include macroeconomic factors (like inflation, Gross National Product (GNP), confidence and risk taking capacity of investor, shifts in the yield curve, etc.) and asset-specific factors (like change in price, market trends, competition level, etc.). The macroeconomic factors can be easily determined as they are known to the general public.

However, identifying the factors that influence a particular asset is crucial for the investors as the impact of a particular factor is different on different assets. For example, the impact of price change in crude oil is more on the share of ExxonMobil than on the share of Colgate.

Thus, in the APT model, an investor needs to identify:

• The factors that influence a specific stock
• The expected returns for each of the identified factors
• The sensitivity of the stock to these factors

The formula for APT model is as follows:

E(rj ) = rf + bj1RP1 + bj2RP2 + … + bjnRPn

Where,

RPk = Risk premium of the factor

Tf = Risk-free rate

bjk = Sensitivity of the jth asset to factor k

In an arbitrage portfolio, the investor tries to maximise return on an asset without any further investment on the asset and keeping the risk at the same level. For instance, an investor has three securities, namely P, Q, and R, in his portfolio and he wants to increase the return from the portfolio without any further investment. So, the investor can change the proportion of the three securities held by him.

For this purpose, he identifies P as the most profitable security for him and provides the maximum return as compared to the other two securities, i.e., Q and R. Thus, the investor will convert Q and R securities in P instead of buying an extra amount of P securities. For example, a shopkeeper has a hundred cream biscuits with three different flavours, i.e., chocolate, pineapple, and strawberry.

He found that the chocolate cream biscuit is sold more as compared to the other two biscuits, i.e., pineapple and strawberry biscuits. As per the arbitrage theory, in order to maximise his returns and minimise the risk of having unsold biscuits in the stock, the shopkeeper start selling pineapple and strawberry biscuits first till their stock lasts and then sell the chocolate biscuits. As a result, all the hundred biscuits were sold.

## Rule of One Price

APT model is based on the rule of one price, which states that in efficient market, security must have a single price, disregarding its origin. Two diverse primary sets of securities create an option for the investor. In such a case, total price for each security would be same or else an arbitrage opportunity would exist. As per this rule, similar type of products should be sold on same prices. For example, the price of one ounce of silver should be same in New York and Paris.

Though this law is not always feasible to put in practice, it could only be practiced if there is no competition, transaction cost, or trade barriers in any industry or country. It is powerfully believed by the leading economists that in liquid financial markets the law of one price can be accomplished because of the probability of arbitrage. The practical aspect of this law is weak, thus, it is voilated many times with the introduction of funds like ADRs, corporate spin-offs and dual share class stocks.

There is another law of one price used in arbitrage pricing theory that is slightly different from the above. This law is based on the fact that two financial instruments or portfolios should have the same cost if they have same return and risk. This is because it takes into consideration only one aspect of financial instruments that it is purchased to earn an expected return while taking a certain amount of risk. Thus, the law of one price states that two financial instruments or portfolios having same return and risk will be sold at the same price.

## Assumption of APT

According to the arbitrage theoretically, the expected returns should be the same for two assets having the same risk. In the case of difference in expected returns, the arbitrageurs can sell the asset with a lower return and can buy the asset offering higher return and make some risk-free arbitrage profit.

The APT model is based on the following assumptions:

• The APT model assumes that the return from an asset is affected by a number of factors; thus, it can be expressed as a multi-factor model.

• This model assumes that the investor holds a number of securities so that the unsystematic risk is eliminated and returns from the securities can be maximised.

• In APT model, it is assumed that the markets are perfect and frictionless. In addition, the investors have same perception about the returns on a particular asset.

• The APT model assumes that the restrictions on short-selling is nil.

• It is also assumed in the APT model that if the condition of market equilibrium is achieved, then there will be no riskless arbitrage opportunities available in the market.

## Factor Model in APT

Let’s take a more statistical approach with the actual behaviour of stock returns and their co-movements. Stocks in a country are affected by the interest rates, changes in technologies, movements in the cost of resources (like men and materials), etc.

Thus, the APT model takes into account various macroeconomic factors that affects the returns from a stock. A factor model is one that represents a set of factors and their relationship with the assets’ return. These factors include common factors (macroeconomic factors) that affect an asset and firm-specific or asset-specific (idiosyncratic) factors.

Factors should possess the following characteristics:

• Every factor should have an impact on the stock returns.

• Factors should also affect the expected returns of the stock. This can be determined by statistically analysing which factor pervasively impacts the stock returns.

• At the beginning of each financial period, it is difficult to predict the risk factors as a whole in the market.

The main point in factor model is the deviation of the factor from the expected value. For example, if the expected rate of inflation is 5% and the actual inflation rate is 4%, then 1% deviation will affect the actual return from the stock.

In APT, the factor model can be expressed as follows:

Where,

bi,j = Represents increase in the asset return (i) due to one unit increase in the factor (j), also called factor loading

el= Denotes idiosyncratic risk

As el is a firm-specific factor, any change in the organisation will affect it, which further impacts the overall return on the asset. For example, if the president of an organisation dies, the value of el will become negative, which when placed in the above formula will lead to a decline in the total return from an asset.

In APT, it is assumed that the expected value of each factor F is zero and so the value of fs in the above equation is the deviation of each factor from this expected value. Let us take an example to understand factor models in APT.

Suppose, The expected return of portfolio A is equal to the risk-free rate as the value of beta is equal to zero. The ratio of risk premium to beta of another portfolio named, portfolio B, is: (12 − 6)/1.2 = 5, while that of portfolio E’s ratio is (8 – 6)/0.6 = 3.33. In such a case, an arbitrage opportunity exists. For instance, a portfolio C is created by clubbing portfolio A and B in equal proportions and taking beta of portfolio E, i.e., 0.6.

Thus, the expected return of portfolio C is:

E(rG) = .5 × 12% + .5 × 6% = 9%

As C and E has same same beta and return, an arbitrage opportunity exists. An investor can earn profit by buying portfolio C and selling equal amount of portfolio E. In such a case, the profit earned by the investor will be equal to:

rG – rE = (9% + .6 × F) – (8% + .6 × F) = 1%

## Use of Apt in Investment Decisions

APT is an strategic approach to portfolio management. It helps the investors to identify the factors that affect most of the securities and their influence on the returns of these securities. This information enables investor to design their portfolio in a manner that its performance is improved. The APT model also help investors to identify the factors to which a particular sector or industry is sensitive to and the degree of this sensitiveness.

For example, utility sector is insensitive to factors like unexpected inflation and change in the growth rate of profits. Similarly, in case of industries, studies shows that retail, mobile, and hotel industry is more sensitive to unexpected inflation while food, shoes, and tire industry are least sensitive to it. Such information helps the investors to make investment decisions and design their portfolio accordingly. A portfolio manager can also forecast the movement of the factor, which affects the return on the asset or securities.

This will further help the manager to design its portfolio in a manner that the impact of the factor is minimum. For doing so, the manager can select the assets that are least sensitive to the factor and sensitive to the factors that are favourable in the present market situation. This will help the manager to enhance the performance of his/her portfolio. Let us take an example of risk arbitrage to understand the importance of the APT model in investment decisions.

Suppose McDonald’s new fat-free burger has more demand than the new fat-free burger of Burger Kings. Based on this assumption, if an arbitrager buys \$10,000 worth of McDonald’s stock and sells \$10,000 worth of Burger King stock, then his/her decision is impacted only by one factor, i.e., the difference between the two companies’ firm-specific movements.

However, the strategy adopted by the arbitrager involves considerable risk by several other factors, such as any change in the management of the organisation, changes in law or legal aspects reslated to the companies, etc. This is an example of risky arbitrage or use of the APT model in investment decisions. The APT model is often used in hedging of funds wherein the investors want to make bets on the relative prices of financial funds which they believe to be inconsistent with respect to economic fundamentals without exposing themselves to other things happening in the markets.