Arbitrage Pricing Theory

Arbitrage pricing theory is one of the tools used by the investors and portfolio managers. The capital asset pricing theory explains the return of the securities on the basis of their respective betas. According to the previous models, the investor chooses the investment on the basis of expected return and variance. The alternative model developed in asset pricing by Stephen Ross is known as Arbitrage Pricing Theory. The APT theory explains the nature of equilibrium in the asset pricing in a less complicated manner with fewer assumptions compared to CAPM.

Arbitrage

Arbitrage is a process of earning profit by taking advantage of differential pricing for the same asset. The process generates riskless profit. In the security market, it is of selling security at a high price and the simultaneous purchase of the same security at a relatively lower price.

Since the profit earned through arbitrage is riskless, the investors have the incentive to undertake this whenever an opportunity arises. In general, some investors indulge more in this type of activities than others. However, the buying and selling activities of the arbitrageur reduce and eliminate the profit margin, bringing the market price to the equilibrium level.

The Assumptions

The investors have homogenous expectations.

The investors are risk averse and utility maximisers.

Perfect competition prevails in the market and there is no transaction cost.

The APT theory does not assume (1) single period investment horizon, (2) no taxes investors can borrow and lend at risk free rate of interest and (4) the selection of the portfolio is based on the mean and variance analysis. These assumptions are present in the CAPM theory.

Arbitrage Portfolio

According to the APT theory an investor tries to find out the possibility to increase returns from his portfolio without increasing the funds in the portfolio. He also likes to keep the risk at the same level. For example, the investor holds A, B and C securities and he wants to change the proportion of the securities without any additional financial commitment. Now the change in proportion of securities can be denoted by X_A, X_B, and X_C. The increase in the investment in security A could be carried out only if he reduces the proportion of investment either in B or C because it has already stated that the investor tries to earn more income without increasing his financial commitment. Thus, the changes in different securities will add up to zero. This is the basic requirement of an arbitrage portfolio. If X indicates the change in proportion,

ΔX_A+  X_B+     _C=0

The factor sensitivity indicates the responsiveness of a security’s return to a particular factor. The sensitiveness of the securities to any factor is the weighted average of the sensitivities of the securities, weights being the changes made in the proportion. For example bA, bB and b are the sensitivities, in an arbitrage portfolio the sensitivities become zero.

h_AΔX_A +b_BΔX_B +b_cΔX_c = 0

The investor holds the A, B and C stocks with the following returns and sensitivity to changes in the industrial production. The total amount invested is ` 1,50,000.

Now the proportions are changed.

The changes are

For an arbitrage portfolio

The sensitivities also become zero

In an arbitrage portfolio, the expected return should be greater than zero.

The investor would increase his investment in stock A and B by selling C. The new compositions of weights are

The portfolio allocation on stocks A, B and C is as follows

The sensitivity of the new portfolio will be

This is same as the old portfolio sensitivity

i.e. .45x.33+ 1.35x.33+ .55x.34= .781

The return of the new portfolio is higher than the old portfolio.

Old portfolio return

The new portfolio return

This is equivalent to the old portfolio return plus the return that occurred due to the change in portfolio

= 15.63% +1.675% = 17.305%

The variance of the new portfolio’s change is only due to the changes in its nonfactor risk. Hence, the change in the risk factor is negligible. From the analysis it can be concluded that

The return in the arbitrage portfolio is higher than the old portfolio.

The arbitrage and old portfolio sensitivity remains the same.

The non-factor risk is small enough to be ignored in an arbitrage portfolio.

Effect on Price

To buy stock A and B the investor has to sell stock C. The buying pressure on stock A and B would lead to increase in their prices. Conversely selling of stock C will result in fall in the price of the stock C. With the low price there would be rise in the expected return of stock C. For example, if the stock “C” at price 100 per share have earned 12 percent return, at 80 per share the return would be 12/80 x 100=15%.

At the same time, return rates would be declining in stock A and B with the rise in price. This buying and selling activity will continue until all arbitrage possibilities are eliminated. At this juncture, there exists an approximate linear relationship between expected returns and sensitivities.

The APT Model

According to Stephen Ross, returns of the securities are influenced by a number of macro economic factors. The macro economic factors are growth rate of industrial production, rate of inflation, spread between long term and short term interest rates and spread between low-grade and high grade bonds. The arbitrage theory is represented by the equation:

The equation is derived from the model

Let us take the two factor model

If the portfolio is a well diversified one, unsystematic risk tends to be zero and systematic risk is represented by b_i1 and b_i2 in the equation.

Let us assume the existence of three well diversified portfolios as shown in the table.

The equation R_i = λ₀ + λ₁b_i1 + λ₂b_i2 + b₂ can be determined with the help of the above mentioned details. By solving the following equations

We can get

The expected return is

The risk is indicated by the sensitivities of the factors

All the portfolios constructed from portfolios A, B and C lie on the plane described A, B and C. Assume there exists a portfolio D with an expected return 14%, b_i1 2.3 and b_i2 = .066. This portfolio can be compared with the portfolio E having equal portion of A, B and C portfolios. Every portfolio would have a share of 33%. The portfolio b are

The risk for portfolio E is identical to the risk on portfolio D. The expected return for portfolio B is

Since the portfolio B lies on the plane described above, the return could be obtained from the equation of the plane.

The portfolio D and B have the same risk but different returns. In this juncture, the arbitrageur enters in and buy portfolio D t selling portfolio B short. Thus buying of portfolio D through the funds generated from selling B would provide riskless profit with no investment and no risk. Let us assume that the investor sells Rsr1000 with of portfolio E and buys Rs1000 worth of portfolio D. The cash flow is as shown in the following table.

The arbitrage portfolio involves zero investment, has no systematic risk (b_il and b_i2) and earns ` 15.4. Arbitrage would continue until portfolio D lies on the same plane.

Arbitrage Pricing Equation

In a single factor model, the linear relationship between the return and sensitivity b can be given in the following form.

The above model is known as single factor model since only one factor is considered. Here, the industrial production alone is considered. The APT one factor model is given in figure.

The risk is measured along the horizontal axis and the return on the vertical axis. The A, B and C stocks are considered to be in the same risk class The arbitrage pricing line intersects the Y axis on which represents riskless rate of interest i e the interest offered for the treasury bills Here, the investments involve zero risk and it is appealing to the investors who are highly risk averse stands for the slope of arbitrage pricing line It indicates market price of risk and measures the risk-return trade off in the security markets. The is the sensitivity coefficient or factor beta that shows the sensitivity of the asset or stock A to the respective risk factor.