Definition of 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 : single period investment horizon, no taxes, investors can borrow and lend at risk free rate of interest and 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 + ΔX_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 b_A, b_B 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

ΔX_A = .02

ΔX_B = .025

ΔX_c = -.225

For an arbitrage portfolio

ΔX_A + ΔX_B + ΔX_c = 0

.2 + .025 -.225 = 0

The sensitivities also become zero

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

ΔX_AR_A+ ΔX_BR_B + ΔX_cR_c> 0

.2 x 20 + .025 x 15 -.225 x 12

4.375—2.7 > 0

i.e. 1675%

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

X_A = 0.53

X_B = 0.355

X_c = 0.115

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

= 1, 50,000 X .53 + 1, 50,000 X .355 + 1, 50,000 x .115

= `79, 500 +53250 +17250

The sensitivity of the new portfolio will be

= .45x.53+ l.35x.355+.55x.115

= .239 + .479 + .063 =.781

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

=20x.33+ 15x.33+ 12x.34

=6.6 + 4.95 + 4,08

=15.63%

The new portfolio return

= 20x.53+ 15x.355+ 12x.115

= 10.6 + 5.325 + 1.38

= 17.305%

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 non-factor 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:

R_i  = λ₀ + λ₁bi₁ + λ₂bi₂ + ……. + λ_jbi_j

R_i  = average expected return

λ_I  = sensitivity of return to b_il

b_il = the beta co-efficient relevant to the particular factor.

The equation is derived from the model

R_i = α₁ + b_il I₁ + b₁₂I₂ ….. + b_ijI_j + e_i

Let us take the two factor model

R_i = λ₀ + λ₁b₁₂ + λ₂b₁₂ + b₂

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

12 = λ₀ + 1λ₁ + 0.5λ₂

13.4 = λ₀ + 3λ₁ + 0.2λ₂

12 = λ₀ + 3λ₁ - 0.5λ₂

We can get

R_i = 10 + 1b_il + 2bi₂

The expected return is

Rp = ∑  NX R

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

b_p1 =1/3 x 1+ 1/3 x 3 + 1/3 x 3 = 2.33

b_p2 =0.5 x l/3 + 0.2 x 1/3 + (-0.5 x 113)=0.066

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

1/3(12) + 1/3(13.4) + 1/3(12)

=12.46

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

R = 10 + 1(2.33)+2(.066)

 = 12.46

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.

R_i = λ₀ + λ_ib_i

R_i = return from stock A

λ₀ = riskless rate of return

b_i = the sensitivity related to the factor

λ_i = slope of the arbitrage pricing line

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.

The Constants of the APT Equation

The existence of the risk free asset yields a risk free rate of return that is a constant.

The asset does not have sensitivity to the factor for example, the industrial production.

If b_i = 0

R_i = λ₀ + λ_iO

R_i = λ₀

In other words, λ₀ is equal to the risk free rate of return. If the single factor portfolio’s sensitivity is equal to one i.e. b₁ = 1 then

R_i = λ₀ + λ_i1

This can be rewritten as

R_i                    = λ₀ + λ_i

R_i - λ₀   = λ_i

Thus λ₁ is the expected excess return over the risk free rate of return for a portfolio with unit sensitivity to the factor. The excess return is known as risk premium.

Factors Affecting the Return

The specification of the factors is carried out by many financial analysts. Chen, Roll and Ross have taken four macro economic variables and tested them. According to them the factors are inflation, the term structure of interest rates, risk premium and industrial production. Inflation affects the discount rate or the required rate of return and the size of the future cash flows. The short term inflation is measured by monthly percentage changes in the consumer price index. The interest rates on long term bonds and short term bonds differ. This difference affects the value of payments in future relative to short term payments. The difference between the return on the high grade bonds and low grade (more risky) bonds indicates the market’s reaction to risk. The industrial production represents the business cycle. Changes in the industrial production have an impact on the expectations and opportunities of the investor. The real value of the cash flow is also affected by it.

Burmeister and McElroy have estimated the sensitivities with some other factors. They are given below

Default risk

Time premium

Deflation

Change in expected sales

The market returns not due to the first four variables.\

The default risk is measured by the difference between the return on long term government bonds and the return on long terms bonds issued by corporate plus one-half of one per cent. Lime premium is measured by the return on long term government bonds minus one month Treasury bill rate one month ahead.

Deflation is measured by expected inflation at the beginning of the month minus actual inflation during the month. According to then, the first four factors accounted 25% of the variation in the Standard and 1or Composite Index and all the four co-efficient were significant.

Salomon Brothers identified five factors in their fundamental factor model. Inflation is the only common factor identified by others. The other factors are given below

Growth rate in gross national product

Rate of interest

Rate of change in oil prices

Rate of change in defence spending

All the three sets of factors have some common characteristics. They all affect the macro economic activities. Inflation and interest rate are identified as common factors. Thus, the stock price is related to aggregate economic activity and the discount rate of future cash flow

APT and CAPM

The simplest form of APT model is consistent with the simple form of the CAPM model. When only one factor is taken into consideration, the APT can be stated as:

R_i = λ₀ + b_iλ_i

It is similar to the capital market line equation

R = R_f + ß (R_m —R_f)

Which is similar to the CAPM model?

APT is more general and less restrictive than CAPM. In APT, the investor has no need to bold the market portfolio because it does not make use of the market portfolio concept. The portfolios are constructed on the basis of the factors to eliminate arbitrage profits. APT is based on the law of one price to hold for all possible portfolio combinations.

The APT model takes into account of the impact of numerous factors on the security. The macro economic factors are taken into consideration and it is closer to reality than CAPM.

The market portfolio is well defined conceptually. In APT model, factors are not well specified. Hence the investor finds it difficult to establish equilibrium relationship. The well defined market portfolio is a significant advantage of the CAPM leading to the wide usage of the model in the stock market.

The factors that have impact on one group of securities may not affect another group of securities. There is a lack of consistency in the measurements of the APT model.