Arbitrage Pricing Theory (APT)

The fundamental foundation for the arbitrage pricing theory is the law of one price, which states that 2 identical items will sell for the same price, for if they do not, then a riskless profit could be made by arbitrage—buying the item in the cheaper market then selling it in the more expensive market. This principle also applies to financial instruments, such as stocks and bonds. For instance, if Microsoft stock is selling for $30 on one exchange, but $30.25 on another exchange, then an arbitrageur could simultaneously buy the stock on the cheaper exchange and sell it short on the more expensive exchange for a riskless profit. (The arbitrage is done simultaneously because the price discrepancy must be taken advantage of immediately; otherwise it will probably disappear by the time of settlement.) The arbitrageur would continue doing this until the price discrepancy disappeared, since buying on the cheaper exchange would increase the demand, and therefore the price, on that exchange, while the short selling on the more expensive exchange would increase supply, thereby reducing its price.

There is another law of one price used in the arbitrage pricing theory that is slightly different from the above examples. It is predicated on the fact that 2 financial instruments or portfolios—even if they are not identical—should cost the same if their return and risk is identical. The justification for this is that the only reason that a financial instrument is purchased is to earn a return for a certain amount of risk—no other aspect of the financial instrument matters. Hence, the law of one price requires that any 2 financial instruments or portfolios that have the same return-risk profile should sell for the same price. If this is not true, then a profit could be made by selling short the security or portfolio with the lower return, and buying the higher return portfolio.

The simplest form of the APT is the one macroeconomic factor model for the i^th security or portfolio:

E(r_i) = λ₀ + λ₁b_i

A graph of this line is the arbitrage pricing line for 1 risk factor. Not that this is similar to the capital allocation line (CAL), with λ₀ as the proportion of the portfolio consisting of the risk-free security and λ₁b_i representing the proportion of the risky asset, with λ₁ representing the risk premium for the macroeconomic factor and b_i representing the sensitivity of the return compared to the market return, just as beta represents the volatility of a stock compared to the market in the capital asset pricing model.

The 1 factor APT is similar to the capital asset pricing model (CAPM). Both models assume that investors:

• prefer more wealth to less;
• are risk-averse;
• have similar expectations;
• and that capital markets are efficient.

However, APT has more general applicability, since it does not assume:

• a 1-period horizon;
• that returns are normally distributed;
• a market portfolio;
• that investors can borrow or lend at the risk-free rate;
• nor is there any need for utility functions.

Example—Portfolio Arbitrage

Consider the following 2 portfolios:

where E(r_p) is the expected return of the portfolio and b_P is the portfolio beta. Both portfolios should have equal lambda factors, which can be found by solving for them simultaneously:

1. λ₀ + λ₁(1.5) = 20%
2. λ₀ + λ₁(1.0) = 10%
3. λ₁(0.5) = 5%
4. λ₁ = 10%
5. Substituting Equation 4 into Equation 1, we find:
6. λ₀ + 0.1(1.5) = 20%
7. λ₀ = 20% - 15% = 5%

From this, we find the equilibrium APT equation:

E(r_p) = 5% + 10%(b_p)

Now consider a portfolio C where E(r_P) = 20% and b_P = 1.2. Since portfolio C yields the same as A, but has a reduced risk factor as evidenced by its lower beta, an arbitrage profit should be possible.

1. Construct a portfolio from A and B that has the same risk as portfolio C:
2. b_C = y_A(b_A) + (1 - y_A)(b_B) -> y_A + y_B = 1
3. 1.2 = y_A(1.5) + (1 - y_A)(1.0)
4. 1.2 = 1.5y_A + 1 - y_A
5. 1.2 = 1 + 0.5y_A
6. 0.2 = 0.5y_A -> Subtracted 1 from both sides.
7. 0.4 = y_A -> Divide both sides by 0.5
8. Amount of portfolio B must be 0.6, since the proportions of both portfolios must sum to 1.
9. Now construct a portfolio D that has the same risk factor as portfolio C:
10. E(r_D) = .4(r_A) + .6(r_B)
11. E(r_D) = .4(20%) + .6(10%) = 8% + 6% = 14%

As you can see, both portfolios have the same beta, and, therefore, the same risk profile. But by selling portfolio D short, and using the proceeds of the short sale to buy portfolio C, an arbitrageur can earn a $60 profit for each $1,000 invested. Continuing to short portfolio D and buying portfolio C will eliminate the arbitrage profits, thereby equalizing the returns of portfolios C and D, which is as it should be, since they have the same risk. This is the essence of the arbitrage pricing theory and the law of one price.

Multifactor Arbitrage Pricing Theory

Implicit to the APT as well as the CAPM is that only macroeconomic risk factors, such as unanticipated changes in interest rates or inflation, or unemployment rates that affect every firm and are the cause of systematic risk, have pricing value. Microeconomics factors, such as the death of key employees or the firm's credit rating, that cause firm-specific risk have no pricing power because such risk can be reduced to zero through diversification. Although the simplest form of the arbitrage pricing theory assumes that there is only 1 macroeconomic factor causing systematic risk, the theory can easily be extended to include any number of macroeconomic factors with associated betas for each factor:

E(r_i) = λ₀ + λ₁b_i1 + λ₂b_i2 + ... + λ_nb_in

where λ_n is the risk premium for that factor and b_in is the beta for that risk factor.

The general solution to solving these multi-factor equations is to solve for these factors simultaneously to see what they equal for a given set of portfolios.