# Arbitrage Pricing Theory (APT)

The fundamental foundation for the **arbitrage pricing theory** (**APT**) 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 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 returns and risks are identical**. The justification for this is that the only reason that a financial instrument is purchased is to earn an expected return in exchange for accepting some risk — no other aspect of the financial instrument matters. Hence, the law of one price requires that any 2 financial instruments or portfolios with the same **return-risk profile** should sell for the same price. If this is not true, then a profit could be made through **risk arbitrage** — selling short the security or portfolio with the lower return, and buying the higher return portfolio. This coheres with the **capital asset pricing model** (**CAPM**), which postulates that the expected return of an asset is proportional to its risk.

## Macroeconomic Factor Risks

Investment risk is composed of systemic risk and firm-specific risk. Systemic risk derives from macroeconomic factors, which affects all investments, including the general state of the economy, the stage of the business cycle, interest rates, and inflation. Firm-specific risk affects only particular firms, such as the death of key employees or the quality of management. APT considers only macroeconomic risks, since these risks cannot be eliminated by through diversification. On the other hand, firm-specific risks can be eliminated through diversification. Therefore, the market offers no risk premium for taking on firm-specific risk, since it can be easily eliminated. However, systemic risk cannot be eliminated through diversification. Therefore, investors will only hold assets with an expected return commensurate with their systemic risk.

Different assets have different sensitivities to systemic risk, which is the beta of the asset. By definition, the beta of the market = 1. Some assets will have a higher beta, meaning that their percentage change in price will usually exceed that of the market, and some assets will have a lower beta, where the percentage change in price will usually be less than the market.

A **factor beta** (aka **factor sensitivity**, **factor loading**) can also be calculated for each type of macroeconomic factor risk, equal to the percentage change in the expected return for each unit change in the macroeconomic risk factor. So if the expected return declined by 1.5% for each 1% increase in interest rates, then the beta for interest rate risk for that particular firm is -1.5. Generally, factor betas can be found through regression analysis of historical changes in the expected return for a given change in the systemic risk factor.

The expected rate of return = the risk-free rate + the risk premium for taking on any systemic risk. Since different systemic risk factors have different risk premiums, the expected rate of return can be further decomposed into the risk-free rate plus the premium for each risk factor multiplied by the beta for that risk factor.

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

E(r_{i}) = r_{f} + F_{1}b_{1}

A graph of this line is the **arbitrage pricing line** for 1 risk factor. Not that this follows the capital allocation line (CAL), with r_{f} as the proportion of the portfolio consisting of the risk-free security and F_{1}b_{1} representing the proportion of the risky asset, with F_{1} representing the risk premium for the macroeconomic factor and b_{1} representing the sensitivity of the return compared to a unit change in the risk factor, just as beta represents the volatility of a stock compared to the market in the CAPM.

Both the APT and the CAPM 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;
- a market portfolio;
- that returns are normally distributed;
- that investors can borrow or lend at the risk-free rate;
- nor is there any need for utility functions.

Additionally, APT assumes unrestricted short-selling, since the arbitrage of portfolios requires this.

### Example: Portfolio Risk Arbitrage

Consider the following 2 portfolios: P_{a} has an expected return of 20% and P_{b} has an expected return of 17%. Both have a beta of 1.8. Thus, a riskless profit could be made with no net investment by buying P_{a} and selling short P_{b}. The proceeds of selling short P_{b} can be used to buy P_{a}, so the $30 income earned for each $1,000 requires no net investment. Note that the betas are equal and opposite, so there is no risk.

Portfolio | Initial Payments | Cash Flow | Beta |
---|---|---|---|

P (long)_{a} | -$1,000 | $200 | 1.8 |

P (short)_{b} | +$1,000 | -$170 | -1.8 |

0 | +$30 | 0 |

Continuing to short P_{b} and buying P_{a} will eliminate the arbitrage profits since the demand for P_{a} will increase, thus increasing its price while the supply of P_{b} will increase, thus decreasing its price, until the returns of P_{a} and P_{b} are equalized, 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 are the cause of systemic risk, and 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 systemic risk, the theory can easily be extended to include any number of macroeconomic factors with associated betas for each factor:

E(r_{i}) = r_{f} + F_{1}b_{i1 } + F_{2}b_{i2} + ... + F_{n}b_{in}

where F_{n} is the risk premium for that factor and b_{in} is the factor 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.