# Arbitrage Pricing Theory Description

The theory refers to an asset pricing model based on the idea that t is possible to project the returns of a particular asset by establishing the relationship existing between the asset and the existing risk factors. Arbitrage Pricing Theory (hereafter referred to as APT) is based on the assumption that the investors hold a portfolio of assets. As a result, the unsystematic risk is minimized. APT is affected by a number of macroeconomic factors such as changes in the yield curve, inflation or deflation, investor confidence, and change in Gross Domestic Product (GDP). The theory specifies the returns of the portfolio in a linear function. The returns of the portfolio are based on the formula outlined below.

E(Ri) = Rf + b1(E(R1) – Rf) + b2(E(R2) – Rf) + b3(E(R3) – Rf) + …+ bn(E(Rn) – Rf)

Where Rf represents the risk-free interest rate, bi =asset sensitivity to the factor i and E (Ri) – Rf) is the risk premium.

## Example

An investor has diversified his investment by developing two portfolios. The portfolios have the following characteristics.

**Portfolio A:**

Expected return represented by (E (RA) is 20%,

Systematic risk ( bA ) =1.5.

**Portfolio B:**

Expected return= 10%

Systematic risk= 1.

**Another portfolio C **has the following characteristics

Expected return = 20%

Systematic risk= 1.2.

From the above information, it is evident that the rate of return for portfolio C is similar to that o A. However, the risk involved differs. Therefore, an investor can make a higher profit from portfolio C. Alternatively, he can create another portfolio, D, by combining portfolios A and B on the proportion of 40% and 60%. Using the APT formula, the systematic risk can be calculated as follows.

E (RD) = .4E (RA) +.6E (RB) = .4.2 + .6.1 = .14 or 14%

bD = .4bA + .6bB = .41.5 +.6 1 = 1.2

This shows that portfolio C and D have the same degree of risk; however, the level of returns are relatively higher. This means that it is possible for investors to identify securities to invest in. APT does not give clear guidance regarding the factors which should be expected n the process of determining risk premium. A correlation must exist between the factors affecting APT and the sources of uncertainty which is a concern to investors. This means that it is important for researchers to establish the relationship between the degree of uncertainty with regard to investment and consumption. Investors should incorporate Industrial Production (IP) indicators in an effort to evaluate changes within the business cycle. Other factors affecting risk premium which should be investigated are related to GDP, the rate of interest, and inflation.