## Monday, January 31, 2011

### Arbitrage Pricing Theory

Though not used commonly due to its complexities but Arbitrage Pricing Theory had attracted the attention of many financial managers and analysts since it has been initiated by Stephen Ross in 1976. It can be described as a theory of asset pricing and has been emerged into a powerful tool in determining the prices of financial assets.

The fundamental idea prevailing in Arbitrage Pricing Theory is that there are various dynamics usually referred to as macro-economic factors which form or comprise the expected returns from financial assets.  Primarily expected return can be characterized as linear function of such factors. Each factor involved in determination of expected return is sensitive to changes and is represented by beta coefficient specific to each factor. The return based on various factors and derived from APT is then used in valuing the assets accurately which is referred to as present value of expected cash flows from the assets till the end, discounted at expected rate of return as determined in APT.

APT Model

The Arbitrage Pricing Model is expressed as follows:

For a financial asset “A

The actual return is:

RA = E(RA) + bA1F1 + bA2F2 + bA3F3 + . . . . + bAnFn + EA

Where,

RA                           = Return on specific Asset
E(RA)                      = Assets Expected Return
F1,2,3…n                = Systematic Factors
bA1,A2,A3…An       = Beta Coefficients (sensitivity of each factor)
EA                          = Distinctive Random Astound of each Factor or error term

Note: There is no correlation exists between random astounds of each asset and factors.

In Arbitrage Pricing Model the expected return of a financial asset is:

E(RA)                 =          Rf + bA1P1 + bA2P2 + bA3P3 + . . . . + bAnPn

Where,

Rf                    =          Risk free rate of return
P1,2,3..n                        =          Risk premium associated with each factor

Note: The E(RA) model works on two assumptions:

Firstly there is perfect competition in the market
Secondly number of assets is more than number of factors

What is arbitrage?

In finance arbitrage refers to the process compelling the prices back into line when they diverge. It may happen that the prices are more or less than the potential and they diverge from the actual position hence differ market to market. Rational investors take advantage of the imbalance of prices among markets and make unusual gains through avoiding risks.

According to APT the investors buy or sell two or more assets. In this process they sell the expensive assets and invest in comparatively economical assets on the same return. Thus, enable them to generate or secure the investment for other opportunities as well.

The divergence of the price is determined when current price of the asset is observed different from the one derived as per APT model. Such divergence is also called out of price. For divergence not to prevail the asset price must equal till the end cash flows discounted at the rate consequent of the APT model. If APT price does not equate the current price the divergence exists and rational investor would Endeavour for to exploit the opportunities. In APT the asset return is the linear function of various factors.

Difference of APT with CAPM

CAPM is another influential theory of asset pricing and is commonly used in practice. Some differences between the APT and CAPM can be explained as follows:

• The assumptions in APT are less restraining.
• APT is descriptive in nature whereas CAPM is statistical.
• APT takes into account individual betas of each factor as compare to CAPM which is based on beta of identical market portfolio. In other words APT is multiple factor model with multiple risks effecting returns whereas CAPM is one factor model where only relevant risk is beta.

Practical Example of APT

Expected return for three securities:

A         =          13%
B         =          12%
C         =          18%

Risk free rate                        =          Rf        =          6%

There are two factors determining the expected return:

Factor 1 risk premium         =          P1       =          11%
Factor 2 risk premium         =          P2       =          3%

Beta coefficients are as follows:

 P1 P2 A 0.75 0.15 B 0.10 1.05 C 1.15 0.35

Required:     Please determine which security is underpriced and which one is overpriced.
Please explain resultant arbitrage process.

Solution:

Using two factor model the required returns as per APT are:

E(RA)             =          0.06 + 0.75 (0.11) + 0.2 (0.03)        =          14.85%

Security A is over priced

E (RB)            =          0.06 + 0.10 (0.11) + 1.05 (0.03)      =          10.25%

Security B is underpriced

E(RC)             =          0.06 + 1.15 (0.11) + 0.35 (0.03 0    =          19.70%

Security C is over priced

Arbitrage process:

An arbitrager will sell or short sell security A and C and buy security B.

Along with the actions of other arbitragers the following will result:

1. Price of security A and C will drive down
2. Expected return of security A and C will rise
3. Price of security C will drive up
4. Expected return of security C will fall

These actions will continue until the expected returns equal the required returns generated by APT factor model.