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:

R_A = E_(RA) + b_A1F₁ + b_A2F₂ + b_A3F₃ + . . . . + b_AnF_n + E_A

Where,

R_{A                           =} Return on specific Asset

E_{(RA)                      =} Assets Expected Return

F_{1,2,3…n                =} Systematic Factors

b_{A1,A2,A3…An       =} Beta Coefficients (sensitivity of each factor)

E_{A                          =} 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)                 =          R_f + b_A1P₁ + b_A2P₂ + b_A3P₃ + . . . . + b_AnP_n

Where,

R_f                    =          Risk free rate of return

P_1,2,3..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.

Insurance

Insurance is a contract where by insurer undertakes to indemnify the insured for any loss suffered due to specific risk. This is essentially a method of averaging the risks. Several people exposed to a particular risk contribute small amounts to an insurance fund from which the unfortunate who actually suffers the risk is compensated. The party which indemnifies the risk is normally an insurance company called insurer and the party which is indemnified called insured. The document containing the term of contract is known as policy.

Depending upon the type of risk, there are several forms of insurance such as:

• Fire Insurance
• Marine Insurance
• Burglary Insurance
• Workmen’s Compensation Insurance
• Life Insurance

Life insurance takes two forms:

• Endowment Policy – sum of money is paid by the insurer to the insured after the attainment of specific age or paid to his or her family after the death of insured.
• Whole Life Insurance – sum of money is paid to the family of insured after his or her dearth

Life insurance is different from other types of insurance in the sense that other forms of insurance are contracts of indemnity whereas life insurance is a contract of assurance. In contract of indemnity those who suffered loss are compensated to the extent of actual loss. In life insurance eventually the amount covered by the policy must be paid, therefore, life insurance is also known as assurance.

It is also difficult to evaluate a loss when someone dies and therefore amount of policy is the agreed amount which the insurance company pays. The main purpose of life insurance is protection of the family of insured either because of old age or death. Apart from the fact it covers the risk the distinctive advantage of life insurance is that it is also a form of investment and an investment that is increasingly preferred because of tax incentives.

In a contract of insurance, however, there is an implied condition that each party must disclose every material fact known to him. This is because all contracts of insurance are contract of uberrimae fidie i.e. contract of utmost good faith.

Investment in and Valuation of Stock

The main objective behind investment in shares is to get dividends and earn capital gains over the periods due to increase in the market price of these shares. Literally in the investment decisions of shares the concept of undervaluation and overvaluation works. A mindful investor always prefers to invest in undervalued shares which mean intrinsic value of a share is greater than its market price and investors foresee more benefits in future. In case of overvaluation a cognizant investor chooses to sell the stock because true value or intrinsic value of a share is less than its market price and he/she gains from selling the share and using the proceed in more advantageous opportunities.

The question arises what does true or intrinsic value of a share means. Ideally true value of a share is the present value of the dividends in years to come which is computed by keeping in view the expected growth pattern of the dividend in future. Under the dividend growth pattern the share valuation models are explained as under:

Zero-Growth-Model (No growth in dividend per share in future)

In case the company is offering constant dividend per share and its policy of constant dividend is expected to remain continue in future the intrinsic value of a share is simply the present value of perpetuity (constant dividend per share) discounted at required rate of return.

Illustration

A firm pays dividend of $4 regularly over the years and declares that the same policy will continue for an indefinite time period. Investors’ required rate of return is 15%.

Requirement: Please compute price per share

Answer:

Price per share = $4 / 0.15 = $26.67

Steady-Growth- Model (Constant growth in dividend per share in future)

Steady-Growth-Model also called Gordon Stock Valuation Model refers the computation of value of a share in the situation where constant percentage growth prevails. In other words companies have a declared parentage increase in dividend per share in future years. The current market price is computed through dividing the expected dividend per share in next period by the difference of required rate of return and constant growth rate.

Illustration

A hypothetical company used to pay dividend at steady growth rate of 5% regularly over the last 4 years and declares that the same policy will continue for an indefinite time period. In the year 5, $10 dividend per share is expected. Investors’ required rate of return is 15%.

Requirement: Please compute price per share

Answer:

Price per share = $10 / 0.15 - 0.05 = $100

Uneven-Growth-Model (Irregular growth in dividend per share in future)

It is not common in practice that companies operate with zero growth or constant growth in dividends. Factually growth rates in dividend change after some time in future. In this case computation of the value of one share in the current period becomes complex. The process of computation is as under:

·         In the first step present values of the dividends per share are computed over the periods the dividend growth rate is expected to keep on changing.

·         A sum total is ascertained of the present values of each year dividend per share computed in step 1.

·         Dividend per share is computed for the year from which the growth is expected to be at the constant rate.

·         Present value the value of share computed in step 3 is determined.

·         Sum the values calculated in step 2 and 4.

·         The resultant value in step 5 is the value per share in current year.

Illustration

Suppose a company pays dividend per share of $5 currently and is expected to grow at 8% for the next 4 years. After that growth rate will decrease to 4% forever.15% is the required rate of return.

Requirement: Please compute price per share

Answer:

Present value of dividends per share for first four years.

End of Year	Dividend Per Share (8% Growth) $	PVIF 1 to 4, 15%	Present Value $
1	5.4	0.87	4.7
2	5.8	0.76	4.4
3	6.3	0.65	4.1
4	6.8	0.57	3.9
			17.1

Dividend in year 5                              =          $6.8 x 1.04                  =          $7.1

Price at the end of 4^th year                  =          $7.1 / 0.15 – 0.04        =          $64.5

Present Value                                      =          $64.5 x 0.57                =          $36.8

Price per share                                     =          $17.1 + $36.8              =          $53.9

End of solution

The purpose of this article is to acquaint a student or common investor with somewhat understanding of the complexities of the calculations of the value per share. It is intended to enable them comprehend the basic concepts involved in the investments in stock and its valuation.