The Arbitrage Pricing Theory

Introduction

The capital asset pricing model that assists the security have different expected return

because they have different beta however they exists an alternative model of asset

pricing that was developed by Stephen Ross it is known as arbitrage pricing model.

Arbitrage pricing theory (APT) is an equilibrium model of security prices, as is the

capital asset pricing model (CAPM). It makes different assumptions than the CAPM

does. APT assumes that security returns are generated by a factor model but does not

identify the factors. An arbitrage portfolio includes long and short positions in securities.

it must have a net market value of zero, no sensitivity to any factor, and a positive

excepted return. Investors will invest in arbitrage portfolios, provided they exist, driving

up the prices of the securities held in long positions until all arbitrage possibilities have

been eliminated. When all arbitrage possibilities have been eliminated, the equilibrium

expected returns on a security will be a linear function of its sensitivities to the factors a

factor-risk premiums is the equilibrium to the factors and no sensitivity to any other

factor APT does not specify the number or identify of the factors that affect excepted

returns or the magnitudes or signs of the risk premiums. Most research into factors has

focused on indicators of aggregate economic activity, inflation and interest rates

Limitations of CAPM formed the Arbitrage pricing theory

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1. It is based on Unrealistic Assumptions

Difficult to find risk free assets

Equality of lending and borrowing rates

2. It is difficult to test the validity of CAPM

3. BETAS do not remain stable over time.

Assumptions of the Arbitrage pricing theory

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1. The investors have homogenous expectations.

2. The investors are risk averse and utility maximize.

3. Perfect competition prevails in the market an there is no transaction cost.

4. Under APT investors borrows and lend at risk free rate.

5. There is no market friction such as transaction cost or restriction on short selling.

6. Investors agree on the number and identify of the factors that are important

symbolically in pricing asset.

Relationship with the capital asset pricing model

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The APT along with the capital asset pricing model (CAPM) is one of two influential

theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in

its assumptions. It allows for an explanatory (as opposed to statistical) model of asset

returns. It assumes that each investor will hold a unique portfolio with its own particular

array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM

can be considered a "special case" of the APT in that the securities market line

represents a single-factor model of the asset price, where beta is exposed to changes in

value of the market.

Additionally, the APT can be seen as a "supply-side" model, since its beta coefficients

reflect the sensitivity of the underlying asset to economic factors. Thus, factor shocks

would cause structural changes in assets' expected returns, or in the case of stocks, in

firms' profitability.

On the other side, the capital asset pricing model is considered a "demand side" model.

Its results, although similar to those of the APT, arise from a maximization problem of

each investor's utility function, and from the resulting market equilibrium (investors are

considered to be the "consumers" of the assets).

Principle

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Arbitrage is the process of earning profit by taking advantage of differential pricing

for the some physical asset or security. As widely applied investment tactic, arbitrage

typically entails the sale of security at a relatively high price and the simultaneous

purchase of the same security at a relatively low price.

Arbitrage is critical element of modern, efficient security markets. Because

arbitrage profits are by definition riskless all investors are motivated to tale advantage

of then whenever they are discovered. Granted some investors are greater resources

and are more inclined to engage in arbitrage than others. It only takes a few of these

active investors to exploit arbitrage than others. It only takes few of these active

investors to exploit arbitrage situations and by their buying and selling actions,

eliminates these profit opportunities.

The nature of arbitrage is clear when discussing different prices for an

individual security. However “almost arbitrage” opportunity involve similar securities or

portfolios. The similarity can be defined in many ways for example in the exposure to

pervasive factor that affect security prices.

Arbitrage and the APT

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Arbitrage is the practice of taking advantage of a state of imbalance between two (or

possibly more) markets and thereby making a risk-free profit; sees rational pricing.

Arbitrage in expectations

The APT describes the mechanism of arbitrage whereby investors will bring an asset

which is mispriced, according to the APT model, back into line with its expected price.

Note that under true arbitrage, the investor locks-in a guaranteed payoff, whereas under

APT arbitrage as described below, the investor locks-in a positive expected payoff. The

APT thus assumes "arbitrage in expectations" - i.e. that arbitrage by investors will bring

asset prices back into line with the returns expected by the model.

Arbitrage mechanics

In the APT context, arbitrage consists of trading in two assets – with at least one being

mispriced. The arbitrageur sells the asset which is relatively too expensive and uses the

proceeds to buy one which is relatively too cheap.

Under the APT, an asset is mispriced if its current price diverges from the price

predicted by the model. The asset price today should equal the sum of all future cash

flows discounted at the APT rate, where the expected return of the asset is a linear

function of various factors, and sensitivity to changes in each factor is represented by a

factor-specific beta coefficient.

A correctly priced asset here may be in fact a synthetic asset - a portfolio

consisting of other correctly priced assets. This portfolio has the same exposure to each

of the macroeconomic factors as the mispriced asset. The arbitrageur creates the

portfolio by identifying x correctly priced assets (one per factor plus one) and then

weighting the assets such that portfolio beta per factor is the same as for the mispriced

asset.

When the investor is long the asset and short the portfolio (or vice versa) he has

created a position which has a positive expected return (the difference between asset

return and portfolio return) and which has a net-zero exposure to any macroeconomic

factor and is therefore risk free (other than for firm specific risk). The arbitrageur is thus

in a position to make a risk-free profit:

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Where today's price is too low:

The implication is that at the end of the period the portfolio would have appreciated at

the rate implied by the APT, whereas the mispriced asset would have appreciated at

more than this rate. The arbitrageur could therefore: Today:

1 short sells the portfolio

2 buy the mispriced asset with the proceeds.

At the end of the period:

1 sells the mispriced asset

2 use the proceeds to buy back the portfolio

3 pocket the difference.

Factor model

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Single-factor modelsBefore using the notion of absence of arbitrage to provide pricing relations, We need a

basis for the generation of stock returns. Within the context of The APT, this basis is

given by the assumption that the population of stockreturns are generated by a factor

model. The simplest factor model, given

below, is a one-factor model:

ri = αi + βi F + εi E(εi) = 0. (3.1)

In equation 3.1, the returns on stock i are related to two main components:

1. The first of these is a component that involves the factor F. This factor is posited to

affect all stock returns, although with differing sensitivities. The sensitivity of stock i’s

return to F is βi. Stocks that have small values for this parameter will react only slightly

as F changes, whereas when βi is large, variations in F cause very large movements in

the return on stocki. As a concrete example, think of F as the return on a market index

(e.g.the S&P-500 or the FTSE-100), the variations in which cause variations in

individual stock returns. Hence, this term causes movements in individual stock returns

that are related. If two stocks have positive sensitivities to the factor, both will tend to

move in the same direction.

2. The second term in the factor model is a random shock to returns, which is assumed

to be uncorrelated across different stocks. We have denoted this term εi and call it the

idiosyncratic return component for stock i. An important property of the idiosyncratic

component is that it is also assumed to be uncorrelated with F, the common factor in

stock returns. In statistical terms we can write the conditions on the idiosyncratic

component as follows:

Cov(εi, εj) = 0 i ≠ j Cov(εi, F) = 0 i

An example of such an idiosyncratic stock return might be the unexpected departure of

a firm’s CEO or an unexpected legal action brought against the company in question.

The partition of returns implied by equation 3.1 implies that all common variation in

stock returns is generated by movements in F (i.e. the correlation between the returns

on stocks i and j derives solely from F). Asthe idiosyncratic components are ncorrelated

across assets they do not bring about covariation in stock price movements.

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Multi-factor modelsA generalisation of the structure presented in equation 3.1 posits k factors

or sources of common variation in stock returns.

ri = αi + β1iF1 + β2iF2 + .... + βkiFk + εi E(εi) = 0. (3.2)

Again the idiosyncratic component is assumed uncorrelated across stocks

and with all of the factors. Further, we’ll assume that each of the factors

has a mean of zero. These factors can be thought of as representing news on economic

conditions, financial conditions or political events. Note that this assumption implies that

the expected return on asset i is just given by the constant in equation 3.2 (i.e. E(ri) =

αi). Each stock has a complement of factor sensitivities or factor betas, which determine

how sensitive the return on the stock in question is to variations in each of the factors.

A pertinent question to ask at this point is how do we determine the return on a portfolio

of assets given the k-factor structure assumed? The answer is surprisingly simple: the

factor sensitivities for a portfolio of assets are calculable as the portfolio weighted

averages of the individual factor sensitivities. The following example will demonstrate

the point. ExampleThe returns on stocks X, Y, and Z are determined by the following two-factor model:

rX = 0.05 + F1 – 0.5F2 + εX

rY = 0.03 + 0.75 F1 + 0.5F2 + εY

rz = 0.04 + 0.25 F1 – 0.3F2 + εz

Given the factor sensitivities in the prior three equations, we wish to derive the factor

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structure followed by an equally weighted portfolio of the three assets (i.e. a portfolio

with one-third of the weights on each of the assets). Following the result mentioned

above, all we need to do is form a weighted average of the stock sensitivities on the

individual assets. Subscripting the coefficients for the equally weighted portfolio with a p

we have:

αp = 1

3 (0.05 + 0.03 + 0.04) = 0.04

β1p = 1

3 (1 + 0.75 – 0.25) = 0.5

β2p = 1

3 (–0.5 + 0.5 –0.3) = –0.1;

and hence; the factor representation for the portfolio return can be written as:

rp = 0.04 + 0.5F1 – 0.1F2 + εp

where the final term is the idiosyncratic component in the portfolio return.

Arbitrage portfolio

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According to APT, an investor will explore the possibility of forming an arbitrage

portfolio in order to increase the expected return of his or her current portfolio without

increasing its risk. Just what is an arbitrage portfolio? There are three characteristic of

an arbitrage portfolio ;-

1. It does not require any additional funds from the investor. If Xi denotes change in

the investors holding of security I, this requirement of a three security arbitrage

portfolio written as

X1+ X2+X3 =0

2. It has no sensitivity to any factor because the sensitive of a portfolio factor is just

a weighted average of the sensitivities of the securities in it ie to that factor, this

requirement of a three security arbitrage portfolio is one factor can be written as

B1X1 +b2X2+ b3X3 =0

3. It has a positive expected return. Mathematically this third and last thing for a

three security arbitrage portfolio is

X1r1+X3r2+X3r3 >0

Implementation of APT

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The implementation of APT involves three steps:

1. Identify the factors

2. Estimate factor loadings of assets

3. Estimate factor premia.

1. Factors. Since the theory itself does not specify the factors, we have to construct the

factors empirically:

(a) Using macroeconomic variables:

• changes in GDP growth

• changes in T-bill yield (proxy for expected inflation)

• changes in yield spread between T-bonds and T-bills

• changes in default premium on corporate bonds

• changes in oil prices (proxy for price level)

(b) Using statistical analysis – factor analysis:

• estimate covariance of asset returns

• extract “factors” from the covariance matrix

(c) Data mining: Explore different portfolios to find those whose

returns can be used as factors.

15.407 Lecture Notes Fall 2003 c_Jiang Wang

Chapter 12 Arbitrage Pricing Theory (APT) 12-15

2. Factor Loadings. Given the factors, we can regress past asset

returns on the factors to estimate factor loadings (bik):

˜rit = ¯ri +bi1 ˜ f1t +· · ·+biK ˜ fKt +uit.

3. Factor Premia. Given the factor loading of individual assets,

we can construct factor portfolios.

Identifying the factors

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Left unanswered by APT are the number and identify of the factors that have value of lambda (λ) that are sufficiently positive or negative in magnitude that they need to be included when estimating excepted returns.; several researchers have investigated stock returns and have estimated that anywhere from three to five factors are “priced” various people attempted to identify those factors nai –fu chen, Richard roll, and Stephen Ross identify the following factors

1. Growth rate in industrial production

2. Rate o f inflation (both expected and unexpected)

3. Spread between long term and short –term interest rates

4. Spread between low-grade and high grade bonds

Michael berry, Edwin Burmeister and Marjorie McElroy identify five factor .three correspond closely to the last three identified by Chen, Roll, and Ross and the other two are the growth rate in aggregate sales in the economy and the rate of return on the S &P 500

Finally, .consider the five factors used by Salomon Brothers (now Salomon Barney) in their fundamental factor model. Only one factor, inflation, is the same as the factors identified by the others. The remaining factors are as follows:

1. Growth rate in gross national product.2. Rate of interest3. Rate of change in oil prices4. Rate of growth in defense spending

It is interesting to note that the three sets of factors have common characteristics, first contain some indication of aggregate economic activity (industrial production. aggregator sales and GDP) second, they include inflation. Third, they contain some type of interest rate factor (either spreads or a rate itself) because stock prices are equal to the discounted value of future dividends, the factors make intuitive sense. Further dividends are related to aggregate economic activity and the discount rate used to determine present value is related to inflation and interest rates

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Using the APT

As with the CAPM, the factor-specific Betas are found via a linear regression of

historical security returns on the factor in question. Unlike the CAPM, the APT, however,

does not itself reveal the identity of its priced factors - the number and nature of these

factors is likely to change over time and between economies. As a result, this issue is

essentially empirical in nature. Several a priori guidelines as to the characteristics

required of potential factors are, however, suggested:

1.their impact on asset prices manifests in their unexpected movements

2.they should represent undiversifiable influences (these are, clearly, more likely to be

macroeconomic rather than firm-specific in nature)

3.timely and accurate information on these variables is required

4.the relationship should be theoretically justifiable on economic grounds

Chen, Roll and Ross (1986) identified the following macro-economic factors as

significant in explaining security returns: surprises in inflation; surprises in GNP as

indicted by an industrial production index; surprises in investor confidence due to

changes in default premium in corporate bonds surprise shifts in the yield curve. As a

practical matter, indices or spot or futures market prices may be used in place of macro-

economic factors, which are reported at low frequency (e.g. monthly) and often with

significant estimation errors. Market indices are sometimes derived by means of factor

analysis. More direct "indices" that might be used are: short term interest rates; the

difference in long-term and short-term interest rates;a diversified stock index such as

the S&P 500 or NYSE Composite Index;oil pricesgold or other precious metal prices

Currency exchange rates.

Differences between APT and CAPM’s

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1. The CAPM can be considered a "special case" of the APT in that the securities

market line represents a single-factor model of the asset price, where beta is

exposed to changes in value of the market.

2. The APT can be seen as a "supply-side" model, since its beta coefficients reflect

the sensitivity of the underlying asset to economic factors

On the other side, the capital asset pricing model is considered a "demand side"

model. Its results, although similar to those of the APT, arise from a maximization

problem of each investor's utility function, and from the resulting market equilibrium

(investors are considered to be the "consumers" of the assets).

3. Under APT investors do not look at expected returns and standard deviation

while in the case of CAPM investors look at the expected return and

accompanying risk measured by standard deviation.

4. Under APT risk return analysis is not the basis while in the case of CAPM risk return analysis is necessary.

5. APT is based on the return generated by factor models, while in the case of CAPM investors maximize wealth for a given level of risk.

Empirical evidence

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The APT has been empirically tested two different approaches. in the first approach, the

technique of factor analysis (a statistical technique) is applied to stock returns to

discover the basic factors. These are then examined to see whether they correspond to

some economic or behavioral variables. Empirical studies done so far suggest that

there is hardly any consistency in terms of:

1. The number of basic factors.

2. The interpretation that may be put on these factors (typically the factors identified

are artificial constructs representing several economic variables).

3. The stability of these factors from test to test.

In the second approach, factors are specified a priori, rather than extracted by analyzing

stock returns. The classical work of Roll and Ross, typifies this approach.

They employ four factors:

1. Industrial production

2. Inflation rate.

3. Term structure of interest rates

4. Default risk premiums.

Sensitivity to unanticipated changes in these factors provides explanations for

differences in excepted returns among stocks in their study.

Strength and Weaknesses of APT

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1. The model gives a reasonable description of return and risk.

2. Factors seem plausible.

3. No need to measure market portfolio correctly.

4. Model itself does not say what the right factors are.

5. Factors can change over time.

6. Estimating multi-factor models requires more data.

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ConclusionThe APT gives us a straightforward, alternative view of the world from the CAPM. The

CAPM implies that the only factor that is important in generating expected returns is the

market return and, further, that expected stock returns are linear in the return on the

market. The APT allows there to be k sources of systematic risk in the economy. Some

may reflect macroeconomic factors, like inflation, and interest rate risk, whereas others

may reflect characteristics specific to a firm’s industry or sector. Empirical research has

indicated that some of the well-known empirical problems with the CAPM are driven by

the fact that the APT is really the proper model of expected return generation.

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