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
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
Assumptions of the Arbitrage pricing theory
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
Relationship with the capital asset pricing model
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
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
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:
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
Single-factor models
Before 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 models
A 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
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. Example
The 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 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
Arbitrage portfolio
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
Implementation of APT
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
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
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 interest
3. Rate of change in oil prices
4. 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
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)
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
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
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
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.
Conclusion
The 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.
