Capital Asset Pricing Model and Arbitrage Pricing

Capital Asset Pricing Model and Arbitrage Pricing Theory:

Capital Asset Pricing Model (CAPM) is an arithmetical theory that describes the relationship between risk and return in a balanced market. The Capital Assets Pricing Model was autonomously and simultaneously developed by William Sharpe, Jan Mossin, and John Litner. The researches of these founders were published in three different and highly respected journal articles between 1964 and 1966. Since its inception, the model has been used in various applications that range from public utility rates to corporate capital budgeting. However, the initial introduction of the model was characterized by suspicious view from the investment community. This was largely because CAPM apparently indicated that professional investment management was hugely a waste of time. Due to its implementation problems and shortcomings associated with its relation to Arbitrage Pricing Theory, Capital Asset Pricing Model has continued to face constant academic attacks.

Overview of Capital Asset Pricing Model:

Since its introduction, the Capital Asset Pricing Model offers a huge portion of the justification for the tendency toward reactive investing in large index mutual funds (Cooper and Cousins, n.d.). After the initial suspicious view of CAPM, investment professionals changed their perspective nearly a decade later to view the model as a vital tool that assist investors to understand risk. Actually, the development of this model not only resulted in the birth of asset pricing theory but it has also been widely used in various applications like calculating capital costs for companies and analyzing the performance of managed portfolios.

This is because the Capital Asset Pricing Model consists of a key element that separates the risk affecting an asset's return into two main classifications i.e. company-specific or unsystematic risk and systematic or general economic risk. Unlike the systematic risk that occurs because of the general economic uncertainty, the long-term average returns for company-specific or unsystematic risk should be zero. According to this model, the return on assets should averagely equal the yield on a risk-free bond within a given period of time. This should also include a premium that is proportional to the amount of systemic risk that the stock contains. Generally, the Capital Assets Pricing Model refines the concepts of systematic and unsystematic risks that were developed in 1950s by Harry M. Markowitz.

In Markowitz model that is commonly known as the mean-variance model, an investor chooses a portfolio at a time (t -- 1) that results in a stochastic return at t. This is based on the assumption that investors are risk reluctant and only care about the mean and variance of their investment return when selecting their portfolio. Consequently, these investors select portfolios that are mean-variance efficient on the basis that the portfolios lessen the variance of their return, given the probable return and lessen this return, given variance (Fama & French, 2004).

The Capital Asset Pricing Model refines the algebraic statement in Markowitz's mean-variance model into a testable prediction regarding the link between risk and probable return. The model turns the initial approach through identifying a portfolio that must be effective for asset prices to clear each asset market.

Consequently, the model states that the risk associated with an asset is calculated in relationships to the risk of the entire market, which is expressed either as correlation to the market average or as the stock's beta. The concept of the Capital Asset Pricing Model is that it hypothesizes a simple linear relationship between the anticipated return and the market risk of a security (Banz, 1981). Since this model provides powerful and spontaneously enjoyable predictions on how to calculate risk and its relationship with expected return, it's an attractive concept, which is the core of investment courses.

Analysis of the Capital Asset Pricing Model:

In order to achieve the equilibrium of the Capital Asset Pricing Model, there are various assumptions that must be defined including the need for investors to capitalize on the expected utility of wealth. The other assumptions are the existence of many investors who behave competitively, lack of taxes and commissions, investors' equal access to all securities, frictionless markets, and investors' use of similar input lists because of uniform expectations. Based on these assumptions, the Capital Asset Pricing Model can be developed and the prevailing equilibrium achieved. Furthermore, the achievement of the CAPM equilibrium based on the defined assumptions results in other elements.

First, all investors will select an optimal market portfolio, M, that incorporates every asset in the economy because all assets are evaluated in the portfolio in proportion to their weight in the economy. Given that all investors use similar input lists and have the same expectations, they are likely to select an identical risky portfolio i.e. The one on the efficient frontier within the tangency line drawn from the risk-free asset (Taylor, 2005). Consequently, the demand of any asset left outside the portfolio would be zero and its price would also approach zero. Once investors note this, they will adjust their portfolios to include this asset until its price reflects its amount or level of risk resulting in inclusion of all assets in M.

Secondly, as the market portfolio, M, is within the efficient frontier and is the tangent to the risk-free asset, it includes all information regarding assets in the market that makes it an efficient portfolio. As a result, every investor will select to distribute his/her wealth between the efficient portfolio, M, and the risk-free asset.

The third element regarding the achievement of CAPM equilibrium is that the market portfolio's risk premium will be relative to its own risk and the risk aversion degree of the average investor. Therefore, every investor selects a proportion, b to invest in the efficient portfolio, M and another fraction 1-b to spend in the risk-free asset. This leads to the next element with which the risk premium on each asset is proportional market portfolio's risk premium and the beta coefficient of the market portfolio's asset.

Arbitrage Pricing Theory:

As a single-period model, Arbitrage Pricing Theory is a model where all investors believes that stochastic returns of capital assets are steady with a factor structure (Huberman & Wang, 2005). The theory is based on the concept that the returns of an asset can be predicted through the relationship between that asset and other common risk factors. Generally, the Arbitrage Pricing Theory was developed as an alternative tool of computing the expected returns on stock. Ross, who primarily developed this theory, states that the probable returns on assets are almost linearly associated with factor loadings if the equilibrium prices don't provide arbitrage opportunities beyond the static portfolios of the assets.

Based on this argument, the Arbitrage Pricing Theory is based on the anticipation of arbitrage. According to Ross' formal proof, the linear pricing relation is an essential condition for equilibrium in a market with which agents maximize specific types of utility. The linear relation between the expected returns and the betas is identical to the determination of the stochastic discount factor.

As an alternative to the Capital Asset Pricing Model, the Arbitrage Pricing Theory emphasizes on the linear relation between the expected returns of assets and their covariance with several random variables. In this case, the covariance is understood as the amount of risk which investors cannot avoid through diversification. Moreover, the slope coefficient in this linear relationship and the covariance are understood as risk premiums.

Overview and Analysis of Arbitrage Pricing Theory:

As previously mentioned, the Arbitrage Pricing Theory is a substitute model that is used to measure equilibrium expected returns on financial assets. As suggested by its name, this theory rests on the concept that financial markets that are operating effectively should be arbitrage-free. Through the use of a factor model of returns of an asset, the concept implies limitations on the relations between the asset returns while generating an equilibrium pricing relationship. Since every stock is sensitive to all factors, the determination of expected returns through the notion of returns on stock requires the common set of factors to be mutually consistent.

An arbitrage strategy that can occur in two major ways is used to help in understanding the need for the factors to be mutually consistent. First, the arbitrage strategy may involve investment in both the buying and selling assets that generate an instant positive cash inflow and a guaranteed profit in the future. An investor who could pay more or less wealth may attempt investing on an infinite scale when faced by an investment strategy that is based on this pay-off structure. Secondly, an arbitrage strategy may be an investment plan that is currently costless though it guarantees positive returns in the future. Sensible investors will take advantage of the likelihood of investing as much as possible in this strategy since it's similar to obtaining something from nothing ("Chapter 3 -- The Arbitrage Pricing Theory," n.d.).

Therefore, the underlying principle behind the Arbitrage Pricing Theory is that investment situations like those described above shouldn't be allowed in financial markets that are functioning well. As a result, if the well-functioning financial markets don't allow the existence of pricing strategies, they result in limitations on the relationship between probable returns because of the factor structure within returns.

As stated by Ross, the return on stocks must adhere to a simple relationship, which is explained with the following formula for Arbitrage Pricing Theory:

Expected Return = rf + b1 * (factor 1) + b2 * (factor 2) & #8230;.. + bn * (factor n)

In this case, the rf is the risk-free interest rate or the interest rate that the investor expects to gain from an investment that is risk-free. While b is the stock's or security's sensitivity to every factor, the factor is the risk premium that is linked with every factor.

The other notable aspect about the Arbitrage Pricing Theory is that the risk premium of stocks is dependent on two factors i.e. The risk premium linked with every factor that has been described above and the sensitivity of the stock to every factor, which is akin to the beta concept. Therefore, a stock risk premium can be described through the following formula:

Risk Premium = r - rf = b (1) * (r factor (1) - rf) + b (2) * (r factor (2) -- rf) & #8230; + bn * (r factor (n) -- rf)

Based on the above formula, investors are likely to sell their stocks if a stock's expected risk premium was less than the determined risk premium. However, the investors would buy stocks until a balance is achieved on both sides of the equation if the risk premium was more than the calculated value. Therefore, arbitrage concept is basically a term that is used to explain how investors can make the equation or formula to balance ("Arbitrage Pricing Theory," n.d.).

While the ATP model can be described through simple formulas, the concept can also be explained through identifying the factors that are used in the model. This is largely because the concept itself doesn't notify the investors the exact factors for certain stocks or assets. As investors are left with the responsibility of identifying all factors for certain stocks, it's not a trivial matter. However, some of the major macro-economic factors include inflation, investor confidence, changes in the Yield Curve, and Gross National Product.

Capital Asset Pricing Model and Arbitrage Pricing Theory:

The contemporary asset pricing theories are based on the concept that the expected return of a certain asset relies only on the underlying component of the complete risk within it that can't be diversified away. Based on its definition, market equilibrium excludes a price system with which diversification earns an incentive. Therefore, the essential question for asset pricing is lessened to the identification and calculation of the associated risk component that has an impact on the expected return of an asset, especially in a world of costless arbitrage.

Based on the Capital Asset Pricing Model, a specific mean-variance efficient portfolio is identified and used as a formalization of fundamental risk in the entire market. Through this, an asset's expected return is linked to its normalized covariance with the market portfolio becoming the beta of the asset. The outstanding component in the complete risk of a certain asset, inessential risk, does not receive and incentive since it can be eradicated by another portfolio with a lower risk level but similar cost and return.

On the contrary, based on the Arbitrage Pricing Theory, specific number of factors is used as formalization of systematic risks in the entire market and an asset's expected return is linked to its exposure of all the factors (Khan & Sun, 1997). The result of the entire formalization of the risks and linking of factors is summarized by a vector of factor loadings. Consequently, the incentive to the outstanding component in the return of a specific asset regardless of whether its idiosyncratic or unsystematic risk can be relatively or arbitrarily small through examining portfolios with a randomly huge number of assets.

The basic principle between the Capital Asset Pricing Model and Arbitrage Pricing Theory is that the two models consist of two divergent sets of risks and tackle the various aspects of the premium-awarding scheme for taking the risks. Through its emphasis on efficient diversification on the basis of a finite number of assets, the Capital Asset Pricing Model neglects unsystematic risks. On the other hand, the Arbitrage Pricing Theory neglects essential risks through its emphasis on markets with large number of assets, naive diversification, and the concept of large numbers. As a result, the Arbitrage Pricing Theory and Capital Asset Pricing Model appear to be inherently disjoint.

One of the significant relations between the Capital Asset Pricing Model and Arbitrage Pricing Theory is that they are the two most influential hypotheses on asset and stock pricing today. This is regardless of the differences between the two models as the Arbitrage Pricing Theory is far less restrictive in its assumptions as compared to the Capital Asset Pricing Model. CAPM and APT are influential models on asset and stock pricing because they have attempted to scientifically calculate the possibility for assets to produce a return or loss (Donovan & Weinraub, 2007). Both models are similar since they attempt to calculate or determine an asset's propensity to follow the general market.

Due to its relation to the Arbitrage Pricing Theory, the Capital Asset Pricing Model is a simplified version of the APT because it only involves the consideration of the risk of a specific stock in relation to the rest of the stock market, which is described as the stock's beta. Therefore, the Capital Asset Pricing Model describes a stock's expected return by explaining the movement of that stock based on the entire stock market. On the other hand, Arbitrage Pricing Theory tends to make more sense because it eliminates the restrictions of the Capital Asset Pricing Model. Unlike CAPM that consists of some restrictions, the Asset Pricing Theory permits the individual investor more liberty. Therefore, the main relationship between these two models is that the Asset Pricing Theory is the basis with which the Capital Asset Pricing Model is developed as a simplified theory.

Shortcomings of the Capital Asset Pricing Model:

As previously mentioned, the Capital Asset Pricing Model has been prone to attacks from academicians and experts since its inception. These numerous attacks have mainly centered on the shortcomings of the model which include:

The Mean-Variance Market Portfolio:

The Capital Asset Pricing Model is based on some simplifying assumptions to guarantee its validity including the mean-variance efficiency of the market portfolio. As stated earlier, the model assumes that investors are risk-averse and are only concerned about the mean and the variance of their investment strategy return when choosing between several portfolios. Roll (1977) states that determining the validity of this model is equivalent to calculating the mean-variance efficiency of the market portfolio. The researcher's criticism is mainly based on the methods to formally measure the statement of the Capital Asset Pricing Model i.e. The equation:

E (Ri) = Rf + ?im[E (Rm) -- Rf]

where E (Ri) represents the expected return of an asset, ?im is the asset covariance, and Rm the market portfolio return.

According to the findings of several other researchers, the mean-variance efficiency of diverse market portfolio proxies are inefficient and don't meet the efficient frontier. Moreover, the findings of some analyses state that efficient portfolios are very sensitive to changes in asset means. In earlier examinations that were conducted from 1926 to 1968 using returns from size-based portfolios, the mean-variance efficiency of the market portfolio shows a high degree of sensitivity to the considered test. Statistical inference is hugely and strongly affected by mis-specified distributional assumptions when testing the mean-variance efficiency of the market portfolio.

The other shortcomings of the Capital Asset Pricing Model under this point is that the model is tautological based on the assumption that the market is mean-variance efficient. Since the market portfolio in practice includes all available assets, the returns on all the probable investment opportunities are unobservable. Without an observation of all the investment opportunities, it's difficult to test the mean-variance efficiency of all portfolios.

Both the mean-variance approach and the Capital Asset Pricing Model are based on the normal distribution assumption. The findings of numerous tests on actual rates of return distributions to normal distribution reject the null hypothesis that distribution of rates of return is ordinary (Levy, 2010). When beta shows minimal descriptive power of the variation in mean returns, the test not only rejects the CAPM theory but it also shows only minimal support for the probable linear risk-return relationship.