Capital Asset Pricing Model (CAPM) and Multifactor Models

Introduction

Capital Asset Pricing Model (CAPM) can be described as a financial model that finance analysts use to show how risk and expected returns are related. CAPM is used when pricing risky securities (Dempsey, 2013:8). In simple terms, CAPM is a reorganization that investors should be compensated for the funds they invest in two ways. First, investors should be compensated for time value of money because the investors have put their money into investments for a given time. The other compensation is concerned with the risk that the investors take by investing their money in projects that they are not certain of the outcomes (Lettau & Ludvigson, 2001:1250). Considering these two elements, the formula that is used to calculate CAPM is as shown below.

R_{i} = R_{f }+ (R_{m} – R_{f})β

Where:

R_{i} = Required rate of return

R_{f }= Risk free rate

R_{m }= Expected market return

β = Beta of the security (Ng, 2004:190).

CAPM is one of the areas in finance that has been explored almost exhaustively, but researchers are still interested in the topic. Every day a new puzzle arises in this topic and researchers are keen to keep on improving what has already been discovered. It is not that scholars do not agree with what the researchers have found and the theories that they have formulated, rather they seek to review the theories critically and identify areas that may need improvement (Mirza & Shabbir, 2005:37). This has been the case with CAPM. As the business world undergoes changes, CAPM seems to hold no longer in the real world and this has led to the emergence of multifactor models. This paper analyzes this in detail by discussing the limitations of CAPM. In addition, the paper highlights how the multifactor approach attempts to overcome these limitations. CAPM is one the most significant areas of financial management and any improvement in this model would be a vital step in the field of financial management. Finance is said to have become ‘a fully-fledged, scientific discipline’ after William Sharpe published CAPM’s derivation in 1986 (Galagedera, 2007:823).

Assumptions of CAPM

CAPM is often criticized because of the assumptions underlying its operation and hence understanding the assumption would be a good foundation of learning its limitations. The first assumption is that the portfolios held by the investors are diversified (Maio, 2013:4960). This assumption implies that investors can ignore the unsystematic risk and focus on the return related to systematic risk. The other assumption is single-period transaction horizon. CAPM assumes a standardized holding period to make it possible to compare returns from different securities. For instance, it is not possible to compare a return over eight months with that of 12 months. Usually, a one-year holding period is used (Ang & Chen, 2007:29). Thirdly, under CAPM, investors can borrow or lend money at the risk-free rate. This assumption comes from the portfolio theory, which formed the basis of CAPM. The assumption provides a minimum return level that investors require. The fourth assumption of CAPM is that the investments are done in a perfect capital market (Ogilvie, 2006:18). A perfect market implies that no taxes and regulations exist in the market. A perfect market also means that there is perfect information in the market, which all investors can access freely. In addition, the assumption implies that the investors’ expectations are the same. The characteristics of the investors are assumed identical in terms of being risk averse, desiring to maximize their utilities. In a perfect capital market as assumed under CAPM, the market has large numbers of sellers and buyers (Fama & French, 2006:2169).

Limitations of CAPM

The assumptions that CAPM makes enable it to provide a relationship between systematic risk and return. However, the ideal world that the assumptions create does not resemble the real world where the investments are made. The difference between the ideal world of the assumptions and the real world of the investments forms the basis of limitations and criticisms of CAPM. For instance, capital markets in the real world are never perfect. Well-developed stock markets show a high level of efficiency, but they never achieve the level of being perfect. Some factors in the market will always render it imperfect (Pandey, 1979:42). CAPM assumes a single-period transaction horizon and this may appear reasonable. However, for investment appraisal purposes, application of a single-period time horizon may insensible. Experience shows that in reality, the variables used in CAPM cannot remain constant in multi-period nature, which is usually the norm in investment appraisal. Only a few investment projects run are completed in a single-time horizon (Roll, 1977:140).

Another problem with the assumptions of CAPM is that it is impossible for investors to lend or borrow at the risk-free rate as CAPM suggests. The risk-free rate is the rate for short-term government debts and it is taken normally as a proxy. In the real world, risk rate for companies and individuals is usually higher that the government rate. The fact that investors cannot lend or borrow at the risk-free rate means that CAPM will not hold in practice. By assuming that the investors can lend or borrow at the risk-free rate, the calculations made using CAPM will be not be accurate (Banz, 1981:12).

When CAPM is being used, certain values must be assigned to the various parameters of the CAPM formula, that is, the risk-free rate, the equity risk premium or return on market, and the beta value. The value that is taken as the risk-free rate is the rate of short-term government debts. The problem with this value is that it keeps on changing as market conditions vary. In the modern business market, market conditions change on an almost daily and hence the risk-free rate should take this into condition. To smooth out the volatility in the risk-free rate, a value that is a takes an average of the short-term period should be taken (Fama, 1996:450).

Another problem with CAPM is that it is difficult to find the value of equity risk premium. To determine the return on stock, average capital is usually added to dividend yield. Sometimes in the short-term, the average dividend yield could be outweighed by the falling share prices’ effects, which could result to a negative return. This calls for long-term average ERP values to be considered instead of short-term values. However, the problem with long-term ERP is that they are never stable. Uncertainty in ERP values renders the values of required return calculated using CAPM inaccurate (Banz, 1981:8).

The assumptions underlying the operation and application of CAPM appear too stringent to hold or be practical in the real world. However, criticisms that befall this asset-pricing model are not attributed solely to the assumptions. Supporters of CAPM believe that the model can hold in the real world even if some of the assumptions were to be relaxed. A study done by Darrat and Park (2011) showed that even if it the assumption that investors can lend or borrow at the risk-free rate is violated, a liner relationship between risk and assets return would be obtained. This type of formulation of the CAPM model is called the zero-beta CAPM. Fama and French (1996) argue that under some assumptions, the single-time transaction horizon is similar to the multi-period utility and CAPM would hold over time. Even if the results appeared to favor multi-period CAPM, the assumptions forming the basis of CAPM appeared more inflexible compared to the CAPM itself, which raises more questions from the issue. Research shows that the assumptions that form the basis of CAPM are not so critical to make the model impractical in the real world. What makes CAPM fail to hold in the real world and be impractical is its nonconformity to reality. In addition, the model shows inherent weaknesses when subjected to empirical tests (Roll, 1977:145).

Under ideal conditions, the CAPM framework looks very simple. As mentioned earlier, the model states that the return on an investment is a positive function of three variables, namely risk-free rate, expected market return, and beta. The CAPM’s equation can be rewritten in another way as a times series model. The regression interpretation would be:

R_{it} –R_{ft} = α_{i }+ β_{i}γ_{it} + e_{it}

Where γ_{it= }R_{mt} –R_{ft }= Market risk premium (Mirza & Shabbir, 2005:39).

If CAPM was to hold as per the above time series model, the regression coefficient α_{i }should be zero. The above equation shows that systematic risk, which can be attributed to sensitivity of macroeconomic factors, is reflected in β* _{i}*. At the same time, non-systematic risk, which comes about because of unexpected events, is reflected in e

_{it}. The expected return depends on the systematic risk only. Irrespective of the total risk of an asset, it only the systematic risk that is relevant in the determining its expected return (Lewellen & Nagel, 2006:295).

Just from the look of things, CAPM is a simple model when estimating beta coefficients or when estimating expected returns. However, it is actually complicated in practice when applied for investment appraisal. Besides being one of the most researched topics in finance, CAPM has been tested severally and extensively. Tests on CAPM main focus on whether the returns on the assets have a positive statistical relationship with the betas (Lehmann & Modest, 1988:240). This makes beta the root of the problem of the relationship between risk and return as suggested by CAPM. The systematic risk that beta measures is, in mathematical terms, the covariance of asset returns and market returns divided by the variance of the market returns. Beta is the most used parameter for systematic risk, but questions still arise of whether it is the most appropriate parameter (Mirza & Shabbir, 2005:47).

The single index model or the market model is another popular method that is used to estimate betas. Studies carried out on stock price behavior reveal that a rise in the market index results to an increase in stock prices. Similarly, stocks tend to lose their value when the market experiences a downside. This observation could suggest that the reason for the correlation in stock returns could be due to a common response to the market of the stocks (Eun, 1994:335). The correlation can be obtained if the stock return is correlated to the market index. In mathematical terms, it can be expressed as:

R_{i} = α_{i }+ β_{i}R_{m} + e_{i }(Mirza & Shabbir, 2005:42)

e_{i }and α_{i} are return of security i, components and do not depend on the market conditions. The variables are random and represent the insensitivity of the returns to market conditions. This single index model can also be related to a portfolio. By using a portfolio of securities P to replace security i, and using the simple index model, the return on the portfolio can be represented as:

R_{P }= α_{P }+ β_{P}R_{M} + e_{P }(Mirza & Shabbir, 2005:42)

If all the stocks in portfolio P are held in equal proportion in the market index, then the return on portfolio P must be. Considering the equation on the single index model, and assuming no standard error e, we can only have a guaranteed R_{P }= R_{M, }using any R_{m} values, if β_{P} is one and α_{P }is zero. the conclusion that can be derived from this interpretation is that the market’s beta is one and the riskiness of stocks depends on the beta. When a stock’ beta is more than one, the stock would be considered more risky, whereas stocks with a beta value less than one will be considered to be less risky compared to the market (Bartholdy & Peare, 2005:420).

Beta measures risk in equilibrium conditions where investors desire to maximize their utilities. The utility functions depend on the variance and mean of the portfolio’s returns. There are two reasons why variance of returns is questioned as a method of measuring the risk of a return. To begin with, variance is not an appropriate risk measure if the return’s underlying distribution is asymmetric. Secondly, the application of variance as a measure of risk is only straightforward if the return’s underlying distribution is normal (Mirza & Shabbir, 2005:45). However, empirical evidence collected from studies shows that both the normality and the symmetry of stock returns is questionable. Studies have also revealed that beta cannot be considered as a stable parameter. The beta that is used in CAPM is derived using historical returns, which makes it a historical beta. CAPM suffers from the limitation of using a historical beta to estimate future beta while it is known that the factors used to calculate the historical beta might not hold in the future (Akdeniz, Altay-Salih & Caner, 2003:22). Some studies have suggested the suspension of beta from use in measuring risk, with some scholars saying that there is not statistical relationship between beta and returns.

A number of methodological problems are associated with the estimation of beta, especially econometric issues. Beta estimates on systematic risk use ex-ante risk premiums, which in the real world cannot be observed directly. The investors are expected to be rational and the drawings are made using probability distributions of assets’ returns. Over a given period, investors may not be rational as expected (Mirza & Shabbir, 2005:43). Linear regression is used in estimation beta and the assumption of normal distribution that is made with regard to the returns may not be necessary. This could give rise to an issue of hetroskedasticity. In addition, there is a problem with market portfolio’s proxy observations. Most of the assets may not be marketable. The proxies used for market portfolio returns do not take include some types of assets such as private businesses, human capital, and private real estate (Carhart, 1997:60).

Rise of Multifactor Models

In spite of the simplicity and elegance exhibited By CAPM, it makes several assumptions that are somehow unrealistic. In addition, the use of ordinary least square method to estimate betas adds to its limitations. The observations of historic returns for specific stocks or portfolios are time series data and involve non-zero correlation (Sinclair, 1987:20). Generally, CAPM has failed in empirical tests especially in the wake of the modern business world. Research shows that CAPM has failed to explain return on size and book-to-market sorted portfolios. Attempts have been to come up with alternative models that will solve the problems associated with CAPM. This is a research area in finance that is currently receiving much attention. With the increased research, a multi-factor approach has been developed. The aim of the multifactor approach is incorporate different market conditions that have a significant impact on the expected return on an investment (Fama, & French, 2006:3). Some of the factors included in the multifactor models include expected and unexpected inflation, yield spread in interest rates and corporate bonds, and industrial production growth.

Proposals and suggestions have been made concerning several multifactor models. Examples of multifactor models include the ad hoc theorizing and the curve fitting models. These models involve running the CAPM model and then finding the anomalies. The models then incorporate factors that will explain the anomalies identified. The after incorporation of the factors, the model is rerun to see if the anomalies have been explained. Fama and French (1996), have a suggested a three-factor model. This model describes average returns in a better way than CAPM. The model has a strong theoretical standing, and has the excess market return as one of the factors that influence risk. The model captures time-series variation in returns. The model uses market premium to explain difference observed between bills and average return on stocks (Connor, & Korajczyk, 1993:70). Just like in CAPM, the market premium is the average return on market above the risk free rate. Another example of a multifactor model that overcomes some of the limitations of CAPM is the Arbitrary Pricing Theory (APT). This model uses multiple factors and does not require the market portfolio to be identified (Connor & Korajczyk, 1995:90). The number of factors to be used in the model can be specified ex-ante or ex-post. Unlike CAPM, APT assumes existence of competitive markets, which is the situation in the real world (Maio & Santa-Clara, 2012:590).

A different group of researchers proposes that CAPM holds only under specific conditions. The researchers argue that even if the cross-sectional CAPM holds at a given time t, it is possible for the unconditional CAPM to fail unconditionally. The result of this assertion is modeling conditional distribution of returns at time t as a function of lagged state variables. A covariance between portfolio returns and the market is derived from the structure of the model. Beta of the model is a function of the state variables. From the model, it is found that there is a high correlation between the stocks’ betas and consumption growth rate. The fact that true betas cannot be observed makes investors to indulge themselves in the learning process (Han, 2006:13). The multifactor model can therefore be referred to as the learning-factor model. In this model, the beta combines both historic and current information. Investors update their beta values on a continuous basis because they cannot observe beta values directly (Maio & Santa-Clara, 2012:602).

Conclusion

CAPM has been a useful tool for financial analysts in investment appraisal. However, with time, CAPM seems no longer to hold in the real world. Most of the limitations of CAPM are associated with its assumptions. The parameters used in the CAPM’s formula are also not sufficient to cover all the market conditions. In addition, use of historic data to estimate future beta values makes CAPM impractical in the real world. Considering the failure of CAPM to hold in the real world, researchers have come up with multifactor models that attempt to overcome CAPM’s limitations. The models use factors that will incorporate prevailing market conditions and try to use current data. The assumptions in the multifactor models are not as stringent as those of CAPM. However, the problem with multifactor models is that no one knows the exact factors to include in the model. As research continues in this interesting topic, there is a possibility that models that are more comprehensive will emerge.

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