An Analysis of Capital Asset Pricing Model (CAPM)

Introduction Harry Markowitz, a Nobel Memorial Prize winning economist, devised the modern portfolio theory in 1952.  Markowitz's theories emphasized the importance of portfolios, risk, the correlations between securities and diversification of securities. His work changed the attitude of investment community. Prior to Markowitz's theories, emphasis was placed on picking single high-yield stocks without any regard to their effects on portfolios as a whole. Markowitz's portfolio theory was a large stepping stone towards the creation of the Capital Asset Pricing Model (CAPM).

Pronounced as though it were spelled "cap-m", this model was originally developed in 1952 by Harry Markowitz and fine-tuned over a decade later by others, including William Sharpe. In fact, this model was introduced by Jack Treynor, William Sharpe, John Lintner and Jan Mossin independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. Sharpe received the Nobel Memorial Prize in Economics (jointly with Markowitz and Merton Miller) for this contribution to the field of financial economics.

The CAPM helps us to calculate investment risk and what return on investment we should expect. CAPM describes the relationship between risk and expected return, and it serves as a model for the pricing of risky securities.

Meaning of CAPMCapital asset pricing model is simply a model for asset pricing and portfolio construction. This model presents a very simple theory that delivers a simple result. According to this theory, the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat our required return, the investment should not be undertaken. This model is used to confirm a theoretically suited necessary rate of return of an asset at a time, when it is about to be added to an existing and good going portfolio.In short, it is a model that describes the relationship between risk and expected return and that is used in the pricing of risky securities.Mathematically,

The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium (rm-rf).

Different Parameters of CAPM

1. Risk and return: This model assumes that investors are risk averse, meaning that given two assets that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher returns must accept more risk.

2. Mean and variance: It is further assumed that investor's risk / reward preference can be described via a quadratic utility function. The effect of this assumption is that only the expected return and the volatility (i.e., mean return and standard deviation) matter to the investor. The investor is indifferent to other characteristics of the distribution of returns, such as its skew (measures the level of asymmetry in the distribution) or kurtosis (measure of the thickness or so-called "fat tail").

3. Diversification: An investor can reduce portfolio risk simply by holding instruments which are not perfectly correlated. In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Diversification will allow for the same portfolio return with reduced risk.

4. The risk-free asset: The risk-free asset is the (hypothetical) asset which pays a risk-free rate. It is usually proxied by an investment in short-dated Government securities. The risk-free asset has zero variance in returns (hence is risk-free); it is also uncorrelated with any other asset (by definition: since its variance is zero). As a result, when it is combined with any other asset, or portfolio of assets, the change in return and also in risk is linear.

5. Efficient frontier: The efficient frontier graph correlates a portfolio's risk profile to possible returns. The curve, or frontier, illustrates the possibility for risk and return. The lower-left hand corner illustrates a low-risk & return portfolio. Moving up along the curve, one enters higher risk territory, where investors may expect higher returns.

Beyond the curve reflects returns impossible under current conditions. Below the curve reflects and inefficient portfolio, that may achieve greater returns in a different portfolio arrangement, with the same risk.

Efficient frontier is a set of portfolios that each maximize expected return for a given level of risk. The CAPM assumes that the risk-return profile of a portfolio can be optimized - an optimal portfolio displays the lowest possible level of risk for its level of return. Additionally, since each additional asset introduced into a portfolio further diversifies the portfolio, the optimal portfolio must comprise every asset, (assuming no trading costs) with each asset value-weighted to achieve the above (assuming that any asset is infinitely divisible). All such optimal portfolios, i.e., one for each level of return, comprise the efficient frontier.

6. Beta: According to CAPM, beta is the only relevant measure of a stock's risk. It measures a stock's relative volatility - that is, it shows how much the price of a particular stock jumps up and down compared with how much the stock market as a whole jumps up and down. If a share price moves exactly in line with the market, then the stock's beta is 1. A stock with a beta of 1.5 would rise by 15% if the market rose by 10%, and fall by 15% if the market fell by 10%.

Measurement of Various Risks

Overall Measure of RiskStandard deviation of a security provides a measure of its overall risk.

Standard deviation takes into account fluctuations of returns of a stock for a given period with the mean returns. This is an indicator of relative fluctuations in the returns given by the stock. This takes into account both systematic and unsystematic risk. The formula for calculating standard deviation is:

√ {[∑(Rt – Rm)2] / (t – 1)}Where, Rt is the return of the given time period, Rm is the mean (average) return during the time period and t is the number of time periods included.

Unsystematic RiskThis is the risk element which is typical to an individual security. The risk of a

possible strike in the factory of an individual company due to the reasons completely personal to the company is an unsystematic risk. Such risk can be diversified by including many other securities in the portfolio. When we start increasing number of securities in the portfolio such risk becomes negligible.

Systematic RiskSystematic risk is the overall market risk of a security which cannot be

diversified. An investor has to consider this risk in designing portfolios as it cannot be diversified. Beta is a measure of this systematic risk. It measures the relative responsiveness of a security’s returns to market returns. For a layman, for a security with a beta of 2 means that the fluctuations in return (in general) of a security are twice that of market. Thus when market return is 10% the security gives a return of 20% and when the market return is -10%, the return from the security is -20%.

β can be calculated with past as well as expected future prices. Most available data sources show beta calculated from past data as it is easily available.β is calculated by the following formula:

β = [Cov(rm, rs)]/σm2

Where, Cov(rm, rs) is the covariance between market and security returns.σm is the dtandard deviation of market returns.β of a security is also given by the formula:

β = (rms σs)/ σm

Where, rms is the coefficient of correlation between market returns and security returns.

Beta value of a portfolio is the weighted average value of all the individual stocks included in the portfolio which is almost equals 1.

The Effect of Beta on CAPM

The big sticking point is beta. When Professors Eugene Fama and Kenneth French looked at share returns on the New York Stock Exchange, the American Stock Exchange and NASDAQ between 1963 and 1990, they found that differences in betas over that lengthy period did not explain the performance of different stocks. The linear relationship between beta and individual stock returns also breaks down over shorter periods of time. These findings seem to suggest that CAPM may be wrong. While some studies raise doubts about CAPM's validity, the model is still widely used in the investment community. Although it is difficult to predict from beta how individual stocks might react to particular movements, investors can probably safely deduce that a portfolio of high-beta stocks will move more than the market in either direction, or a portfolio of low-beta stocks will move less than the market. This is important for investors - especially fund managers - because they may be unwilling to or prevented from holding cash if they feel that the market is likely to fall. If so, they can hold low-beta stocks instead. Investors can tailor a portfolio to their specific risk-return requirements, aiming to hold securities with betas in excess of 1 while the market is rising, and securities with betas of less than 1 when the market is falling. Not surprisingly, CAPM contributed to the rise in use of indexing - assembling a portfolio of shares to mimic a particular market - by risk averse investors. This is largely due to CAPM's message that it is only possible to earn higher returns than those of the market as a whole by taking on higher risk (beta).

Practical Difficulties in the Application of CAPM

The model assumes that asset returns are (jointly) normally distributed random variables. It is however frequently observed that returns in equity and other markets are not normally distributed. As a result, large swings (3 to 6 standard deviations from the mean) occur in the market more frequently than the normal distribution assumption would expect.

The model assumes that the variance of returns is an adequate measurement of risk. This might be justified under the assumption of normally distributed returns, but for general return distributions other risk measures (like coherent risk measures) will likely reflect the investors' preferences more adequately.

The model does not appear to adequately explain the variation in stock returns. Empirical studies show that low beta stocks may offer higher returns than the model would predict. Some data to this effect was presented as early as a 1969 conference in Buffalo, New York in a paper by Fischer Black, Michael Jensen, and Myron Scholes. Either that fact is itself rational (which saves the Efficient Market Hypothesis but makes CAPM wrong), or it is irrational (which saves CAPM, but makes the EMH wrong – indeed, this possibility makes volatility arbitrage a strategy for reliably beating the market).

The model assumes that, given a certain expected return investors will prefer lower risk (lower variance) to higher risk and conversely given a certain level of risk will prefer higher returns to lower ones. It does not allow for investors who will accept lower returns for higher risk. Casino gamblers clearly pay for risk, and it is possible that some stock traders will pay for risk as well.

The model assumes that all investors have access to the same information and agree about the risk and expected return of all assets (homogeneous expectations assumption).

The model assumes that there are no taxes or transaction costs, although this assumption may be relaxed with more complicated versions of the model.

The market portfolio consists of all assets in all markets, where each asset is weighted by its market capitalization. This assumes no preference between markets and assets for individual investors, and that investors choose assets solely as a function of their risk-return profile.

It also assumes that all assets are infinitely divisible as to the amount which may be held or transacted. The market portfolio should in theory include all types of assets that are held by anyone as an investment (including works of art, real estate, human capital, etc.). In practice, such a market portfolio is unobservable and people usually substitute a stock index as a proxy for the true market portfolio. Unfortunately, it has been shown that this substitution is not innocuous and can lead to false inferences as to the validity of the CAPM, and it has been said that due to the inobservability of the true market portfolio, the CAPM might not be empirically testable. This was presented in greater depth in a paper by Richard Roll in 1977, and is generally referred to as Roll's critique.

ConclusionIn finance, the CAPM is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset’s non-diversifiable risk. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systemic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset. The capital asset pricing model is by no means a perfect theory. But the spirit of CAPM is correct. It provides a usable measure of risk that helps investors determine what return they deserve for putting their money at risk.