Definition of Capital Asset Pricing Theory

Introduction

Investors are interested in knowing the systematic risk when they search for efficient portfolios. They would like to have assets with low beta co-efficient i.e. systematic risk. Investors would opt for high beta co-efficient only if they provide high rates of return. The risk averse nature of the investors is the underlying factor for this behavior. The capital asset pricing theory helps the investors to understand the risk and return relationship of the securities. It also explains how assets should be priced in the capital market.

The CAPM Theory

Markowitz, William Sharpe, John Lintner and Jan Mossin provided the basic structure for the CAPM model. It is a model of linear general equilibrium return. In the CAPM theory, the required rate return of an asset is having a linear relationship with asset’s beta value i.e. undiversifiable or systematic risk.

Assumptions

An individual seller or buyer cannot affect the price of a stock. This assumption is the basic assumption of the perfectly competitive market.

Investors make their decisions only on the basis of the expected returns, standard deviations and co variances of all pairs of securities.

Investors are assumed to have homogenous expectations during the decision-making period.

The investor can lend or borrow any amount of funds at the riskless rate of interest. The riskless rate of interest is the rate of interest offered for the treasury bills or Government securities.

Assets are infinitely divisible. According to this assumption, investor could buy any quantity of share i.e. they can even buy ten rupees worth of Reliance Industry shares.

There is no transaction cost i.e. no cost involved in buying and selling of stocks.

There is no personal income tax. Hence, the investor is indifferent to the form of return either capital gain or dividend.

Unlimited quantum of short sales is allowed. Any amount of shares an individual can sell short.

Lending and Borrowing

Here, it is assumed that the investor could borrow or lend any amount money at riskless rate of interest. When this opportunity is given to the investors, they can mix risk free assets with the risky assets in a portfolio to obtain a desired rate of risk-return combination.

R_p        =          Portfolio return

X_f        =          the proportion of funds invested in risk free assets

The expected return on the combination of risky and risk free combination is

R_p        =          R_fX_f + R_m(1 – X_f)

This formula can be used to calculate the expected returns for different situations, like mixing ri assets with risky assets, investing only in the risky asset and mixing the borrowing with risky assets.Now, let us assume that borrowing and lending rate to be 12.5% and the return from the risky assets to be 20%. There is a trade off between the expected return and risk. If an investor invests in risk free assets and risky assets, his risk may be less than what he invests in the risky asset alone. But if he borrows to invest in risky assets, his risk would increase more than he invests his own money in the risky assets. When he borrows to invest, we call it financial leverage. If he invests 50% in risk free assets and 50% in risky assets, his expected return of the portfolio would be

If there is a zero investment in risk free asset and 100% in risky asset, the return is

If - .5 in risk free asset and 1.5 in risky asset, the return is

The variance of the above mentioned portfolio can be calculated by using the equation.

σ²_p       =          σ²_fX²_f + σ ²_m(1 – X_f)² + 2Cov_fmX_f(1 – X_f)

The previous example can be taken for the calculation of the variance. The variance of the risk free asset is in. The variance of the risky asset is assumed to be 15. Since the variance of the risky asset is zero, the 1,rtfolio risk solely depends on the portion of investment on risky asset.

The risk is more in the borrowing portfolio being 22.5% and the return is also high among the three alternatives. In the lending portfolio, the risk is 7.5% and the return is also the lowest. The risk premium is proportional to risk, where the risk premium of a portfolio is defined as the difference between R_p - R_f i.e. the amount by which a risky rate of return exceeds the riskless rate of return.

Risk - Return Trade Off

The risk-return proportionality ratio is a constant .5, indicating that one unit of risk premium is accompanied by 0.5 unit of risk.

The Concept

According to CAPM, all investors hold only the market portfolio and riskless securities. The market portfolio is a portfolio comprised of all stocks in the market. Each asset is held in proportion to its market value to the total value of all risky assets. For example, if Reliance Industry share represents 20% of all risky assets, then the market portfolio of the individual investor contains 20% of Reliance industry shares. At this stage, the investor has the ability to borrow or lend any amount of money at the riskiness rate of interest. The efficient frontier of the investor is given in figure.

The figure shows the efficient frontier of the investor. The investor prefers any point between B and C because, with the same level of risk they face on line BA, they are able to get superior profits. The ABC line shows the investor’s, portfolio of risky assets. The investors can combine riskless asset either by lending or borrowing. This is shown in Figure.

The line R_fS represents all possible combination of riskless and risky asset. The ‘S’ portfolio does not represent any riskless asset but the line RS gives the combination of both. The portfolio along the path RS is called lending portfolio that is some money is invested in the riskless asset or may be deposited in the bank for a fixed rate of interest. If it crosses the point S. it becomes borrowing portfolio. Money is borrowed and invested in the risky asset. The straight line is called capital market line (CML). It gives the desirable set of investment opportunities between risk free and risky investments. The CML represents linear relationship between the required rates of return for efficient portfolios and their standard deviations.

E(R_p) = portfolio’s expected rate of return R_m =

expected return on market portfolio σ_m =

standard deviation of market portfolio σ_p =

standard deviation of the portfolio

For a portfolio on the capital market line, the expected rate of return in excess of the risk free rate is in proportion to the standard deviation of the market portfolio. The price of the risk is given by the slope of the line. The slope equals the premium for the market portfolio R_m – R _f divided by the risk or standard deviation of the market portfolio. Thus, the expected return of an efficient portfolio is

Expected return = Price of time + (Price of risk . Amount of risk)

Price of time s the risk free rate of return. Price of risk is the premium amount higher and above the risk free return.

Security Market Line

The risk-return relationship of an efficient portfolio is measured by the capital market line. But, it does not show the risk-return trade off for other portfolios and individual securities. Inefficient portfolios lie below the capital market line and the risk-return relationship cannot be established with the help of the capital market line. Standard deviation includes the systematic and unsystematic risk. Unsystematic risk can be diversified and it is not related to the market. If the unsystematic risk is eliminated, then the matter of concern is systematic risk alone. This systematic risk could be measured by beta. The beta analysis is useful for individual securities arid portfolios whether efficient or inefficient.

When an additional security is added to the market portfolio, an additional risk is also added to it. The variance of a portfolio is equal to the weighted sum of the co-variances of the individual securities in the portfolio.

If we add an additional security to the market portfolio, its marginal contribution to the variance of the market is the covariance between the security’s return and market portfolio’s return. If the security i am included, the covariance between the security and the market measures the risk. Covariance can be standardized by dividing it by standard deviation of market portfolio coy i_m/σ_m. This shows the systematic risk of the security. Then, the expected return of the security i is given by the equation:

R_i – R_f = (R_m – Rf/σ_m) Coy i_m/σ_m

This equation can be rewritten as follows

R_i – R_f = Coy i_m/σ²_m (R_m – Rf)

The first term of the equation is nothing but the beta coefficient of the stock. The beta coefficient of the equation of SML is same as the beta of the market (single index) model. In equilibrium, all efficient and inefficient portfolios lie along the security market line. The SML line helps to determine the expected return for a given security beta. In other words, when betas are given, we can generate expected returns for the given securities. This is explained in fig.

If we assume the expected market risk premium to be 8% and the risk free rate of return tube 7%, we can calculate expected return for A, B, C and D securities using the formula

The same can be found out easily from the figure too. All we have to do is, to mark the beta on the horizontal axis and draw a vertical line from the relevant point to touch the SML line. Then from the point of intersection, draw another horizontal line to touch the Y axis. The expected return could be very easily read from the Y axis. The securities A and B are aggressive securities, because their beta values are greater than one. When beta values are less than one, they are known as defensive securities. In our example, security C has the beta value less than one.

Evaluation of Securities

Relative attractiveness of the security can be found out with the help of security market line. Stocks with high risk factor are expected to yield more return and vice-versa. But the investor would be interested in knowing whether the security is offering return more or less proportional to its risk.

The figure provides an explanation for the evaluation. There are nine points in the diagram. A, B and C lie on the security market line, R, S and T above the SML and U, V and W below the SML. ARU have the same beta level of, 9. Likewise beta values of SBV = 1.00 and TCW = 1.10. The stocks above the SML yield higher returns for the same level of risk. They are underpriced compared to their beta value. With the simple rate of return formula, we can prove that they are undervalued.

P_i is the present price P₀ - the purchase price and Div - Dividend. When the purchase price is low i.e. when the denominator value is low, the expected return could be high. Applying the same principle the stocks U, V and W can be classified as overvalued securities and are expected to yield lower returns than stocks of comparable risk. The denominator value may be high i.e. the purchase price may be high. The prices of these scripts may fall and lower the denominator. There by, they may increase the returns on securities.

The securities A, B and C are on the line. Therefore considered to be appropriately valued. They offer returns in proportion to their risk. They have average 4oclc performance, since they are neither undervalued nor overvalued.

Market Imperfection and SML

Information regarding the share price an4 market condition may not be immediately available to all investors. Imperfect information may affect the valuation of securities. In a market with perfect information, all securities should lie on SML. Market imperfections would lead to a band of SML rather than a single line. Market imperfections affect the width of the SML to a band. If imperfections are more, the width also would be larger. SML in imperfect market is given in figure.

SML in Imperfect Market

Empirical Tests of the CAPM

In the CAPM, beta is used to estimate le systematic of the security and reflects the future volatility of the stock in relation to the market. Future volatility of the stock is estimated only through historical data. Historical data are used to plot the regression line or the characteristic line and calculate beta. If historical betas are stable over a period of time, they would be good proxy for their ex-ante or expected risk.

Robert A. Levy, Marshall B. Blume and others have studied the question of beta stability in depth. I calculated betas for both Individual securities and portfolios. His study results have provided the following conclusions

The betas of individual stocks are unstable; hence the past betas for the individual securities are not good estimators of future risk.

The betas of portfolios of ten or more randomly selected stocks are reasonably stable, hence the portfolio betas are good estimators of future portfolio volatility. This is because of the errors in the estimates of individual securities’ betas tend to offset one another in a portfolio.

Various researchers have attempted to find out the validity of the model by calculating beta and realized rate of return. They attempted to test (1) whether the intercept is equal to i.e. risk free rate of interest or the interest rate offered for treasury bills (2) whether the line is linear and pass through the beta = 1 being the required rate of return of the market. In general, the studies have showed the following results.

The studies generally showed a significant positive relationship between the expected return and t systematic risk. But the slope of the relationship is usually less than that of predicted by the CAPM.

The risk and return relationship appears to be linear. Empirical studies give no evidence of significant curvature in the risk/return relationship.

The attempts of the researchers to assess the relative importance of the market and company risk have yielded definite results. The CAPM theory implies that unsystematic risk is not relevant, but unsystematic and systematic risks are positively related to security returns. Higher returns are needed to compensate both the risks. Most of the observed relationship reflects statistical problems rather than the true nature of capital market.

According to Richard Roll, the ambiguity of the market portfolio leaves the CAPM untestable. The practice of using indices as proxies is loaded with problems. Different indices yield different betas for the same security.

If the CAPM were completely valid, i4 should apply to all financial assets including bonds. But, when bonds are introduced into the analysis, they do not fall on the security market line.

Present Validity of CAPM

The CAPM is greatly appealing at an intellectual level, logical and rational. The basic assumptions on which the model is built raise, some doubts in the minds of the investors. Yet, investment analysts have been more creative in adapting CAPM for their uses.

The CAPM focuses on the market risk, makes the investors to think about the riskiness of the assets in general. CAPM provides basic concepts which are truly of fundamental value.

The CAPM has been useful in the selection of securities and portfolios. Securities with higher returns are considered to be undervalued and attractive for buy. The below normal expected return yielding securities are considered to be overvalued and Suitable for sale.

In the CAPM, it has been assumed that investors consider only the market risk. Given the estimate of the risk free rate, the beta of the firm, stock and the required market rate of return, one can find out the expected returns for a firm’s security. This expected return can be used as an estimate of the cost of retained earnings.

Even though CAPM has been regarded as a useful tool to financial analysts, it has its own critics too. They point out, when the model is ex-ante, the inputs also should be ex-ante, i.e. based on the expectations of the future. Empirical tests and analyses have used ex-post i.e. past data only.