Question

Explain the following issues of the Capital Asset Pricing Model:

1. Explain the separation theorem of the CAPM.
2. What does the beta (β) measure?
3. What is the alpha (α) of a stock?
4. Describe the security market line (SML).
5. Describe the capital market line (CML).

Answer 1

CAPM (Capital asset pricing Model)

Diversification through a combination of securities that are not perfectly/positively correlated helps to lessen the risk of the portfolio. Total portfolio risk has 2 components:- systematic & unsystematic risk. Systematic risk is caused by those factors that affect the overall market/ all securities (for e.g. war, inflation, political turmoil, natural calamity etc.). The unsystematic risk is unique to the particular company/ industry/security (for e.g. strikes, regulatory action etc.). This kind of risk can be reduced & even sometimes can be eliminated thorugh efficient diversification. Therefore the important risk is systematic risk. Investors can expect compensation for bearing this type of risk. thus there is a trade off between risk & return. This logic is behind the formulation of CAPM theory.

CAPM model describes the trade off between risk and return. It explains the behaviour of the security prices & provides a mechanism to assess the impact of a proposed security investment on investor's overall portfolio risk & return.

Assumptions of the model

The basic assumptions of the model are related to :- the efficiency of the markets & the investor preferences. In an efficient capital market the investors are well informed, transaction costs are low,there are negligible restrictions on investments, no investor is large enough to influence the market price of a share & investors are in general agreement about the likely performance of the individual securities & their expectations are based on common 1 year holding period.

Investors are assumed to prefer to invest in securities with highest return for a given level of the risk or the lowest risk for a given level of return, return & risk being measured in terms of expected value & standard deviation respectively.

The CAPM model links the relevant (systematic) risk & return of all securities and assets. The measure/ index of systematic risk is Beta coefficient  It measures the sensitivity of the return of a security to changes in the retrurns of the market portfolio. In other words beta coefficient is an index of the degree of responsiveness of security return with market return. For e.g.- In the Indian Stock exchange the index or the beta is measured in terms of the Nifty, Sensex etc. Beta is a measure of a stock's volatility in relation to the overall market. By definition, the market, such as the S&P 500 Index, has a beta of 1.0, and individual stocks are ranked according to how much they deviate from the market. A stock that swings more than the market over time has a beta above 1.0.

From the point of risk individual investments may be risk free whose return over the holding period is known with certainty Such as treasury bills (risk free asset). The other types of investments are the shares on the return is uncertain. The expected return on such investments less the risk free return is the excess return. Beta coefficient represents the change in the expected return on the market portfolio. The beta for the market portfolio is equal to 1 by definition. It is thus an index of the systematic risk of an individual security relative to that of the market portfolio.

The = 1 is that excess return for the security vary proportionately with excess returns for the market portfolio that is the security has the same systematic risk as that of the market portfolio. A   >1 means more unavoidable risk for the security than the market as a whole. A  <1 means the security has less systematic risk vis-a-vis the market portfolio.

To calculate the beta of a security, the covariance between the return of the security and the return of the market must be known, as well as the variance of the market returns.

Beta = Variance / Covariance​

where: - Covariance = Measure of a stock’s return relative to that of the market
Variance = Measure of how the market moves relative to its mean​

Covariance measures how two stocks move together. Variance, on the other hand, refers to how far a stock moves relative to its mean. For example, variance is used in measuring the volatility of an individual stock's price over time. Covariance is used to measure the correlation in price moves of two different stocks.

Alpha is a measure of the performance of an investment as compared to a suitable benchmark index, such as the S&P 500. "Alpha" (the Greek letter α) is a term used in investing to describe a strategy's ability to beat the market. Alpha is thus also often referred to as “excess return” or “abnormal rate of return,” which refers to the idea that markets are efficient, and so there is no way to systematically earn returns that exceed the broad market as a whole. Alpha is often used in conjunction with beta.
Alpha is used in finance as a measure of performance, indicating when a strategy, trader, or portfolio manager has managed to beat the market return over some period of time. Alpha, often considered the active return on an investment, measures the performance of an investment against a market index or benchmark that is considered to represent the market’s movement as a whole. The excess return of an investment relative to the return of a benchmark index is the investment’s alpha. Alpha may be positive or negative and is the result of active investing.

Alpha is commonly used to rank active mutual funds as well as all other types of investments. It is often represented as a single number (like +3.0 or -5.0), and this typically refers to a percentage measuring how the portfolio or fund performed compared to the referenced benchmark index (i.e., 3% better or 5% worse).

The CAPM is used to calculate the amount of return that investors needs to get to compensate for a particular level of risk. It subtracts the risk-free rate from the expected rate and weighs it with a factor, beta, to get the risk premium. It then adds the risk premium to the risk-free rate of return to get the rate of return an investor expects as compensation for the risk.
The CAPM formula is expressed as follows:-

R = Rf + beta (Rm – Rf) + Alpha

Therefore,

Alpha= R – Rf – beta (Rm-Rf)

Where:

• R represents the portfolio return
• Rf represents the risk-free rate of return
• Beta represents the systematic risk of a portfolio
• Rm represents the market return, as per a benchmark stock index

For example, lets assume that the actual return of the mutual fund is 30, the risk-free rate is 8%, beta is 1.1, and the benchmark index return is 20%, alpha is calculated as:

Alpha = (0.30-0.08) – 1.1 (0.20-0.08)

= 0.088 or 8.8%

The above result shows that the fund in has outperformed the benchmark index by 8.8%.

Security market line (SML) is the graphical representation of the Capital Asset Pricing Model (CAPM) and gives the expected return of the market at different levels of systematic or market risk. It is also called ‘characteristic line’ where the x-axis represents beta or the risk of the assets and y-axis represents the expected return. The "market risk premium" of a given security is determined by where it is plotted on the chart relative to the SML.

Above is an example of a security market line graphed. The y-intercept of this line is the risk-free rate (the ROI of an investment with beta value of 0), and the slope is the premium that the market charges for taking the systematic risk.

The security market line is an investment evaluation tool derived from the CAPM model. SML is a good representation of investment opportunity cost which provides the combination of risk-free asset and the market portfolio. Zero-beta security or zero-beta portfolio has an expected return on the portfolio which is equal to the risk-free rate. The slope of the Security Market Line is determined by market risk premium. Higher the market risk premium steeper the slope and vice-versa.

The assets which are above the SML are undervalued as they give the higher expected return for a given amount of risk. The assets which are below the SML are overvalued as they have lower expected returns for the same amount of risk.

The Capital Market Line is a graphical representation of all the portfolios that optimally combine risk and return. CML is a theoretical concept that gives optimal combinations of a risk-free asset and the market portfolio. The CML is superior to Efficient Frontier in the sense that it combines the risky assets with the risk-free asset.

• The slope of the Capital Market Line(CML) is the Sharpe Ratio of the market portfolio.
• The efficient frontier represents combinations of risky assets.
• If we draw a line from the risk-free rate of return which is tangential to the efficient frontier, we get the Capital Market Line. The point of tangency is the most efficient portfolio.
• Moving up the CML will increase the risk of the portfolio and moving down will decrease the risk. Subsequently, the return expectation will also increase or decrease respectively. All investors will choose the same market portfolio given a specific mix of assets and the associated risk with them.

Separation theorem of the CAPM

In the CAPM model it has been established thateach investor will hold combinations of risk free asset & the tangency portfolio obtained from the point of touch between the CML & the efficient frontier, which is the Market Portfolio. Because CAPM model assumes homogeneous expectations of the investors, in equilibrium all investors will determine the same tangency portfolio. Also all investors agree on the risk-free asset.

Borrowing and lending possibilities combined with one portfolio of risky assets M, offer an investor whatever risk - expected return combination he or she seeks. An investor could :-

• Invest 100 percent of investable funds in the risk-free asset, providing an expected return of RF and zero risk,

• Invest 100 percent of investable funds in risky-asset portfolio-M

• Invest in any combination of return and risk between the two points on the Capital Market Line; obtained by varying the proportion of investment in the risk-free asset.

Different investors will choose different portfolios because of their risk preferences (i.e. they have different indifference curves), but they will choose the same combination of risky securities as denoted by the tangency point M. Investors will then borrow or lend to achieve various positions on the linear trade-off (capital market line), between expected return and risk. It is not necessay to match each client's indifference curves with a particular efficient portfolio because only one efficient portfolio is held by all investors. Rather each client will use his or her indifference curve to determine where along the efficient frontier h/she should be. Each client must determine how much of the investible funds should be lend or borrowed at risk-free rate and how much should be invested in the portfolio M. This theory is called seperation theorem.