CAPM

As far as mathematics refers to reality,

they are not certain; and as far as they are certain,

they do not refer to reality.

Albert Einstein, 1879 – 1955

The capital market theory builds on portfolio theory and leads to the capital asset pricing model. This theory begins with the efficient frontier, which depicts the return-risk relationship for portfolios consisting of risky assets. The capital market theory, because it builds on Markowitz’s work, has the same assumptions underlying it, plus the following additional assumptions.

• Many investors participate in the capital market, each of whose wealth is small compared to the entire wealth in the market. Moreover, investors behave as though their actions have no impact on the market.

• All investors have the same holding period time horizon. This shortsighted behavior leads to suboptimal investment decisions.

• Investors can borrow and lend at the same nominal risk-free rate.

• There are no taxes or transaction costs.

• All investors want to be on the efficient frontier, which means they are mean-variance optimizers. Their exact location depends on the investor’s risk-return utility function.

• All investors have homogeneous expectations. Thus, they all estimate the same probability distributions for future rates of return and covariances among assets. Therefore, all investors base decisions on the same efficient frontier and optimal risky portfolio.

• All investments are infinitely divisible. Thus, fractional shares are possible, with the result being continuous curves possibilities for all combinations of assets.

• There is no inflation (or it is fully anticipated) or any change in interest rates.

• Capital markets are in equilibrium, which means investments are priced properly given their risk.

A Risk-free Asset

A key factor in the development of the capital market theory is the concept of a risk-free asset. The theory states that there is an alternative to investing only in risky assets; it is to invest in a riskless asset, defined as an asset whose return (R_F) has no standard deviation ( = 0).

Given the correlation r_xy between two assets, x and y, and their respective standard deviations, the covariance is:

COV_xy = r_xy_x_y

The risk-free asset, which we will call x, has a known, certain return. This is the result of its standard deviation being 0 ( = 0). Thus, the covariance of the risk-free asset with any risky asset is zero. With covariance of zero, the correlation (r_xy) between any risky asset y and the risk-free asset is zero.

Adding the risk-free asset to a risky asset portfolio that lies on the efficient frontier will have several effects. The expected return for a portfolio that includes a risk-free asset and a risky asset portfolio is the weighted average of the two returns. The formula is as follows:

E(R_portfolio) = w_RfE(R_F) + (1 – w_Rf)E(R_i)

where:

The variance and standard deviation for this new portfolio can be derived and are as follows:

²_portfolio = w²_Rf²_Rf + (1 – w_Rf)²²_i + 2w_Rf(1 – w_Rf)COV_Rf,i

Replace ²_Rf and COV_Rf,i with 0

²_portfolio = w²_Rf0 + (1 – w_Rf)²²_i + 2w_Rf(1 – w_Rf)0

Results in

²_portfolio = (1 – w_Rf)²²_i

_portfolio = (1 – w_Rf)_i

As the last equation shows, the standard deviation for a portfolio consisting of a risky assets portfolio and a risk-free asset is equal to the linear proportion of the standard deviation of the risky asset portfolio.

As a result, any combination of two assets (the risk-free asset and a portfolio of risky assets) will be a straight line, the Capital Allocation Line (CAL) or the Capital Market Line (CML). The figure below shows how this touches the efficient frontier.

The tangency portfolio is market portfolio, which comprises all risky assets (domestic and international stocks, bonds, options, antiques, real estate and so forth). It is by definition, fully diversified. The assets are weighted based on their market values.

The market portfolio is a completely diversified portfolio. Thus, all risk unique to individual assets has been diversified away. These unique individual asset risks are known as unsystematic risks. Because unsystematic risk can be diversified away, investors are not rewarded for assuming it. What is left is systematic risk, which is the variability in all risky assets caused by changes in macroeconomic factors and it is measured as the standard deviation of returns of the market portfolio. Systematic risk does change over time in response to changes in macroeconomic variables.

Diversification

A perfectly diversified portfolio will have a correlation with the market portfolio of +1.00. This is true, as a perfectly diversified portfolio will have eliminated all unsystematic risk, thus leaving only systematic risk, which is the same as the market portfolio. Studies have shown that a stock portfolio requires 30 or more stocks to be well diversified. As more securities are added, assuming imperfect correlations among securities, the average covariance of the portfolio declines.

The CML indicates that all investors should invest in the same risky asset portfolio, the market portfolio (M). Effectively, the only relevant portfolio is the market portfolio and what is important is where the investors will be on the CML. More risk-averse investors will use a portion of their assets to buy the risk-free asset (thereby lending) and then invest the rest in the market portfolio. Thus they will be on the portion of the CML that is to the left of the tangency portfolio. An investor willing to take on extra risk will borrow funds and use this leverage to buy more of the market portfolio. Thus, this investor’s portfolio will be to the right of tangency portfolio.

A Risk Measure for the CML

The relevant risk to consider when adding a security to a portfolio is its average covariance with all other assets in the portfolio. Thus, the relevant risk measure for an individual risky asset is its covariance with the market portfolio, also known as its systematic risk.

Since all individual risky assets are a part of the market portfolio, another conclusion that can be drawn is that the return for an individual risky asset can be found using the following linear model:

R_i,t = α_i + b_iR_M,t + 

where:

The variance using the same base equation can be found as follows:

Var(R_i,t) = Var(α_i) + Var(b_iR_M,t) + Var()

The Var(b_iR_M,t) is the variance of the asset’s return related to the variance of the market portfolio. This is the systematic variance or risk. The Var() is the residual variance, which is not related to the market portfolio. Thus, it is the unsystematic risk. But, unsystematic risk can be diversified away and, therefore, unsystematic risk is not relevant to investors. The only relevant risk is systematic, which cannot be diversified away and is due to macroeconomic factors.

The Capital Asset Pricing Model (CAPM)

With the understanding of the capital market theory, including the key concept that the only relevant measure of risk is an asset’s covariance with the market portfolio, the next step is to find an appropriate expected rate of return on a risky asset. The capital asset pricing model (CAPM) indicates what should be the expected or required rates of return for risky assets.

The CAPM is useful in determining an appropriate discount rate or in comparing the estimated rate of return to the required rate of return. Thus, the CAPM is useful in identifying whether an asset is undervalued, overvalued, or properly valued.

Security Market Line (SML)

The security market line is a visual representation of the relationship between risk and the expected or required return on an asset. The relevant risk measure of an individual risky asset is its covariance with the market portfolio (COV_i,M).

The covariance of the market with itself is its variance _M². The equation for the SML can be expressed as follows.

E(R_i) = R_f + (R_m – R_f)(COV_i,M)/ _M²

If β_i = COV_i,M / _M² , then

E(R_i) = R_f + β_i(R_m – R_f)

which has the following graph:

This is a simple linear equation, with slope β_i and y-intercept R_f. Note that market portfolio has a beta of 1. (β_M = 1).

Contrasting the CML and SML

The capital market line and the security market line are similar, yet different concepts. The capital market line is the relationship between the required returns on efficient portfolios (R_P) and their total risk (_P); the security market line is the relationship between the expected returns on individual assets (R_S) and their risk as measured by their covariance with the market portfolio (COV_SM), or their normalized risk relative to the market, as measured by their betas(β_S). This is because the relevant measure of risk for an individual security held as a part of a well-diversified portfolio is not the security’s standard deviation or variance, but the contribution that it makes to the overall portfolio variance, measured by the asset’s beta.

If the expected return-beta relationship is true for any individual security, it must also be true for any combination of securities. Therefore, the security market line is valid for both efficient portfolios and individual securities. Thus, in equilibrium, all securities and efficient portfolios must lie on the security market line (SML). But, only efficient portfolios lie on the capital market line (CML) because standard deviation is a measure of risk for efficiently diversified portfolios that are candidates for an investor’s overall portfolio.

Relaxing the Assumptions of the CAPM

In theory there is no difference between theory and practice. In practice there is.
Yogi Berra

The assumptions of CAPM are not realistic. Alternative models have been developed to relax many of the assumptions.

• Unique Risk-Free Rate. Violations of this assumption include situations where (1) no risk-free asset exists, (2) investors can lend but not borrow at the risk-free rate, and (3) investors can borrow at a rate higher than the risk-free rate. Fischer Black and others developed a models to address these situations.

• Common Investment Time Horizon. Relaxing this constraint by assuming that investor preferences are stable over the long-term and that changes in the risk-free rate and the distribution of returns on risky assets are assumed to be predictable. (You are replacing one unrealistic assumption with another.)

• No Transaction or Trading Costs. This is a proxy for liquidity. Illiquid securities have higher trader costs (bid-ask spreads). The investor with the longer time horizon can amortize the transaction costs of illiquid securities over a longer period. A liquidity premium can be added to the CAPM to account for the liquidity effect.

Sharpe Ratio

The goal of active portfolio management is to maximize the ex-ante Sharpe ratio. This ratio measures the amount of excess return per unit of risk taken. Excess return is E(r_P) – r_f, the difference between the portfolio’s expected return and the risk-free rate. Risk is measured as the portfolio’s standard deviation of return, _P.

S_P = (E(r_P) – r_f) / _P.

Notice that this ratio is the slope of the Capital Allocation Line (CAL). The active portfolio manager earns their fees by maximizing the slope of the CAL.

CAPM Exercises 1

The security market line can be used to determine the required return and the price of a stock, as the expected return for a risky asset is derived from the risk-free rate and the asset’s risk premium. The components of an asset’s risk premium are its beta and the prevailing market risk premium.

1. The expected return on the stock market is 10% with a standard deviation of 20%. The risk-free rate is 6%. The Acme Corporation common stock’s returns are 40% correlated with the stock market return and has a standard deviation of 80%. What is the expected return for Acme stock?

2. An investor has already calculated the betas for three stocks: β_A = .70, β_B = 1.00, β_C = 1.40. Using CAPM, what are the required rates of returns for these three stocks assuming the market risk premium is 6% and the risk-free rate is 4%?

3. The same three stocks as above are being considered for purchase. An investor has determined the following information:

Determine whether each stock is undervalued, overvalued, or properly valued.

The SML and Stock Selection

It is possible to determine, via conventional analysis (such as dividend discount models), the expected return on individual stocks and to relate these returns to the individual stock betas. A regression of expected stock returns on stock betas produces a security market line that represents the average risk-return relationship in the market. Stocks that plot above the security market line are those with above-average expected returns given their beta and are therefore attractive purchase candidates. Stocks that plot below the security market line are sale candidates.

It should be noted that in a perfectly efficient market, all stocks and all efficient portfolios should plot on the security market line. This is so because the act of buying the stocks that lie above the security market line will tend to raise their prices, thereby reducing their expected returns, while the process of selling the stocks that lie below the line will force their prices down and their expected returns up. In the end, the buying and selling of securities should cease when equilibrium is reached, that is, when all stocks lie on the SML.

CAPM Summary

• A single factor model

• The factor is usually “the market”.

• Based on several simplifying assumptions

• Premised on a linear risk-return trade-off.

• Attempts to relate the returns on an individual stock to the returns on the market (or index).

• An equilibrium model: the entire range of efficient portfolios can be shown to be expressible as the weight sum of only two portfolios.

Beta

• Beta is typically calculated by looking at past historical data.

• Can get an idea of beta “reliability” through the use of statistics (R²).

• Beta is often referred to as measuring a stock’s risk, but it is just a measure of association (a standardized correlation.)

• Theory tells us if markets are efficient, there must be a risk-return trade-off.

• Risk is always bad and return is always good. The proxy for risk is volatility.

• Excess returns are proportional to beta.

Myth: High Beta is “Bad”

• Imagine your portfolio has a beta of 2.45. Is this risky?

• If everyone “knows” the market will continue to skyrocket (grow at an annual rate of over 20%), then you would expect your portfolio to exhibit awesome returns (annual returns of over 49%).

• Where’s the risk?

Myth: Low Beta = Low Risk

• Consider a portfolio consisting of gold stocks.

• Historically, gold (as an asset) has had a low beta.

• This implied it will not move with “the market”,

• Gold volatility could be huge, yet this portfolio could exhibit a beta of .10.

• Can things be too uncorrelated?

Sources of Beta’s

• Value-line (5 year, weekly)

• S&P 500 (5 year, monthly)

• Merrill Lynch (5 year, monthly)

• BARRA (calculated from factor model covariance matrix)

• Bloomberg (historical 60 month, fundamental predicted forecast, both long term 5 year and short term 3 month)

• DataStream (5 year, monthly)

• Ibbotson (OLS, Sum, other see www.ibbotson.com)

Note: beta is not a constant. Below are two Bloomberg screens for Cisco’s beta for different periods.

Alpha

• When regressing a stock’s return against the market, the slope is beta and the y-intercept is alpha.

• A security is undervalued if its alpha is positive.

• Alternatively, a security is undervalued if it is located above the Security Market Line.

• Alpha is sometimes defined as the difference between a security’s implied return and its required return.

• Investment managers seek to construct portfolios with large alpha’s.

• The alpha for the portfolio is the weighted average of the alphas of the securities which constitute that portfolio.

• Alpha is sometimes called a portfolio’s net risk-adjusted return premium.

CAPM Exercises 2

1. You are planning to invest $100, with a portion in a risky asset and the rest in a risk-free asset. The risky asset has an expected return of 12% and a standard deviation of 15%, while the risk-free asset has an expected return of 5%. What allocation between these assets will result in a portfolio with an expected return of 9%? What will that portfolio’s variance be?

2. What is the expected return for each asset class in the following situation?

3. Assuming a market variance of .5, what is the beta of a stock with a covariance relative to the market 0.075?

4. The expected return on the stock market is 12% with a standard deviation of 37.4%. If the risk-free rate is 3%, calculate the expected return of a stock that has a covariance of 0.14 with the market return.

5. You hold the following portfolio:

If the stock market index produced a return that was 10% greater than the risk-free rate, how much would you expect your portfolio to outperform the risk-free rate?

6. A portfolio consists of $10,000 in bonds and $40,000 in stocks. The expected return on bonds is 5% and its standard deviation is 1%. The expected return on stocks is 12% and its standard deviation is 6%. Assuming that the bonds and stocks are uncorrelated, determine the standard deviation of this portfolio.

7. The correlation between hedge fund returns and common stocks is .25. The respective standard deviations are 15% and 13.4%. What is the covariance between these two asset classes?

8. Assume a portfolio is formed by investing equally in four assets: stock X, stock Y, the risk-free asset, and the market portfolio. The beta of stock X is 0.80 and the beta of stock Y is 1.60. What is the value of beta for the portfolio?

9. Given the following information determine the expected return on the XYZ mutual fund, using the capital market line model.

10. The S&P has a standard deviation of 18%. If stock ABC has a standard deviation of 15% and a correlation coefficient (r) of .6 with the S&P, what is the beta for the stock?