BKM Chapter 9 - The Capital Asset Pricing Model

Question 1

Capital Market Line

Answer 1

When we combine the market portfolio with the risk free asset, our restuling return lie along the capital market line. It goes through the risk free rate on the y-axis and the market portfolio point (σ_M, r_M)

Question 2

Risk Premium of Market Portfolio

Answer 2

Each individual investor chooses a proportion y, allocated to the optimal portfolio M, such that y = E(Rm)/Aσ_m²

Since investors can lend and borrow at the risk-free rate, any borrowing position must be offset by a lending position and this results in a net borrowing and lending position of zero across all investors. Thus, the average position in the risky portfolio is 100%, or y = 1.

Setting y = 1 in the equation above, the risk premium on the market portfolio is related to its
variance by the average degree of risk aversion E(Rm) = A(bar)σ_m²

Question 3

Market Price of Risk

Answer 3

The reward to risk ratio for investment in the market portfolio is E(R_M)/σ_M²
and is known as the market price of risk because it quantifies the extra return demanded to bear portfolio risk.

Question 4

Deriving CAPM, Simple Method

Answer 4

we know that any complete portfolio that investors would be willing to own (i.e. efficient portfolios) must have an expected return along the line connecting the risk free rate and the market portfolio—the Capital Market Line. Using basic point slope form, this line can be written as:

E(r_c) = r_f + { [E(r_m) - r_f]/σ_M }*σ_c

                                           = r<sub>f</sub> + Market Reward to Risk Ratio \* Amount of Risk

Note that the formula for the expected return as a linear function of the market price of risk and the asset’s standard deviation does NOT apply to all stocks or portfolios. It only applies to the efficient portfolios that lie along the Capital Market Line.

We cannot do this because for individual stocks the total standard deviation „i… is not a good measure of risk. For investors with large portfolios, their risk depends on the covariances, or the systematic risk, not the total risk of the stocks in the portfolio. Therefore, what we would prefer to use is a measure of the marginal risk that any one stock would add to our existing portfolio. This means we need to find the derivative of the portfolio risk with respect to the
weight we put on stock i.

Remember that if we already assume that investors own the market portfolio, then their total portfolio variance can be written as:

σ²_M = Σw_i²σ_i² + ΣΣw_iw_jσ_ij

What is the marginal increase in risk from a small change in the weight for security i? Defining risk as the standard deviation of returns, then the marginal risk is given by taking derivative of σ_M w.r.t. w_i

Marginal Risk = { w_iσ_i² + Σw_jσ_ij } / σ_M = Cov(r_i, r_m)/σ_M

This should be a more appropriate risk measure for an individual stock, so we can plug that in as the amount of risk in the Capital Market Line equation to get:

E(r_i) = rf + { {E(r_m)-r_f}/σ_M }*{ Cov(r_i, r_m)/σ_M }

Define ß = Cov(r_i, r_m)/σ_M²,

E(r_i) = r_f + ß * [E(r_m)-r_f] ⇒ Capital Asset Pricing Model

Question 5

Security Market Line

Answer 5

The graph of the CAPM equation is called the Security Market Line (SML).

If a stock’s expected return-beta relationship does not lie along the SML, then we say that it has a non-zero alpha, which is the difference between the fair and actual expected returns. If assets are priced fairly, then they will lie along the SML. If their prices are too high, their expected returns will be too low and will lie below the line; if their prices are too low then they will lie above the SML. This is an important perspective since it suggests that the goal of portfolio management should be to identify assets with positive alpha.

Question 6

CAPM Assumptions

Answer 6

1. There are many investors, each with wealth that is small compared to the total wealth of all invesots
2. All investors have the same identical holding period
3. Investors are limited to the universe of publicly traded finacial assets and risk-free borrowing and lending arragement.
4. Investors pay no taxes on returns and incur no fees on trades in securities
5. All investors are rational mean-variance optimizers, meaning they all use the Markowitz portfolio selection
6. All investors analyze securities in the same way and share the same economic view of the world.

Question 7

Extensions of CAPM - Zero Beta CAPM

Answer 7

key assumptions in the standard CAPM that are modified: investors cannot borrow or lend at the risk-free rate, which leads to a model similar in form to CAPM but with a zero-beta portfolio used in place of the risk-free rate.

Zero beta portfolio: Every portfolio on the efficient frontier has a “companion” portfolio on the bottom half (the inefficient part) of the minimum-variance frontier with which it is uncorrelated. The companion portfolio is the zero-beta portfolio of the efficient portfolio.

The key result: E(r_i) = E(r_z) + ß [E(r_m) - E(r_z)}

Notice that E(r_z) > E(r_f). This will produce a SML with a higher intercept and a lower slop. This could make it appear as though low beta stocks have positive alphas and high beta stocks have negative alphas.

Question 8

Extentions of CAPM - Labor Income and Non-traded Assets

Answer 8

The standard CAPM assumes that the market portfolio contains all risky assets, which rules out the existence of future labor income (earnings from your job, referred to as human capital) or privately held businesses (non-traded assets).

Including those in the model means that investors will avoid holding risky assets that are highly correlated with their own individual non-traded assets and/or labor income. This has the effect of increasing the betas of low-beta stocks and decreasing the betas of high-beta stocks, which eliminates that apparent positive alphas for low-beta stocks and negative alphas for high-beta stocks.

Question 9

Extensions of CAPM - Intertemporal CAPM (ICAPM)

Answer 9

The ICAPM assumes that investors care about risk and return over multiple periods and recognizes the betas, market risk premiums and other factors that could impact their consumption over time can change, which is itself another source of risk.

This cause investors to bid up the prices of assets that can be used to hedge against this risk. Similarly, they pay less for assets that are correlated with these risks and demand higher expected returns.

This results in a model in which returns are a function of betas and risk premiums for various risk factors besides just the market risk.

Question 10

Extensions of CAPM - Consumption CAPM

Answer 10

While CAPM assumes that the only thing investors care about is their end of period wealth and that this is driven entirely by the returns on the market portfolio, the Consumption CAPM focuses directly on the real reason people want to accumulate wealth — to buy things.

Returns on risky assets will then depend on how they are correlated with consumption instead of market returns. The Consumption CAPM is really just a way of collapsing the multiple factors in the ICAPM into a single factor.

Question 11

Extensions of CAPM - Liquity Adjustment

Define Illiquidity Premium

Answer 11

Investors prefer more liquid assets with lower transaction costs, so all else equal, investors require higher expected returns from relatively illiquid assets. Thus an illiquidity premium must be impounded into the price of each asset.

Question 12

Extensions of CAPM - Liquity Adjustment

Answer 12

One such model discussed in the text adds liquidity considerations to the standard CAPM in several ways.

First, it includes the expected cost of liquidity as a component of the expected return, with an adjustment to reflect the amortization of this cost over the average holding period.

Second, it measures the market risk premium net of the average market liquidity premium.

And third, it adds three additional liquidity betas (with different signs) to reflect the following sources of systematic liquidity risk:

1. Sensitivity of the security’s illiquidity to market illiquidity — This reflects the fact that investors will want more compensation for liquidity risk if the security becomes illiquid when general liquidity is low.
2. Sensitivity of the stock’s return to market illiquidity — This reflects the fact that investors will accept a lower expected return on stocks whose returns are higher when market illiquidity is greater.
3. Sensitivity of the security illiquidity to the market’s rate of return — This reflects the fact that investors will expect a lower expected return on securities that can be sold more readily when the market declines and they are poorer (need more liquidity).
4. Overall adjusted ß = Cov (r-c, r_m-c_m)/Var(r_m-c_m)