BKM Chapter 9: The Capital Asset Pricing Model
Key Notation
Notation for the CAPM Extensions is listed within the notes for that section, because it’s
generally not used elsewhere and can be conflicting.
Key Formulas
CAPM Formulas
βi = Cov(ri,RM)/ σM2
E(ri)-rf = βi [E(rM) – rf]
2
Market Price of Risk = [E(rM) – rf] / σM
Stock Contribution to Market Variance = wi Cov(ri,rM)
E(rM) – rf = A* σM2, where A* is the average risk aversion for all investors
Formulas for the CAPM extensions are discussed with their notation in the other notes section.
Key Definitions
Homogenous Expectations – investors come up with the same input list for the Markowitz
selection model because they have the same views on risks and expected returns.
Market Portfolio – Portfolio that holds all traded assets in proportion to their market
capitalization
Security Market Line – Graph of E(r) vs. β, shows the returns available to investors according to
CAPM. The slope of the SML is the market risk premium.
Zero-beta Portfolio – A companion policy on the inefficient part of the frontier that is
uncorrelated with a portfolio on the efficient frontier.
Liquidity – the speed and ease an asset can be traded for fair market value
Illiquidity – measured by the discount an investor must take off fair market value to sell an
asset quickly
Other Notes
CAPM Assumptions
1. There are many investors with wealth that is small compared to total wealth. Investors are
price-takers; security prices are unaffected by their own trades.
2. All investors plan for an identical, single holding period.
3. Investments are limited to publicly traded financial assets and risk-free borrowing and
lending. Investors can borrow or lend any amount at the risk-free rate.
4. No taxes on returns and no transaction costs.
5. All investors are rational mean-variance optimizers. All use the Markowitz portfolio selection
model.
6. Homogeneous Expectations assumption.
CAPM Implications
1. All investors will choose to hold a portfolio of risky assets in proportions identical to the
market portfolio.
2. The Market Portfolio M is the optimal risky portfolio. The Capital Market Line is the best
attainable Capital Allocation Line. Investors only differ in their risk aversion, and therefore the
weight they put on the risk-free asset.
3. The Risk Premium on M is proportional to market risk and average degree of risk aversion.
4. An individual asset risk premium is proportional to the market risk premium and the assets’
beta.
These implications mean that the passive strategy is efficient, and that there is no benefit to
security analysis. In reality, this is not true, however, it does give investors who do not wish to
do/pay for security analysis a starting point of assuming all stocks follow the expected returnbeta relationship.
The expected-return beta relationship holds for any combination of assets, as long as it holds
for all assets individually, so we can write:
E(rP)-rf = βP [E(rM) – rf]
If we perform a security analysis to get expected returns, then plot those expected returns on
the same graph as the SML, overpriced stocks will plot below the SML, and underpriced stock
will plot above the SML. The vertical distance is the stock’s alpha.
Implementation of CAPM
Actual vs Expected Returns
CAPM makes projections about the expected returns, which can not be measured or observed.
We can only measure the realized returns after they have happened. Therefore, we must make
additional assumptions to use and test CAPM.
CAPM vs Index Model
The Index Model from Chapter 8 uses the formula Ri = αi + βiRM + ei, where the βi is the same as
in the CAPM. CAPM therefore predicts that αi = 0 for all stocks. Again, this is a projection on
expected returns, so realized alphas should have the following properties:
1) Average out to 0 over a large sample
2) The alphas should be unpredictable.
Long term tests of mutual funds show their average alphas tend to be near zero, which is
evidence for the CAPM.
Testing the CAPM
Normative Tests test the validity of the assumptions. Few, if any, models would pass these
types of tests, because we can’t solve models that accurately reflect the real world. Most of
the CAPM assumptions are clearly violated.
Positive Tests test the robustness of the predictions when the assumptions are violated. They
ask if the predictions are still reasonable under typical real world situations.
Predictions of the CAPM
1. Market Portfolio is efficient
This is not testable, because the market portfolio includes foreign assests, private
businesses, and human capital, all of which are illiquid or non publicly traded.
2. The Security Market Lines describes the risk-return tradeoff for all assets.
CAPM fails these tests, as there are clearly some classes of stocks where α<>0.
Extensions of the CAPM
There are additional adjustments people have tried to make to the CAPM to account for the
non-zero alphas. 5 of these are discussed below.
Zero Beta Model
1) A combination of two efficient frontier portfolios (P & Q) is on the efficient frontier
2) Any asset return can be expressed as linear combination of P & Q:
E(ri) – E(rQ) = [E(rP) – E(rQ)] [ Cov(ri,rP) – Cov(rP,rQ)] / [σP2 - Cov (rP,rQ)]
3) Every portfolio on the efficient frontier has a zero-beta portfolio on the inefficient frontier
with which it is uncorrelated. Let Z be the zero-beta portfolio of M:
E(ri) – E(rZ) = [E(RM) – E(RZ)] [ Cov(ri,rM) – Cov(rM,rZ)] / [σM2 - Cov (rM,rZ)]
and since M and Z are uncorrelated, we have
E(ri) – E(rZ) = βi [E(RM) – E(RZ)]
This is the CAPM equation that results if investors face restrictions on borrowing and/or lending
at the risk-free rate. This model does not save CAPM from empirical rejection.
Labor Income & Non-traded Assets
There are two key types of non-traded assets:
1. Human Capital/Labor Income
2. Privately held businesses
Privately held businesses are not that big of a problem for CAPM. They can be sold at will (with
some illiquidity issues) and their value can be borrowed against.
Labor income is more problematic, because borrowing against it is typically limited to things
like a home mortgage, and it can only be partially hedged by something like life insurance. It is
not portable through time, and it is difficult to hedge with traded securities. These issues
should cause the demand for labor-intensive stocks to be lower, which would generate positive
alphas in the normal CAPM.
This gives a new CAPM equation:
E(Ri) = E(RM) { [Cov(Ri,RM) + PH/PM Cov(Ri,RH)] / [σM2 + PH/PM Cov(RM,RH)]}
Multiperiod Model (ICAPM)
In a multiperiod model, there are additional sources of risk to consider:
1. Changes in the parameters of investment opportunity (risk free rates, expected
returns, risk of the market portfolio)
2. Inflation Risks
If we can identify K portfolios that hedge against K different extra-market risks, then CAPM
becomes:
E(Ri) = βiM E(RM) + Σ βikE(Rk) where k is the kth hedging portfolio
Consumption CAPM
Investors value additional income more during poor economic times than affluent times. They
will consider assets riskier if they are highly correlated with consumption growth.
E(Ri) = βiC E(RC) where C is a consumption tracking portfolio.
This model has more success predicting returns than traditional CAPM.
Liquidity Adjusted CAPM
E(Ri) = kE(Ci) + λ (β + βL1 – βL2 – βL3)
Where:
k = adjustment for average holding periods
E(Ci) = expected cost of illiquidity
λ = E(RM-CM) = market returns net of liquidity costs
β = measure of systematic market risk
βL1 = measures security illiquidity sensitivity to market illiquidity
βL2 = measures security return sensitivity to market illiquidity
βL3 = measures security illiquidity sensitivity to market return