Dick Sweeney
A Review of Asset-Pricing Models and the CAPM
Asset pricing models are built on the idea that a handful of risk factors drive the overall economy. Examples of risk factors are the rate of return on the overall asset market, surprises to interest rates, exchange-rate surprises, inflation surprises, surprises in GDP growth. The
economy as a whole, or the economic system, cannot avoid these surprises. These surprises are systematic, meaning they apply to the economic system as a whole. The economy as a whole cannot diversify away these systematic risks. By definition, they are risks the system has to bear. Any one individual can avoid these systematic risks, but the economy as a whole must bear them —and the investors who bear these risks demand risk premia as compensation.
Individual firms are subject to these systematic risk factors. But many of the surprises that happen to individual firms are not due to these systematic risk factors. Instead, they are due to non-systematic risk, risk that can be diversified away. By definition, the non-systematic risk that affects a single firm is does not have to be borne by the economy as a whole. For the
economy as a whole, this risk is diversified away—it is diversifiable, not systematic risk.
These ideas are best understood in the CAPM, but they apply to more general asset-pricing models.
The Capital Asset Pricing Model
The CAPM assumes that the single systematic risk factor is the rate of return on the market. The CAPM explains expected rate of return on a share of Intel equity as equal to the risk-free rate (rf) plus the risk premium on Intel (RPI),
ERI = rf + RPI
or explains the expected excess rate of return as
ERI - rf = RPI.
In the CAPM, the risk premium is the product of Intel’s market beta (I) times the risk premium
on the market (ERM - rf), or
RPI = I (ERM - rf),
and thus
ERI - rf = I (ERM - rf).
[For example, if the market risk premium is 6% and Intel’s beta is 1.5, Intel’s risk premium is 9% = 1.5 x 6%. For equilibrium, Intel’s expected excess rate of return must be 9%. With a risk-free rate of 5%, this implies the expected rate of return on Intel’s shares must be 14%.]
The surprise in Intel’s rate of return is
Surprise = (beta x market surprise) + non-systematic error
RI - ERI = I (RM - ERM) + eI,
where eI is the non-systematic error. Of this surprise, the non-systematic error is diversified
away in a well-diversified portfolio. But the component I (RM - ERM) is the systematic risk that
cannot be diversified away. The CAPM says that competition drives down expected returns to the level where the investor is paid just enough to make bearing the risk worthwhile. The bare minimum risk premium that will induce a well-diversified investor to bear this market risk is I
(ERM - rf), just what the investor is paid.
[For example, if the expected rate of return on the market is 11% and the actual rate of return on the market is 13%, then the market surprise is 2%. Because Intel’s beta is 1.5, this 2% market surprise contributes and extra 3% to Intel’s return above the expected rate. The best guess about the non-systematic error is zero. But if the error is in fact 8%, then the total surprise is 11%, 3% from the market surprise plus 8% from non-systematic risk. The contribution of Intel to the total surprise in a well-diversified portfolio is only the 3% from the market surprise—the
non-systematic surprise is diversified away. The risk premium on Intel is 9%, and is just adequate to compensate a well-diversified investor for the risk from market surprises that Intel is exposed to.]
The CAPM assumes that the marginal investor, the investor whose decisions dominate asset markets, understands the difference between systematic and non-systematic risk, and can borrow and lend at the risk-free rate. If the investor sees an opportunity where say
(ERI - rf) > I (ERM - rf),
or the expected excess rate of return is larger than the risk premium, she will buy Intel, drive up its price, and stop only when Intel becomes so expensive that ERI - rf = I (ERM - rf). Thus, on
average, ERI - rf = I (ERM - rf). Of course, investors can be wrong about ERI and I, but the
investor who catches on first will be ahead of the herd and will profit when the herd catches on.
The CAPM thus plays two roles. As an equilibrium model, it tells the expected rate of return that holds when competition among investors drives markets to equilibrium. But it is also a model that describes how the investor chooses assets in hopes of getting a larger rate of return than the bare minimum that just compensates for risk. Investors looking for these opportunities provide the competitive forces that eventually drive the market to equilibrium.
When the investor is evaluating a project or a stock, she forms a guess about its expected excess rate of return, and she compares it with the risk premium. The risk premium gives the required excess rate of return,
RRI - rf = I (ERM - rf).
where RRI is the required rate of return on Intel’s equity. In equilibrium, of course, investors
will bid up or down the value of an asset until they drive ERI to be equal to RRI. But before this
happens, ERI >/< RRI = rf + I (ERM - rf). The investor wants to take those projects (buy those
assets) where ERI > RRI and reject those where ERI < RRI and is indifferent when ERI = RRI.
The point of taking a project where ERI > RRI is that other investors will catch on and drive its
price up until ERI fall to the point where ERI = RRI = I (ERM - rf). This equilibrium condition
ends up holding on average as an unintended result of the buy-sell decisions of investors who understand the difference between systematic and non-systematic risk.
[For example, the investor considers a project that has an internal rate of return of 15%, or with a risk-free rate of 5%, has an expected excess rate of return of 10%. The investor thinks this project has average market risk, or its = 1. With a risk premium on the market of 6% = (ERM -
rf), the project has a required excess rate of return of RR - rf = 6% = (ERM - rf). But the project
has ER - rf = 10% > RR - rf = (ERM - rf) = 6%. The investor knows that if she is correct in her
analysis, when the market catches on to how great this project is, the market value will be so high that it drives down the internal rate of return to 11%. When the internal rate of return is 11%, the expected excess rate of return is 6% so that ER - rf = 6% = RR - rf = (ERM - rf).]
These considerations show up in two diagrams. In the first, the marginal investor examines an efficient frontier found from optimal combinations of all the risky assets available in the market. She combines the efficient frontier with the risk-free rate by finding a straight line that goes through the risk-free rate and is tangent to the EF. The investor can choose any position on or below this line—the Capital Market Line—and of course chooses to be on this line. In a CAPM world, the tangency portfolio is the market portfolio: the investor chooses a portfolio on the CML that is a weighted average of investments in the risk-free asset and in the overall market.
Given that the investor is on a CML, the Security Market Line follows mathematically (many advanced corporate finance books provide the derivation). The equation of the SML is just the standard CAPM equation,
ERj - rf = j (ERM - rf).
Given the asset’s j, equilibrium means the expected excess rate of return ERj - rf must have a
value that makes ERj - rf = j (ERM - rf) hold. If the investor stumbles on an asset or project with
ERj above the SML and thus with (ERj - rf) > j (ERM - rf) or ERj > RRj, this project is a good
one. Acceptable projects are above the SML or on it (borderline), unacceptable projects are below the SML.
Multi-factor Asset-Pricing Models
Sweeney and Warga (1986) examine a model with two risk factors, the rate of return on the market and the change in long-term government bond yields. In this model, the expected excess rate of return on Consolidated Edison is
ERC - rf = M,C (ERM - rf) + I,C RPI.
There are two betas for Con Ed, the first on the market and the second on yield changes, M,C and
I,C. The premium on market risk is (ERM - rf) as in the CAPM, and the premium on the risk of
yield changes is RPI. The risk premium on the market is say 6% as above, and the yield risk
premium is say -1%. (How can a risk premium be negative?! Stay tuned for more.) Con Ed’s market beta is say 0.5 and its beta on yield changes is -1.0—increases in government bond yields are associated with declines in Con Ed’s stock price. Then,
ERC - rf = M,C (ERM - rf) + I,C RPI. = 0.5 (6%) - 1.0 (-1%) = 3.0% + 1% = 4.0%.
With rf = 5%, the expected rate of return on Con Ed is 9.0% = 5% + 4.0%. Because Con Ed is
subject to yield-change risk, then its required rate of return is 100 basis points higher than if only its market risk is considered.
Sweeney and Warga point out that the betas on yield-change risk are measures of
sensitivity above or below the average. The average firm may well be hurt by yield changes, but this average effect shows up in the market and is captured in the firm’s market beta. They show that the average of the yield-change betas has to be zero—the average market beta has to be 1.0, but the average beta on non-market, systematic risk factors has to be 0.0. Empirically, they find that the majority of industries and firms show little sensitivity to yield changes. The big
exception, they find, is the utility industry; others have also reported that financial firms show sensitivity. Utilities in the U.S. use exceptionally high debt ratios. When the yield on government bond rises, share prices of utilities go down on average.
As Sweeney and Warga argue, many firms and industries have betas on non-market risk factors that are very small and are not significantly different from zero. In these cases, the non-market systematic risk factors can be ignored without materially affecting the estimate for the required (and in equilibrium, the expected) rate of return. Similarly, for many projects, the required rate of return can be adequately estimated without considering non-market risk factors. The Con Ed example shows that in some cases, ignoring yield-change risk can cause serious mistakes. Similarly, for some firms or projects, ignoring exchange-rate risk may lead to serious error.
It is worthwhile to check for sensitivity to non-market factors. But it is not necessary to apply a general model in all cases—in many cases, focusing on market risk, as in the CAPM, is adequate. In fact, given that estimates of betas on non-market risk factors can be fairly
