CAPM and the security market line explained

CM2 · Financial Engineering & Loss Reserving · 11 min read

The idea in one sentence

An investor is paid for taking risk that cannot be avoided, and not paid for taking risk that could have been diversified away for free.

That sentence is the whole of the Capital Asset Pricing Model, and everything else is bookkeeping. If some of an asset's risk disappears when you hold it alongside everything else, then nobody needs to be compensated for bearing it, because nobody has to bear it. Only the part that survives diversification — the part that moves with the market as a whole — earns a return.

Why one number is enough

Mean-variance portfolio theory, which comes just before this in the syllabus, ends with an awkward result: to choose a portfolio you need every variance and every covariance between every pair of assets. For 500 assets that is over 124,000 covariances, and they must all be estimated.

The Capital Asset Pricing Model escapes that by adding an assumption: every investor holds the same risky portfolio, the market portfolio, differing only in how much risk-free asset they hold alongside it. If that is true, then the only thing that matters about an individual asset is how it behaves relative to the market, and that is one number per asset rather than one per pair.

The number is beta:

β_i = Cov(R_i, R_M) / Var(R_M) = ρ_i,M · σ_i / σ_M

In words: beta is the sensitivity of the asset's return to the market's return. A beta of 1.6 says that when the market moves one percentage point, this asset tends to move 1.6.

The second form of the formula is the useful one in a calculation, because questions usually give a correlation and two standard deviations rather than a covariance. The two forms are the same thing: Cov = ρ σ_i σ_M, and dividing by σ_M² leaves one σ_M underneath.

The security market line

E[R_i] = r_f + β_i(E[R_M] − r_f)

In words: the expected return on an asset is the risk-free rate, plus its beta multiplied by the extra return the market as a whole is expected to earn over the risk-free rate.

The bracket is the market risk premium, and it is the price of one unit of beta. Read the equation as a straight line with beta on the horizontal axis: it starts at the risk-free rate where beta is zero and rises at a constant slope. Every asset, and every portfolio of assets, must sit on that line if the model holds.

Three points on the line are worth fixing in your head, because they are free marks and they anchor everything else:

Beta zero gives the risk-free rate. An asset uncorrelated with the market earns no risk premium at all, however volatile it is on its own.
Beta one gives the market return. By definition the market portfolio has a beta of one against itself.
Beta above one gives more than the market, and beta below one less.

The security market line is not the capital market line

These two are confused constantly, and the difference is one word.

The capital market line plots expected return against total risk, measured by standard deviation. Only efficient portfolios sit on it. Its slope is the market price of risk.
The security market line plots expected return against systematic risk, measured by beta. Every asset sits on it, efficient or not.

The reason an individual asset lies on one and below the other is exactly the point of the model. It carries specific risk that the market does not pay for, so it earns less than its total risk would suggest — below the capital market line — while still earning precisely what its systematic risk deserves — on the security market line.

The slope of the capital market line has a name of its own:

market price of risk = (E[R_M] − r_f) / σ_M

In words: the extra expected return per unit of standard deviation available from the market. It is the exchange rate between risk and return in this market, and it is the figure a question means when it asks how much reward is on offer for taking risk.

Splitting an asset's risk in two

A single-index model writes the return on an asset as a market part plus a part specific to the asset. Taking variances, and using the assumption that the specific part is uncorrelated with the market:

σ_i² = β_i²σ_M² + σ_ε,i²
(total) = (systematic) + (specific)

The proportion of an asset's variance that is systematic turns out to be ρ², the squared correlation with the market. That identity is one of the cleanest checks available in this topic: compute the systematic share from betas and variances, compute the correlation, square it, and the two must match. The worked example below does exactly that.

What the model assumes, and what goes wrong

The assumptions are strong, and a question asking for limitations is asking you to pick the ones that matter rather than recite all of them.

All investors hold the market portfolio. Nobody does. Real portfolios are concentrated in a home market and a few sectors, so the relevant beta is not the beta of the model.
One period, one risk-free rate, borrowing and lending at it. Investors borrow at more than they lend, which bends the line.
No taxes or transaction costs. Both change what an investor nets, and taxes differ between investors, so there is no single optimal portfolio for everyone.
Everyone agrees on the inputs. If investors held the same expectations there would be far less trading than there is.
The market portfolio is unobservable. This is the one with practical bite. A real calculation substitutes an index, which is not the market, and beta is then measured relative to the substitute.
Beta is estimated from the past. It is estimated with error, and it moves over time with the company's leverage and business mix, so a historical beta is a forecast and should be treated as one.

Where marks are lost

Multiplying the market return by beta instead of the market risk premium. The line is r_f + β(E[R_M] − r_f), not β · E[R_M]. The second form gives the risk-free asset a return of zero and an asset with beta one the right answer, so it can survive a careless check.

Forgetting that portfolio betas are value-weighted. The beta of a portfolio is the weighted average of its constituents' betas, weighted by market value, not equally. Equal weighting is the most frequent slip in a question that gives market values, and the value-weighted beta of the whole market must come to exactly 1.

Using total risk where systematic risk belongs. An asset's own standard deviation does not appear anywhere in the security market line. If a question gives you one, it is either for a beta calculation through the correlation, or for a variance split, or as a distractor.

Confusing the two lines. Covered above, and worth checking before writing a sentence about either.

Treating a positive specific return as evidence against the model. An asset outperforming its security market line prediction over one year says almost nothing. The model is about expectations, and specific risk is exactly the part that is expected to average out and does not, over any particular year.

Questions students ask

Can beta be negative?

Yes, and the model handles it without modification: a negative beta gives an expected return below the risk-free rate. That is not a mistake. An asset that reliably rises when the market falls is valuable insurance, and an investor will accept a low expected return to hold it. Gold is often quoted as an approximate example. Such assets are rare, which is itself the reason they are prized.

If the market portfolio cannot be observed, is the model useless?

No, but it has to be used with that limitation in view. It is still the clearest available statement of the idea that diversifiable risk is not rewarded, and it still gives a defensible discount rate when an index is used as a proxy. What it cannot do is be tested cleanly, because a test of the model using an index is also a test of the index.

Why does specific risk earn nothing when it is clearly risk?

Because it can be removed at no cost. Hold enough assets and the specific parts largely cancel, since they are unrelated to one another. An investor who holds a single share is bearing specific risk by choice, and the market does not pay people for choosing not to diversify. Whether that is quite true of a real investor who cannot afford to diversify is a fair criticism of the model, not of the logic.

How is this different from a multifactor model?

Only in the number of things an asset is allowed to be sensitive to. The Capital Asset Pricing Model allows one source of systematic risk; a multifactor model allows several — an index, interest rates, inflation, a sector — each with its own beta and its own risk premium. The structure of the equation is unchanged: expected return is the risk-free rate plus a sum of sensitivities multiplied by premiums. More factors fit history better and need more parameters estimated, which is the trade-off every factor model lives with.

A worked example

A market contains three risky assets and a risk-free asset. The risk-free rate is 4% a year. The standard deviation of the market portfolio's return is 10%.

All the figures below are invented for this example. They are not real companies and not a real market.

Asset	Market value (₹ crore)	Beta	Expected return
A	300	0.60	7.0%
B	500	?	9.0%
C	200	1.60	?

It is also known that asset A has a standard deviation of return of 15% and a correlation with the market portfolio of 0.40.

Step 1: the expected return on the market

Asset A sits on the security market line, so one equation in one unknown:

7.0 = 4.0 + 0.60 × (E[R_M] − 4.0)
3.0 = 0.60 × (E[R_M] − 4.0)
E[R_M] − 4.0 = 5.0, so E[R_M] = 9.0%

The market risk premium is 5.0 percentage points, and that is the slope of the line.

Step 2: the beta of asset B

9.0 = 4.0 + β_B × 5.0, so β_B = 1.00

Which is the answer you should have expected before dividing: asset B's expected return equals the market's, and the only point on the line where that happens is beta 1.

Step 3: the expected return on asset C

E[R_C] = 4.0 + 1.60 × 5.0 = 12.0%

Step 4: the market price of risk

(9.0 − 4.0) / 10.0 = 0.50

Half a percentage point of extra expected return for each percentage point of market standard deviation borne.

The first check: the market's beta must be exactly 1

The three assets are the market here, so their value-weighted beta has to come to 1. Total market value is ₹1,000 crore, giving weights of 0.30, 0.50 and 0.20.

β_M = 0.30(0.60) + 0.50(1.00) + 0.20(1.60)
= 0.18 + 0.50 + 0.32 = 1.00

Exactly 1. Had it come to 1.22, one of the three betas would be wrong and nothing downstream could be trusted.

The second check: the value-weighted return must be the market return

The same weights applied to the three expected returns must reproduce the 9.0% found in step 1 — and this is a genuinely independent route, because step 1 used only asset A.

0.30(7.0) + 0.50(9.0) + 0.20(12.0) = 2.10 + 4.50 + 2.40 = 9.0%

It does. Two checks, each using numbers the original calculation did not, both landing exactly.

The third check: beta from the covariance

Asset A's beta was given, but it can be rebuilt from the standard deviation and the correlation:

β_A = ρ_A,M · σ_A / σ_M = 0.40 × 15 / 10 = 0.60

Or the long way, through the covariance: Cov(A, M) = 0.40 × 15 × 10 = 60, and 60 / 10² = 0.60. The same 0.60, which confirms that the correlation, the standard deviation and the beta in the question are mutually consistent.

Splitting asset A's risk

total variance = 15² = 225
systematic variance = β²σ_M² = 0.60² × 10² = 0.36 × 100 = 36
specific variance = 225 − 36 = 189

So only 36 / 225 = 16% of asset A's variance is systematic. The remaining 84 per cent is specific, earns nothing, and vanishes in a diversified portfolio. Check it against the identity from the body of this page:

ρ² = 0.40² = 0.16

16% both ways. That is why holding asset A on its own is a poor decision even though its expected return is exactly fair: the investor bears 225 units of variance and is paid for 36 of them.

What the answer tells you

Asset C expects 12% against asset A's 7%, and the entire five-point difference is explained by beta, with no reference to either asset's own volatility. Someone who knew only that asset A has a 15% standard deviation and asset C a smaller one would have predicted the ranking backwards. That is the single most useful habit this model teaches: when you are asked what return an asset should earn, look at how it moves with everything else, not at how much it moves.

Where this fits in CM2

This page explains one idea from CM2 Financial Engineering & Loss Reserving. The full CM2 syllabus, with what each part is examined on, sits on the course page, and past-paper style mock exams with full solutions are where you find out whether the idea has stuck under time pressure.

CAPM and the security market line: beta and the market price of risk