CAPM, beta and the security market line · RiskAtoms Academy

By the end you will be able to

1Derive the capital asset pricing model from the tangency portfolio and the capital market line, and use it to find an expected return.
2Calculate beta as cov(Ri, Rm) ÷ var(Rm) and read what it says.
3Split an asset's variance into systematic and specific parts, and explain why only the systematic part is rewarded.
4Use the security market line to judge whether an asset is underpriced or overpriced, and contrast it with the capital market line.

Why it matters

The capital market line tells you how to build the best portfolio. It does not tell you what return to expect from one share, one bond or one new project. CAPM does, with a single number per asset: its beta. Sixty years on it is still the first answer most firms give to “what return should we demand?”

Core explanation

From the best portfolio to the price of every asset

L3.1.08: The risk-free asset and the capital market line showed that with a risk-free asset, every investor's best portfolio is a mix of the risk-free asset and one tangency portfolio, along the capital market line (CML). Now add one more assumption: all investors agree on expected returns, volatilities and correlations. Then they all hold the same tangency portfolio. Since together they hold every risky asset, the tangency portfolio must be the market portfolio, M.

The question for one asset. What matters about a single asset is not its own volatility but how much it adds to the risk of the market portfolio everyone holds. That contribution is its covariance with the market (L3.1.06: Many assets and the covariance matrix). Scaled by the market's variance, it is beta, which you met as a sensitivity in L2.5.04: Sensitivities.

The capital asset pricing model

\mathbb{E}[R_i] = r_f + \beta_i\,\big(\mathbb{E}[R_m] - r_f\big), \qquad \beta_i = \frac{\operatorname{cov}(R_i, R_m)}{\operatorname{var}(R_m)}

Expected return = risk-free rate + beta × market risk premium. Plotted against beta it is a straight line, the security market line (SML).

Two kinds of risk. Write an asset's return as a market part plus its own news, $R_i = \alpha_i + \beta_i R_m + \varepsilon_i$ . Its variance splits in two:

\sigma_i^2 = \underbrace{\beta_i^2\,\sigma_m^2}_{\text{systematic}} + \underbrace{\sigma_\varepsilon^2}_{\text{specific}}

Specific risk cancels out in a large portfolio, so an investor can be rid of it for nothing. A market will not pay a premium for a risk that anyone can remove for free. Only systematic risk, measured by beta, earns a premium.

ExploreDiversify away the specific risk

Equal amounts in n shares, each with β = 1 and specific volatility 25%, their specific news independent. Market volatility 16%.

Number of shares n1

■ systematic variance β²σm² · ■ specific variance σε²/n

Portfolio volatility29.68%

Systematic variance0.0256

Specific variance0.0625

Specific share70.9%

With one share the portfolio's volatility is 29.68%; with 10 shares 17.85%; with 100, 16.19%. It never falls below the market's 16%: that floor is systematic risk.

The SML against the CML. Both lines start at the risk-free rate and pass through the market, but they plot different things and hold for different assets:

	Capital market line	Security market line
Across	Total volatility σ	Beta β
Slope	Market Sharpe ratio (E[Rm] − rf) ÷ σm	Market risk premium E[Rm] − rf
Who lies on it	Efficient portfolios only	Every asset and portfolio (if CAPM holds)
Single shares	Below the line: their specific risk is not paid	On the line
Use	Choosing how much risk to take	Pricing one asset; spotting mispricing

Under- and overpriced. If analysts forecast a return above the SML, the asset offers more than its beta requires: its alpha is positive and it is cheap. Investors buy, its price rises, and its expected return falls back to the line. Below the line, the reverse.

Sort itAbove, on or below the SML?

Risk-free rate 3%, market expected return 7%. Where does each forecast sit?

β = 1.0, forecast 7%.
β = 0.5, forecast 6%.
β = 2.0, forecast 10%.
β = 0, forecast 3%.
β = −0.5, forecast 2%.

The formal version

Derivation 1: from the tangency condition. Recall from L3.1.08: The risk-free asset and the capital market line that the tangency portfolio solves $\mathbf{w}_T \propto \Sigma^{-1}(\boldsymbol\mu - r_f\mathbf 1)$ . Write the constant explicitly and multiply both sides by $\Sigma$ :

\begin{aligned} \mathbf{w}_T &= \frac{1}{k}\,\Sigma^{-1}(\boldsymbol\mu - r_f\mathbf 1), \qquad k = \mathbf 1'\Sigma^{-1}(\boldsymbol\mu - r_f\mathbf 1) \\[4pt] \boldsymbol\mu - r_f\mathbf 1 &= k\,\Sigma\,\mathbf{w}_T \end{aligned}

Row $i$ of $\Sigma\mathbf w_T$ is $\sum_j \sigma_{ij} w_{T,j} = \operatorname{cov}(R_i, R_T)$ , the covariance of asset $i$ with the portfolio (L3.1.06: Many assets and the covariance matrix). So each asset satisfies one equation; weighting those equations by $w_{T,i}$ and adding gives one more for the portfolio itself:

\begin{aligned} \mu_i - r_f &= k\,\operatorname{cov}(R_i, R_T) && \text{for each asset } i \\ \textstyle\sum_i w_{T,i}(\mu_i - r_f) = \mu_T - r_f &= k\,\mathbf w_T'\Sigma\,\mathbf w_T = k\,\sigma_T^2 && \text{weighting by } w_{T,i} \\[4pt] \frac{\mu_i - r_f}{\mu_T - r_f} &= \frac{\operatorname{cov}(R_i, R_T)}{\sigma_T^2} = \beta_{i} && \text{dividing} \\[4pt] \mu_i &= r_f + \beta_i\,(\mu_T - r_f) \end{aligned}

This is an identity: with respect to the tangency portfolio, every asset lies on a line in beta. The economics comes in one step: if all investors hold $T$ , then $T = M$ , and the line is CAPM. Note that $\sum_i w_{T,i}\beta_i = 1$ : the market's beta with itself is 1.

Illustrative dataRiskAtoms Group's investment portfolio

The group invests £1bn of surplus assets across five asset classes. The expected returns, volatilities and correlations are round, made-up long-run assumptions in pounds, a year, not estimates from market data. The risk-free rate (a one-year Treasury bill) is 3.0%. The same numbers run through the whole module.

Asset class	Expected return	Volatility	Sharpe ratio
Global equities	7.0%	16%	0.250
Emerging-market equities	8.5%	22%	0.250
Government bonds	3.5%	6%	0.083
Corporate bonds	4.5%	8%	0.188
Property	6.0%	18%	0.167

Check it on the module's five asset classes. The tangency portfolio has expected return 5.51% and volatility 8.72%, with k = 3.3054. Each asset's beta against it puts it exactly on the line:

Asset class	Weight in T	cov with T	β = cov ÷ σT²	rf + β(μT − rf)	Assumed μ
Global equities	29.4%	0.01210	1.5931	7.00%	7.0%
Emerging-market equities	18.3%	0.01664	2.1906	8.50%	8.5%
Government bonds	41.8%	0.00151	0.1991	3.50%	3.5%
Corporate bonds	13.1%	0.00454	0.5974	4.50%	4.5%
Property	−2.6%	0.00908	1.1949	6.00%	6.0%

σT² = 0.00760. The tangency portfolio holds a small short position in property (−2.6%). In a real CAPM equilibrium that cannot last: someone must hold every property, so property's price would fall, and its expected return rise, until investors wanted to hold all of it. That price adjustment is exactly how CAPM's equilibrium is reached.

Derivation 2: Sharpe's argument. William Sharpe's 1964 proof uses only the picture: mix a little of any asset into the market portfolio and look at the curve this traces near M.

Numerically: with the market at 7% expected return and 16% volatility, the CML slope is 0.2500. RiskAtoms Bank plc's mix with the market has slope 0.2500 at M if its expected return is the CAPM 8.2%, but 0.5208 if it is the analysts' 9.5%: steeper than the CML, so a little more of the bank would beat the market. That cannot be true in equilibrium.

Systematic and specific risk. From the regression $R_i = \alpha_i + \beta_i R_m + \varepsilon_i$ with $\operatorname{cov}(R_m, \varepsilon_i) = 0$ :

\begin{aligned} \operatorname{var}(R_i) &= \operatorname{var}(\beta_i R_m) + \operatorname{var}(\varepsilon_i) + 2\beta_i\operatorname{cov}(R_m, \varepsilon_i) \\ \sigma_i^2 &= \beta_i^2\sigma_m^2 + \sigma_\varepsilon^2 \\[4pt] \operatorname{cov}(R_i, R_m) &= \beta_i\sigma_m^2 \quad\Longrightarrow\quad \beta_i = \frac{\operatorname{cov}(R_i, R_m)}{\sigma_m^2} = \rho_{im}\frac{\sigma_i}{\sigma_m} \\[4pt] R^2_i &= \frac{\beta_i^2\sigma_m^2}{\sigma_i^2} = \rho_{im}^2 \end{aligned}

For an equally weighted portfolio of $n$ assets with independent specific risks, the specific variance is $\frac{1}{n^2}\sum_i \sigma_{\varepsilon_i}^2 = \bar\sigma_\varepsilon^2/n$ , which goes to zero, while the systematic part $\bar\beta^2\sigma_m^2$ stays.

Pricing. If an asset is expected to be worth $\mathbb{E}[X_1]$ in a year (price plus dividend), its CAPM value today is $P_0 = \mathbb{E}[X_1]/(1 + r_f + \beta_i(\mathbb{E}[R_m] - r_f))$ . A forecast return above the SML means today's price is below that value.

Try it

Start with RiskAtoms Bank plc: it sits above the SML, so it looks cheap, but on the CML chart it sits well below the line, because a single share carries specific risk. Set its specific volatility to 0% and its forecast to the CAPM return, and it lands on both lines: it has become the market, scaled. Then try the software company, which sits below the SML.

Try itPlace an asset on the SML and the CML

Start from

Beta β1.30

Specific volatility σε22.0%

Analysts' forecast return9.5%

Security market line: return against beta

Capital market line: return against total volatility

CAPM required return8.20%

Forecast9.50%

Alpha1.30%

Total volatility σ30.28%

Systematic share of variance47.2%

CML return at this σ10.57%

Underpriced: above the SML

Fair price of a £100 share£101.20

The forecast payoff £109.50 discounted at the required 8.20%.

Worked example

RiskAtoms Bank plc on the SML

The equity market: risk-free rate 3%, market expected return 7%, market volatility 16% (the module's global equities). RiskAtoms Bank plc's covariance with the market is 0.03328 and its specific volatility 22%. Analysts forecast a return of 9.5% next year. Illustrative numbers. Try each step before revealing it.

Beta, risk split, required return and verdict

Step 1 of 6

1
Beta
$\beta = 0.03328 \div 0.16^2 = 0.03328 \div 0.0256 = 1.30$ . When the market moves 1%, the bank tends to move 1.3%.

Share	β	σ	Required	Forecast	α	Fair price of £100 share
RiskAtoms Bank plc	1.30	30.3%	8.2%	9.5%	1.3%	£101.20
A water utility	0.55	17.4%	5.2%	6.0%	0.8%	£100.76
A software company	1.50	38.4%	9.0%	8.0%	−1.0%	£99.08

Market risk premium used throughout: 4%.

Regulatory lens · looking ahead

No rule sets expected returns, but beta runs through daily risk management. Market risk systems map a single-stock position to an equivalent index position by its beta (L3.2.03: Equity risk), and capital rules for equity risk come in M4.3: Market risk capital (FRTB) and CVA. L3.1.10: Using and testing CAPM shows how regulators and firms turn CAPM into a cost of equity.

Spark Atoms

Spark AtomFour discoverers

CAPM was found independently by William Sharpe (1964), John Lintner (1965) and Jan Mossin (1966), and in unpublished notes by Jack Treynor in the early 1960s. Sharpe shared the 1990 Nobel memorial prize in economics with Harry Markowitz and Merton Miller.

Spark AtomNegative beta, lower return

An asset whose returns rise when the market falls has a negative beta. CAPM says investors should accept an expected return below the risk-free rate for it, because it pays out when everything else is losing: it works like insurance, and insurance has a price.

Key takeaways

If all investors hold the tangency portfolio, it is the market portfolio, and every asset satisfies E[Ri] = rf + βi(E[Rm] − rf).
Beta = cov(Ri, Rm) ÷ var(Rm) = ρ σi ÷ σm: the asset's contribution to market risk, per unit of market variance.
Variance splits as σi² = βi²σm² + σε². Specific risk diversifies away and earns no premium; systematic risk does not and is paid.
The CML (return against σ) holds only for efficient portfolios; the SML (return against β) holds for every asset.
A forecast above the SML means positive alpha: the asset is underpriced. Below it, overpriced.

Reaction

Check what stuck

Question 1 · objective 1

The risk-free rate is 3% and the market is expected to return 8%. What expected return does CAPM give a share with beta 1.2, in %?

%

Question 2 · objective 2

A share's covariance with the market is 0.018 and the market's volatility is 20%. What is the share's beta, to two decimal places?

Question 3 · objective 3

A share has total volatility 35% and beta 0.9; the market's volatility is 20%. What is its specific volatility σε, in %, to one decimal place?

%

Question 4 · objective 4

Analysts expect a share with beta 0.8 to return 10%. The risk-free rate is 3% and the market risk premium 6%. What does CAPM say?

AOverpriced: its alpha is negativeBUnderpriced: it offers 2.2% more than CAPM requiresCFairly priced, because its beta is below 1DCAPM cannot say without its total volatility

Question 5 · objective 3

Two shares have the same total volatility. One has beta 1.4, the other beta 0.6. Under CAPM:

AThey have the same expected return, because their risk is the sameBThe beta 1.4 share has the higher expected return: only systematic risk is rewardedCThe beta 0.6 share has the higher expected return, because more of its risk is specificDNeither earns more than the risk-free rate

Glossary

CAPM: The capital asset pricing model: every asset's expected return is the risk-free rate plus its beta times the market risk premium.
Beta: Covariance of the asset's return with the market's, divided by the market's variance: how much of the market's move the asset follows.
Market portfolio: The portfolio of all risky assets, each held in proportion to its market value. In CAPM it is the tangency portfolio.
Market risk premium: The market's expected return above the risk-free rate.
Systematic risk: Risk that comes from the market as a whole and cannot be diversified away: β²σm².
Specific risk: Risk from news about the asset itself, which cancels out in a large portfolio: σε². Also called idiosyncratic risk.
Security market line: The line of expected return against beta, through the risk-free rate and the market. Under CAPM every asset lies on it.
Alpha: The gap between an asset's forecast expected return and the return the SML requires for its beta.

Deep Dives & Origins

Deep Dives

Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill, the chapter on the capital asset pricing model. Intermediate.
E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, Wiley, the chapters on the standard CAPM and its derivation. Advanced.

Origins

W. F. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk”, Journal of Finance 19(3), 1964.
J. Lintner, “The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets”, Review of Economics and Statistics 47(1), 1965.
J. Mossin, “Equilibrium in a Capital Asset Market”, Econometrica 34(4), 1966.
Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill.

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