Finance-Theory · Unit 18 · Video 2 · Interactive Practice

From the Tangency Line to Beta: Pricing Assets That Are Not Efficient

IKey Formulas

Formula	Name	Applies to
$\mathsf{E}[R_p] = R_f + \dfrac{\sigma_p}{\sigma_m}\left(\mathsf{E}[R_m] - R_f\right)$	Tangency line (capital market line)	Efficient portfolios only: mixes of T-bills and the market portfolio $M$
$\beta_i = \dfrac{\operatorname{Cov}[R_i, R_m]}{\operatorname{Var}[R_m]} = \dfrac{\rho_{im}\,\sigma_i}{\sigma_m}$	Market beta	Any asset; the second form uses $\operatorname{Cov}[R_i, R_m] = \rho_{im}\sigma_i\sigma_m$
$\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right)$	Sharpe–Lintner CAPM	Any asset: a stock, a project, a portfolio, efficient or not
$\beta_p = \dfrac{\sigma_p}{\sigma_m}$	Efficient special case	A portfolio on the tangency line, where the CAPM reduces to the line's equation

Key Insight: The tangency line prices risk by $\sigma_p/\sigma_m$ , which is right only for portfolios on the line. For any asset the multiplier on the market risk premium is $\beta_i$ , which counts only the part of its risk that moves with the market, so beta, not sigma, sets the cost of capital.

IIThe Tangency Line's Equation

Along the line, each point of volatility adds the slope $(\mathsf{E}[R_m] - R_f)/\sigma_m$ to expected return above $R_f$ .

Inputs σ_p (%) 30

IIIBeta Prices What Sigma Cannot

A stock rarely lies on the tangency line; the CAPM places it by beta, $\beta_i = \rho_{im}\sigma_i/\sigma_m$ , instead.

ρ with market 0.6 σ_i (%) 40

IVThe Efficient Special Case

For a mix of T-bills and $M$ , beta equals $\sigma_p/\sigma_m$ , so the CAPM collapses back to the tangency line.

Weight ω in M 0.5 Step 1

Step 1 — The efficient portfolio

R_p = \omega R_m + (1 - \omega) R_f

💡 Sigma is the right risk measure only in this special case; for every other asset the CAPM rate $R_f + \beta_i(\mathsf{E}[R_m] - R_f)$ is the discount rate an NPV calculation needs.

VQuiz Questions

Problem 1 · An Efficient Portfolio's Expected Return

Given: $R_f = 3\%$ , the market risk premium $\mathsf{E}[R_m] - R_f = 8\%$ and $\sigma_m = 20\%$ , all per year. An efficient portfolio (T-bills plus $M$ ) has $\sigma_p = 25\%$ per year. Find its expected return per year.

✅ Correct! The portfolio is

1.25

times as risky as the market, so it earns

1.25

times the market premium on top of

R_f

❌ Not quite.

11\%

\mathsf{E}[R_m]

, the return at

\sigma_p = \sigma_m

. Here

\sigma_p/\sigma_m = 1.25

, above 1, so the portfolio earns more than the market.

❌ Not quite.

1.25 \times 8\% = 10\%

is only the risk premium; the base return

R_f = 3\%

still has to be added.

❌ Close, but… the multiplier applies to the market's excess return,

8\%

, not to

\mathsf{E}[R_m] = 11\%

3\% + 1.25 \times 11\% = 16.75\%

counts

R_f

twice.

Show solution

A portfolio of T-bills and $M$ lies on the tangency line, so its expected return comes from the line's equation:

\mathsf{E}[R_p] = R_f + \frac{\sigma_p}{\sigma_m}\left(\mathsf{E}[R_m] - R_f\right) = 3\% + \frac{25\%}{20\%}\times 8\% = 3\% + 1.25 \times 8\% = 3\% + 10\% = 13\%

The slope is $8\%/20\% = 0.4$ : each percentage point of volatility adds $0.4$ points of expected return, and $25 \times 0.4 = 10$ points above $R_f$ .

Problem 2 · A Stock Is Not on the Line

Given: $R_f = 3\%$ , $\mathsf{E}[R_m] - R_f = 8\%$ , $\sigma_m = 20\%$ per year. A stock has volatility $\sigma_i = 36\%$ per year and correlation $\rho_{im} = 0.5$ with the market. Find its expected return per year under the CAPM.

✅ Correct! Only the co-moving part of the stock's volatility,

\rho_{im}\sigma_i = 18\%

, is priced:

\beta_i = 0.9

❌ That is the tangency-line answer.

3\% + (36/20) \times 8\% = 17.4\%

treats the stock as efficient. A single stock is not on the line, so its multiplier is

\beta_i

, not

\sigma_i/\sigma_m

❌ Not quite. The beta is right, but it multiplies the premium

8\%

, not

\mathsf{E}[R_m] = 11\%

❌ Not quite. That uses

\rho_{im}\sigma_m/\sigma_i \approx 0.28

. Beta scales the asset's volatility by the market's:

\rho_{im}\sigma_i/\sigma_m

Show solution

Beta from correlation and volatilities:

\beta_i = \frac{\rho_{im}\,\sigma_i}{\sigma_m} = \frac{0.5 \times 36\%}{20\%} = \frac{18\%}{20\%} = 0.9

Then the CAPM, which holds for any asset:

\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right) = 3\% + 0.9 \times 8\% = 3\% + 7.2\% = 10.2\%

The tangency line would claim $17.4\%$ at $\sigma = 36\%$ ; the stock sits $7.2$ points below it, because, with a correlation of one half, only half its volatility counts toward its market risk.

Problem 3 · Beta from Covariance

Given: with returns written as decimals per year, $\operatorname{Cov}[R_i, R_m] = 0.012$ , $\sigma_m = 20\%$ and $\sigma_i = 45\%$ . Again $R_f = 3\%$ and $\mathsf{E}[R_m] - R_f = 8\%$ per year. Find the stock's beta and its CAPM expected return.

What is $\beta_i$ ?

What is $\mathsf{E}[R_i]$ per year?

✅ Correct! A very volatile stock with a low beta: most of its

45\%

volatility does not move with the market, and the CAPM pays only for the part that does.

❌ Not quite.

0.012/0.20 = 0.06

divides by

\sigma_m

. Beta divides by the market's variance,

\sigma_m^2 = 0.04

❌ Not quite.

0.012/(0.45 \times 0.20) \approx 0.13

is the correlation

\rho_{im}

, not beta. Beta divides the covariance by

\operatorname{Var}[R_m]

alone.

❌ Not quite.

\sigma_i/\sigma_m = 2.25

is the tangency-line multiplier, valid only for efficient portfolios. Use

\operatorname{Cov}[R_i, R_m]/\operatorname{Var}[R_m]

❌ Check the beta.

\beta_i = \operatorname{Cov}[R_i, R_m]/\operatorname{Var}[R_m]

, with

\operatorname{Var}[R_m] = 0.20^2

❌ That is the tangency-line answer.

3\% + 2.25 \times 8\% = 21\%

prices the stock as if it were efficient. Use

\beta_i = 0.3

❌ Not quite.

3\% + 0.3 \times 11\% = 6.3\%

multiplies

\mathsf{E}[R_m]

; beta multiplies the premium,

8\%

❌ Not quite.

3\% + 0.06 \times 8\% = 3.48\%

uses

\beta_i = 0.012/0.20 = 0.06

, which divides the covariance by

\sigma_m

instead of

\operatorname{Var}[R_m] = 0.04

❌ Check the expected return. Use

\mathsf{E}[R_i] = R_f + \beta_i \times 8\%

with

\beta_i = 0.3

Show solution

Step 1: Beta. The market's variance is $\operatorname{Var}[R_m] = 0.20^2 = 0.04$ , so

\beta_i = \frac{\operatorname{Cov}[R_i, R_m]}{\operatorname{Var}[R_m]} = \frac{0.012}{0.04} = 0.3

Step 2: CAPM.

\mathsf{E}[R_i] = 3\% + 0.3 \times 8\% = 3\% + 2.4\% = 5.4\%

Check: the correlation is $\rho_{im} = 0.012/(0.45 \times 0.20) \approx 0.133$ , and $\rho_{im}\sigma_i/\sigma_m = 0.012/0.04 = 0.3$ — the same beta. The tangency line would have asked $21\%$ of this stock, almost four times the CAPM rate.

Problem 4 · Same Beta, Same Rate

Given: $R_f = 3\%$ , $\mathsf{E}[R_m] - R_f = 8\%$ , $\sigma_m = 20\%$ per year. Portfolio $P$ is a mix of T-bills and $M$ with $\sigma_p = 14\%$ . A firm is also pricing a project with $\beta = 0.7$ and $\sigma = 35\%$ per year. Find the expected return on $P$ and the project's discount rate (cost of capital).

Expected return on $P$

The project's discount rate

✅ Correct!

P

has

\beta_p = \sigma_p/\sigma_m = 0.7

, the project's beta, so both earn

8.6\%

even though the project is

2.5

times as volatile.

❌ Not quite. Only the efficient portfolio with

\sigma_p = \sigma_m

earns

\mathsf{E}[R_m] = 11\%

;

P

holds

\omega = 14/20 = 0.7

M

and the rest in T-bills.

❌ Not quite.

0.7 \times 8\% = 5.6\%

is only the risk premium; the base return

R_f = 3\%

still has to be added.

❌ Not quite.

3\% + 0.7 \times 11\% = 10.7\%

multiplies

\mathsf{E}[R_m]

; the multiplier (

\sigma_p/\sigma_m

for

P

\beta

for the project) applies to the premium,

8\%

❌ Check $P$ . On the tangency line,

\mathsf{E}[R_p] = R_f + (\sigma_p/\sigma_m) \times 8\%

: keep the

R_f

and multiply the premium, not

\mathsf{E}[R_m]

❌ That prices the project by sigma.

3\% + (35/20) \times 8\% = 17\%

is the tangency-line rate, valid only for efficient portfolios. A project is priced by its beta.

❌ Check the rate. The cost of capital is the CAPM return

R_f + \beta(\mathsf{E}[R_m] - R_f)

, with the

3\%

base included and the

8\%

premium multiplied.

Show solution

Portfolio $P$ is efficient, so the tangency line applies:

\mathsf{E}[R_p] = 3\% + \frac{14\%}{20\%} \times 8\% = 3\% + 0.7 \times 8\% = 3\% + 5.6\% = 8.6\%

The project is not an efficient portfolio, so its discount rate comes from the CAPM:

r = R_f + \beta\left(\mathsf{E}[R_m] - R_f\right) = 3\% + 0.7 \times 8\% = 8.6\%

The two answers agree because $\beta_p = \sigma_p/\sigma_m = 0.7$ : the CAPM gives both the same rate since they share a beta, whatever their volatilities. Pricing the project by $\sigma$ instead would demand $17\%$ and reject projects that earn their true cost of capital.

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