Finance-Theory · Unit 18 · Video 3 · Interactive Practice

Why a Risky Asset Can Earn Less Than a Treasury Bill: The Security Market Line

IKey Formulas

Formula	Name	What it prices
$\mathsf{E}[R_p] = R_f + \dfrac{\sigma_p}{\sigma_m}\left(\mathsf{E}[R_m] - R_f\right)$	Capital market line	Efficient portfolios only: expected return against standard deviation
$\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right)$	Security market line (CAPM)	Every asset, efficient or not: expected return against beta
$\beta_i = \dfrac{\text{Cov}[R_i, R_m]}{\text{Var}[R_m]}$	Beta	Sensitivity to market movements; only this risk earns a premium
$\mathsf{E}[R_i] < 0 \iff \beta_i < -\dfrac{R_f}{\mathsf{E}[R_m] - R_f}$	Negative expected return	Where the security market line crosses zero

Key Insight: $\beta_i = 1$ gives $\mathsf{E}[R_i] = \mathsf{E}[R_m]$ ; $\beta_i = 0$ gives $\mathsf{E}[R_i] = R_f$ however volatile the asset is; $\beta_i < 0$ puts $\mathsf{E}[R_i]$ below $R_f$ , because a hedge against market risk is worth paying for.

IIThree Special Cases on One Line

With $R_f = 2\%$ and a $5.5\%$ premium, beta alone fixes an asset's expected return, from the market's $7.5\%$ down past zero.

Beta, β_i 1.00

IIIZero Beta Is Not Zero Risk

A zero-beta asset can swing as widely as the market, yet its expected return is only the T-bill's $2\%$ ; it is not a risk-free asset. The path below is illustrative, built so that its 40-year average lands exactly on that expected return.

Asset Year 40

IVA Hedge Costs More Today

Discounting an expected $100 payoff in one year at the security-market-line rate: the lower the beta, the higher the price.

Beta, β_i 1.00

💡 Measured betas are almost all positive and cluster between $0.5$ and $1.5$ ; gold mining shares, with betas near zero, are about the closest real case to a hedge, so assets on the left of this curve are very hard to find.

VQuiz Questions

Problem 1 · A Point on the Line

Given (illustrative, annual): $R_f = 2\%$ and a market risk premium $\mathsf{E}[R_m] - R_f = 5.5\%$ . A stock has $\beta_i = 1.2$ . Find its expected return according to the security market line.

✅ Correct!

1.2

premiums of

5.5\%

on top of

2\%

: the stock sits

1.1

points above the market's

7.5\%

, because it moves more than one for one with the market.

❌ Missing the intercept.

1.2 \times 5.5\% = 6.6\%

is only the premium term; the line starts at

R_f = 2\%

, so add it back.

❌ Wrong multiplicand.

2\% + 1.2 \times 7.5\% = 11\%

multiplies beta by

\mathsf{E}[R_m]

; beta multiplies the premium

\mathsf{E}[R_m] - R_f = 5.5\%

, otherwise

R_f

is counted twice.

❌ Not quite.

1.2 \times 7.5\% = 9\%

scales the whole market return; the security market line is

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

❌ Not quite. Start at

R_f

and add

\beta_i

times the market risk premium.

Show solution

\mathsf{E}[R_i] = R_f + \beta_i\left(\mathsf{E}[R_m] - R_f\right) = 2\% + 1.2 \times 5.5\% = 2\% + 6.6\% = 8.6\%

The market itself ( $\beta = 1$ ) earns $2\% + 5.5\% = 7.5\%$ ; a beta of $1.2$ adds a fifth of a premium more, $1.1$ points.

Problem 2 · Volatile but Zero Beta

Given (illustrative, annual): $R_f = 2\%$ and a market risk premium of $5.5\%$ . Asset $Z$ has a standard deviation of $28\%$ a year, and its covariance with the market is zero. Find how $Z$ 's CAPM expected return compares with a T-bill's.

✅ Correct!

\beta_Z = 0/\text{Var}[R_m] = 0

, so

\mathsf{E}[R_Z] = R_f + 0 \times 5.5\%

: none of

Z

's risk moves with the market, and none of it is paid.

❌ Volatility is not what is priced. The security market line pays only for covariance with the market;

Z

28\%

swings are unrelated to the market's, so they earn no premium.

❌ That needs a negative beta. A hedge rises when the market falls: negative covariance,

\beta < 0

. Zero covariance neither adds to nor offsets market risk.

❌ That is the capital market line's restriction. The capital market line prices only efficient portfolios; the security market line prices every asset, efficient or not.

❌ Not quite. Compute

\beta_Z

from the covariance, then place it on the security market line.

Show solution

\beta_Z = \frac{\text{Cov}[R_Z, R_m]}{\text{Var}[R_m]} = \frac{0}{\text{Var}[R_m]} = 0

\mathsf{E}[R_Z] = R_f + 0 \times \left(\mathsf{E}[R_m] - R_f\right) = R_f = 2\%

$Z$ sits at the line's intercept, level with the T-bill. The two are still different assets: the T-bill's return is known in advance, while $Z$ 's typically swings by around $28\%$ either way; it earns the risk-free return only on average.

Problem 3 · Past the Crossing Point

Given (illustrative, annual): $R_f = 2\%$ and a market risk premium of $5.5\%$ . An asset has $\beta_i = -0.4$ . Find its expected return, and the beta below which the security market line's expected return turns negative.

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

Below which beta is $\mathsf{E}[R_i] < 0$ ?

✅ Correct!

-0.4

lies just past the crossing point

-\tfrac{4}{11} \approx -0.364

, so this risky asset is expected to lose

0.2\%

a year: investors pay to hold it because it hedges market risk.

❌ Sign dropped.

2\% + 0.4 \times 5.5\% = 4.2\%

treats the beta as

+0.4

; with

\beta_i = -0.4

the premium term is

-2.2\%

❌ Missing the intercept.

-0.4 \times 5.5\% = -2.2\%

is only the premium term; add

R_f = 2\%

❌ Wrong multiplicand.

2\% - 0.4 \times 7.5\% = -1\%

multiplies beta by

\mathsf{E}[R_m]

instead of the premium

5.5\%

❌ Check $\mathsf{E}[R_i]$ . Use

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

with

\beta_i = -0.4

❌ That is where it drops below $R_f$ . Any negative beta puts

\mathsf{E}[R_i]

below

2\%

, but it stays positive until the premium term outweighs the whole

2\%

intercept.

❌ Wrong denominator.

-2/7.5

divides by

\mathsf{E}[R_m]

; the line falls

5.5

points per unit of beta, so the threshold is

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

❌ Upside down.

-5.5/2

inverts the ratio; solve

2\% + \beta_i \times 5.5\% < 0

for

\beta_i

❌ Check the threshold. Set

R_f + \beta_i(\mathsf{E}[R_m] - R_f) < 0

and solve for

\beta_i

Show solution

\mathsf{E}[R_i] = 2\% + (-0.4)(5.5\%) = 2\% - 2.2\% = -0.2\%

The expected return is negative when the premium term outweighs the intercept:

2\% + \beta_i \times 5.5\% < 0 \iff \beta_i < -\frac{2}{5.5} = -\frac{4}{11} \approx -0.364

Since $-0.4 < -0.364$ , the asset's expected return is below zero, consistent with the $-0.2\%$ above. A beta between $-0.364$ and $0$ would earn less than $2\%$ but still more than nothing.

Problem 4 · Pricing a Hedge

Given (illustrative, annual): $R_f = 2\%$ and a market risk premium of $5.5\%$ . Assets $H$ ( $\beta_H = -0.4$ ) and $S$ ( $\beta_S = 1.2$ ) are each expected to pay $1,000 in one year and nothing else. Find each price today, discounting the expected payoff for one year at its security-market-line rate.

What is the price of $H$ today?

How much more does $H$ cost than $S$ today?

✅ Correct! The same expected $1,000 costs about $81 more as a hedge than as a high-beta stock, and the hedge even costs more than its expected payoff: buying pushes its price up until its expected return is

-0.2\%

❌ Beta ignored. $1,000 / 1.02 discounts at

R_f

, the rate for a zero-beta asset;

H

's rate is

2\% - 0.4 \times 5.5\% = -0.2\%

❌ Sign dropped. $1,000 / 1.042 uses

+4.2\%

; a negative beta lowers the discount rate below

R_f

, here to

-0.2\%

❌ Missing the intercept. $1,000 / 0.978 discounts at

-2.2\%

, the premium term alone; add

R_f

to get

-0.2\%

❌ Check $H$ 's price. Find

H

's rate on the security market line, then divide $1,000 by one plus that rate.

❌ Same payoff, different risk. Equal expected payoffs have equal prices only if they are discounted at the same rate;

H

and

S

have different betas.

❌ Not discounted. $1,000 × (8.6% − (−0.2%)) = $88 is the gap in the rates applied to the payoff; compute each price, $1,000 / (1 + rate), and subtract.

❌ Wrong rates. $1,000 / 0.99 − $1,000 / 1.11 uses

R_f + \beta\,\mathsf{E}[R_m]

; the rates are

-0.2\%

for

H

and

8.6\%

for

S

❌ Check the difference. Price

S

2\% + 1.2 \times 5.5\% = 8.6\%

and subtract it from

H

's price.

Show solution

Discount rates from the security market line:

r_H = 2\% + (-0.4)(5.5\%) = -0.2\%, \qquad r_S = 2\% + 1.2 \times 5.5\% = 8.6\%

Prices, one year of discounting:

$H$ : $1,000 / 0.998 ≈ $1,002.00

$S$ : $1,000 / 1.086 ≈ $920.81

Difference: $1,002.00 − $920.81 ≈ $81.19

$H$ 's price is above its expected payoff, so its expected return is negative: investors accept that because $H$ tends to pay off when the market falls.

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