Finance-Theory · Unit 18 · Video 7 · Interactive Practice

Why Markets Pay Only for the Risk You Cannot Diversify Away

IKey Formulas

Formula	Name	What it says
$\beta_i = \dfrac{\text{Cov}[R_i, R_m]}{\text{Var}[R_m]} = \rho_{im}\,\dfrac{\sigma_i}{\sigma_m}$	Beta	The part of a stock's risk that stays in your portfolio however widely you diversify
$\sigma_p = \omega\,\sigma_m, \quad \beta_p = \omega$	Efficient portfolio: $\omega$ in M, the rest, $(1 - \omega)$ , in T-bills	T-bills add no risk, so all of $\sigma_p$ is market risk
$\beta_p = \dfrac{\sigma_p}{\sigma_m}$	Efficient portfolios only	$\sigma$ and $\beta$ rank efficient portfolios in the same order
$\text{E}[R_i] = R_f + \beta_i\,(\text{E}[R_m] - R_f)$	Security market line	Only $\beta$ is paid; firm-specific risk earns nothing

Key Insight: Variance $\sigma_i^2$ counts all of a stock's risk, the avoidable and the unavoidable together; beta counts only the systematic part that no number of stocks removes, and that is the only part the market pays for.

IIDiversification Leaves Only Market Risk

Each added stock shrinks the firm-specific risk; the market risk underneath it never diversifies away.

Stocks held, N 1

Illustration, not market data: $N$ equally weighted stocks, each with $\sigma = 40\%$ a year and $\beta = 1$ , and every pair of stocks with covariance $\sigma_m^2$ , where $\sigma_m = 16\%$ . The equal-weight formula from the portfolio-theory units, $\sigma_p^2 = \dfrac{\sigma^2}{N} + \left(1 - \dfrac{1}{N}\right)\sigma_m^2$ , then falls toward $\sigma_m^2$ as $N$ grows: the covariances dominate.

IIIEfficient Portfolios: $\sigma_p$ Is a Beta Ruler

On the capital market line, $\sigma_p$ and $\beta_p$ are tied by $\beta_p = \sigma_p / \sigma_m$ , with $\sigma_m = 16\%$ .

Weight in M, ω 0.60

IVSame Beta, Same Pay, More Risk

A stock and the efficient portfolio with its $\beta$ earn the same CAPM return; the stock's extra $\sigma$ is unpaid.

Stock σ 40% Stock β 0.60

VQuiz Questions

Problem 1 · A Levered Efficient Portfolio

Given: An efficient portfolio puts $\omega = 1.4$ of your wealth in the market portfolio M, borrowing the other 0.4 at the T-bill rate. The market's standard deviation is $\sigma_m = 16\%$ a year. Find the portfolio's standard deviation and its beta.

What is σ_p?

What is β_p?

✅ Correct!

\sigma_p = 1.4 \times 16\% = 22.4\%

and

\beta_p = 1.4

, so

\beta_p = \sigma_p / \sigma_m = 22.4 / 16 = 1.4

: for an efficient portfolio the two measures agree.

❌ Not quite. The borrowed 0.4 times 16% gives 6.4%, which counts only the borrowed part. The whole 1.4 sits in M, and the loan itself is riskless, so

\sigma_p = 1.4\,\sigma_m

❌ Not quite. Borrowing at a riskless rate adds no risk of its own, but it scales up your holding of M: every swing in M now moves 1.4 times your wealth.

❌ Not quite.

16 / 1.4 \approx 11.4

divides by the weight. T-bills (or a T-bill loan) carry no risk, so the portfolio's standard deviation is the weight in M times

\sigma_m

❌ Check σ_p. T-bills (or a T-bill loan) carry no risk, so the portfolio's standard deviation is the weight in M times

\sigma_m

❌ Close, but…

\omega^2 = 1.4^2 = 1.96

is how the variance scales. Covariance with the market is linear in the weight:

\text{Cov}[R_p, R_m] = 1.4\,\text{Var}[R_m]

, so

\beta_p = 1.4

❌ Not quite.

\beta = 1

belongs to M itself. Holding 1.4 of your wealth in M scales your exposure:

\text{Cov}[R_p, R_m] = 1.4\,\text{Var}[R_m]

, so

\beta_p = 1.4

❌ Not quite. 0.4 is the borrowed fraction. The T-bill loan has no covariance with the market; the beta comes from the full 1.4 held in M.

❌ Check β_p. Divide the portfolio's covariance with the market,

\omega\,\text{Var}[R_m]

, by

\text{Var}[R_m]

Show solution

T-bills have no risk and no covariance with anything, so only the $\omega = 1.4$ in M contributes:

\sigma_p = \omega\,\sigma_m = 1.4 \times 16\% = 22.4\%

\beta_p = \frac{\text{Cov}[R_p, R_m]}{\text{Var}[R_m]} = \frac{1.4\,\text{Var}[R_m]}{\text{Var}[R_m]} = 1.4

Check: $\sigma_p / \sigma_m = 22.4 / 16 = 1.4 = \beta_p$ . An efficient portfolio has no firm-specific risk left, so its standard deviation is pure market risk.

Problem 2 · Volatility Is Not Beta

Given: Stock K has $\sigma_K = 32\%$ a year and correlation $\rho_{Km} = 0.35$ with the market; $\sigma_m = 16\%$ a year. A classmate says that since K is twice as volatile as the market, its beta must be 2. Find $\beta_K$ .

✅ Correct!

\beta_K = 0.35 \times 32 / 16 = 0.7

. K is twice as volatile as the market yet carries less market risk than the market itself.

❌ Close, but…

\sigma_K / \sigma_m = 2

would be the beta only if K moved in lockstep with the market (

\rho = 1

), like an efficient portfolio. Most of K's volatility is firm-specific, and the correlation strips it out.

❌ Not quite. The correlation alone ignores how big K's swings are relative to the market's. Scale it by

\sigma_K / \sigma_m

❌ Not quite. The ratio is upside down:

0.35 \times 16 / 32 = 0.175

. Use

\beta_K = \rho_{Km}\,\sigma_K / \sigma_m

, with the stock's volatility on top.

Show solution

Write the covariance as $\text{Cov}[R_K, R_m] = \rho_{Km}\,\sigma_K\,\sigma_m$ and divide by $\text{Var}[R_m] = \sigma_m^2$ :

\beta_K = \frac{\rho_{Km}\,\sigma_K\,\sigma_m}{\sigma_m^2} = \rho_{Km}\,\frac{\sigma_K}{\sigma_m} = 0.35 \times \frac{32}{16} = 0.7

The classmate's $\sigma_K / \sigma_m = 2$ treats all of K's volatility as market risk. Only the correlated part stays in a diversified portfolio, so only it enters beta.

Problem 3 · The Efficient Twin

Given: Stock Q has $\sigma_Q = 34\%$ a year and correlation $\rho_{Qm} = 0.4$ with the market; $\sigma_m = 16\%$ a year. Find $\beta_Q$ , then the standard deviation of the efficient portfolio (a mix of M and T-bills) that the CAPM gives the same expected return as Q.

What is β_Q?

σ of the efficient portfolio with the same CAPM expected return?

✅ Correct!

\beta_Q = 0.85

, so the twin holds

\omega = 0.85

in M with

\sigma_p = 13.6\%

. Q carries

34\%

volatility but is priced like a

13.6\%

-volatility efficient portfolio: the extra volatility is risk the market does not reward.

❌ Close, but…

\sigma_Q / \sigma_m = 34 / 16 = 2.125

assumes

\rho = 1

. Multiply by

\rho_{Qm} = 0.4

❌ Not quite.

0.4

is the correlation alone. Scale it by how large Q's swings are relative to the market's,

\sigma_Q / \sigma_m = 34 / 16

❌ Not quite.

0.4 \times 16 / 34 \approx 0.19

turns the ratio upside down. Beta is

\rho_{Qm}\,\sigma_Q / \sigma_m

, with the stock's volatility on top.

❌ Not quite.

16\%

is the market itself, the efficient portfolio with

\beta = 1

. Q's twin needs

\beta_p = \beta_Q

, so

\omega = \beta_Q

and

\sigma_p = \beta_Q\,\sigma_m

❌ Not quite.

0.4 \times 16\% = 6.4\%

multiplies the correlation by

\sigma_m

. The twin's weight in M is

\beta_Q

, not

\rho_{Qm}

, so

\sigma_p = \beta_Q\,\sigma_m

❌ Check β_Q. Use

\beta_Q = \rho_{Qm}\,\sigma_Q / \sigma_m

❌ Not quite. The CAPM matches expected returns through

\beta

, not

\sigma

. The twin needs

\beta_p = \beta_Q

, and for an efficient portfolio that means

\sigma_p = \beta_Q\,\sigma_m

❌ Check the twin. Same

\beta

⇒ same SML return. An efficient portfolio's beta is its weight

\omega

in M, and its

\sigma_p = \omega\,\sigma_m

Show solution

Step 1: beta.

\beta_Q = \rho_{Qm}\,\frac{\sigma_Q}{\sigma_m} = 0.4 \times \frac{34}{16} = 0.85

Step 2: same expected return means same beta. On the security market line, $\text{E}[R] = R_f + \beta\,(\text{E}[R_m] - R_f)$ depends only on $\beta$ , so the twin needs $\beta_p = 0.85$ . An efficient mix has $\beta_p = \omega$ , so $\omega = 0.85$ (the other 0.15 in T-bills).

Step 3: its risk.

\sigma_p = \omega\,\sigma_m = 0.85 \times 16\% = 13.6\%

Q and the twin earn the same expected return, but Q's $\sigma$ is 2.5 times the twin's. Q's extra volatility is the Irish jig: risk Q's holder chose to bear and will not be paid for.

Problem 4 · Selling the Employer's Stock

Given: An engineer holds all of her savings in her employer's stock, which has $\beta = 1.4$ and $\sigma = 45\%$ a year. She sells it and buys an equally weighted portfolio of 60 stocks from many different industries, each with $\beta = 1.4$ . According to the CAPM, what happens?

✅ Correct! She keeps all the priced risk and sheds only the unpriced kind: same expected return, much less volatility. Holding the single stock was the jig.

❌ Not quite. Portfolio beta is the weighted average of the holdings' betas:

\beta_p = \sum_i w_i\,\beta_i = 1.4

. Diversification removes firm-specific risk, not market exposure.

❌ Not quite. The risk she shed was firm-specific, which the market never paid for. Her expected return depends only on

\beta

, which is still 1.4.

❌ Not quite. Beta is unchanged, but

\sigma

is not: 60 stocks from different industries carry firm-specific risks that largely cancel, which a single stock's cannot.

Show solution

Beta. Covariance with the market is linear in the weights, so

\beta_p = \sum_{i=1}^{60} \tfrac{1}{60}\,(1.4) = 1.4

Expected return. The security market line gives $\text{E}[R_p] = R_f + 1.4\,(\text{E}[R_m] - R_f)$ , the same as for the employer's stock alone.

Risk. With 60 stocks the firm-specific swings of the individual companies largely average out, so $\sigma_p$ falls well below 45%. What remains is the market risk that $\beta = 1.4$ measures.

She gives up no expected return and removes risk she was never paid to bear, which is why a concentrated position in your employer's stock is hard to defend from a portfolio point of view.

Solved: 0 / 4

Why Markets Pay Only for the Risk You Cannot Diversify Away

IKey Formulas

IIDiversification Leaves Only Market Risk

IIIEfficient Portfolios: σp\sigma_pσp​ Is a Beta Ruler

IVSame Beta, Same Pay, More Risk

VQuiz Questions

IIIEfficient Portfolios: $\sigma_p$ Is a Beta Ruler