Finance-Theory · Unit 19 · Video 2 · Interactive Practice

Paid for Beta, Not Volatility: Multiple Factors and Equilibrium

IKey Formulas

Formula	Name	Applies to
$\beta_i = \dfrac{\text{Cov}[R_i, R_m]}{\text{Var}[R_m]} = \rho_{im}\,\dfrac{\sigma_i}{\sigma_m}$	Beta: a covariance over a variance	Any security; equals $\sigma_i / \sigma_m$ only when $\rho_{im} = 1$
$\text{E}[R_i] = R_f + \beta_i\left(\text{E}[R_m] - R_f\right)$	Security market line (SML)	Any security or portfolio, efficient or not
$\text{E}[R_p] = R_f + \dfrac{\sigma_p}{\sigma_m}\left(\text{E}[R_m] - R_f\right)$	Capital market line (CML)	Efficient portfolios only
$\text{efficient } p:\ \beta_p = \dfrac{\sigma_p}{\sigma_m}$	Where the two lines agree	$\rho_{pm} = 1$

Key Insight: Volatility $\sigma$ counts all of a stock's risk, including the firm-specific part that diversification removes. The market pays only for the risk that survives diversification, and $\beta$ measures exposure to it. Richer models add further betas (book-to-market, liquidity, trading volume; Fama and French's five factors), and the factors worth using in practice are ones you can trade.

IIVisualization 1 — Two Sorts of the Same Stocks

Over 1926–2004, ten portfolios sorted by beta line up by return; ten sorted by total volatility do not.

💡 The bars are realised averages read off the video's charts, so they estimate expected returns rather than measure them; the beta-sorted bars rise overall, not in a straight line.

IIIVisualization 2 — Same Pay, Very Different Volatility

The CAPM pays for $\beta = \rho_{im}\,\sigma_i / \sigma_m$ , so every stock on one curve earns the same, whatever its volatility.

Stock σ (volatility) 48% ρ (with the market) 0.25

IVVisualization 3 — CML and SML: Where They Agree

The CML multiplies the premium by $\sigma_p / \sigma_m$ , the SML by $\beta_i$ ; the two coincide only when $\rho = 1$ .

σ (volatility) 27% ρ (with the market) 0.50

💡 Every earlier price in the course (bonds, stocks, futures, forwards, options) needed only no free lunch. The CAPM also needs equilibrium, supply equal to demand, to identify the tangency portfolio with the market portfolio $M$ that both panels use.

VQuiz Questions

Problem 1 · Beta from Correlation

Given: annual $R_f = 3\%$ , market risk premium $\text{E}[R_m] - R_f = 7\%$ , $\sigma_m = 18\%$ . Stock K has volatility $\sigma_K = 40\%$ and correlation $\rho = 0.45$ with the market. Find its CAPM expected return.

✅ Correct!

\beta_K = 0.45 \times 40 / 18 = 1

, so the CAPM gives K exactly the market's expected return,

3\% + 7\% = 10\%

, although K is more than twice as volatile as the market.

❌ That uses $\sigma_K / \sigma_m$ as the beta.

40/18 \approx 2.22

is the CML multiplier, valid only for an efficient portfolio (

\rho = 1

). For a single stock,

\beta_K = \rho\,\sigma_K / \sigma_m

❌ That treats the correlation as the beta.

\rho = 0.45

must be scaled by

\sigma_K / \sigma_m = 40/18

to become a covariance over a variance.

❌ That is only the risk premium.

\beta_K(\text{E}[R_m] - R_f) = 7\%

is what K earns above

R_f

; add

R_f = 3\%

Show solution

Beta is a covariance over a variance, $\beta_K = \rho\,\sigma_K / \sigma_m$ :

\beta_K = 0.45 \times \frac{40\%}{18\%} = \frac{18}{18} = 1

The SML then gives

\text{E}[R_K] = 3\% + 1 \times 7\% = 10\%

K's volatility is $40\%$ , against $18\%$ for the market, yet its expected return is the market's. Most of K's risk does not move with the market, and that part is not paid.

Problem 2 · The More Volatile Stock

Given: the same market ( $R_f = 3\%$ , premium $7\%$ , $\sigma_m = 18\%$ ). Stock X has $\sigma_X = 55\%$ and $\rho_X = 0.20$ ; stock Y has $\sigma_Y = 22\%$ and $\rho_Y = 0.90$ . Find which one the CAPM says should earn more, and by how much.

✅ Correct! X is two and a half times as volatile, but its beta is only about

0.61

against Y's

1.10

, so Y earns about

3.42

points more.

❌ That ranks the stocks by volatility. Using

\sigma_i / \sigma_m

as the beta gives X about

24.39\%

and Y about

11.56\%

. The market pays for

\beta = \rho\,\sigma_i / \sigma_m

, and X's low correlation shrinks its beta.

❌ That uses the correlations as betas.

7\% \times (0.90 - 0.20) = 4.90\%

ignores each stock's

\sigma_i / \sigma_m

; the betas are

0.20 \times 55/18

and

0.90 \times 22/18

❌ The size is right, the direction is not. Higher volatility does not mean a higher CAPM return: X's beta is about

0.61

, below Y's

1.10

Show solution

Compute each beta as $\rho\,\sigma_i / \sigma_m$ :

\beta_X = 0.20 \times \frac{55}{18} = \frac{11}{18} \approx 0.611 \qquad \beta_Y = 0.90 \times \frac{22}{18} = \frac{19.8}{18} = 1.10

Then the SML:

\text{E}[R_X] = 3\% + \tfrac{11}{18} \times 7\% \approx 7.28\% \qquad \text{E}[R_Y] = 3\% + 1.10 \times 7\% = 10.7\%

The gap is $7\% \times \left(1.10 - \tfrac{11}{18}\right) = 7\% \times \tfrac{8.8}{18} \approx 3.42$ percentage points, in Y's favour. X is the more volatile stock, but most of its risk is firm-specific and diversifies away.

Problem 3 · Same Volatility, Two Lines

Given: the same market ( $R_f = 3\%$ , premium $7\%$ , $\sigma_m = 18\%$ ). Portfolio P is efficient, with $\sigma_P = 22\%$ . Stock Z has the same volatility, $\sigma_Z = 22\%$ , but correlation $\rho_Z = 0.45$ with the market. Find the expected return of each.

Expected return of the efficient portfolio P

Expected return of stock Z

✅ Correct! Equal volatility, unequal pay: P's multiplier is

\sigma_P / \sigma_m \approx 1.22

, Z's beta only

0.55

, so Z is expected to earn about

4.71

points less.

❌ That is only P's risk premium.

\tfrac{22}{18} \times 7\% \approx 8.56\%

sits on top of

R_f = 3\%

❌ That squares the ratio.

(22/18)^2

is a ratio of variances; the CML multiplier is the ratio of standard deviations,

\sigma_P / \sigma_m = 22/18

❌ That is stock Z's return. P is efficient, so its correlation with the market is

1

and its multiplier is

\sigma_P / \sigma_m

, not

0.45 \times \sigma_P / \sigma_m

❌ The CML does not price a single stock. Its multiplier

\sigma / \sigma_m

applies only to efficient portfolios. Z has

\rho = 0.45

, so it is priced on the SML with

\beta_Z = 0.45 \times 22/18

❌ That uses $\rho = 0.45$ as the beta. Scale it by

\sigma_Z / \sigma_m = 22/18

\beta_Z = 0.55

❌ That is only Z's risk premium.

0.55 \times 7\% = 3.85\%

is earned on top of

R_f = 3\%

Show solution

P (efficient, on the CML):

\text{E}[R_P] = 3\% + \frac{22}{18} \times 7\% = 3\% + \frac{154}{18}\% \approx 3\% + 8.56\% \approx 11.56\%

Z (a single stock, on the SML):

\beta_Z = 0.45 \times \frac{22}{18} = \frac{9.9}{18} = 0.55 \qquad \text{E}[R_Z] = 3\% + 0.55 \times 7\% = 6.85\%

Both have $\sigma = 22\%$ , yet Z is expected to earn about $11.56\% - 6.85\% \approx 4.71$ points less. Only the part of Z's risk that moves with the market is paid; for P, perfectly correlated with the market, all of it is.

Problem 4 · What the CAPM Rests On

Given: the bond, futures and binomial option prices of earlier units needed only no free lunch. A fund manager now uses the CAPM to set a discount rate, and her fund carries more market beta than her target. Find the extra condition the CAPM needs, and the exposure she can cut directly with a trade.

The extra condition the CAPM needs to identify the tangency portfolio with the market

The exposure she can cut directly with a trade

✅ Correct! The CAPM needs equilibrium, a stronger assumption than no free lunch, and market beta is a financially relevant factor: it is tied to securities she can trade.

❌ That is the no-free-lunch assumption itself. Every earlier pricing argument already used it; the CAPM needs something stronger.

❌ The CAPM assumes no such thing. Stocks share market risk, and that common risk is exactly what beta measures.

❌ The CAPM allows any volatilities. It says how expected returns differ across securities through their betas; what it adds is that the market for each security must clear.

❌ Economically relevant, not financially relevant. Unemployment has some power to explain stock returns, but there is no liquid contract to sell; if you cannot trade it, you cannot manage it.

❌ An index of opinions is not a security. It may relate to returns, but she has nothing to sell that removes the exposure.

❌ A macroeconomic factor again. GDP growth may help explain returns, yet it is not a portfolio of securities she can trade.

Show solution

Condition: bonds, stocks, futures, forwards and options were priced from no free lunch alone, the assumption that people prefer more money to less. The CAPM invokes equilibrium, supply equal to demand, and it is that condition which identifies the tangency portfolio with the market portfolio $M$ . In return, it lets you read a discount rate off the market's own prices.

Trade: factors used in financial applications are portfolios of securities, or tradable in some other way. Selling S&P 500 futures lowers the fund's market beta directly. Unemployment, consumer confidence and GDP growth may be economically relevant, but none can be traded as easily, so none can be managed this way.

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