Finance-Theory · Unit 18 · Video 5 · Interactive Practice

Same Return, Same Volatility, Different Alpha: Judging Managers by Beta

IKey Formulas

Formula	Name	What it says
$\text{E}[R_i] = R_f + \beta_i(\text{E}[R_m] - R_f)$	Security market line (SML)	The return any asset, project or fund is required to earn; the risk measure is $\beta_i$ , not $\sigma_i$
$\alpha_i = \text{E}[R_i] - \{R_f + \beta_i(\text{E}[R_m] - R_f)\}$	Alpha	Return delivered minus return required: the vertical distance from the SML, positive above it and negative below it
$\beta_i = \dfrac{\text{Cov}[R_i, R_m]}{\text{Var}[R_m]} = \rho_{im}\,\dfrac{\sigma_i}{\sigma_m}$	Beta from correlation	Equal volatilities can carry very different betas when the correlations with the market differ
$\rho = 1 \;\Rightarrow\; \beta_p = \dfrac{\sigma_p}{\sigma_m}$	Efficient portfolios	A mix of T-bills and $M$ has $\rho = 1$ , so the SML and the capital market line give the same required return

Key Insight: Two funds with the same average return and the same volatility can have opposite alphas: the SML charges each for its beta, and volatility that is not correlated with the market earns no reward.

IIAlpha as a Vertical Distance

Alpha is the vertical gap between a fund's average return and the SML's required return at its beta.

Fund beta β 0.8 Average return, % per year 9%

IIISame Volatility, Different Beta

With volatility fixed at $25\%$ , correlation with the market alone sets beta, and with it the required return.

Correlation with the market ρ 0.56

IVQuiz Questions

Problem 1 · Alpha of a High-Beta Manager

Given (per year): $R_f = 3.5\%$ and a market risk premium $\text{E}[R_m] - R_f = 6.5\%$ . A manager has averaged $11\%$ with $\beta = 1.4$ . Find the manager's alpha against the security market line.

✅ Correct! The line requires

3.5\% + 1.4 \times 6.5\% = 12.6\%

for this much market risk;

11\%

falls

1.6

points short, so the manager sits below the SML.

❌ The intercept is missing.

11\% - 1.4 \times 6.5\% = 1.9\%

leaves out

R_f

; the required return is

R_f

plus the risk premium,

3.5\% + 9.1\% = 12.6\%

❌ Check the sign. Alpha is delivered minus required,

11\% - 12.6\%

, not the other way round; a fund below the line has negative alpha.

❌ Premium, not market return.

11\% - 1.4 \times 10\% = -3\%

multiplies

\beta

by the whole market return

\text{E}[R_m] = 10\%

and drops

R_f

;

\beta

multiplies the premium

6.5\%

, and

R_f

is added on top.

❌ Not quite. Compute the required return

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

first, then subtract it from

11\%

Show solution

The SML gives the return a $\beta = 1.4$ investment is required to earn:

R_f + \beta(\text{E}[R_m] - R_f) = 3.5\% + 1.4 \times 6.5\% = 3.5\% + 9.1\% = 12.6\%

Alpha is what the manager delivered minus that requirement:

\alpha = 11\% - 12.6\% = -1.6\%

The manager underperforms by $1.6$ percentage points a year given the market risk the fund carries.

Problem 2 · Equal on Return and Risk?

Given (per year): $R_f = 3.5\%$ , market risk premium $6.5\%$ . Three managers each averaged $11\%$ with volatility $\sigma = 25\%$ ; their betas are $0.4$ , $1.1$ and $1.4$ . Which statement is right?

✅ Correct! A

\beta

0.4

requires only

3.5\% + 2.6\% = 6.1\%

, so

11\%

beats the line by

4.9

points; the other two have

\alpha = +0.35\%

and

\alpha = -1.6\%

❌ Same $\sigma$ , not same risk. The Sharpe ratios are equal, but the SML prices risk by

\beta

: the three are required to earn

6.1\%

10.65\%

and

12.6\%

, so their alphas differ.

❌ Backwards. More market risk raises the required return to

12.6\%

, so the same

11\%

is a shortfall:

\alpha = -1.6\%

❌ On the line is par, not skill.

\alpha = 11\% - 10.65\% = +0.35\%

means this manager earns about what its

\beta

requires; anyone holding T-bills and the market could do that.

❌ Not quite. Compute each manager's required return from its

\beta

, then compare alphas.

Show solution

Required return $3.5\% + \beta \times 6.5\%$ and alpha $11\% - \text{required}$ for each manager:

$\beta = 0.4$ : $3.5\% + 2.6\% = 6.1\%$ , so $\alpha = 11\% - 6.1\% = +4.9\%$
$\beta = 1.1$ : $3.5\% + 7.15\% = 10.65\%$ , so $\alpha = 11\% - 10.65\% = +0.35\%$
$\beta = 1.4$ : $3.5\% + 9.1\% = 12.6\%$ , so $\alpha = 11\% - 12.6\% = -1.6\%$

Measured by $\sigma$ the three are identical, with the same Sharpe ratio $7.5/25 = 0.3$ . Measured against the SML, only the low-beta manager adds return beyond what its market risk requires.

Problem 3 · From Beta Back to Correlation

Given (per year): the market's volatility is $\sigma_m = 17.5\%$ and each of the managers above has $\sigma_p = 25\%$ . Find the correlation with the market implied by $\beta = 0.4$ and by $\beta = 1.4$ .

Correlation for $\beta = 0.4$ ?

Correlation for $\beta = 1.4$ ?

✅ Correct!

\rho = \beta\,\sigma_m/\sigma_p = \beta \times 0.7

0.28

and

0.98

. Same volatility, very different co-movement with the market.

❌ Beta is not correlation.

\beta = \rho\,\sigma_p/\sigma_m

also scales by the volatility ratio

25/17.5

; a correlation can never exceed

1

, while a beta can.

❌ Ratio upside down. Solving

\beta = \rho\,\sigma_p/\sigma_m

for

\rho

gives

\rho = \beta\,\sigma_m/\sigma_p

, with

\sigma_m = 17.5\%

on top; a value above

1

is a sure sign of the flip.

❌ Beta left out.

17.5/25 = 0.7

is the volatility ratio alone, which is the same for every manager; multiply it by each manager's

\beta

❌ Check the first correlation. Use

\rho = \beta\,\sigma_m/\sigma_p

with

\beta = 0.4

❌ Check the second correlation. Use

\rho = \beta\,\sigma_m/\sigma_p

with

\beta = 1.4

Show solution

From $\beta = \rho\,\dfrac{\sigma_p}{\sigma_m}$ , solve for the correlation:

\rho = \beta\,\frac{\sigma_m}{\sigma_p} = \beta \times \frac{17.5}{25} = \beta \times 0.7

\beta = 0.4:\ \rho = 0.4 \times 0.7 = 0.28 \qquad \beta = 1.4:\ \rho = 1.4 \times 0.7 = 0.98

Both funds swing by $25\%$ a year, but only the second moves almost in lockstep with the market. The low-beta fund's volatility is mostly its own, and the SML pays nothing for it.

Problem 4 · The One Case Where Volatility Sets the Price

Given (per year): $R_f = 3.5\%$ , market risk premium $6.5\%$ , $\sigma_m = 17.5\%$ . An efficient portfolio, a mix of T-bills and the market portfolio $M$ , has $\sigma_p = 25\%$ . Find its beta and its required expected return.

What is its beta?

What expected return is it required to earn?

✅ Correct! An efficient portfolio has

\rho = 1

, so

\beta_p = 25/17.5 \approx 1.43

and the SML and the capital market line agree at about

12.79\%

. The three managers share this

\sigma

but have

\rho < 1

, so none of them is required to earn

12.79\%

: their betas, not their volatility, set the bar.

❌ Ratio upside down. With

\rho = 1

\beta_p = \sigma_p/\sigma_m = 25/17.5

;

17.5/25 = 0.7

is the inverse.

❌ Not the market itself. Only a portfolio fully in

M

has

\beta = 1

; a volatility of

25\%

above the market's

17.5\%

means about

143\%

M

, financed by borrowing at

R_f

❌ The whole position counts.

0.43

is only the borrowed part; all of the roughly

143\%

held in

M

carries market risk, so

\beta_p = 25/17.5 \approx 1.43

❌ That is the market's return.

3.5\% + 6.5\% = 10\%

is the requirement for

\beta = 1

; this portfolio carries more market risk than

M

❌ The intercept is missing.

\tfrac{25}{17.5} \times 6.5\% \approx 9.29\%

is only the risk premium; add

R_f = 3.5\%

❌ Ratio upside down.

3.5\% + 0.7 \times 6.5\% = 8.05\%

uses

\sigma_m/\sigma_p

; the multiplier on the premium is

\sigma_p/\sigma_m

❌ Check the beta. For an efficient portfolio

\rho = 1

, so

\beta_p = \sigma_p/\sigma_m

❌ Check the required return. Use

\text{E}[R_p] = R_f + \dfrac{\sigma_p}{\sigma_m}(\text{E}[R_m] - R_f)

Show solution

A mix of T-bills and $M$ moves exactly with the market, so $\rho = 1$ and

\beta_p = \rho\,\frac{\sigma_p}{\sigma_m} = 1 \times \frac{25}{17.5} = \frac{10}{7} \approx 1.43

The capital market line and the SML give the same required return:

\text{E}[R_p] = 3.5\% + \frac{10}{7} \times 6.5\% = 3.5\% + \frac{65}{7}\% \approx 3.5\% + 9.29\% = 12.79\%

The managers in Problem 2 have the same $\sigma_p = 25\%$ but correlations below $1$ , so they are not efficient. For them $\sigma$ overstates the priced risk, and the SML with each manager's own $\beta$ is the right yardstick.

Solved: 0 / 4