Finance-Theory · Unit 18 · Video 6 · Interactive Practice

Hedge Fund Alpha or Hidden Beta? Fund XYZ's 771 Basis Points

IKey Formulas

Formula	Name	What it says
$\text{E}[R_i] = R_f + \beta_i\,(\text{E}[R_m] - R_f)$	Required return (SML)	What the market beta alone calls for; with $\beta_i < 0$ it is below $R_f$
$\alpha_i = R_i - \big(R_f + \beta_i\,(\text{E}[R_m] - R_f)\big)$	Alpha	Return earned minus return required: the vertical distance from the SML (positive above it)
$(1 - \beta_i)\,R_f + \beta_i\,\text{E}[R_m] = R_f + \beta_i\,(\text{E}[R_m] - R_f)$	Do-it-yourself benchmark	T-bills plus the index, mixed to the same beta, earn exactly the required return
$1\% = 100 \text{ bp}$	Basis points	Fund XYZ: $12.54\% - 4.83\% = 7.71\%$ , or $771$ bp a year

Key Insight: The CAPM benchmark prices one risk, the market beta. Liquidity, currency, commodity or any other risk the market portfolio does not represent shows up as alpha: beta dressed up as alpha. A large alpha is the signal to ask which risks produced it.

IIAlpha Is the Gap Above the Line

With $R_f = 2.5\%$ and a $5\%$ premium, the fund's beta fixes its required return; alpha is the vertical gap.

Fund β 0.50 Fund return 9.20%

IIIA Benchmark You Could Build Yourself

Holding $1 - \beta$ in T-bills and $\beta$ in the index earns exactly the SML's required return, at any beta.

Target β −0.40

IVSmooth Is Not the Same as Alpha

In simulated market histories, T-bills are the smoothest line of all and earn $\alpha = 0$ ; volatility alone is not priced risk.

Year 18

💡 The paths are simulated, not Fund XYZ's real record: a hypothetical fund like the one in the quizzes, with $\beta = 0.15$ and an expected return of $9.2\%$ a year, beside an index with an expected return of $7.5\%$ a year and $15\%$ volatility. Which line ends higher changes from history to history; the alpha built into the fund, measured against the SML, is the same in every history.

VQuiz Questions

Problem 1 · The Required Return

Given: a hedge fund with $\beta = 0.15$ , a risk-free rate $R_f = 2.5\%$ and a market risk premium $\text{E}[R_m] - R_f = 5\%$ , all per year. Find the return the CAPM requires of the fund.

✅ Correct!

2.5\% + 0.15 \times 5\% = 3.25\%

a year: a low beta calls for little more than the risk-free rate.

❌ Beta multiplies the premium, not the market return.

0.15 \times 7.5\%

counts the risk-free rate a second time; use

\text{E}[R_m] - R_f = 5\%

❌ That is only the risk premium.

0.15 \times 5\% = 0.75\%

is what the beta adds; the required return starts from

R_f = 2.5\%

❌ That is the market's own return.

7.5\%

is what a beta of

1

requires; this fund's beta is

0.15

❌ Not quite. Use the SML:

R_f

plus beta times the market risk premium.

Show solution

The security market line gives the return the fund's market beta calls for:

\text{E}[R] = R_f + \beta\,(\text{E}[R_m] - R_f) = 2.5\% + 0.15 \times 5\% = 2.5\% + 0.75\% = 3.25\%

The fund is required to earn $3.25\%$ a year, only $0.75$ percentage points above T-bills, because it carries little market risk.

Problem 2 · Alpha, Not Excess Return

Given: the same fund ( $\beta = 0.15$ , $R_f = 2.5\%$ , premium $5\%$ ) averaged a return of $9.2\%$ a year. Find its CAPM alpha.

✅ Correct!

9.2\% - 3.25\% = 5.95\%

, which is

595

basis points a year above the SML.

❌ That is the excess return over T-bills.

9.2\% - 2.5\%

ignores the

0.75\%

the fund's beta also calls for; alpha subtracts the full required return.

❌ That benchmarks the fund against the market.

9.2\% - 7.5\%

would be right only for

\beta = 1

; with

\beta = 0.15

the required return is far lower.

❌ The risk-free rate is missing.

9.2\% - 0.75\%

subtracts only the beta's premium; the required return is

2.5\% + 0.75\%

❌ Not quite. Alpha is the return earned minus the SML-required return.

Show solution

First the required return from the SML, then the gap:

\text{E}[R] = 2.5\% + 0.15 \times 5\% = 3.25\%

\alpha = 9.2\% - 3.25\% = 5.95\% = 595 \text{ bp per year}

The fund sits $5.95$ percentage points above the SML at $\beta = 0.15$ , measured against market risk only.

Problem 3 · The Do-It-Yourself Benchmark

Given: $R_f = 2.5\%$ , $\text{E}[R_m] = 7.5\%$ (the index), and a fund with $\beta = 0.15$ . Find the mix of T-bills and the index with the fund's beta, and the index's contribution to that mix's expected return.

What are the weights?

What does the index contribute?

✅ Correct!

2.125\% + 1.125\% = 3.25\%

: the mix earns exactly the SML's required return, so alpha is measured against a portfolio any investor could build.

❌ Weights reversed. The index weight equals the beta,

0.15

; the rest,

1 - 0.15

, goes to T-bills.

❌ The weights must sum to one.

100\% + 15\%

invests

115\%

of the money; the T-bill weight is

1 - \beta = 85\%

❌ Sign error. A short index position matches a negative beta, as for fund XYZ; this fund's beta is

+0.15

❌ Check the weights. The index weight is

\beta

and the T-bill weight is

1 - \beta

❌ That is the premium the beta adds, not the index's contribution. The index position earns the whole index return,

0.15 \times 7.5\%

❌ That is the T-bills' contribution. The index's piece is

0.15 \times 7.5\%

❌ The benchmark holds the index, not the fund. Use the index's expected return,

7.5\%

❌ Check the contribution. Multiply the index weight by the index's expected return.

Show solution

A mix with weight $w$ in the index and $1 - w$ in T-bills has beta $w \times 1 + (1 - w) \times 0 = w$ , so matching $\beta = 0.15$ needs $w = 0.15$ :

\text{T-bills: } 1 - 0.15 = 85\%, \qquad \text{index: } 15\%

Contributions to the expected return:

0.85 \times 2.5\% = 2.125\%, \qquad 0.15 \times 7.5\% = 1.125\%

2.125\% + 1.125\% = 3.25\% = 2.5\% + 0.15 \times 5\%

The mix earns the SML's required return, so the fund's $595$ bp alpha is what it earned beyond a portfolio with the same market exposure.

Problem 4 · Beta Dressed Up as Alpha

Given: the fund's CAPM alpha is $5.95\%$ a year (return $9.2\%$ , required $3.25\%$ ). A later study estimates that $2.2$ percentage points a year of its return is payment for bearing liquidity risk, which the market portfolio does not represent. Find the most of the alpha that can still be credited to skill.

✅ Correct!

5.95\% - 2.2\% = 3.75\%

at most; other unseen risks, such as currency or commodity exposure, could shrink it further.

❌ The CAPM never charged for liquidity risk. Its benchmark adjusts for market beta only, so the

2.2

points sit inside the

5.95\%

alpha.

❌ Wrong direction. Payment for a risk the benchmark missed is part of the measured alpha; it comes out of the alpha, not on top of it.

❌ The required return is missing.

9.2\% - 2.2\%

still contains the

3.25\%

the market beta calls for; start from the alpha,

5.95\%

❌ Not quite. The liquidity payment was counted as alpha; remove it from the alpha.

Show solution

Measured against the market alone, everything above the SML counts as alpha, including payment for risks the market portfolio does not represent:

5.95\% = \underbrace{2.2\%}_{\text{liquidity risk}} + \underbrace{3.75\%}_{\text{everything else}}

At most $3.75\%$ a year, $375$ bp, is left for skill and insight, and only if the fund bears no other unpriced risk. The $2.2$ points are beta dressed up as alpha.

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