Learn alpha and benchmark comparison — Stock-Market notes

WHAT is a benchmark?

WHY we need one: Absolute returns hide two things — (1) the risk-free return you get just for lending money, and (2) the market return you get just for being invested. Skill is what's left over.

HOW alpha is defined — derive it from CAPM

We build alpha from first principles using the Capital Asset Pricing Model (CAPM), which predicts the fair expected return for a given amount of market risk.

Step 1 — The building blocks.

$R_f$ = risk-free rate (return with no risk, e.g. T-bills).
$R_m$ = return of the market/benchmark.
$R_p$ = return of your portfolio.
$\beta$ = sensitivity of your portfolio to market moves.

Step 2 — Excess returns. We only care about return above the risk-free rate, because $R_f$ is free to everyone. Define excess returns: $R_p - R_f \quad\text{(your reward for risk)}, \qquad R_m - R_f \quad\text{(market risk premium)}$

Why this step? Subtracting $R_f$ removes the "free" baseline so we compare only the risky parts.

Step 3 — CAPM's fair prediction. CAPM says your expected excess return should be exactly $\beta$ times the market's excess return: $E[R_p - R_f] = \beta\,(R_m - R_f)$

Why this step? If your portfolio swings 1.5× as hard as the market ( $\beta=1.5$ ), you deserve 1.5× the risk premium — no more, no less. That's the fair price of the risk you took.

Step 4 — Alpha is the shortfall or surplus. Whatever return you got beyond CAPM's fair prediction is skill (or luck). Rearranging:

Why the $\beta$ term matters: A naive comparison is $R_p - R_m$ (raw outperformance). But if you took more risk ( $\beta>1$ ) to beat the market, that extra return was bought, not earned. Alpha corrects for exactly that by scaling the market premium by $\beta$ .

Figure — Learn alpha and benchmark comparison

The chart above is the Security Market Line (SML): CAPM's fair line. Points above the line have positive alpha (vertical distance = $\alpha$ ); points below it have negative alpha.

WHY alpha ≠ raw outperformance

Worked examples

Common mistakes

Forecast-then-verify

Flashcards

What is a benchmark?

A reference portfolio (e.g. an index) used to judge performance relative to a default investable alternative.

Define Jensen's alpha in one formula.

\alpha = (R_p - R_f) - \beta(R_m - R_f)

Why isn't

R_p - R_m

a good measure of skill?

It ignores risk — extra return from higher

\beta

(leverage) is free, not skill.

What does

\beta(R_m-R_f)

represent?

The fair excess return CAPM says you deserve for the market risk you took.

A fund beats the market by 5% but has

\beta=2

with

R_m-R_f=5\%

. Alpha?

Fair excess =

2\times5=10\%

; raw excess =

5+5=10\%

⇒

\alpha=0

. No skill.

On the Security Market Line, what does positive alpha look like?

The portfolio point plots above the SML; alpha is the vertical distance.

Can a fund trailing its index still have positive alpha?

Yes — if it has low

\beta

and beat its (smaller) fair return, e.g. a defensive fund.

What is the alpha of a

\beta=0

market-neutral fund?

Its entire excess return over

R_f

, since it bears no market risk.

Recall Feynman: explain to a 12-year-old

Imagine a race where everyone gets a rolling walkway that moves them forward for free (that's the market). If you finish ahead, that's not impressive if the walkway did the work. Alpha is how far you'd finish ahead of where the walkway alone would carry someone as fast as you. If you ran on a faster walkway (more risk), we expect you further ahead — so beating others isn't enough; you must beat what your walkway should have given you.

Connections

Capital Asset Pricing Model (CAPM) — supplies the fair-return prediction alpha is measured against.
Beta and Systematic Risk — the $\beta$ term that scales the market premium.
Security Market Line — the visual line; alpha = vertical distance from it.
Sharpe Ratio — alternative risk-adjusted measure using total risk (σ) instead of $\beta$ .
Risk-free Rate — the baseline subtracted before computing excess returns.
Efficient Market Hypothesis — the claim that consistent positive alpha is very hard to earn.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, jab tumhara portfolio 15% return deta hai, to sabse pehla sawaal yeh hai — "yeh accha hai ya nahi?" Iska jawab tabhi milega jab tum ise kisi benchmark (jaise Nifty 50 ya S&P 500) se compare karo. Agar market bhi 15% chadha, to tumne koi extra skill nahi dikhaayi — bas lehar pe surf kar liya. Jo return market explain nahi kar sakta, wahi hai alpha — tumhari asli kamaayi.

Formula simple hai: $\alpha = (R_p - R_f) - \beta(R_m - R_f)$ . Pehle risk-free rate $R_f$ hataao, kyunki wo to sabko free milta hai. Fir CAPM kehta hai ki jitna market risk tumne liya (yani jitna tumhara $\beta$ hai), utna fair return tumhe milna hi chahiye. Us fair return se jo upar ho, wahi alpha — tumhari skill. Neeche ho to tumne risk ke hisaab se underperform kiya.

Sabse bada trap yeh hai: "Maine market ko beat kiya, matlab mera alpha positive!" Galat! Agar tumne $\beta = 2$ le liya (double risk, jaise leverage), to market ko beat karna to easy hai — CAPM already double return expect karta hai. Isliye sirf $R_p - R_m$ dekhna dhokha hai; hamesha $\beta$ se adjust karo. Aur ulta bhi — ek defensive fund jo index se peeche reh gaya lekin risk aadha liya, uska alpha positive ho sakta hai.

Yaad rakhne ka trick: "Alpha = apni allotted line se kitna Aage." Security Market Line hai tumhari fair line, aur alpha hai us line se upar ki vertical doori. Yeh concept isliye important hai kyunki real investing mein consistent positive alpha nikaalna bahut mushkil hai — isiliye log index funds pasand karte hain.

Test yourself — Portfolio Theory

Connections