Understand Sortino and Treynor ratios — Stock-Market notes

1. Starting point — recall the Sharpe ratio

WHAT each ratio changes: keep the numerator (excess return), swap the denominator.

Ratio	Denominator	Risk it measures
Sharpe	$\sigma_p$ (total std)	all volatility
Sortino	$\sigma_d$ (downside deviation)	only bad volatility
Treynor	$\beta_p$ (portfolio beta)	only market/systematic risk

2. Sortino Ratio — derive the downside deviation

HOW we build it from scratch. Ordinary variance is: $\sigma^2 = \frac{1}{N}\sum_{i=1}^{N}(R_i - \bar{R})^2$ Every squared term is positive whether $R_i$ is above or below the mean — that's the problem. We only want to penalize shortfalls. Define the shortfall: $\min(0,\; R_i - T)$ This is $0$ when $R_i \ge T$ (good) and negative when $R_i < T$ (bad). Square it, average, root:

3. Treynor Ratio — why $\beta$ instead of $\sigma$

Recall $\beta$ from CAPM: $\beta_p = \dfrac{\text{Cov}(R_p, R_m)}{\sigma_m^2}$ — how much the portfolio moves per unit of market move.

Figure — Understand Sortino and Treynor ratios

4. Worked examples

Worked example Sortino from raw returns

Monthly returns: $\{+8\%, -4\%, +6\%, -10\%, +2\%\}$ . Target $T = 0\%$ . Mean $R_p = \frac{8-4+6-10+2}{5} = 0.4\%$ .

Step 1 — shortfalls only. Keep only returns $<0$ : $-4$ and $-10$ . Why this step? Upside ( $+8,+6,+2$ ) contributes $0$ to downside risk. Step 2 — square & average over all N. $\sigma_d^2 = \frac{(-4)^2 + (-10)^2}{5} = \frac{16+100}{5}=23.2$ . Why divide by N=5, not by 2? Convention: we still average over the full sample so the frequency of losses matters, not just their size. Step 3 — root. $\sigma_d = \sqrt{23.2} \approx 4.82\%$ . Step 4 — ratio. $\text{Sortino} = \frac{0.4 - 0}{4.82} \approx 0.083$ .

5. Common mistakes (steel-manned)

6. Active recall

Recall Test yourself (open after attempting)

Numerator is the same for all three ratios — what is it? → excess return $R_p - R_f$ (or $R_p - T$ ).
Sortino denominator? → downside deviation $\sigma_d$ .
Treynor denominator? → beta $\beta_p$ .
When is Treynor the right choice? → when the portfolio is well-diversified.
Why prefer Sortino over Sharpe? → it ignores upside volatility, which investors don't fear.

Recall Feynman: explain to a 12-year-old

Imagine grading a rollercoaster on how "scary" it is. Sharpe counts every bump as scary — even the fun ones that fling you upward. That's silly, you want the up-flings! Sortino only counts the scary drops. Treynor says: some bumps happen to every ride in the park (that's the "market"), and some bumps you could avoid by picking a smarter track — Treynor only charges you for the park-wide bumps you can't dodge. All three still ask: "how much fun (extra return) do I get for each unit of scariness?"

7. Flashcards

What do Sortino, Treynor and Sharpe all share in the numerator?

The excess return

R_p - R_f

(Sortino may use a target

T

Sortino ratio formula

(R_p - T)/\sigma_d

where

\sigma_d

is downside deviation.

Downside deviation formula

\sigma_d=\sqrt{\frac{1}{N}\sum[\min(0,R_i-T)]^2}

Why does Sortino only use downside deviation?

Upside volatility helps investors, so only bad (below-target) volatility should count as risk.

Treynor ratio formula

(R_p - R_f)/\beta_p

Why does Treynor use beta not total std?

Only systematic (undiversifiable) risk should be rewarded; diversifiable risk can be removed for free.

Decomposition of total variance

\sigma_p^2 = \beta_p^2\sigma_m^2 + \sigma_\varepsilon^2

(systematic + diversifiable).

When is Treynor the appropriate ratio?

For well-diversified portfolios where unsystematic risk ≈ 0.

In downside deviation, do you divide by N or by number of losses?

By total N (standard convention).

If Sortino > Sharpe for a fund, what does that imply?

Its volatility is mostly upside, so Sharpe understated performance.

Definition of beta

\beta_p=\mathrm{Cov}(R_p,R_m)/\sigma_m^2

, sensitivity of portfolio to market moves.

8. Connections

Sharpe Ratio — parent formula; Sortino & Treynor are variants.
CAPM — source of $\beta$ used in Treynor.
Systematic vs Unsystematic Risk — justifies Treynor's denominator.
Diversification — why unsystematic risk vanishes.
Standard Deviation and Variance — foundation for downside deviation.
Risk-Adjusted Performance Measures — the family these belong to.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, teeno ratios — Sharpe, Sortino, Treynor — ek hi sawal poochte hain: "har unit risk ke badle mujhe kitna extra return mil raha hai?" Numerator sabme same hota hai: excess return, yaani $R_p - R_f$ . Farq sirf denominator ka hai, matlab "risk" ko define kaise karte hain.

Sortino kehta hai ki upar jaana (upside) toh khushi ki baat hai, use risk kyun ginein? Toh yeh sirf downside deviation count karta hai — jo returns target se neeche gire, unhi ko square karke risk maanta hai. Formula ka $\min(0, R_i - T)$ ka jugaad yahi karta hai — upar wale returns ko zero bana deta hai. Isliye agar kisi fund ka volatility zyada upside ki wajah se hai, toh Sortino uski asli value dikha deta hai jo Sharpe chhupa deta hai.

Treynor ka logic thoda alag hai. Total risk ko do part me todo: systematic (market wala, jise $\beta$ pakadta hai) aur unsystematic (jise diversification se free me hata sakte ho). Ab jab tum diversify karke unsystematic risk ko zero kar sakte ho, toh reward sirf market risk ke liye milna chahiye. Isliye Treynor denominator me $\beta$ use karta hai, $\sigma$ nahi. Yaad rakho — Treynor tabhi valid hai jab portfolio well-diversified ho.

Yaad rakhne ka trick: "Same upar, alag neeche." Sortino = Shortfall, Treynor = The market (beta), Sharpe = Standard deviation. Bas denominator badalta hai, kaam ka concept ek hi rehta hai.

Test yourself — Portfolio Theory

Connections