Understand Sortino and Treynor ratios — Hinglish notes

5.5.9 · Stock-Market › Portfolio Theory

Intuition Bada picture

Dono ratios wahi core question ka jawab dete hain jo Sharpe ratio poochta hai — "maine risk ke har unit ke badle kitna extra return kamaya?" — lekin ye dono "risk" ki definition ko denominator mein badal dete hain.

Sortino: sirf downside risk (buri volatility) count karta hai, kyunki upside surprises tumhe hurt nahi karte.
Treynor: sirf systematic risk ( $β$ , market risk) count karta hai, kyunki diversifiable risk free mein eliminate ho sakta hai.

YE KYUN MATTER KARTA HAI: Sharpe ek investment ko upar jaane ke liye utna hi punish karta hai jitna neeche jaane ke liye. Ye galat lagta hai — tumhe upside pasand hai! Aur Sharpe total volatility use karta hai, lekin ek well-diversified investor sirf us risk ki parwah karta hai jo wo diversify nahi kar sakta. Ye do ratios un do blind spots ko fix karte hain.

1. Starting point — recall the Sharpe ratio

HAR RATIO KYA CHANGE KARTA HAI: numerator (excess return) wahi rakho, denominator badlo.

Ratio	Denominator	Kaun sa risk measure karta hai
Sharpe	$σ_{p}$ (total std)	saari volatility
Sortino	$σ_{d}$ (downside deviation)	sirf buri volatility
Treynor	$β_{p}$ (portfolio beta)	sirf market/systematic risk

2. Sortino Ratio — downside deviation derive karo

HUM ISSE SCRATCH SE KAISE BANATE HAIN. Ordinary variance hai: $σ^{2} = \frac{1}{N} \sum_{i = 1}^{N} (R_{i} - \overset{ˉ}{R})^{2}$ Har squared term positive hai chahe $R_{i}$ mean se upar ho ya neeche — yahi problem hai. Hum sirf shortfalls ko penalize karna chahte hain. Shortfall define karo: $min (0, R_{i} - T)$ Ye $0$ hai jab $R_{i} \geq T$ (acha) aur negative hai jab $R_{i} < T$ (bura). Ise square karo, average karo, root lo:

3. Treynor Ratio — $σ$ ki jagah $β$ kyun

CAPM se $β$ yaad karo: $β_{p} = \frac{Cov ( R _{p} , R _{m} )}{σ _{m}^{2}}$ — portfolio market ke har unit move par kitna move karta hai.

Figure — Understand Sortino and Treynor ratios

4. Worked examples

5. Common mistakes (steel-manned)

6. Active recall

Recall Khud test karo (try karne ke baad kholo)

Numerator teeno ratios mein same hai — wo kya hai? → excess return $R_{p} - R_{f}$ (ya $R_{p} - T$ ).
Sortino denominator? → downside deviation $σ_{d}$ .
Treynor denominator? → beta $β_{p}$ .
Treynor sahi choice kab hai? → jab portfolio well-diversified ho.
Sortino ko Sharpe se prefer kyun karein? → ye upside volatility ignore karta hai, jo investors ko darr nahi lagata.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho ek rollercoaster ko "kitna scary hai" ke basis par grade kar rahe ho. Sharpe har bump ko scary count karta hai — chahe wo fun wala upar ka jhonka ho. Ye silly hai, tumhe upar ke jhonke pasand hain! Sortino sirf scary girna count karta hai. Treynor kehta hai: kuch bumps park ki har ride mein hote hain (ye "market" hai), aur kuch bumps tum smarter track chunkar avoid kar sakte ho — Treynor tumhe sirf un park-wide bumps ke liye charge karta hai jo tum dodge nahi kar sakte. Teeno phir bhi yehi poochhte hain: "mazze (extra return) ke liye mujhe scariness ke har unit par kitna milta hai?"

7. Flashcards

Sortino, Treynor aur Sharpe teeno numerator mein kya share karte hain?

Excess return

R_{p} - R_{f}

(Sortino ek target

T

bhi use kar sakta hai).

Sortino ratio formula

(R_{p} - T) / σ_{d}

jahan

σ_{d}

downside deviation hai.

Downside deviation formula

σ_{d} = \frac{1}{N} \sum [min (0, R_{i} - T)]^{2}

Sortino sirf downside deviation kyun use karta hai?

Upside volatility investors ki help karta hai, isliye sirf buri (target se neeche) volatility risk count honi chahiye.

Treynor ratio formula

(R_{p} - R_{f}) / β_{p}

Treynor total std ki jagah beta kyun use karta hai?

Sirf systematic (undiversifiable) risk ke liye reward milna chahiye; diversifiable risk free mein remove ho sakta hai.

Total variance ka decomposition

σ_{p}^{2} = β_{p}^{2} σ_{m}^{2} + σ_{ε}^{2}

(systematic + diversifiable).

Treynor appropriate ratio kab hai?

Well-diversified portfolios ke liye jahan unsystematic risk ≈ 0 ho.

Downside deviation mein N se divide karte hain ya losses ki sankhya se?

Total N se (standard convention).

Agar kisi fund ke liye Sortino > Sharpe ho, toh iska kya matlab hai?

Uski volatility zyaatar upside hai, isliye Sharpe ne performance understate ki.

Beta ki definition

β_{p} = Cov (R_{p}, R_{m}) / σ_{m}^{2}

, portfolio ki market moves ke saath sensitivity.

8. Connections

Sharpe Ratio — parent formula; Sortino & Treynor variants hain.
CAPM — $β$ ka source jo Treynor mein use hota hai.
Systematic vs Unsystematic Risk — Treynor ke denominator ko justify karta hai.
Diversification — kyun unsystematic risk khatam ho jaata hai.
Standard Deviation and Variance — downside deviation ki neenv.
Risk-Adjusted Performance Measures — wo family jinse ye belong karte hain.