5.5.8
Sharpe ratio yeh jawab deta hai: "Har ek unit risk lene ke badle mujhe kitna extra return milta hai?" Yeh reward-to-variability ratio hai. Do portfolios dono 12% return kar sakte hain, lekin agar ek bahut zyada upar-neeche jaata hai aur doosra stable rehta hai, toh stable wala behtar hai — Sharpe uski tarif karta hai.
Intuition Yeh kis problem ko solve karta haiRaw return ek jhooth bolta hai . Ek fund "20% last year!" advertise kare toh yeh nahi bataata ki raat ko chain ki neend aayegi ya nahi. Sirf return do cheezein ignore karta hai:
Risk — woh 20% sab kuch gambling mein laga ke bhi aa sakta tha.Wo free lunch jo tune skip kiya — tu cash ko T-bills mein risk-free rate r f zero risk ke saath rakh sakta tha. Sirf woh return jo r f zyada hai, wahi risk lene ka asli reward hai.
Toh humein ek aisa number chahiye jo bataye: khatra ke har unit ke badle, manager ne actually kitna bonus return deliver kiya?
Ek portfolio ka Sharpe ratio S R p r f σ p S = σ p R p − r f
Numerator R p − r f excess return (a.k.a. risk premium).
Denominator σ p volatility (returns ka standard deviation) — yeh hamaara total risk ka measure hai.
Intuition Geometric meaningEk graph par jisme y-axis par expected return aur x-axis par risk ho, Sharpe ratio risk-free point ( 0 , r f ) ( σ p , R p ) line ka slope hai. Jitni steep line ⇒ utna achha deal. Efficient frontier ko touch karne wali sabse steep achievable line Capital Market Line aur tangency portfolio deti hai.
Worked example Example 1 — Basic comparisonFund A: R p = 12% σ p = 10% R p = 12% σ p = 20% r f = 2%
S A = 10 12 − 2 = 1.0 Kyun? 10% excess return over 10% risk = 1 unit reward per unit risk.
S B = 20 12 − 2 = 0.5 Kyun? Same reward, double wobble ⇒ aadhi quality.
Verdict: Same return, lekin A do guna behtar hai. Sharpe woh cheez expose karta hai jo raw return chupaata hai.
Worked example Example 2 — Jab zyada return bura hoFund C: R p = 25% σ p = 40% R p = 8% σ p = 6% r f = 2%
S C = 40 25 − 2 = 0.575 S D = 6 8 − 2 = 1.0
D kyun jeeta? Uska return chhota hai lekin almost risk-free hai . Khature ke har unit ke badle, D kahin zyada deta hai. Ek leveraged D (scale up karne ke liye borrow karo) C ka return kam risk par achieve kar sakta hai.
Worked example Example 3 — AnnualizingEk strategy ka monthly excess return 0.8% σ = 3% S = 0.8/3 = 0.267 S annual = 0.267 × 12 = 0.267 × 3.464 = 0.924 Kyun? Barah mahine ka edge noise se tez stack hota hai, 12
Common mistake "Zyada return ka matlab hamesha zyada Sharpe hota hai."Kyun sahi lagta hai: Return hi goal hai, toh zyada return hona chahiye behtar. Galti: Sharpe risk se divide karta hai. 25% return wild swings ke saath 8% steady return se haar sakta hai (Example 2). Fix: hamesha ratio dekho, sirf numerator kabhi nahi.
Common mistake "Denominator mein varianceσ p 2 Kyun sahi lagta hai: Variance stats mein standard risk measure hai. Galti: variance ke squared units hote hain, toh ratio dimensionless nahi hota aur reward-per-unit-risk interpret nahi hota. Fix: standard deviation σ p
r f Kyun sahi lagta hai: Log ek step bachane ke liye R p / σ p Galti: tum manager ko us return ka credit de doge jo waise bhi risk-free milta . Fix: pehle hamesha r f excess hai.
Common mistake "Monthly Sharpe ko annualize karne ke liye bas 12 se multiply karo."Kyun sahi lagta hai: Returns ×12 se annualize hote hain, toh Sharpe bhi hoga. Galti: denominator (σ 12 Fix: 12
Intuition Jahan Sharpe toot jaata hai
Yeh maanta hai ki returns roughly symmetric/normal hain — yeh upside volatility ko utna hi penalize karta hai jitna downside ko. Skewed strategies ke liye Sortino ratio use karo (sirf downside deviation).
Aisi strategies se gamed ho sakta hai jo tail risk bechti hain (steady gains, rare crashes).
Meaningless ho jaata hai agar excess return negative ho (ranking ulti taraf se kaam karti hai).
Recall Khud test karo (answers chupaao)
Sharpe ratio ke numerator mein kya represent hota hai, aur r f
Denominator mein variance nahi, standard deviation kyun?
Do funds 12% return karte hain. Phir bhi ek "behtar" kaise ho sakta hai?
Monthly Sharpe ratio ko annualize kaise karte hain? Square root kyun?
Return-vs-risk plot par geometrically Sharpe ratio kya hai?
Recall Feynman: ek 12-saal ke bache ko samjhaoSocho do ice-cream waale dono ₹100 profit roz kmaate hain. Ek calmly bechta hai aur hamesha lagbhag ₹100 kamata hai. Doosra kabhi ₹300 kamata hai aur kabhi ₹100 lose karta hai — bahut upar-neeche. Dono average ₹100 karte hain, lekin calm wala smarter hai kyunki tum us par trust kar sakte ho. Sharpe ratio ek score hai: jo extra paise tumne ek bilkul safe piggy bank se zyada kamaye, use lo, phir divide karo se ki tumhari kamayi kitni "jumpy" thi. Bada score = tune accha paisa kamaya bina darne wale rollercoaster ke.
"Excess over Wobble." Top = safe se zyada extra return (R p − r f σ p "Root the periods" (× n
Capital Market Line — r f Tangency Portfolio — woh portfolio jiska maximum Sharpe ratio hota hai.Efficient Frontier — Sharpe uska best point pick karta hai.Standard Deviation as Risk — denominator ki foundation.Risk-Free Rate — woh baseline jisse excess return measure hoti hai.Sortino Ratio — Sharpe ka downside-only cousin.Capital Asset Pricing Model — jahan excess-return thinking generalize hoti hai.
Sharpe ratio formula S = ( R p − r f ) / σ p
Risk-free rate kyun subtract karte hain? Sirf r f r f
Denominator mein variance nahi std dev kyun? Std dev return ke same units mein hai, toh ratio dimensionless hota hai (reward per unit risk).
Do funds, same 12% return — ek behtar kaise? Jiska σ p
Monthly Sharpe ratio ko annualize kaise karte hain? 12 se multiply karo (generally
n periods/year ke liye
n ).
Annualizing mein n n Return
n ke saath scale hota hai, std dev
n ke saath; ratio
n / n = n ke saath scale hota hai.
Sharpe ratio ka geometric meaning? Return-vs-risk plot par ( 0 , r f ) ( σ p , R p )
Kaunsa portfolio Sharpe ratio maximize karta hai? Tangency portfolio (efficient frontier ko touch karta hai, Capital Market Line define karta hai).
Skewed returns ke liye Sharpe ka alternative? Sortino ratio, jo sirf downside deviation se divide karta hai.
Sharpe ki main assumption/weakness? Symmetric (normal) returns assume karta hai; upside volatility ko penalize karta hai aur tail-risk selling se gamed ho sakta hai.
misleading hai, risk ignore karta hai
frontier ko touch karta hai
Risk-adjusted metric chahiye
