Understand performance metrics (CAGR, max DD)

6.2.9 · Stock-Market › Backtesting Frameworks

Hum Kya Measure Kar Rahe Hain?

Jab tum ek trading strategy ko backtest karte ho, tumhe time ke saath portfolio values ka ek sequence milta hai: $V_{0}, V_{1}, V_{2}, \dots, V_{T}$ . "Portfolio ₹100,000 se ₹250,000 ho gaya" jaise raw numbers bina context ke bekar hain:

Kitna time laga?
Kya yeh smoothly grow hua ya terrifying drops ke saath?

Performance metrics results ko standardize karte hain taaki tum strategies compare kar sako, indices ke against benchmark kar sako, aur risk-adjusted returns assess kar sako.

CAGR (Compound Annual Growth Rate)

Derivation First Principles Se

Compound interest formula se shuru karo. Agar tum $V_{0}$ invest karte ho constant annual rate $r$ par $n$ saal ke liye:

$V_{n} = V_{0} (1 + r)^{n}$

Hume pata hai $V_{0}$ (starting capital), $V_{n}$ (ending capital), aur $n$ (saal). $r$ ke liye solve karo:

$\frac{V _{n}}{V _{0}} = (1 + r)^{n}$

$n$ -th root lo:

$(1 + r) = (\frac{V _{n}}{V _{0}})^{1/ n}$

Isliye:

1 kyun subtract karte hain? Kyunki $(1 + r)$ mein original capital bhi shaamil hai; 1 subtract karne se sirf rate of return alag ho jaati hai.

Exponent $1/ n$ kyun? Hum compounding process ko reverse kar rahe hain—geometric mean le rahe hain.

Figure — Understand performance metrics (CAGR, max DD)

Worked Example 1: Simple CAGR Calculation

Setup: Tumhara portfolio 3 saal mein ₹1,00,000 se ₹1,80,000 ho gaya.

Step 1: Values identify karo.

$V_{start} = 100000$
$V_{end} = 180000$
$n = 3$

Step 2: Formula apply karo. $CAGR = (\frac{180000}{100000})^{1/3} - 1$

Yeh step kyun? Hume ending aur starting value ka ratio chahiye—yahi total growth multiplier hai.

Step 3: Calculate karo. $= (1.8)^{0.333} - 1 = 1.2164 - 1 = 0.2164$

Result: CAGR = 21.64% per year.

Interpretation: Bhale hi portfolio har saal exactly 21.64% nahi bada (shayad +30%, +10%, +25%), lekin equivalent smooth annual rate 21.64% hai.

Worked Example 2: Fractional Years

Setup: Ek strategy 520 days tak chalti hai (years ki neat sankhya nahi). Portfolio: ₹50,000 → ₹68,000.

Step 1: Days ko years mein convert karo. $n = \frac{520}{365} = 1.4247 years$

Yeh step kyun? CAGR ek annual rate hai, isliye time years mein hona chahiye.

Step 2: Formula apply karo. $CAGR = (\frac{68000}{50000})^{1/1.4247} - 1$

$= (1.36)^{0.7020} - 1 = 1.2376 - 1 = 0.2376$

Result: CAGR ≈ 23.76%.

Maximum Drawdown (Max DD)

Yeh Kyun Matter Karta Hai

CAGR seductive ho sakta hai—"30% annual returns!" bahut acha lagta hai. Lekin agar yeh achieve karne ke liye 70% drawdown survive karna padta, toh zyaadatar retail investors panic mein exit kar dete. Max DD emotional pain aur capital preservation risk ko capture karta hai.

Derivation: Max DD Kaise Calculate Karein

Apne equity curve mein har point $i$ ke liye:

Running Maximum: Abhi tak dekhi gayi sabse zyada portfolio value track karo: $Peak_{i} = max (V_{0}, V_{1}, \dots, V_{i})$
Drawdown at $i$ : Hum peak se kitna neeche hain abhi? $DD_{i} = \frac{V _{i} - Peak _{i}}{Peak _{i}} = \frac{V _{i}}{Peak _{i}} - 1$

Yeh formula kyun? Drawdown ek percentage loss from the peak hai. Agar peak ₹100 thi aur current ₹80 hai, toh DD = -20%.
Maximum Drawdown: Sabhi $i$ mein sabse zyada negative drawdown: $Max DD = min (DD_{0}, DD_{1}, \dots, DD_{T})$

(Hum $min$ lete hain kyunki drawdowns negative hote hain.)

Worked Example 3: Max DD Step-by-Step Compute Karna

Setup: 7 days mein daily portfolio values:

Day	Value (₹)
0	100,000
1	105,000
2	110,000
3	102,000
4	98,000
5	108,000
6	106,000

Step 1: Running peak compute karo.

Day	Value	Peak
0	100000	100000
1	105000	105000
2	110000	110000
3	102000	110000
4	98000	110000
5	108000	110000
6	106000	110000

Peak kyun track karein? Hume pata hona chahiye us point tak ki sabse zyada value taaki drop measure kar sakein.

Step 2: Har day par drawdown compute karo.

$DD_{i} = \frac{V _{i} - Peak _{i}}{Peak _{i}}$

Day	Value	Peak	Drawdown
0	100000	100000	0.00%
1	105000	105000	0.00%
2	110000	110000	0.00%
3	102000	110000	-7.27%
4	98000	110000	-10.91%
5	108000	110000	-1.82%
6	106000	110000	-3.64%

Yeh step kyun? Har drawdown dikhata hai hum recent peak se kitna gire hain.

Step 3: Maximum drawdown dhundo.

$Max DD = max (∣ - 0.00∣, ∣ - 0.00∣, ∣ - 0.00∣, ∣ - 7.27∣, ∣ - 10.91∣, ∣ - 1.82∣, ∣ - 3.64∣) = 10.91%$

Result: Max DD = 10.91% (Day 4 par hua).

CAGR aur Max DD Ko Connect Karna

Recall Ek 12-Saal Ke Bachche Ko Explain Karo

Socho tum ek paudha ugaa rahe ho. CAGR aise poochhna hai: "Agar mera paudha har saal same amount badha, toh woh yearly growth kya hoti?" Bhale hi ek saal zyada bada aur dusre saal kam, CAGR sab smooth karke ek steady number deta hai—jaise kisi race ki average speed nikalte hain.

Max Drawdown sabse dara dene wala moment hai. Shayad tumhara paudha 12 inch lamba tha, phir ek toofan mein 8 inch par toot gaya. Woh 4 inch ki girawat (12 ka 33%) tumhara "maximum drawdown" hai—recover hone se pehle ki sabse buri damage. Bhale hi paudha baad mein 20 inch tak bada, phir bhi tum woh 8 inch wala dara dene wala moment yaad rakhte ho. Max DD yahi track karta hai—raaste mein tumhe aayi sabse buri dar.

Connections

6.2.01-Introduction-to-backtesting — Standardized metrics ki zaroorat kyun hai
6.2.08-Equity-curve-analysis — Max DD equity curves se derive hota hai
6.3.01-Sharpe-ratio-and-risk-adjusted-returns — CAGR/Max-DD Calmar tak le jaata hai; Sharpe volatility use karta hai
7.1.04-Position-sizing-and-Kelly-criterion — Max DD batata hai kitna leverage safe hai
8.2.02-Psychological-trading-discipline — Max DD survive karne ke liye mental toughness chahiye

#flashcards/stock-market

CAGR ka full form kya hai aur yeh kaun sa sawaal answer karta hai?

Compound Annual Growth Rate. Yeh answer karta hai: "Kaun sa fixed yearly return mera total growth replicate karta agar returns smoothly compound hote?"

CAGR formula kya hai?

CAGR = (V_end / V_start)^(1/n) - 1, jahaan n saalon ki sankhya hai.

CAGR formula mein 1 kyun subtract karte hain?

Kyunki (V_end/V_start)^(1/n) (1+r) deta hai, jisme principal bhi hai. 1 subtract karne se rate of return r alag ho jaati hai.

Ek portfolio 2 saal mein ₹50,000 se ₹80,000 ho jaata hai. CAGR kya hai?

(80000/50000)^(1/2) - 1 = (1.6)^0.5 - 1 = 1.2649 - 1 = 0.2649 = 26.49%.

Maximum Drawdown kya measure karta hai?

Backtest period mein portfolio value ki sabse badi peak-to-trough decline, percentage mein. Yeh kisi bhi prior high point se sabse bura loss measure karta hai.

Time i par drawdown kaise calculate karte hain?

DD_i = (V_i - Peak_i) / Peak_i, jahaan Peak_i time i tak ki sabse zyada portfolio value hai.

CAGR ke aage Maximum Drawdown kyun important hai?

CAGR returns dikhata hai lekin volatility aur dard chhupa leta hai. Max DD worst loss reveal karta hai jo tumhe emotionally survive karna padega, jo decide karta hai ki strategy psychologically tradeable hai ya nahi.

Calmar Ratio kya hai?

Calmar Ratio = CAGR / |Max DD|. Yeh risk-adjusted return measure karta hai: zyada better hai. Example: 20% CAGR with 10% Max DD se Calmar = 2.0 milta hai.

Annual returns ko arithmetically average karne ki jagah CAGR use na karne mein kya galat hai?

Returns add nahi, compound hote hain. Arithmetic mean compounding ignore karta hai. Example: +50% phir -50% arithmetic mean 0% deta hai, lekin actual result -25% hai (100→150→75). CAGR geometric mean use karta hai.

Portfolio values: ₹100k → ₹120k (peak) → ₹90k → ₹110k. Max DD kya hai?

Peak = 120k. Lowest = 90k. DD = (90-120)/120 = -25%. Max DD = 25%.

Agar ek strategy 730 days tak chali, toh CAGR ke liye ise years mein kaise convert karein?

n = 730/365 = 2.0 years.

Sach ya jhooth: Max DD = 40% matlab portfolio end mein overall 40% lose kar gaya.

Jhooth. Max DD sabse bura moment hai, final result nahi. Portfolio recover ho sakta hai ya overall gain bhi kar sakta hai, jabki kisi point par 40% DD experience kiya ho.