Understand duration and interest rate sensitivity

2.1.8 · Stock-Market › Equity & Fixed Income

WHAT is Duration?

WHY do? Macaulay batata hai ki average mein aapka paisa kab wapas aayega (ek time). Modified us time ko ek price-change rate mein convert karta hai — jo cheez traders actually use karte hain.

HOW: Price, phir Duration, scratch se derive karo

Step 1 — Bond ko price karo (first principles). Ek cash flow $C_{t}$ jo $t$ years mein aane wala hai, aaj $\frac{C _{t}}{( 1 + y ) ^{t}}$ worth hai (discounting). Yeh step kyun? Baad ka paisa abhi ke paison se kam valuable hota hai; $(1 + y)^{t}$ se divide karne par woh growth hat jaati hai jo market otherwise deta.

$P = \sum_{t = 1}^{n} \frac{C _{t}}{( 1 + y ) ^{t}}$

Step 2 — Poochho ki jab $P$ change hota hai toh $y$ kaise change hota hai. $P$ ko $y$ ke respect mein differentiate karo:

$\frac{d P}{d y} = \sum_{t = 1}^{n} \frac{- t C _{t}}{( 1 + y ) ^{t + 1}} = - \frac{1}{1 + y} \sum_{t = 1}^{n} \frac{t C _{t}}{( 1 + y ) ^{t}}$

Yeh step kyun? Har term $C_{t} (1 + y)^{- t}$ differentiate hokar $- t C_{t} (1 + y)^{- t - 1}$ banta hai (power rule). $\frac{1}{1 + y}$ factor out karne par ek jaana-pehchaana sum appear hota hai.

Step 3 — Macaulay sum ko pehchano. $P$ se divide karo:

$\frac{1}{P} \frac{d P}{d y} = - \frac{1}{1 + y} = D_{ma c} \frac{\sum t C _{t} ( 1 + y ) ^{- t}}{P}$

Convexity — correction term (WHY line bend karti hai)

Upar wala formula ek straight-line (linear) approximation hai. Sacchi price–yield curve convex hoti hai (origin ki taraf bowed). 2nd-order Taylor expansion deta hai:

$\frac{Δ P}{P} \approx - D_{m o d} Δ y + \frac{1}{2} C (Δ y)^{2}, C = \frac{1}{P} \frac{d ^{2} P}{d y ^{2}}$

Figure — Understand duration and interest rate sensitivity

Worked Example 1 — Ek 3-year bond

Face $= 100$ , annual coupon $= 5%$ (toh $C = 5$ ), yield $y = 5%$ . Cash flows: $5, 5, 105$ .

$t$	$C_{t}$	$P V = C_{t} /1.0 5^{t}$	$t \cdot P V$
1	5	4.7619	4.7619
2	5	4.5351	9.0703
3	105	90.7029	272.109
Σ		100.00	285.94

Yeh step kyun? Kyunki coupon = yield, toh price = par = 100 (ek useful sanity check).

$D_{ma c} = \frac{285.94}{100} = 2.859 yrs, D_{m o d} = \frac{2.859}{1.05} = 2.723$

Agar yields badhein $Δ y = + 0.01$ : $\frac{Δ P}{P} \approx - 2.723 \times 0.01 = - 2.72%$ . Price ≈ $100 \times (1 - 0.0272) = 97.28$ .

Worked Example 2 — Zero-coupon bond

Ek zero sirf $t = n$ par face $100$ deta hai. Tab sirf ek hi cash flow hai time $n$ par, toh $D_{ma c} = n$ exactly. Kyun? Ek single date ka weighted average wahi date hoti hai. 10-year zero ka $D_{ma c} = 10$ hota hai → bahut sensitive. $D_{m o d} = 10/ (1 + y)$ .

Common Mistakes

Forecast-then-Verify

Recall Answer padhne se pehle: Ek 20-year zero-coupon bond,

$y = 4%$ . $D_{m o d}$ aur $Δ y = + 1%$ ke liye price drop predict karo. $D_{ma c} = 20$ . $D_{m o d} = 20/1.04 = 19.23$ . Price drop $\approx 19.23%$ . (Convexity ki wajah se actual drop thoda kam hoga.)

Macaulay duration kya hai (words mein)?

Ek bond ke cash flows milne tak ka present-value-weighted average time (years mein).

Modified duration formula Macaulay ke terms mein?

D_{m o d} = D_{ma c} / (1 + y)

Price-sensitivity ka rule of thumb?

Δ P / P \approx - D_{m o d} \cdot Δ y

Yield badhne par bond price kyun girti hai?

Future cash flows bade factor

(1 + y)^{t}

se discount hote hain, toh unki present value kam ho jaati hai.

Maturity

n

ke zero-coupon bond ki duration?

Exactly

n

years (sirf ek cash flow, time

n

par).

Higher coupon ka duration par effect?

Duration kam hoti hai — cash jaldi wapas aata hai, average payback time kam ho jaata hai.

Convexity kya correct karta hai?

Woh curvature jo linear duration estimate miss karta hai;

+ \frac{1}{2} C (Δ y)^{2}

add karta hai.

Holder ke liye convexity acchi hai ya buri?

Acchi — actual prices duration line se behtar hoti hain (bade gains, chhote losses).

Macaulay vs Modified duration ki units?

Macaulay years mein; Modified fractional price change per unit yield ke roop mein.

Recall Feynman: ek 12-saal ke bacche ko explain karo

Socho tumne doston ko paisa udhar diya aur woh alag-alag dino mein wapas karte hain. Duration woh average din hai jab tumhara paisa wapas aata hai. Agar zyada paisa jaldi wapas aaye, toh interest rates change hone par zyada fark nahi padta — lekin agar saalon tak wait karna pade, toh rate change hone se us door ke paison ki aaj ki value bahut zyada hilti hai. Toh "lamba wait" = "hili hui price," "chhota wait" = "stable price."

Connections

Bond Pricing and Present Value
Yield to Maturity (YTM)
Interest Rate Risk
Convexity
Coupon Rate vs Yield
Term Structure of Interest Rates