Understand duration and interest rate sensitivity — Stock-Market notes

WHAT is Duration?

WHY two? Macaulay tells you when on average your money comes back (a time). Modified converts that time into a price-change rate — the thing traders actually use.

HOW: Derive the price, then duration, from scratch

Step 1 — Price a bond (first principles). A cash flow $C_t$ arriving in $t$ years is worth $\dfrac{C_t}{(1+y)^t}$ today (discounting). Why this step? Money later is worth less than money now; dividing by $(1+y)^t$ removes the growth the market would otherwise give you.

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

Step 2 — Ask how $P$ changes when $y$ changes. Differentiate $P$ w.r.t. $y$ :

$\frac{dP}{dy} = \sum_{t=1}^{n} \frac{-t\,C_t}{(1+y)^{t+1}} = -\frac{1}{1+y}\sum_{t=1}^{n} \frac{t\,C_t}{(1+y)^{t}}$

Why this step? Each term $C_t(1+y)^{-t}$ differentiates to $-t\,C_t(1+y)^{-t-1}$ (power rule). Factoring out $\frac{1}{1+y}$ makes a familiar sum appear.

Step 3 — Recognise the Macaulay sum. Divide by $P$ :

$\frac{1}{P}\frac{dP}{dy} = -\frac{1}{1+y}\underbrace{\frac{\sum t\,C_t(1+y)^{-t}}{P}}_{=\;D_{mac}}$

Convexity — the correction term (WHY the line bends)

The formula above is a straight-line (linear) approximation. The true price–yield curve is convex (bowed toward the origin). A 2nd-order Taylor expansion gives:

$\frac{\Delta P}{P} \approx -D_{mod}\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2,\qquad C=\frac{1}{P}\frac{d^2P}{dy^2}$

Figure — Understand duration and interest rate sensitivity

Worked Example 1 — A 3-year bond

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

$t$	$C_t$	$PV=C_t/1.05^t$	$t\cdot PV$
1	5	4.7619	4.7619
2	5	4.5351	9.0703
3	105	90.7029	272.109
Σ		100.00	285.94

Why this step? Since coupon = yield, price = par = 100 (a useful sanity check).

$D_{mac}=\frac{285.94}{100}=2.859\text{ yrs},\qquad D_{mod}=\frac{2.859}{1.05}=2.723$

If yields rise $\Delta y=+0.01$ : $\dfrac{\Delta P}{P}\approx -2.723\times0.01=-2.72\%$ . Price ≈ $100\times(1-0.0272)=97.28$ .

Worked Example 2 — Zero-coupon bond

A zero pays only face $100$ at $t=n$ . Then the only cash flow is at time $n$ , so $D_{mac}=n$ exactly. Why? The weighted average of a single date is that date. A 10-year zero has $D_{mac}=10$ → very sensitive. $D_{mod}=10/(1+y)$ .

Common Mistakes

Forecast-then-Verify

Recall Before reading the answer: A 20-year zero-coupon bond,

$y=4\%$ . Predict $D_{mod}$ and the price drop for $\Delta y=+1\%$ . $D_{mac}=20$ . $D_{mod}=20/1.04=19.23$ . Price drop $\approx 19.23\%$ . (Convexity means the actual drop is a bit less.)

Macaulay duration is (in words)?

The present-value-weighted average time (in years) until a bond's cash flows are received.

Modified duration formula in terms of Macaulay?

D_{mod}=D_{mac}/(1+y)

The price-sensitivity rule of thumb?

\Delta P/P \approx -D_{mod}\cdot\Delta y

Why does bond price fall when yields rise?

Future cash flows are discounted by a larger factor

(1+y)^t

, so their present value shrinks.

Duration of a zero-coupon bond of maturity n?

Exactly

n

years (only one cash flow, at time

n

Effect of higher coupon on duration?

Lower duration — cash returns sooner, reducing average payback time.

What does convexity correct?

The curvature the linear duration estimate misses; adds

+\tfrac12 C(\Delta y)^2

Is convexity good or bad for the holder?

Good — actual prices beat the duration line (bigger gains, smaller losses).

Units of Macaulay vs Modified duration?

Macaulay in years; Modified as fractional price change per unit yield.

Recall Feynman: explain to a 12-year-old

Imagine you lend friends money and they pay you back on different days. Duration is the average day you get your money back. If you get most of it back soon, you don't care much when interest rates change — but if you have to wait years, a change in rates shakes up how much that faraway money is worth today a lot. So "long wait" = "shaky price," "short wait" = "steady price."

Connections

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, ek bond basically future cash flows ka bundle hai — coupons aur end mein face value. Aaj ki price nikaalne ke liye har future payment ko discount karte hain, yaani $(1+y)^t$ se divide. Jab market interest rate (yield $y$ ) badhta hai, to har future paisa aaj ke hisaab se sasta ho jaata hai, isliye bond ki price girti hai. Duration yahi batata hai ki 1% yield badhne par price kitna girega — yeh bond ki interest-rate sensitivity hai, aur ise saalon (years) mein naapte hain.

Do type hain: Macaulay duration matlab average kitne saal mein tumhara paisa wapas aata hai (PV ke weights ke saath). Aur Modified duration = $D_{mac}/(1+y)$ , jo seedha price change deta hai: $\Delta P/P \approx -D_{mod}\times\Delta y$ . Jaise agar $D_{mod}=7$ hai aur yield 1% badhta hai, to price lagbhag 7% girega. Zero-coupon bond ka duration exactly uski maturity ke barabar hota hai, kyunki saara paisa ekdum end mein aata hai — isliye woh sabse zyada sensitive hote hain.

Ek important baat: high coupon wale bonds ka duration kam hota hai, kyunki paisa jaldi wapas aa jaata hai — average waiting time chhoti ho jaati hai. Yeh cheez log ulti samajhte hain, isliye yaad rakho: "Longer wait, harder shake."

Aur convexity — yeh duration ki straight-line approximation ki galti ko sudharta hai. Asli price–yield curve bent (convex) hoti hai, aur woh hamesha tangent line ke upar rehti hai. Iska matlab convexity tumhaara dost hai: gains thode bade, aur losses thode chhote ho jaate hain. Bade yield changes ke liye sirf duration se kaam nahi chalta, convexity zaroor add karo.

Test yourself — Equity & Fixed Income

Connections