Understand zero-coupon bonds — Hinglish notes

2.1.10 · Stock-Market › Equity & Fixed Income

WHAT is a zero-coupon bond?

Face value (par) $F$ : maturity par jo rakam milti hai.
Price $P$ : aaj aap kya pay karte ho.
Maturity $T$ : kitne saal baad $F$ milega.
Yield $y$ : price mein baki hua annual return.

WHY do these exist? Kyunki kabhi kabhi ek investor ek single, certain cash amount ek jaani-pehchani future date par chahta hai (jaise 15 saal mein bachche ki college ke liye paise). Koi coupons nahi matlab koi reinvestment ki chinta nahi — aapko bilkul pata hai kya milega aur kab.

HOW is the price derived (from first principles)

Core idea: aaj ka paisa kal ke paise se zyada keemti hai, kyunki aaj ka paisa interest kama sakta hai. Toh ek future $F$ ko aaj tak discount karna padega.

Step 1 — Aage badhao (Grow forward). Agar main $P$ aaj annual rate $y$ par invest karta hoon, toh ek saal baad mere paas hoga: $P (1 + y)$ Kyun? Kyunki main $P$ par interest $y \cdot P$ kamata hoon.

Step 2 — $T$ saal ke liye compound karo. Har saal $(1 + y)$ se multiply hota hai: $P (1 + y)^{T}$ Kyun? Interest par interest milta hai — yahi compounding hai.

Step 3 — Demand karo ki yeh payout ke barabar ho. Ek fair price wo hoti hai jab badha hua amount face value $F$ ke barabar ho: $P (1 + y)^{T} = F$

Step 4 — Price ke liye solve karo. $P = \frac{F}{( 1 + y ) ^{T}}$

WHY the price moves opposite to yield

Dekho $P = F / (1 + y)^{T}$ . Yield $y$ denominator mein hai.

Agar $y$ ↑ (rates badhte hain), denominator bada hoga → $P$ ↓.
Agar $y$ ↓ (rates girte hain), denominator chota hoga → $P$ ↑.

Yeh inverse relationship saari bond math ki dhadkan hai. Zero-coupon bonds isse sabse zyada strongly feel karte hain kyunki saara cash unke dur ke end par aata hai, isliye unki price discount rate ke liye bahut sensitive hoti hai (high duration ≈ maturity $T$ ).

Worked examples

Common mistakes (steel-manned)

Active recall

Recall Quick self-test (answers chhupaao, pehle andaza lagao)

Zero face value se neeche kyun bika jaata hai? → kyunki return discount se aata hai; $P < F$ .
Price formula likho. → $P = F / (1 + y)^{T}$ .
Rates badhte hain — price upar jaati hai ya neeche? → down (yield denominator mein hai).
Long-maturity zeros sabse zyada kyun move karte hain? → saara cash door aata hai; duration ≈ maturity, toh high rate sensitivity.
Price se yield kaise nikaalte ho? → $y = (F / P)^{1/ T} - 1$ .

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek magic gift card hai jo 5 saal mein exactly ₹100 ki hogi. Koi bhi aapko aaj ₹100 mein nahi bechega — yeh toh bewakoofi hogi, kyunki aap 5 saal wait karoge kuch gain ke liye nahi. Isliye woh isse saste mein bechte hain, shayad ₹75 mein. Aap wait karte ho, aur baad mein yeh ₹100 ban jaata hai. Extra ₹25 aapka wait karne ka inaam hai. Woh "abhi sasti price, baad mein poori value" wali card hi zero-coupon bond hai. Agar log chahte hain ki unka paisa kahin aur tezi se badhe (rates badhte hain), toh woh aapki card aur bhi saste mein khareedeinge — toh uski price giregi.

Flashcards

$1000 f a ce,$ 6% $y i e l d, 5 y r z er o k i p r i ce ? :::$ 1000/1.06^5 \approx $747.26 $. Y i e l d s ba d hn e p a r z er o k i p r i ce k y u n g i r t ihai ? ::: Y i e l d$ F/(1+y)^T $k e d e n o mina t or m e inhai; ba d a d e n o mina t or c h o t i p r i ce d e t ahai . L o n g z er os m e in s ab sez y a d a p r i cer i s k k y u nh o t ahai ? ::: S aa r a c a s hma t u r i t y p a r aa t ahai, t o h d u r a t i o n \approx ma t u r i t y \to d i sco u n t r a t e k e l i y e hi g h se n s i t i v i t y . S e mi - ann u a l p r i ce f or m u l a ? :::$ P = F/(1+y/m)^{mT} $j i s m e$ m=2 $. Z er o k i co n t in u o u s - co m p o u n d in g p r i ce ? :::$ P = F e^{-yT} $. A g a r co u p o n s nahihain t o h z er o k a " in t er es t " k ahan se aa t ahai ? ::: D i sco u n t se — p r i ce k i$ P $se$ F $t ak a ccr e t i o n se . K y ah e l d - t o - ma t u r i t y z er o k a p a y o u t cer t ainh o t ahai (d e f a u l t - f r ee i ss u er)? ::: H aan,$ F$ fixed hai; lekin interim market price rates ke saath fluctuate karti hai.

Zero-coupon bond kya hota hai?

Ek bond jo koi periodic coupons nahi deta, face value se discount par becha jaata hai aur maturity par poori face value par redeem hota hai; return = price appreciation.

Zero-coupon price formula (annual)?

P = F / (1 + y)^{T}

Price se yield (annual)?

y = (F / P)^{1/ T} - 1

Connections

Bond Yield and Yield to Maturity
Duration and Interest-Rate Risk
Present Value and Time Value of Money
Coupon Bonds vs Zero-Coupon Bonds
Treasury Bills (T-bills as short zeros)
Compounding Frequency and Effective Annual Rate