WHY do these exist? Because sometimes an investor wants a single, certain cash amount on a known future date (e.g. money for a child's college in 15 years). No coupons means no reinvestment worry — you know exactly what you get and when.
Look at P=F/(1+y)T. The yield y is in the denominator.
If y ↑ (rates rise), the denominator grows → P ↓.
If y ↓ (rates fall), the denominator shrinks → P ↑.
This inverse relationship is the heartbeat of all bond math. Zero-coupon bonds feel it most strongly because all their cash arrives at the far end, so their price is very sensitive to the discount rate (high duration ≈ maturity T).
Why is a zero sold below face value? → because the return comes from the discount; P<F.
Write the price formula. → P=F/(1+y)T.
Rates rise — does price go up or down? → down (yield is in the denominator).
Why do long-maturity zeros move most? → all cash arrives far out; duration ≈ maturity, so high rate sensitivity.
How do you get yield from price? → y=(F/P)1/T−1.
Recall Feynman: explain to a 12-year-old
Imagine a magic gift card that will be worth exactly ₹100 in 5 years. Nobody sells it to you for ₹100 today — that would be silly, because you'd wait 5 years for nothing. Instead they sell it cheaper, maybe ₹75. You wait, and later it becomes ₹100. The ₹25 extra is your reward for waiting. That "cheaper price now, full value later" card is a zero-coupon bond. If people suddenly want their money to grow faster elsewhere (rates rise), they'll only buy your card even cheaper — so its price drops.
Price of 1000face,6%yield,5yrzero?:::1000/1.06^5 \approx $747.26.Whydoesazero′spricefallwhenyieldsrise?:::YieldisinthedenominatorofF/(1+y)^T;abiggerdenominatorgivesasmallerprice.Whydolongzeroshavethemostpricerisk?:::Allcasharrivesatmaturity,soduration≈maturity→highsensitivitytothediscountrate.Semi−annualpriceformula?:::P = F/(1+y/m)^{mT}withm=2.Continuous−compoundingpriceofazero?:::P = F e^{-yT}.Wheredoesazero′s"interest"comefromiftherearenocoupons?:::Fromthediscount—accretionofpricefromPuptoF.Isaheld−to−maturityzero′spayoutcertain(default−freeissuer)?:::Yes,F$ is fixed; but interim market price fluctuates with rates.
What is a zero-coupon bond?
A bond paying no periodic coupons, sold at a discount to face value and redeemed at full face value at maturity; return = price appreciation.
Dekho, zero-coupon bond ka funda bilkul simple hai: yeh bond beech mein koi interest (coupon) nahi deta. Aap ise aaj sasti price pe kharidte ho, aur maturity pe poora face value milta hai. Jaise ₹1000 wali cheez aaj ₹747 mein mili, aur 5 saal baad seedha ₹1000 mil gaye. Woh ₹253 ka farak hi aapka profit hai — isko discount ya price accretion kehte hain. Coupon nahi milta iska matlab return nahi hai, aisa socho mat; return chhupa hua hai us discount mein.
Formula bhi khud derive ho jaata hai: agar aaj P invest karo rate y pe, to T saal baad woh P(1+y)T ban jaayega. Fair price wahi hai jahan yeh face value F ke barabar ho, yaani P(1+y)T=F, isliye P=F/(1+y)T. Bas denominator dekho — yield upar (rates badhe) to price neeche, yield neeche to price upar. Yeh see-saw relationship har bond ka dil hai.
Ek important baat: zero-coupon bond ka saara cash end mein aata hai, isliye jab rates hilte hain to iski price sabse zyada hilti hai (high duration). Agar aap maturity tak hold karoge to F pakka milega (default-free issuer ho to), lekin beech mein bechna pade to market price rate ke hisaab se upar-neeche hoti rehti hai. Isliye long-maturity zeros zyada risky feel karte hain price ke terms mein.
Exam aur real investing dono mein yeh cheez kaam aati hai — T-bills bhi basically short zeros hain, aur present value / time value of money ka pura concept isi ek formula mein samaaya hua hai. Yaad rakho: "ZERO cash beech mein, ALL discount mein."
