WHAT is happening: A bond promises you a stream of unchangeable cash — coupons plus the face value at maturity. The market's required rate of return (the yield) is the discount rate you apply to that fixed cash.
WHY inverse: Price is the present value of fixed future cash. Discounting means dividing by (1+y) terms. If y grows, each denominator grows, so every present value shrinks, so total price falls. The cash on top (numerator) never changes — only the discount factor changes.
Step 1 — What is one future dollar worth today?
If you can earn y per period, then $1 today grows to (1+y) next period. So $1 received next period is worth 1+y1 today.
Why this step? This is the atom of discounting; everything else is just summing atoms.
Step 2 — Discount each coupon.
Coupon at time t is worth (1+y)tC today.
Step 3 — Discount the face value (paid once, at time n): (1+y)nF.
Step 4 — Add them all up:
P=∑t=1n(1+y)tC+(1+y)nF
WHY this proves the inverse relationship: every term has y in a denominator. Raise y ⇒ each term drops ⇒ P drops. Mathematically, the derivative is always negative:
dydP=−∑t=1n(1+y)t+1tC−(1+y)n+1nF<0
Since every summand is positive and there's a minus sign, dydP<0always. Price is a strictly decreasing function of yield. That's the proof, not just a picture.
What is a bond's price mathematically? → the present value of its fixed cash flows.
Why must price fall when yield rises? → y sits in every denominator; bigger y shrinks every present value.
Sign of dP/dy? → always negative.
Yield = coupon rate implies price = ? → face value (par).
Why is a yield drop better than an equal yield rise is bad? → convexity.
Recall Feynman: explain to a 12-year-old
A bond is a promise to give you the same amount of pocket money every week and a big lump at the end — no matter what. If you decide you want to make more money on your deal, the only way is to buy the promise cheaper today. So "I want more return" always means "I'll pay less now." That's why the price goes down when the return (yield) goes up. They're on a seesaw.
Price is the present value of fixed cash flows; yield is the discount rate in every denominator, so a higher yield shrinks every present value and lowers price.
What is the sign of dP/dy for any standard bond?
Always negative — price is strictly decreasing in yield.
Price formula for a coupon bond?
P=C⋅y1−(1+y)−n+(1+y)nF
When does a bond trade at par?
When yield equals the coupon rate, so P = face value.
Yield > coupon rate implies the bond trades at a...?
Discount (price below face value).
Yield < coupon rate implies the bond trades at a...?
Premium (price above face value).
Why is the price–yield curve convex not linear?
The second derivative d²P/dy² > 0, so a yield fall raises price more than an equal yield rise lowers it.
Zero-coupon bond price?
P=(1+y)nF — only the discount factor, no coupons.
Does an interest-rate rise make a held-to-maturity bond worthless?
No — you still receive all coupons and face value; the price drop only matters if you sell early.
Dekho, bond ek fixed promise hai — woh tumhe har period ek fixed coupon deta hai aur maturity par face value. Ye cash flows kabhi nahi badalte. Ab "yield" ka matlab hai — market abhi is bond se kitna return demand kar raha hai, yaani discount rate. Jab tum in fixed cash flows ko zyada rate par discount karte ho, toh unki aaj ki value (present value) kam ho jaati hai. Isliye yield badhe toh price gire — ye ek seesaw jaisa relationship hai.
Sabse simple example: zero-coupon bond jo 1 saal baad $100 dega. Agar tum 5% return chahte ho toh aaj $95.24 doge; agar 10% chahte ho toh sirf $90.91 doge. Same $100 ka promise, par zyada return maangne ka matlab hai kam paisa aaj dena. Yahi poori kahani hai.
Ek important baat: agar yield exactly coupon rate ke barabar hai, toh bond par par bikta hai (price = face value). Yield coupon se zyada ho toh discount (price kam), aur yield kam ho toh premium (price zyada). Yaad rakho — coupon fixed feature hai, yield market ka mood hai; price yield ke ulta chalta hai.
Aur ek pro-tip: ye curve seedhi line nahi, convex (katori jaisi) hai. Iska matlab yield gire toh price jitna badhta hai, utna yield badhne par price nahi girta — ye asymmetry bondholders ke liye faydemand hai. Aur agar tum bond ko maturity tak hold karte ho toh price gir bhi jaaye tension mat lo — saare coupons aur face value tumhe milenge hi.
