Learn the inverse price-yield relationship — Hinglish notes

2.1.5 · Stock-Market › Equity & Fixed Income

YE relationship KYUN exist karti hai?

KYA ho raha hai: Ek bond tumhe unchangeable cash ka ek stream promise karta hai — coupons plus maturity par face value. Market ka required rate of return (yaani yield) woh discount rate hai jo tum us fixed cash par apply karte ho.

KYUN inverse hai: Price future fixed cash ka present value hai. Discounting ka matlab hai $(1 + y)$ terms se divide karna. Agar $y$ badhta hai, toh har denominator badhta hai, isliye har present value shrinks ho jaati hai, toh total price girta hai. Upar wala cash (numerator) kabhi nahi badlta — sirf discount factor badlta hai.

Price formula ko first principles se derive karna

Step 1 — Aaj ek future dollar kitna worth hai? Agar tum $y$ per period earn kar sakte ho, toh aaj ka $1 agli period mein $(1 + y)$ ban jaata hai. Toh agli period mein mila $1, aaj $\frac{1}{1 + y}$ worth hai. Ye step kyun? Ye discounting ka atom hai; baaki sab sirf atoms ka sum hai.

Step 2 — Har coupon ko discount karo. Time $t$ par coupon aaj $\frac{C}{( 1 + y ) ^{t}}$ worth hai.

Step 3 — Face value ko discount karo (ek baar, time $n$ par): $\frac{F}{( 1 + y ) ^{n}}$ .

Step 4 — Sab ko add karo:

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

WHY ye inverse relationship prove karta hai: har term mein $y$ denominator mein hai. $y$ badhao ⇒ har term gire ⇒ $P$ gire. Mathematically, derivative hamesha negative hai:

$\frac{d P}{d y} = - \sum_{t = 1}^{n} \frac{t C}{( 1 + y ) ^{t + 1}} - \frac{n F}{( 1 + y ) ^{n + 1}} < 0$

Kyunki har summand positive hai aur ek minus sign hai, $\frac{d P}{d y} < 0$ hamesha hoga. Price yield ki strictly decreasing function hai. Ye proof hai, sirf ek picture nahi.

Figure — Learn the inverse price-yield relationship

Curve ek straight line nahi hai — ye convex hai

Worked examples

Worked example Example 3 — Teen states ka summary

Situation	Price vs Face
yield < coupon rate	Premium (P > F)
yield = coupon rate	Par (P = F)
yield > coupon rate	Discount (P < F)
Kyun: agar market coupon se kam return maangti hai, toh coupon "generous" lagta hai, isliye buyers price upar bid karte hain — aur vice versa.

Common mistakes (steel-manned)

Active-recall

Recall Quick self-quiz (answers cover karo)

Bond ki price mathematically kya hai? → uske fixed cash flows ka present value.
Price kyun girni chahiye jab yield rise kare? → $y$ har denominator mein hai; bada $y$ har present value ko shrink karta hai.
$d P / d y$ ka sign? → hamesha negative.
Yield = coupon rate matlab price = ? → face value (par).
Yield drop kyun utna achha hai jitna equal yield rise bura hai? → convexity.

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

Ek bond ek promise hai jo tumhe har hafte same pocket money aur end mein ek bada lump deta hai — chahe kuch bhi ho jaye. Agar tum decide karo ki tum apne deal par zyada paisa banana chahte ho, toh iska sirf ek tarika hai — promise ko aaj saste mein kharido. Toh "main zyada return chahta hun" ka matlab hamesha hai "main abhi kam pay karunga." Isliye price neeche jaati hai jab return (yield) upar jaata hai. Ye ek seesaw pe hain.

Connections

Present Value & Discounting
Bond Duration & Interest-Rate Risk
Convexity of Bonds
Coupon Rate vs Yield to Maturity
Par, Premium & Discount Bonds
Zero-Coupon Bonds
Interest Rate Risk in Fixed Income

Bond prices aur yields inversely kyun move karte hain?

Price fixed cash flows ka present value hai; yield har denominator mein discount rate hai, toh higher yield har present value ko shrink karke price ko lower karta hai.

Kisi bhi standard bond ke liye dP/dy ka sign kya hai?

Hamesha negative — price yield mein strictly decreasing hai.

Coupon bond ka price formula?

P = C \cdot \frac{1 - ( 1 + y ) ^{- n}}{y} + \frac{F}{( 1 + y ) ^{n}}

Bond par kab trade karta hai?

Jab yield coupon rate ke equal ho, toh P = face value.

Yield > coupon rate matlab bond ... par trade karta hai?

Discount (price face value se neeche).

Yield < coupon rate matlab bond ... par trade karta hai?

Premium (price face value se upar).

Price–yield curve convex kyun hai, linear kyun nahi?

Second derivative d²P/dy² > 0 hai, isliye yield fall price ko utna zyada raise karta hai jitna equal yield rise use kam karta hai.

Zero-coupon bond price?

P = \frac{F}{( 1 + y ) ^{n}}

— sirf discount factor, koi coupons nahi.

Kya interest-rate rise ek held-to-maturity bond ko worthless bana deta hai?

Nahi — tum abhi bhi saare coupons aur face value receive karte ho; price drop tabhi matter karta hai jab tum early sell karo.