Understand yield and yield-to-maturity (YTM)

2.1.4 · Stock-Market › Equity & Fixed Income

Hum KISKA measurement kar rahe hain?

KYUN teen alag "yields"? Kyunki har ek zyada reality capture karta hai:

Coupon rate price ko bilkul ignore karta hai — yeh compare karne ke liye useless hai ki aap kya pay kar rahe ho.
Current yield price add karta hai lekin maturity par capital gain/loss ignore karta hai.
YTM sab kuch capture karta hai: coupons aur pull-to-par (agar discount par kharida toh gain, premium par kharida toh loss) aur time value of money.

Bond price (aur isliye YTM) ko scratch se KAISE derive karein

Step 1 — cash flows list karo. $n$ periods wala bond pay karta hai:

coupon $C = c F$ times $t = 1, 2, \dots, n$ par
face value $F$ time $n$ par.

Step 2 — har ek ko aaj tak discount karo. Kyun? Kyunki aaj ki value = har future dollar ki aaj ki worth ka sum.

$P = \frac{C}{( 1 + y ) ^{1}} + \frac{C}{( 1 + y ) ^{2}} + \dots + \frac{C}{( 1 + y ) ^{n}} + \frac{F}{( 1 + y ) ^{n}}$

Step 3 — coupon sum ko collapse karo. Coupons ek geometric series banaate hain ratio $r = \frac{1}{1 + y}$ ke saath. Geometric series kyun? Kyunki har term pichli term ko usi factor $\frac{1}{1 + y}$ se multiply karke milti hai.

Sum of $r + r^{2} + \dots + r^{n} = r \cdot \frac{1 - r ^{n}}{1 - r}$ . $r = \frac{1}{1 + y}$ substitute karke aur simplify karke:

$annuity t = 1 \sum n \frac{C}{( 1 + y ) ^{t}} = C \cdot \frac{1 - ( 1 + y ) ^{- n}}{y}$

Figure — Understand yield and yield-to-maturity (YTM)

Par / premium / discount rule

Worked Example 1 — YTM di gayi ho toh price solve karna

Bond: $F = 1000$ , coupon rate $c = 8%$ annual, $n = 3$ years, market YTM $y = 10%$ .

Coupon cash: $C = 0.08 \times 1000 = 80$ . Kyun? Coupon rate face par apply hota hai, price par nahi.
Annuity part: $80 \cdot \frac{1 - ( 1.10 ) ^{- 3}}{0.10}$ . $y$ se kyun divide? Woh geometric-sum collapse hai. $(1.10)^{- 3} = 0.7513$ , toh $= 80 \cdot \frac{0.2487}{0.10} = 80 \times 2.4869 = 198.9$ .
Face PV: $\frac{1000}{( 1.10 ) ^{3}} = 751.3$ . Kyun? $1000 teen saal baad aayega, use discount karo.
Price: $198.9 + 751.3 = 950.3$ .

Rule check karo: $c = 8% < y = 10%$ , toh price $<$ par ✓ (discount).

Worked Example 2 — current yield vs YTM (Feynman-style comparison)

Wahi bond $P = 950.3$ par kharida.

Current yield $= \frac{80}{950.3} = 8.42%$ . 8% se zyada kyun? Aapne par se kam pay kiya, toh fixed $80 aapki chhhoti outlay ka bada % hai.
YTM $= 10%$ . Current yield se bhi zyada kyun? Kyunki YTM $49.7 c a p i t a l g ain * * a dd * * k a r t ahai (aa p k o$ 1000 wapas milenge lekin aapne $950.3 pay kiye), teen saal mein spread out karke. Current yield woh gain miss kar deta hai.

Discount bond ka ordering: $coupon rate < current yield < YTM$ . Premium bond mein inequality ulti ho jaati hai.

Worked Example 3 — price se YTM estimate karna (Forecast-then-Verify)

Bond: $F = 1000$ , $C = 60$ , $n = 2$ , price $P = 1035$ . $y$ nikalo.

Forecast: Price > par → premium → $y < c = 6%$ . Guess $y = 4%$ .
Verify $y = 4%$ : $60 \cdot \frac{1 - 1.0 4 ^{- 2}}{0.04} + \frac{1000}{1.0 4 ^{2}} = 60 (1.8861) + 924.6 = 113.2 + 924.6 = 1037.8$ . Thoda zyada hai.
Try $y = 4.15%$ : $\approx 1035.0$ deta hai ✓.
YTM ≈ 4.15%, aur yeh 6% coupon se neeche hai, jo premium rule se match karta hai.

Common mistake Classic errors ko steel-man karna

Mistake A — "YTM = coupon rate." Sahi lagta hai kyunki coupon bond par printed headline number hai. Fix: coupon rate face value se fixed hai; YTM us price par depend karta hai jo aap pay karte ho. Woh tabhi equal hain jab price = par ho.

Mistake B — "Higher current yield = better return." Sahi lagta hai kyunki yeh return % jaisa dikhta hai. Fix: current yield pull-to-par ignore karta hai. Ek premium bond ka high current yield ho sakta hai lekin ek lower YTM jab maturity par capital loss count ki jaaye.

Mistake C — "Agar yields rise karein, mere bond ka coupon rise karega." Association se sahi lagta hai. Fix: coupon contractually fixed hai. Jab market yields rise karein, toh aapke existing bond ki price girti hai taaki uska YTM market se match kare.

Mistake D — reinvestment blind spot. YTM assume karta hai ki aap har coupon ko $y$ par reinvest karte ho. Agar reality mein reinvestment lower ho, toh aapka realized return < YTM.

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

Socho tum ek dukaandaar ko $100 u d e d e t e h o a u r w o h p r o mi se k a r t ahaik iha r mahin ee k c hh o t i t i p d e g aa u r e k s aa l baa d$ 100 wapas karega. Ab socho koi tumhe unki jagah is deal mein bechta hai — lekin woh $100 nahi,$ 95 maangta hai. Tumhe phir bhi tips aur pura $100 w a p a s mi l e g a, t o h t u ma c t u a l l y t h o d a e x t r ak ama t e h o . * * Y i e l d * * ba sy e hhai : " M ainj o p ai s a l a g a t ah oo n, u s p a r a c t u a l l y s aa l aanak i t nak amaa t ah oo n ? " * * Y T M * * i s k a c a r e f u l v er s i o nhaij o * d o n o * co u n t k a r t ahai — t i p s * a u r * w o h c hh o t ab o n u so f g e tt in g$ 100 back after paying only $95 — sab aaj ke paison mein measure karke.

Active Recall

$1000 f a ce, 8% co u p o nb o n d p a r co u p o n c a s h ? :::$ 80 per year (coupon rate × face value, price nahi).

Yield-to-maturity (YTM) kya hai?

Woh single discount rate jo ek bond ke saare future cash flows (coupons + face) ki present value ko uske current market price ke barabar kar de; maturity tak hold karo toh IRR.

Yield y ke terms mein bond price ka formula?

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

, jahan

C = c F

Bond prices aur yields inversely kyun move karte hain?

Yield discount denominators mein hoti hai; zyada y har future cash flow ko zyada heavily discount karta hai, present value (price) ko kam karta hai.

Current yield formula?

annual coupon / current price.

Bond par kab at par trade karta hai?

Jab coupon rate = YTM ho.

Premium vs discount par kab?

Premium jab coupon rate > YTM (price > face); discount jab coupon rate < YTM (price < face).

Discount bond ke liye coupon rate, current yield, YTM ka order?

coupon rate < current yield < YTM.

YTM coupons ke baare mein kya key assumption karta hai?

Ki har coupon ko maturity tak usi rate y par reinvest kiya jaayega.

Current yield ek incomplete measure kyun hai?

Yeh capital gain/loss ko ignore karta hai jo bond ke maturity par par pull karne se aata hai, aur time value of money ko bhi.

Connections

Time Value of Money — YTM directly discounting ka application hai.
Present Value and Discounting — price formula ke peeche geometric-series engine.
Bond Pricing — same formula, $y$ ki jagah $P$ ke liye solve kiya.
Duration and Interest-Rate Risk — quantify karta hai ki yield move hone par price kitna move karti hai.
Coupon vs Zero-Coupon Bonds — $C = 0$ set karo toh $P = F / (1 + y)^{n}$ milta hai.
Internal Rate of Return (IRR) — YTM ek bond ke cash flows par apply kiya hua IRR hai.