Understand yield and yield-to-maturity (YTM) — Stock-Market notes

WHAT are we even measuring?

WHY three different "yields"? Because each captures more reality:

Coupon rate ignores price entirely — useless for comparing what you pay.
Current yield adds price but ignores the capital gain/loss at maturity.
YTM captures everything: coupons and the pull-to-par (gain if bought at discount, loss if bought at premium) and the time value of money.

HOW to derive the bond price (and hence YTM) from scratch

Step 1 — list the cash flows. A bond with $n$ periods pays:

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

Step 2 — discount each to today. Why? Because value today = sum of what each future dollar is worth now.

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

Step 3 — collapse the coupon sum. The coupons form a geometric series with ratio $r=\frac{1}{1+y}$ . Why a geometric series? Because each term is the previous one multiplied by the same factor $\frac{1}{1+y}$ .

Sum of $r + r^2 + \cdots + r^n = r\cdot\dfrac{1-r^n}{1-r}$ . Substituting $r=\frac{1}{1+y}$ and simplifying:

$\underbrace{\sum_{t=1}^{n}\frac{C}{(1+y)^t}}_{\text{annuity}} = C\cdot\frac{1-(1+y)^{-n}}{y}$

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

The par / premium / discount rule

Worked Example 1 — solving for price given YTM

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

Coupon cash: $C = 0.08\times1000 = 80$ . Why? Coupon rate applies to face, not price.
Annuity part: $80\cdot\dfrac{1-(1.10)^{-3}}{0.10}$ . Why divide by $y$ ? That's the geometric-sum collapse. $(1.10)^{-3}=0.7513$ , so $=80\cdot\dfrac{0.2487}{0.10}=80\times2.4869=198.9$ .
Face PV: $\dfrac{1000}{(1.10)^3}=751.3$ . Why? The $1000 arrives in 3 years, discount it.
Price: $198.9 + 751.3 = \mathbf{950.3}$ .

Check the rule: $c=8\% < y=10\%$ , so price $<$ par ✓ (discount).

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

Same bond bought at $P=950.3$ .

Current yield $=\dfrac{80}{950.3}=8.42\%$ . Why higher than 8%? You paid less than par, so the fixed $80 is a bigger % of your smaller outlay.
YTM $=10\%$ . Why even higher than current yield? Because YTM adds the $49.7 capital gain (you get$ 1000 back but paid $950.3), spread over 3 years. Current yield misses that gain.

Ordering for a discount bond: $\text{coupon rate} < \text{current yield} < \text{YTM}$ . For a premium bond the inequality flips.

Worked Example 3 — estimating YTM from a price (Forecast-then-Verify)

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

Forecast: Price > par → premium → $y < c = 6\%$ . Guess $y=4\%$ .
Verify $y=4\%$ : $60\cdot\frac{1-1.04^{-2}}{0.04} + \frac{1000}{1.04^2} = 60(1.8861)+924.6 = 113.2+924.6=1037.8$ . Slightly high.
Try $y=4.15\%$ : gives $\approx1035.0$ ✓.
YTM ≈ 4.15%, and indeed below the 6% coupon, matching the premium rule.

Common mistake Steel-manning the classic errors

Mistake A — "YTM = coupon rate." Feels right because the coupon is the headline number printed on the bond. Fix: coupon rate is fixed to face value; YTM depends on the price you pay. They're equal ONLY when price = par.

Mistake B — "Higher current yield = better return." Feels right because it looks like a return %. Fix: current yield ignores the pull-to-par. A premium bond can have a high current yield but a lower YTM once the capital loss at maturity is counted.

Mistake C — "If yields rise, my bond's coupon rises." Feels right by association. Fix: the coupon is contractually fixed. When market yields rise, it's the price of your existing bond that falls so its YTM matches the market.

Mistake D — reinvestment blind spot. YTM assumes you reinvest each coupon at $y$ . If reality reinvests lower, your realized return < YTM.

Recall Feynman: explain to a 12-year-old

Imagine you lend a shopkeeper $100 and he promises to pay you a small tip every month and give the$ 100 back after a year. Now suppose someone sells you their place in this deal — but they ask $95, not$ 100. You still get the tips and the full $100 back later, so you actually earn a bit extra. **Yield** is just "how much do I really earn per year on the money I put in?" **YTM** is the careful version that counts *both* the tips *and* that little bonus of getting$ 100 back after paying only $95 — all measured in today's money.

Active Recall

Coupon cash on a $1000 face, 8\% coupon bond? :::$ 80 per year (coupon rate × face value, not price).

What is yield-to-maturity (YTM)?

The single discount rate that makes the present value of all a bond's future cash flows (coupons + face) equal to its current market price; the IRR if held to maturity.

Formula for bond price in terms of yield y?

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

, where

C=cF

Why do bond prices and yields move inversely?

Yield sits in the discount denominators; a higher y discounts every future cash flow more heavily, lowering present value (price).

Current yield formula?

annual coupon / current price.

When does a bond trade at par?

When coupon rate = YTM.

When at a premium vs discount?

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

For a discount bond, order coupon rate, current yield, YTM.

coupon rate < current yield < YTM.

What key assumption does YTM make about coupons?

That every coupon is reinvested at the same rate y until maturity.

Why is current yield an incomplete measure?

It ignores the capital gain/loss from the bond pulling to par at maturity and the time value of money.

Connections

Time Value of Money — YTM is a direct application of discounting.
Present Value and Discounting — the geometric-series engine behind the price formula.
Bond Pricing — same formula, solved for $P$ instead of $y$ .
Duration and Interest-Rate Risk — quantifies how much price moves when yield moves.
Coupon vs Zero-Coupon Bonds — set $C=0$ to get $P=F/(1+y)^n$ .
Internal Rate of Return (IRR) — YTM is IRR applied to a bond's cash flows.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, bond ka matlab hai ek promise: aaj tum paisa doge, aur badle mein tumhe har saal ek fixed coupon milega, aur end mein tumhara face value (jaise $1000) wapas mil jayega. Ab twist ye hai ki market mein tum us bond ko$ 1000 pe nahi, kabhi $950 pe ya kabhi$ 1080 pe kharid sakte ho. Yield ka sawaal simple hai — "jo price maine aaj pay kiya, uspe mera actual annual return kitna hai?"

YTM (Yield-to-Maturity) sabse complete measure hai. Ye woh single interest rate hai jo saare future cash flows (coupons + face value) ki aaj ki value ko exactly aaj ke price ke barabar bana deta hai. Formula ka core idea sirf itna hai: future ka paisa aaj kam value rakhta hai, isliye har cash flow ko $(1+y)^t$ se divide (discount) karo, aur sabko jod do. Coupons ek geometric series banate hain, isliye woh $C\cdot\frac{1-(1+y)^{-n}}{y}$ mein simplify ho jate hain.

Sabse important baat yaad rakho — price aur yield see-saw ki tarah ulte chalte hain. Yield formula ke denominator mein hai, to yield badhi to har future dollar zyada discount hua, price gir gaya. Aur ek pyaari si rule: agar coupon rate = YTM, bond par pe; agar coupon > YTM, premium (price zyada); agar coupon < YTM, discount (price kam).

Common galti: log samajhte hain coupon rate hi return hai. Nahi! Coupon rate face value pe fix hai, lekin tumhara real return depend karta hai ki tumne kitne mein khareeda. Discount bond mein current yield coupon se zyada hoti hai, aur YTM sabse zyada — kyunki maturity pe $950 de kar$ 1000 wapas milne ka bonus bhi count hota hai. Isliye exam mein aur investing mein hamesha YTM dekho, sirf coupon ya current yield nahi.

Test yourself — Equity & Fixed Income

Connections