Noesis
2.1.2Equity & Fixed Income

Understand bonds - coupon, maturity, face value

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🇮🇳 Yeh note Hinglish mein padho →

The three pillars

Figure — Understand bonds - coupon, maturity, face value

Deriving what a bond is actually WORTH (from scratch)

Step 1 — List the cashflows. A bond with face value FF, annual coupon rate cc, and nn years to maturity pays:

  • Coupon C=c⋅FC = c \cdot F at the end of years 1,2,…,n1, 2, \dots, n.
  • Face value FF at year nn.

Step 2 — Discount each one. A dollar received in year tt is worth 1(1+r)t\frac{1}{(1+r)^t} today. Why this step? Because \1todaygrowstotoday grows to$(1+r)^tinint$ years, so reversing it divides.

Step 3 — Add them up:

P=∑t=1nC(1+r)t⏟coupons+F(1+r)n⏟face valueP = \underbrace{\sum_{t=1}^{n} \frac{C}{(1+r)^t}}_{\text{coupons}} + \underbrace{\frac{F}{(1+r)^n}}_{\text{face value}}

Step 4 — Collapse the coupon sum. It's a geometric series with first term C1+r\frac{C}{1+r} and ratio 11+r\frac{1}{1+r}. Using ∑t=1nxt=x1−xn1−x\sum_{t=1}^{n} x^t = x\frac{1-x^n}{1-x} with x=11+rx=\frac{1}{1+r}:

P=C[1−(1+r)−nr]+F(1+r)n\boxed{P = C\left[\frac{1-(1+r)^{-n}}{r}\right] + \frac{F}{(1+r)^n}}

Worked examples

Common mistakes (steel-manned)

Active recall

What is a bond in one sentence?
A tradeable IOU where you lend money for periodic coupon payments plus return of face value at maturity.
Face value is...
The principal amount repaid to the holder at maturity (often $1,000).
Coupon payment formula?
Coupon rate × face value (paid periodically).
What does maturity mean?
The date the bond expires; issuer repays face value and coupons stop.
When does a bond trade at par?
When market rate r equals coupon rate c, so P = F.
Why does a bond trade at a discount?
Because market rate r > coupon rate c, so its fixed coupons are less attractive than new bonds.
If interest rates rise, what happens to existing bond prices?
They fall (inverse relationship).
Bond price formula?
P = C·(1−(1+r)^−n)/r + F/(1+r)^n, with C = c·F.
Is the coupon rate your actual return?
No — actual return is the yield to maturity, which depends on the price you paid.
Recall Feynman: explain to a 12-year-old

Imagine you lend your friend 100andhepromises:"EverymonthI′llgiveyou100 and he promises: "Every month I'll give you 5 as a thank-you, and after a year I'll give your whole 100back."The100 back." The 100 is the face value, the 5monthlythank−youisthe∗∗coupon∗∗,and"afterayear"isthe∗∗maturity∗∗.Nowsupposeyourotherfriendsstartoffering5 monthly thank-you is the **coupon**, and "after a year" is the **maturity**. Now suppose your other friends start offering 6 thank-yous — your first friend's deal looks worse, so if you wanted to sell your IOU to someone else, they'd only pay you less than $100 for it. That's why bond prices go up and down!

Connections

  • Yield to Maturity (YTM) — the true return that ties price to coupons.
  • Interest Rates and Bond Prices — the inverse see-saw.
  • Present Value & Discounting — the engine behind bond pricing.
  • Duration & Interest Rate Risk — how sensitive price is to rate moves.
  • Equity vs Debt — bonds (lending) vs stocks (ownership).
  • Zero-Coupon Bonds — special case where C=0C=0.

Concept Map

issued because

alternative to

has

pays

expires at

formula

uses

repaid at

coupons stop at

worth found by

gives

r = c

r > c

r < c

Bond = IOU / loan contract

Issuer needs cash now

Selling stock ownership

Face Value / Principal

Coupon periodic interest

Maturity date

C = coupon rate x face value

Discount cashflows at rate r

Bond Price P

Par: P = F

Discount: P < F

Premium: P > F

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, bond basically ek IOU hai — ek loan jo tum kisi company ya government ko dete ho. Isme teen cheezein yaad rakho: face value (jo paisa maturity pe wapas milega, jaise ₹1000), coupon (jo fixed interest har saal milta hai, jaise face value ka 5% = ₹50), aur maturity (jis din bond khatam hoga aur tumhara principal wapas aayega). Ye teeno milke ek promise banate hain.

Ab important baat: bond ka price aur face value alag cheez hai. Face value fix hai, lekin market mein bond ka price upar-neeche hota rehta hai. Kyun? Kyunki future ka paisa aaj ke paisa se kam value ka hota hai — isliye hum har future cashflow ko discount karte hain market rate rr se. Formula banta hai P=C⋅1−(1+r)−nr+F(1+r)nP = C\cdot\frac{1-(1+r)^{-n}}{r} + \frac{F}{(1+r)^n}. Ye koi ratta nahi hai — ye bas saare future payments ka aaj ka value jodna hai.

Sabse zyada confuse karne wali baat: rates up ho jaayein, toh purane bond ka price DOWN hota hai. Socho tumhare paas 5% wala bond hai, aur market mein naye bond 8% dene lage — ab tumhara bond kam attractive hai, toh log usse discount pe hi kharidenge. Isliye "Rates Up, Prices Down" — ye see-saw hamesha yaad rakho.

Aur ek galti se bacho: coupon rate tumhara actual return NAHI hai. Actual return hai yield to maturity, jo tumhare paid price pe depend karta hai. Agar discount pe kharida toh return coupon se zyada, premium pe kharida toh kam. Bas yahi core samajh lo, toh fixed income ka half chapter clear ho jayega!

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Connections

Understand yield and yield-to-maturity (YTM)Stock-Market · 2.1.4Understand duration and interest rate sensitivityStock-Market · 2.1.8Understand zero-coupon bondsStock-Market · 2.1.10