Consider a bond with a coupon rate of 10% charged annually. The par value is ₹2,000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?

Question

Accepted Answer

> [!TLDR]
> The bond’s value is the present value of all future coupon payments plus the present value of the par value at maturity, discounted at the yield to maturity of 11%. For a 5‑year, 10% annual coupon bond with ₹2,000 par, the value is **₹1,926.10**.

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### Why this approach works

A bond is simply a stream of cash flows: you receive a fixed coupon each year (here 10% of ₹2,000 = ₹200), and at the end of 5 years you also get back the par value of ₹2,000. The “value” of the bond today is what those future cash flows are worth *right now*, given the current market interest rate (the yield to maturity, YTM). Since the YTM (11%) is higher than the coupon rate (10%), the bond should trade at a **discount** — below par. We’ll confirm that numerically.

The standard formula for bond valuation is:

> [!FORMULA]
> $$ 	ext{Bond Value} = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n} $$
> where $C$ = annual coupon, $F$ = face value (par), $r$ = YTM per period, $n$ = number of years.

We’ll apply it step by step.

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### Step‑by‑step calculation

**1. Identify the inputs**

- Coupon rate = 10% annually → Coupon payment $C = 10\% 	imes 2000 = ₹200$
- Par value $F = ₹2,000$
- Years to maturity $n = 5$
- Yield to maturity $r = 11\% = 0.11$

**2. Present value of the coupon annuity**

You receive ₹200 at the end of each year for 5 years. This is an ordinary annuity. Its present value is:

$$ PV_{	ext{coupons}} = C 	imes \frac{1 - (1+r)^{-n}}{r} $$

Plug in:

$$ PV_{	ext{coupons}} = 200 	imes \frac{1 - (1.11)^{-5}}{0.11} $$

First compute $(1.11)^{-5}$:

$$ (1.11)^5 = 1.11^2 	imes 1.11^2 	imes 1.11 = 1.2321 	imes 1.2321 	imes 1.11 $$
$$ 1.2321 	imes 1.2321 = 1.51807 \quad (	ext{approx}) $$
$$ 1.51807 	imes 1.11 = 1.68506 $$

So $(1.11)^{-5} = \frac{1}{1.68506} \approx 0.59345$.

Now:

$$ \frac{1 - 0.59345}{0.11} = \frac{0.40655}{0.11} \approx 3.6959 $$

Multiply by 200:

$$ PV_{	ext{coupons}} \approx 200 	imes 3.6959 = ₹739.18 $$ …

Q.Consider a bond with a coupon rate of 10% charged annually. The par value is ₹2,000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?

Concept understanding — Bond Valuation

Bond Valuation: The Intuition First

The Precise Statement

Why This Works: The Logic

A Worked Example

Why this approach works

Step‑by‑step calculation

Step 1 — set up the valuation

Step 2 — compute the discount factors (14%)

Step 3 — combine and solve

Step 1 — Identify the cash flows

Step 2 — Discount each cash flow

Step 3 — Add them up

Q.Consider a bond with a coupon rate of 10% charged annually. The par value is ₹2,000 and the bond has 5 years to maturity. The yield to maturity is 11%. What is the value of the bond?

Why this approach works

Step‑by‑step calculation

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