Suppose that a bond has a face value of ₹1,000 and will mature in 10 years. The annual coupon rate is 5%, the

Exercise 7.2 · Q4

Q. Suppose that a bond has a face value of ₹1,000 and will mature in 10 years. The annual coupon rate is 5%, the bond makes semi-annual coupon payments. With a price of ₹950, what is the bond's YTM?

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Concept understanding — Bond Valuation

Bond Valuation: The Intuition First

Imagine you lend ₹1,000 to a friend. You both agree that every year, your friend will pay you ₹80 as interest (8% of ₹1,000). After 5 years, your friend returns your ₹1,000. That's a bond in its simplest form.

A bond is just a loan that can be traded. The person who buys the bond is lending money. The bond's face value (₹1,000) is the amount you get back at the end. The coupon rate (8%) determines the annual interest payment. The maturity (5 years) is when you get your principal back.

Now, here's the key question: What is this bond worth today?

If you could earn 10% interest elsewhere, would you pay ₹1,000 for a bond that only pays 8%? No — you'd demand a discount. If you could only earn 6% elsewhere, that same 8% bond becomes more valuable, and you'd pay a premium.

Bond valuation is simply the process of calculating the fair price of a bond today, given the current interest rates in the market.

The Precise Statement

\text{Bond Price} = \sum_{t=1}^{n} \frac{C}{(1+r)^t} + \frac{F}{(1+r)^n}

Where:

$C$ = annual coupon payment (face value × coupon rate)
$F$ = face value (principal repaid at maturity)
$r$ = market interest rate (yield to maturity)
$n$ = number of years to maturity

This formula says: The bond's price equals the present value of all future cash flows — both the periodic coupon payments and the final face value repayment.

Why This Works: The Logic

You're essentially asking: "If I want to earn a return of $r$ per year, what should I pay today for a stream of future payments?"

Each future payment is discounted back to today using the market rate $r$ . The sum of these discounted values gives the fair price.

Note

When the coupon rate equals the market rate, the bond sells at par (face value). When the coupon rate is higher than the market rate, the bond sells at a premium (above face value). When lower, it sells at a discount (below face value).

A Worked Example

Consider a 3-year bond with face value ₹1,000, coupon rate 10%, and market interest rate 8%.

Step 1: Identify the cash flows

Annual coupon: $C = 0.10 \times 1000 = ₹100$
Year 1: ₹100
Year 2: ₹100 …

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The YTM is the discount rate that makes the present value of the coupons plus face value equal the ₹950 price. Solving the semi-annual bond equation gives a semi-annual rate of about 2.83%, so the annual YTM is approximately 5.66%.

A bond's YTM is the single rate that equates the present value of its future cash flows to its current price. Coupons are semi-annual, so we work in half-years: semi-annual coupon $=\dfrac{5\%}{2}\times1000=₹25$ , $n=20$ periods, face value ₹1,000, price ₹950.

Pricing equation (let $r$ be the semi-annual YTM): $950=25\times\frac{1-(1+r)^{-20}}{r}+1000(1+r)^{-20}.$ …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CA Foundation 2026Set jan-20261 markMCQ
$Q. An investor intends to purchase a three years bond at a price of ₹ 907.125 having nominal interest rate of 10%. What is the par value of the bond if it matures at par and the investor requires returns at the rate of 14%? (A) ₹ 1,200 (B) ₹ 1,100 (C) ₹ 900 (D) ₹ 1,000$
›Reveal solutionSolution

Price $=0.10F\cdot(\text{PV annuity, 3 yr, 14\%})+F\cdot(1.14)^{-3}=0.907135F$ ; setting this to 907.125 gives $F=₹1,000$ .

Step 1 — set up the valuation

Let $F$ be the par (face) value. Coupon $=10\%$ of $F$ each year for 3 years; redemption $F$ at year 3; required return $14\%$ :
$907.125=0.10F\!\left(\frac{1}{1.14}+\frac{1}{1.14^2}+\frac{1}{1.14^3}\right)+\frac{F}{1.14^3}.$
Step 2 — compute the discount factors (14%)
$\frac{1}{1.14}=0.877193,\quad \frac{1}{1.14^2}=0.769468,\quad \frac{1}{1.14^3}=0.674972.$
Sum of the three (annuity factor) $=2.321633$ .

Step 3 — combine and solve
$907.125=0.10F(2.321633)+0.674972F=F(0.232163+0.674972)=0.907135F.$
$F=\frac{907.125}{0.907135}\approx ₹1,000.$ …
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CA Foundation 2024Set sep-20241 markMCQ
$Q. An investor intends to purchase a three year ₹ 1,000 par value bond having nominal interest rate of 10%. At what price the bond may be purchased now, if it matures at par and the investor requires a rate of return of 14% ? (A) ₹ 904 (B) ₹ 907.125 (C) ₹ 905.25 (D) ₹ 909$
›Reveal solutionSolution

Bond price = PV of coupons + PV of par, discounted at the required return 14% = ₹907.125.

Step 1 — Identify the cash flows

Coupon = 10% of ₹1,000 = ₹100 at end of years 1, 2, 3, plus ₹1,000 par returned at year 3. The required return $i = 14\%$ .
$P = \sum_{t=1}^{3}\dfrac{100}{(1.14)^t} + \dfrac{1000}{(1.14)^3}$
Step 2 — Discount each cash flow
$\dfrac{100}{1.14} = 87.72,\quad \dfrac{100}{1.2996} = 76.95,\quad \dfrac{100}{1.481544} = 67.50$ $\dfrac{1000}{1.481544} = 674.97$
Step 3 — Add them up
$P = 87.72 + 76.95 + 67.50 + 674.97 \approx ₹907.13$
Why the other options are wrong: (A) ₹904 and (D) ₹909 use rounded/wrong factors, and (C) ₹905.25 miscomputes one term; the exact PV at 14% is ₹907.125. …
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Q.Suppose that a bond has a face value of ₹1,000 and will mature in 10 years. The annual coupon rate is 5%, the bond makes semi-annual coupon payments. With a price of ₹950, what is the bond's YTM?

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