Question 303 marksShort AnswerSection C

A company XYZ Ltd. has issued a bond having a face value of Rs 10,000 paying annual dividend at 8.5% p.a. The bond will be redeemed at par at the end of 10 years. Find the purchase value of this bond, if the investor wishes a yield rate of 8%. [Given : (1.08)−10=0.46319349(1.08)^{-10} = 0.46319349]

Financial Mathematics
Valuation of Bonds

Official answer

The purchase value is the present value of all future coupons plus the present value of the redemption amount.


Given data:

  • Face value F = Rs 10,000; coupon rate = 8.5%, so annual coupon C = Rs 850.
  • Redemption at par = Rs 10,000 after n=10n = 10 years.
  • Yield rate i=8%=0.08i = 8\% = 0.08; (1.08)−10=0.46319349(1.08)^{-10} = 0.46319349.

Present value of coupons (annuity):

  • PV(coupons)=C×[1−(1+i)−n]/i=850×(1−0.46319349)/0.08PV(\text{coupons}) = C \times [1 - (1 + i)^{-n}] / i = 850 \times (1 - 0.46319349)/0.08
  • = 850×6.71008850 \times 6.71008 = Rs 5,703.57.

Present value of redemption:

  • PV(redemption)=10,000×0.46319349PV(\text{redemption}) = 10,000 \times 0.46319349 = Rs 4,631.93.

Purchase value:

  • V=5,703.57+4,631.93V = 5,703.57 + 4,631.93 = Rs 10,335.50 (approximately).
Answer keywords:bond valuationpurchase valuepresent valuecoupon annuityyield rateredemption at parpremium bondRs 10,335.50

Marking scheme

  1. 1.0.5 mark: annual coupon C = Rs 850 identified.
  2. 2.1.5 marks: present value of coupon annuity = 850×(1−0.46319349)/0.08850 \times (1 - 0.46319349)/0.08 = Rs 5,703.57.
  3. 3.1 mark: present value of redemption = Rs 4,631.93 and total purchase value = Rs 10,335.50 (accept Rs 10,335 to Rs 10,336).

Hint

Coupon = 8.5% of 10,000 = Rs 850. Discount the coupon annuity and the Rs 10,000 redemption at 8%, then add them.

Quick oral answer

I discount the Rs 850 annual coupons and the Rs 10,000 redemption at the 8% yield; the coupon annuity is Rs 5,703.57 and the redemption present value is Rs 4,631.93, so the purchase value is about Rs 10,335.50, a premium since the coupon rate exceeds the yield.

Common mistakes

  1. 1.Using the yield rate 8% to compute the coupon instead of the coupon rate 8.5%.
  2. 2.Forgetting to add the present value of the redemption (face) value to the coupon annuity.
  3. 3.Discounting the coupons at 8.5% instead of the required yield rate 8%.

Analysis and explanation

Bond valuation is a straight present-value exercise: the fair price today equals the discounted value of every future cash flow at the investor's required yield.


Concept:

  • A bond pays a fixed annual coupon (here 8.5% of face value = Rs 850) for its life and returns the face value at redemption. Discounting these two streams at the desired yield of 8% gives the maximum price the investor should pay.

Why price exceeds face value:

  • The coupon rate (8.5%) is higher than the yield rate (8%), so the bond pays more than the market demands. Such a bond trades at a premium, which is why the purchase value Rs 10,335.50 is above the Rs 10,000 face value.

Exam trap:

  • Confusing coupon rate with yield rate when computing either the annual coupon or the discount factor. The coupon uses 8.5%, the discounting uses 8%.

Real-world link:

  • This is exactly how government securities and corporate bonds are priced on exchanges; when market yields fall below the coupon, bond prices rise above par, and vice versa.

Interesting facts

A bond selling above its face value, like this one, is called a premium bond, and it happens whenever the coupon rate exceeds the market yield.

The relationship 'yield up, price down' from this present-value formula is why central-bank rate changes instantly move bond markets worth trillions worldwide.

Question details

Q30 of 44 questions

Board
CBSE
Class
Class 12
Subject
Applied Mathematics
Year
2026
Section
Section C
Marks
3
Type
Short Answer
Chapter
Financial Mathematics
Topic
Valuation of Bonds
Bloom's level
apply
Syllabus topic
Unit 7: Financial Mathematics — Valuation of Bonds (Present Value of Coupons plus Redemption)
Author
CBSE
Reviewed by
Padhantu Academic Team

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Frequently asked questions

Why is the purchase value more than the Rs 10,000 face value?

The bond's coupon rate (8.5%) is higher than the investor's required yield (8%). Because the bond pays more than the market demands, it is worth a premium over par, so its fair price of Rs 10,335.50 is above the Rs 10,000 face value.

Which rate do I use for the coupon and which for discounting?

Use the coupon rate 8.5% to find the annual interest payment (Rs 850). Use the required yield rate 8% to discount both the coupon annuity and the redemption amount to present value. Swapping these rates is the most common mistake.