Corporate Accounting and Financial Management · Security Analysis

Valuation of Bonds and Debentures: Formula, YTM and Duration

Updated 11 October 2026 · Fact-checked

Bond valuation finds the present value of a bond's future interest and redemption amount, discounted at the investor's required return. Value = interest × annuity factor + redemption value × discount factor. Yield to maturity is the discount rate that makes this value equal the market price. Duration measures how sharply the price moves when rates change.

Understand Valuation of Bonds and Debentures

A bond or debenture is a promise to pay fixed interest each year and to repay the face value at maturity. Its worth to you today is what those future cash flows are worth in today's rupees. So you discount each cash flow at the required rate of return (also called market yield) and add them up. This is the same time value of money idea you used for annuities and single sums.

The interest rate printed on the bond is the coupon rate. It is fixed. The required return changes with the market. If the required return is above the coupon rate, the bond is worth less than face value (it sells at a discount). If it is below the coupon rate, the bond is worth more (it sells at a premium). If both are equal, value equals face value. This is why bond prices and interest rates move in opposite directions.

Yield to maturity (YTM) is the return you earn if you buy at today's market price and hold to maturity. It is the discount rate at which the present value of all cash flows equals the price. You find it by trial and error with interpolation, or by a quick approximation formula. Current yield is simpler: annual interest divided by market price. It ignores the gain or loss at redemption, so it is not the full return.

Duration is the weighted average time to receive a bond's cash flows, where each year is weighted by the present value of the cash flow in that year. It is measured in years. A higher duration means the price is more sensitive to interest rate changes. For a 2-year bond of face value ₹100 with 10% coupon and 10% yield, the present values are ₹9.091 (year 1) and ₹90.909 (year 2). Duration = (1 × 9.091 + 2 × 90.909) ÷ 100 = 1.909 years.

For the same maturity, a lower coupon gives a longer duration. A zero-coupon bond has duration equal to its maturity. Longer maturity generally means higher duration and greater price risk.

Key rules to remember

Value of a redeemable bond: V0 = Σ [I ÷ (1 + kd)^t] + M ÷ (1 + kd)^n, t = 1 to n; I = annual interest (coupon rate × face value), kd = required return, M = redemption value, n = years to maturity.
Annuity-factor form: V0 = I × PVIFA(kd, n) + M × PVIF(kd, n); Use the present value tables given in the exam. This is the fastest way to compute the value.
Irredeemable (perpetual) debenture: V0 = I ÷ kd; Use when there is no maturity date and interest continues forever.
Zero-coupon bond: V0 = M ÷ (1 + kd)^n; No interest is paid. The whole return comes from the gap between price and redemption value.
Semi-annual interest: V0 = (I ÷ 2) × PVIFA(kd ÷ 2, 2n) + M × PVIF(kd ÷ 2, 2n); Halve the interest and the rate, and double the periods. Use only when interest is paid twice a year.
Current yield: Current yield = (Annual interest ÷ Current market price) × 100; Ignores the capital gain or loss at maturity.
YTM (approximate): YTM ≈ [I + (M − P) ÷ n] ÷ [(M + P) ÷ 2]; P = current price. A close estimate, not exact. Use it as a starting rate for trial and error.
YTM by interpolation: YTM = r1 + [(V1 − P) ÷ (V1 − V2)] × (r2 − r1); V1 is the value at the lower rate r1 and V2 the value at the higher rate r2. P must lie between V1 and V2.
Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt); The denominator is the bond's present value (its price at that yield). t is the year of the cash flow.
Modified duration and price change: Modified duration = D ÷ (1 + y); % change in price ≈ − Modified duration × change in yield; A linear estimate that works for small changes in yield. The minus sign shows prices fall when yields rise.

How to solve Valuation of Bonds and Debentures questions

Use this method for any valuation, yield or duration question on bonds and debentures.

1Read the question and list the facts: face value, coupon rate, years to maturity, redemption value (par, premium or discount), required return or market price, and how often interest is paid.
2Work out the annual interest in rupees: coupon rate × face value. Do not use the market price for this.
3Decide what is asked: value (discount at the required return), YTM (find the rate for a given price), current yield, or duration.
4For value, write the cash flow line year by year, with interest in years 1 to n and interest plus redemption value in year n. Apply the given PVIFA and PVIF factors and add.
5For YTM, compute the approximate YTM first. Test two rates on either side of it, find the value at each, and interpolate. Check that the price lies between the two values.
6For duration, make a table: year, cash flow, discount factor, present value, and year × present value. Divide the total of the last column by the total of present values.
7Compare value with market price if asked for a decision. Buy if value is above price, avoid if value is below price.
8Write a one-line conclusion with units (₹ or %) and state any assumption you made.

Quickest way: Annuity-factor shortcut with a YTM bracket

When to use it: Use this when the question gives PVIFA and PVIF tables and asks for the value of a redeemable bond or its YTM.

Calculate I = coupon rate × face value.
Value = I × PVIFA + M × PVIF in one line. Do not build a year-by-year table unless the question asks for it.
For YTM, compute the approximate formula. Round it to the nearest whole percent.
Test that rate and the next rate on the other side of the price. Aim for one value above the price and one below.
Interpolate and stop. Write the YTM to one or two decimals.
If the bond sells at a discount, YTM must be above the coupon rate and current yield. If at a premium, below. Use this to catch errors.

Common mistakes in Valuation of Bonds and Debentures

Using the market price or the required return to calculate the annual interest.
Students confuse the coupon rate with the discount rate, since both are percentages.
Fix: Interest is always coupon rate × face value. The required return is used only for discounting.
Adding the redemption value in every year, or forgetting to add it in the last year.
The year-n cash flow has two parts, and students include only one of them.
Fix: Treat interest as an annuity for n years and the redemption value as a single sum at year n.
Treating current yield as YTM.
Both are percentages related to price, and current yield is quick to calculate.
Fix: Current yield ignores the gain or loss at redemption. YTM includes it. They are equal only when the bond is irredeemable or when price equals face value and redemption is at par.
Interpolating when the price does not lie between the two values tested.
Students pick rates without checking where the price falls.
Fix: After computing both values, confirm that the price is between them. If not, shift the trial rates.
Not adjusting for semi-annual interest.
Students halve the interest but keep the full annual rate and number of years.
Fix: Halve the interest and the rate, and double the periods. State that the resulting yield is per half-year.
Saying that prices rise when interest rates rise.
Students mix up the movement of rates and of discounting.
Fix: A higher discount rate lowers present values, so bond prices fall when market rates rise and rise when they fall. Longer duration means a bigger move.

Worked examples

Example 1

A debenture of face value ₹1,000 carries 10% interest paid annually and is redeemable at par after 5 years. Investors require a return of 12%. Find the value of the debenture. (PVIFA at 12%, 5 years = 3.6048; PVIF at 12%, year 5 = 0.5674.)

Show the solution

Annual interest I = 10% × ₹1,000 = ₹100.
Redemption value M = ₹1,000 (at par), received at the end of year 5.
Present value of interest = ₹100 × 3.6048 = ₹360.48.
Present value of redemption = ₹1,000 × 0.5674 = ₹567.40.
Value = ₹360.48 + ₹567.40 = ₹927.88.
Required return 12% exceeds coupon 10%, so a value below face value is as expected.

Answer: The value of the debenture is ₹927.88, a discount to its face value of ₹1,000.

Example 2

A bond of face value ₹1,000 pays 9% interest annually and is redeemable at par after 5 years. It is available in the market at ₹920. Calculate (a) the current yield and (b) the yield to maturity. (PVIFA at 11%, 5 years = 3.6959; PVIF at 11%, 5 = 0.5935; PVIFA at 12%, 5 = 3.6048; PVIF at 12%, 5 = 0.5674.)

Show the solution

Annual interest I = 9% × ₹1,000 = ₹90.
(a) Current yield = ₹90 ÷ ₹920 × 100 = 9.78%.
(b) Approximate YTM = [90 + (1,000 − 920) ÷ 5] ÷ [(1,000 + 920) ÷ 2] = (90 + 16) ÷ 960 = 106 ÷ 960 = 11.04%. So test 11% and 12%.
Value at 11% = 90 × 3.6959 + 1,000 × 0.5935 = 332.63 + 593.50 = ₹926.13.
Value at 12% = 90 × 3.6048 + 1,000 × 0.5674 = 324.43 + 567.40 = ₹891.83.
The price ₹920 lies between ₹926.13 and ₹891.83, so interpolate.
YTM = 11% + [(926.13 − 920) ÷ (926.13 − 891.83)] × 1% = 11% + (6.13 ÷ 34.30) × 1% = 11% + 0.18% = 11.18%.
Check: the bond sells at a discount, so YTM (11.18%) is above current yield (9.78%) and the coupon rate (9%).

Answer: Current yield is 9.78%. Yield to maturity is about 11.18% (the approximate formula gives 11.04%).

Exam tips

Write the cash flow line first. Even if you make an arithmetic slip, you earn marks for the correct method.
Use the PV factors given in the question. Do not compute your own factors unless none are supplied.
Write the interpolation formula and show both trial values. The marks are for the two trial values and the interpolation.
Always add a one-line interpretation: discount or premium, buy or do not buy, or whether price risk is higher or lower.
For duration questions, show the table of year, cash flow, present value and year × present value. State that duration is in years.

Practice questions from Security Analysis

Valuation of Bonds and Debentures in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Valuation of Bonds and Debentures: frequently asked questions

What is the formula for bond valuation in CS Executive?

Value = I × PVIFA(kd, n) + M × PVIF(kd, n). I is annual interest, M is redemption value, kd is the required return and n is the years to maturity. For an irredeemable debenture, value = I ÷ kd.

How do I calculate yield to maturity of a bond in the exam?

First calculate the approximate YTM using [I + (M − P) ÷ n] ÷ [(M + P) ÷ 2]. Then compute the bond value at two rates on either side of it and interpolate so that the value equals the market price.

What is the difference between current yield and YTM?

Current yield is annual interest divided by market price. It ignores any gain or loss at redemption. YTM is the full return if you hold to maturity, including that gain or loss and the timing of all cash flows.

How are bond duration and bond price related?

Duration measures price sensitivity to interest rate changes. The price change is approximately minus modified duration times the change in yield. A bond with a longer duration loses more when rates rise and gains more when rates fall.

Why does a bond sell at a discount?

A bond sells at a discount when the market's required return is higher than its coupon rate. The discount makes up the difference, so a buyer at that price earns the required return.