Advanced Financial Management · Security Analysis

Bond Valuation and Yield Measures for CA Final AFM

Q: What is the difference between current yield and YTM?

Current yield is annual coupon divided by market price. It ignores the redemption gain or loss and the timing of cash flows. YTM is the discount rate that equates all remaining cash flows to the price, so it includes coupons, redemption value and time value.

Q: How do I calculate YTM in the CA Final exam?

Find the approximate YTM first. Use it and one more rate as trial rates to value the bond with PV tables. Then interpolate so the bond value equals the market price.

Q: Why does a bond price fall when interest rates rise?

The coupons and redemption value are fixed. When the required yield rises, they are discounted at a higher rate, so their present value falls. That present value is the bond's price.

Q: What does convexity add to duration?

Duration gives a linear estimate of price change. The actual price-yield relationship is curved, so duration errs for large yield changes. Convexity corrects for that curve and gives a better estimate.

Q: Which bond has more interest rate risk?

Generally the bond with the higher duration. Longer maturity and a lower coupon rate raise duration. A zero-coupon bond has duration equal to its maturity.

Updated 5 October 2026 · Fact-checked

Bond valuation finds a bond's price by discounting its coupons and redemption value at the investor's required yield. Yield to maturity is the discount rate that makes that present value equal the market price. To solve: list cash flows, discount them, then compute current yield, YTM, duration and convexity as asked.

Understand Bond Valuation and Yield Measures

A bond is a promise to pay fixed coupons and a redemption value on fixed dates. Its value today is the present value of those promises. The discount rate is the return investors demand for bonds of similar risk and maturity.

Price and yield move in opposite directions. When market yields rise, the present value of fixed cash flows falls, so the price falls. When the required yield equals the coupon rate, the bond sells at par. If the required yield is above the coupon rate, it sells at a discount. If it is below, it sells at a premium.

You will meet three yield measures. Current yield looks only at annual coupon income against price. It ignores the gain or loss at redemption. Yield to maturity (YTM) is the single rate that equates the present value of all remaining cash flows to the price. It assumes you hold to maturity and reinvest coupons at the same rate. Yield to call works the same way, using the call date and call price.

Interest rate risk is how much the price changes when yields change. Macaulay duration is the weighted average time to receive the cash flows, with present values as weights. Modified duration converts it into a percentage price sensitivity. Duration gives a straight-line estimate. The true price-yield curve is bent, so duration alone is slightly off for large yield moves. Convexity measures that bend and corrects the estimate.

Longer maturity and lower coupon raise duration, so those bonds carry more interest rate risk. A zero-coupon bond has duration equal to its maturity.

Key rules to remember

Bond price: P = Σ [C ÷ (1 + y)^t] + M ÷ (1 + y)^n; C = coupon per period, M = redemption value, y = required yield per period, n = number of periods. For semi-annual bonds use half the coupon, double the periods and half the annual yield.
Perpetual bond price: P = C ÷ y; Use for irredeemable bonds with fixed coupon.
Current yield: Current yield = Annual coupon ÷ Market price × 100; Ignores capital gain or loss and time value.
Approximate YTM: YTM ≈ [C + (M − P) ÷ n] ÷ [(M + P) ÷ 2]; Quick estimate. Use it as the first trial rate for interpolation.
YTM by interpolation: YTM = Lower rate + [(PV at lower − Price) ÷ (PV at lower − PV at higher)] × (Higher rate − Lower rate); Choose two rates so that the price lies between the two present values.
Macaulay duration: D = Σ [t × PV of cash flow at t] ÷ Σ PV of cash flows; Σ PV of cash flows equals the bond price when discounted at the YTM. Measured in years.
Modified duration: MD = D ÷ (1 + y); y is the yield per period matching the compounding of D. Annual compounding: use the annual YTM.
Price change using duration: ΔP ÷ P ≈ − MD × Δy; Linear estimate. Put Δy in decimals: a 1% rise is Δy = 0.01. The minus sign shows price falls when yield rises.
Convexity: Convexity = [1 ÷ (P × (1 + y)^2)] × Σ [t × (t + 1) × CF_t ÷ (1 + y)^t]; y is in decimals (10% = 0.10). Use consistently with the adjustment formula below.
Price change with convexity: ΔP ÷ P ≈ − MD × Δy + ½ × Convexity × (Δy)^2; Valid with the convexity definition above. Δy must be in decimals (0.01 for 1%), so (Δy)^2 = 0.0001 for a 1% move. The result is a fraction; multiply by 100 for a percentage.
Zero-coupon bond duration: D = Maturity in years; Single cash flow at maturity.

How to solve Bond Valuation and Yield Measures questions

Use this order for any bond question. It keeps your working clear and earns step marks even if the final number slips.

1Read the data and note face value, coupon rate, redemption value, years to maturity, market price and required yield. Check whether coupons are annual or semi-annual.
2Write the cash flow timeline. Coupon is on face value, not on price. Add redemption value in the last period.
3Identify what is asked: price, YTM, current yield, duration, convexity or price change.
4For price, discount each cash flow at the required yield using the PV factor tables. For YTM, compute the approximate YTM, then test two rates and interpolate.
5For duration, build a table: period, cash flow, PV factor, PV, and PV × t. Divide the total of PV × t by the total PV. Divide by (1 + y) for modified duration.
6For price change, apply ΔP ÷ P ≈ − MD × Δy, with Δy in decimals. Add the convexity term if convexity is given or asked.
7State the answer with units (₹, %, years) and one line of interpretation, such as discount or premium bond, or which bond is riskier.

Quickest way: Shortcut: approximate YTM, then bracket and interpolate

When to use it: Use when the question asks for YTM and gives PV tables, and you have limited time.

Compute approximate YTM using [C + (M − P) ÷ n] ÷ [(M + P) ÷ 2].
Take the nearest whole-number rate as the first trial and one rate on the other side of it as the second.
Find the bond value at both rates. The market price must lie between them.
Interpolate. Check that the answer lies between your two trial rates.
If the price is below face value, expect YTM above coupon rate. If above face value, expect YTM below coupon rate. Use this as a sanity check.

Common mistakes in Bond Valuation and Yield Measures

Calculating coupon as a percentage of market price instead of face value.
Students link the coupon with the amount they paid.
Fix: Coupon = coupon rate × face value. Market price is used only in current yield and YTM.
Using current yield as YTM.
Both are called yield, and current yield is quicker to compute.
Fix: Current yield ignores redemption gain or loss and the time value. Use it only when the question asks for it. It equals YTM for a perpetual bond. For a bond trading at par, current yield equals the coupon rate and the YTM.
Forgetting to halve the coupon and rate and double the periods for semi-annual bonds.
The data is quoted in annual terms.
Fix: Convert everything to the payment period first. Then state whether the final yield is per period or annualised.
Using duration without dividing by (1 + y) when estimating price change.
Macaulay and modified duration are confused.
Fix: Price change uses modified duration. Divide Macaulay duration by (1 + y) first.
Interpolating when the price does not lie between the two trial values.
The second trial rate is chosen without checking the direction.
Fix: Remember that a higher rate gives a lower value. If both values are above the price, raise the rate again and recompute.
Ignoring the sign of the price change.
Students drop the minus sign in − MD × Δy.
Fix: A rise in yield means a fall in price. Write the sign and state the direction in words.

Worked examples

Example 1

A 10% bond of face value ₹1,000 pays annual coupons and is redeemable at par after 5 years. Investors require a return of 12%. (a) Find the value of the bond. (b) If it trades at that value, find the current yield and the approximate YTM. Use PVAF(12%, 5) = 3.6048 and PVF(12%, 5) = 0.5674.

Show the solution

Annual coupon = 10% × ₹1,000 = ₹100. Redemption = ₹1,000 at year 5.
PV of coupons = ₹100 × 3.6048 = ₹360.48.
PV of redemption = ₹1,000 × 0.5674 = ₹567.40.
Value = ₹360.48 + ₹567.40 = ₹927.88.
Required yield 12% is above the coupon rate 10%, so the bond sells at a discount. This agrees with ₹927.88 < ₹1,000.
Current yield = ₹100 ÷ ₹927.88 × 100 = 10.78%.
Approximate YTM = [100 + (1,000 − 927.88) ÷ 5] ÷ [(1,000 + 927.88) ÷ 2] = (100 + 14.42) ÷ 963.94 = 114.42 ÷ 963.94 = 11.87%.
The approximate YTM (11.87%) is an estimate of the exact YTM, which is 12% because the price was derived at 12%. Current yield (10.78%) is lower than YTM because it ignores the ₹72.12 gain at redemption (₹1,000 − ₹927.88).

Answer: Value ≈ ₹927.88 (discount bond). Current yield ≈ 10.78%. Approximate YTM ≈ 11.87%, an estimate of the exact YTM of 12%. Current yield is lower because it ignores the ₹72.12 redemption gain.

Example 2

A 3-year bond of face value ₹100 pays a 10% annual coupon and is redeemable at par. Its YTM is 10%, so it trades at ₹100. Find the Macaulay duration, modified duration and convexity. Estimate the new price if the yield rises by 1 percentage point, first using duration only and then using duration with convexity.

Show the solution

Cash flows: ₹10 at year 1, ₹10 at year 2, ₹110 at year 3. Discount at 10%.
PV year 1 = 10 ÷ 1.10 = 9.0909. PV year 2 = 10 ÷ 1.21 = 8.2645. PV year 3 = 110 ÷ 1.331 = 82.6446.
Total PV = 100.00, which matches the price.
PV × t: 1 × 9.0909 = 9.0909; 2 × 8.2645 = 16.5289; 3 × 82.6446 = 247.9339. Total = 273.5537.
Macaulay duration = 273.5537 ÷ 100 = 2.7355 years.
Modified duration = 2.7355 ÷ 1.10 = 2.4868.
Duration only: ΔP ÷ P ≈ − 2.4868 × 0.01 = − 0.024868, i.e. − 2.4868%. New price ≈ 100 × (1 − 0.024868) = ₹97.51.
Convexity sum: t(t+1)CF ÷ (1.10)^t = 2 × 10 ÷ 1.10 + 6 × 10 ÷ 1.21 + 12 × 110 ÷ 1.331 = 18.1818 + 49.5868 + 991.7355 = 1,059.5041.
Convexity = 1,059.5041 ÷ (100 × 1.21) = 1,059.5041 ÷ 121 = 8.756.
Convexity term = ½ × 8.756 × (0.01)^2 = ½ × 8.756 × 0.0001 = 0.000438, i.e. + 0.0438%.
Price change with convexity ≈ − 0.024868 + 0.000438 = − 0.024430, i.e. − 2.443%. New price ≈ 100 × (1 − 0.02443) = ₹97.56.
The convexity-corrected price (₹97.56) is slightly higher than the duration-only price (₹97.51), because duration ignores the curve of the price-yield relationship.

Answer: Macaulay duration ≈ 2.74 years. Modified duration ≈ 2.49. Convexity ≈ 8.76. A 1 percentage point rise in yield lowers the price by about 2.49% using duration only (≈ ₹97.51), and by about 2.44% after the convexity adjustment (≈ ₹97.56).

Exam tips

Write the cash flow table first. Even a wrong final answer can earn method marks when the table is clear.
For YTM, always show the approximate YTM and the two trial rates. Examiners expect to see this path.
Add a one-line interpretation: discount or premium, riskier or safer bond, and the direction of the price move.
In case-scenario MCQs, check the sign of the price change and whether coupons are semi-annual before calculating. Wrong MCQ answers carry no negative marking, so always attempt them.
When comparing two bonds, use duration. The bond with longer duration is more sensitive to yield changes. Use convexity when the yield change is large.

Practice questions from Security Analysis

Bond Valuation and Yield Measures in other exams

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

Bond Valuation and Yield Measures: frequently asked questions

What is the difference between current yield and YTM?

Current yield is annual coupon divided by market price. It ignores the redemption gain or loss and the timing of cash flows. YTM is the discount rate that equates all remaining cash flows to the price, so it includes coupons, redemption value and time value.

How do I calculate YTM in the CA Final exam?

Find the approximate YTM first. Use it and one more rate as trial rates to value the bond with PV tables. Then interpolate so the bond value equals the market price.

Why does a bond price fall when interest rates rise?

The coupons and redemption value are fixed. When the required yield rises, they are discounted at a higher rate, so their present value falls. That present value is the bond's price.

What does convexity add to duration?

Duration gives a linear estimate of price change. The actual price-yield relationship is curved, so duration errs for large yield changes. Convexity corrects for that curve and gives a better estimate.

Which bond has more interest rate risk?

Generally the bond with the higher duration. Longer maturity and a lower coupon rate raise duration. A zero-coupon bond has duration equal to its maturity.