Bond valuation is the fixed-income module's foundation. It is also the module where the mechanical calculations are deceptively straightforward — anyone can compute a bond price with a discount function — but where markers reward students who understand what each number actually means. A bond price is a snapshot; yield to maturity is a promise conditional on assumptions; duration is a linear approximation with a known bias; and convexity exists specifically because that approximation fails.
This guide takes you through what a bond valuation assignment actually tests, the mechanics of bond pricing and YTM by interpolation, a full duration and convexity walkthrough on the same bond, and the interpretation moves that separate competent answers from top ones. The worked example runs continuously through all five calculations so you can see how they fit together.
What a Bond Valuation Assignment Actually Tests
A bond valuation assignment tests four connected skills. Almost every brief hits at least three of them.
- Pricing — Given a coupon, face value, maturity, and required yield, can you price the bond correctly?
- Yield calculation — Given the market price, can you back-solve for the yield to maturity, typically by interpolation or an iterative method?
- Interest rate risk — Can you compute Macaulay and Modified duration, apply modified duration to estimate price change from a yield move, and add the convexity adjustment where appropriate?
- Interpretation — Can you distinguish coupon rate, current yield, and YTM; explain why a bond trades at discount, par, or premium; and identify the limitations of duration-based risk analysis?
Read the brief for what's being asked: "Value this bond" means compute the price. "Compute the YTM" means back-solve from a given price. "Compute the interest rate risk" means duration analysis. Some briefs ask all four; others isolate one calculation. Identify which type of question before starting.
Bond Mechanics — Three Rates You Must Not Confuse
The single most common conceptual error in this module is treating coupon rate and yield to maturity as if they were the same thing. They are not, and the difference determines whether a bond trades at par, discount, or premium.
| Rate | What It Is | When It Changes |
|---|---|---|
| Coupon rate | The annual interest payment as a percentage of face value. Fixed at issue. | Never (for a fixed-coupon bond). Written into the bond contract. |
| Current yield | Annual coupon divided by current market price. | Whenever the market price moves. Ignores capital gain/loss to maturity. |
| Yield to maturity (YTM) | The single discount rate that sets the PV of all cash flows equal to market price. The true total return if held to maturity and coupons reinvested at YTM. | Continuously, as market prices move. The rate embedded in the current price. |
The relationship between coupon rate and YTM determines the bond's price relative to par:
- If coupon rate = YTM → bond trades at par (price = face value)
- If coupon rate < YTM → bond trades at a discount (price < face)
- If coupon rate > YTM → bond trades at a premium (price > face)
The intuition: if the market demands more return (YTM) than the bond promises (coupon rate), the price has to fall so the investor makes up the shortfall as a capital gain to maturity. This is also why bond prices and yields move in opposite directions — the inverse relationship that drives everything downstream.
A Worked Example — Bond Pricing
The examples below run continuously through a single bond so you can see how the calculations connect. The figures are constructed for teaching; the method is identical for any bond.
Coupon rate: 6% annual → £6 per year
Maturity: 5 years
Coupon frequency: Annual
Required YTM: 8%
Question: what should the bond be priced at?
Where:
P = bond price
C = annual coupon payment
y = YTM (per period)
n = number of periods to maturity
F = face value
Year 2: £6.00 ÷ (1.08)2 = £6.00 × 0.8573 = £5.1440
Year 3: £6.00 ÷ (1.08)3 = £6.00 × 0.7938 = £4.7630
Year 4: £6.00 ÷ (1.08)4 = £6.00 × 0.7350 = £4.4102
Year 5: £6.00 ÷ (1.08)5 = £6.00 × 0.6806 = £4.0835
Sum of PV of coupons = £23.9563
Bond price P = £23.9563 + £68.0583 = £92.01
Interpretation: The bond trades at a discount to par (£92.01 vs £100) because the coupon rate (6%) is below the YTM (8%). The market is compensating for the sub-market coupon by pricing the bond low enough that the investor's total return — coupons plus £7.99 capital gain over five years — equals the required 8% YTM.
Yield to Maturity by Interpolation
YTM cannot be solved algebraically for a coupon bond — the equation is a polynomial with no closed-form solution. Financial calculators use iterative methods; in an undergraduate exam, linear interpolation between two trial rates that bracket the true YTM is the standard technique.
Given: market price = £91.99
Question: What is the YTM?
The trial rates need to bracket the true YTM: one produces a price above the target, one below. Since the target price (£91.99) is well below par (£100), the true YTM is well above the 6% coupon — start with 7% and 9%.
PV coupons + PV face at 7% = £24.5993 + £71.2986 = £95.90
Trial YTM 2 = 9%:
PV coupons + PV face at 9% = £23.3372 + £64.9931 = £88.33
£91.99 sits between the two → interpolation is valid.
= 7% + (3.91 ÷ 7.57) × 2%
= 7% + 0.5162 × 2%
= 7% + 1.03%
≈ 8.03%
Interpretation: The bond's YTM is approximately 8.03%. Interpolation is a linear approximation of a non-linear function — the price-yield curve is convex — so the interpolated YTM tends to overshoot the true rate slightly. The error is 0.03pp on this bond, which is negligible for teaching purposes. Narrower trial rates (say 7.5% and 8.5%) reduce the error further.
Fixed income assignment deadline closing in?
Our finance specialists build full bond valuation assignments — pricing, YTM, duration, convexity, and interpretation — to your university's marking criteria and your deadline.
Duration — Interest Rate Sensitivity
Duration answers the question every fixed-income investor asks: "if yields move, how much will my bond price move?" There are two flavours — Macaulay (the concept) and Modified (the working number) — and students routinely mix them up.
Macaulay Duration
The weighted average of times to each cash flow, weighted by the present value of that cash flow relative to the bond price. Expressed in years.
Macaulay duration is a conceptual measure: the average waiting time to receive the bond's cash flows in present-value terms. A zero-coupon bond has Macaulay duration equal to its maturity (all cash flow at the end). A coupon bond has Macaulay duration less than its maturity because some cash flow arrives earlier as coupons.
1 £6.00 £5.5556 £5.5556
2 £6.00 £5.1440 £10.2881
3 £6.00 £4.7630 £14.2890
4 £6.00 £4.4102 £17.6407
5 £106.00 £72.1418 £360.7091
Σ (t × PV) = £408.48
P = £92.01
DMac = £408.48 ÷ £92.01 = 4.44 years
Critical detail: the Year 5 cash flow is coupon plus face value — £6 + £100 = £106, not just £6. Forgetting to add the face is the single most common numerical error in duration calculations at undergraduate level. It changes the answer significantly.
Modified Duration
Modified duration is the number you actually use for price-change estimates. It expresses how much the bond price changes (in percent) for a 1% change in yield.
Applying Duration — Estimating Price Change
Percentage price change is approximately negative modified duration times the yield change. The minus sign encodes the inverse price-yield relationship.
%ΔP ≈ −4.11 × 1.00% = −4.11%
In £: −4.11% × £92.01 = −£3.78
Estimated new price: £92.01 − £3.78 = £88.23
Interpretation: If yields rise 100bp from 8% to 9%, duration predicts the bond price falls by 4.11% to £88.23. Recomputing the price directly at YTM 9% gives an actual price of £88.33. The duration estimate is off by £0.10 — it underestimates the actual price (i.e. overstates the drop). This is not a mistake in the duration formula; it is the well-known limitation of a linear approximation to a convex function. The correction is convexity.
Convexity — The Second-Order Correction
Duration draws a straight line through the current price-yield point. The actual price-yield relationship is convex (curved upward). For small yield changes, the linear approximation is fine. For larger yield changes, the straight line diverges from the curve — and always in the same direction: duration always overstates the price decrease when yields rise, and understates the price increase when yields fall. Convexity is the second-order term that corrects for this.
Where Convexity = Σ [ t × (t+1) × PV(CFt) ] ÷ [ P × (1+y)² ]
Adjustment for 100bp move:
½ × 21.91 × (0.01)² = +0.11%
Adjusted %ΔP = −4.11% + 0.11% = −4.00%
Adjusted new price: £92.01 × (1 − 4.00%) = £88.33
Interpretation: Adding the convexity term reduces the estimated price drop from 4.11% to 4.00%, matching the true new price of £88.33 within pence. The convexity adjustment is small for a 100bp move on a five-year bond; it becomes materially larger for longer maturities and bigger yield moves — which is precisely why traders and portfolio managers report both duration and convexity.
2:2 vs First: The Interest Rate Risk Paragraph
Both students compute duration correctly. The gap between grade bands is in what they say about what duration does and does not measure.
"The bond's modified duration is 4.11 years, meaning the price will fall 4.11% for every 1% rise in yield. Duration is a good measure of interest rate risk and is widely used in fixed income portfolios."
"Modified duration of 4.11 provides a first-order estimate: a 100bp yield rise implies a 4.11% price fall. Empirically, the actual price falls only 4.00% — duration always overstates loss on rises and understates gain on falls, because it linearises a convex relationship. The convexity adjustment (0.11% here) closes most of the gap; for longer-maturity bonds or larger yield moves, that adjustment becomes material. Two further limitations of duration-only analysis: it assumes parallel yield curve shifts, whereas real yield curves twist and steepen (key rate duration decomposes this exposure); and it says nothing about credit risk — a corporate bond's price can move on issuer credit deterioration even when the risk-free curve is unchanged. For a complete risk picture, duration is the first tool, not the only one.
Five Mistakes That Cost Students Marks
Confusing coupon rate with YTM. The coupon rate is fixed at issue; YTM moves with the market. Using coupon rate as the discount factor prices every bond at par, which is wrong except by coincidence.Fix: Always discount using YTM (or the required return), never the coupon rate. Coupon rate only determines the size of the £ cash flow.
Forgetting the face value in the final year's cash flow in duration. The Year n cash flow is coupon + face, not coupon alone. Missing this understates duration significantly.Fix: Write out the cash flow table with the final year's row labelled "coupon + face" explicitly, before computing the t × PV column.
Using Macaulay duration for price-change estimates. Macaulay is the conceptual measure; Modified is the number you apply to yield changes.Fix: For any %ΔP ≈ −D × Δy calculation, use Modified duration (Macaulay ÷ 1+y). Compute Macaulay first, then divide.
Reporting duration without the convex-bias caveat. Duration is a linear approximation to a convex function. It always errs in the same direction. Not saying so treats duration as exact.Fix: Add one sentence noting that duration overstates loss on yield rises and understates gain on yield falls; add the convexity term when yield moves are large or maturities long.
No engagement with credit risk on corporate bonds. Duration measures interest rate risk only. A corporate bond can lose value on issuer credit deterioration even when the risk-free curve is unchanged.Fix: If the brief involves corporate bonds, add a paragraph on credit spread risk. Reference credit ratings, spread duration, or CDS spreads as complementary measures. This one addition consistently lifts a grade band.
Frequently Asked Questions
What is a bond valuation assignment usually asking me to do?
What is the difference between coupon rate, current yield, and YTM?
What is the difference between Macaulay and Modified duration?
When does convexity matter?
How do I structure a bond valuation assignment?
My deadline is close — what should I prioritise on a bond valuation assignment?
Need Your Bond Valuation Assignment Done Right?
Our expert writers deliver fully worked fixed-income assignments — bond pricing, YTM by interpolation, duration and convexity analysis — with the interpretation that wins marks. To your university's marking criteria and your exact deadline.
Get Expert Help Today →
