Bond Pricing Primer: YTM, Duration, Convexity & the Yield Curve
Bonds are tested in ICWIM Chapter 3 (asset classes), Chapter 5 (financial mathematics), CFA Level 1 (fixed income), and the CISI Diploma in Securities & Investment Unit 4. The same mechanics show up everywhere: present value of cash flows, sensitivity to yields, and the term structure of interest rates. This guide covers the essentials with worked examples.
The basic bond pricing formula
Where:
- P = price of the bond today
- C = periodic coupon payment
- y = yield per period (= YTM if annual; YTM/2 if semi-annual)
- F = face value (par)
- n = number of periods until maturity
P = 4/(1.05)1 + 4/(1.05)2 + 4/(1.05)3 + 4/(1.05)4 + 4/(1.05)5 + 100/(1.05)5
= 3.81 + 3.63 + 3.46 + 3.29 + 3.13 + 78.35 = £95.67
The bond trades at a discount because YTM (5%) > coupon rate (4%).
Discount, par, and premium
| Relationship | Price vs face | Why |
|---|---|---|
| Coupon rate < YTM | Discount (P < F) | Market demands higher yield → coupon insufficient → price drops to compensate |
| Coupon rate = YTM | Par (P = F) | Coupon exactly matches market yield demand |
| Coupon rate > YTM | Premium (P > F) | Coupon better than market yield → bond more valuable than face |
This relationship runs both ways. If YTM rises after issuance, price falls (the existing coupon now looks less attractive). If YTM falls, price rises. Price and yield move in opposite directions — the foundational bond market identity.
Three yield measures (and which to use when)
Current Yield
Simple income yield. Ignores capital gains/losses to redemption — so it overstates total return for discount bonds and understates for premium bonds.
Yield to Maturity (YTM)
The IRR of holding the bond to maturity. It's the single discount rate that makes PV of cash flows equal current price. The "true" return assuming hold to maturity AND reinvesting coupons at YTM.
No closed-form formula — solved iteratively (financial calculator or spreadsheet). For exams, you typically work with given YTMs or test direction-of-change (rates up → YTM up).
Yield to Worst (YTW)
For callable bonds: the lower of yield-to-maturity and yield-to-call. The "worst case" yield assuming the issuer exercises the option that's worst for the holder.
Clean vs dirty price
Bond prices are quoted "clean" but settled "dirty":
- Clean price: the quoted price, excluding accrued interest
- Accrued interest: the proportion of the next coupon earned by the seller (days since last coupon ÷ days in period)
- Dirty (or "full") price: what the buyer actually pays at settlement
Accrued = 5 × (80/365) = £1.10
Dirty (settlement) price = 98 + 1.10 = £99.10
Duration: measuring interest-rate sensitivity
Macaulay Duration
The weighted-average time to receive cash flows, weighted by their present value. Measured in years. A bond's "average maturity" — accounting for the fact that coupons arrive before the final principal repayment.
Properties:
- Zero-coupon bonds: Macaulay duration = maturity (only one cash flow at the end)
- Coupon bonds: Macaulay duration < maturity (coupons arrive earlier)
- Higher coupon → lower duration (more weight on near-term flows)
- Higher yield → lower duration (later flows discounted more)
Modified Duration
The percentage change in price for a 1% change in yield. Dimensionless. The practical sensitivity measure used by bond portfolio managers.
ΔP/P ≈ −7.2 × 0.005 = −3.6%
A £100,000 holding would lose ~£3,600.
Convexity
Duration assumes a linear relationship between price and yield. The true relationship is convex — curved. Convexity is the second-order correction:
Positive convexity is good for bondholders:
- When yields rise: actual price loss is LESS than duration predicts
- When yields fall: actual price gain is MORE than duration predicts
Long-duration bonds with regular coupons have positive convexity. Callable bonds have lower (sometimes negative) convexity because the call option caps the upside.
The yield curve
The yield curve plots yields against maturities for bonds of equivalent credit quality (typically government). Its shape carries macroeconomic signals.
Four standard shapes
| Shape | What it means | Typical context |
|---|---|---|
| Upward-sloping (normal) | Long-term yields > short-term yields | Expanding economy; inflation expected to rise |
| Flat | Yields roughly equal across maturities | Transition phase; uncertainty |
| Inverted | Short-term yields > long-term yields | Strong recession signal; market expects rate cuts |
| Humped | Mid-maturity yields highest | Specific market conditions; less common |
Three classical theories of the yield curve
| Theory | Core claim |
|---|---|
| Pure Expectations | Long-term rates = average of expected future short rates |
| Liquidity Preference | Long-term rates = expected short rates + risk premium (compensation for tying up capital longer) |
| Market Segmentation | Supply/demand for each maturity is independent (driven by different investor groups — pension funds prefer long, money markets prefer short) |
In practice, all three forces operate. The yield curve as observed reflects expectations + risk premia + segment-specific supply-demand.
Why the inverted yield curve matters
Inverted yield curves have preceded most US recessions in recent decades. The mechanism: short rates above long rates suggest the market expects the central bank to cut sharply in the near future — and they typically only do that when growth disappoints. The 10y minus 3-month spread is the most-tracked inversion gauge.
Credit risk & credit spreads
Yields on corporate bonds = government yield + credit spread. The spread compensates for default risk and is correlated with the credit rating:
| Rating tier | Typical spread (over government, indicative bps) |
|---|---|
| AAA | 10–40 |
| AA | 30–70 |
| A | 60–130 |
| BBB | 100–250 |
| BB & below (high yield / junk) | 300–800+ |
Spreads widen in recessions (default risk priced higher) and tighten in expansions. "Credit spread compression" is a sign of late-cycle exuberance.
Key bond market vocabulary
| Term | Meaning |
|---|---|
| Par | Face value / principal repaid at maturity |
| Coupon | Periodic interest payment |
| Tenor | Time remaining to maturity |
| Callable | Issuer can redeem before maturity (option held by issuer) |
| Putable | Holder can sell back to issuer before maturity (option held by holder) |
| Convertible | Holder can convert into the issuer's equity |
| Zero-coupon | No coupons; sold at deep discount, redeemed at face |
| FRN (floating rate) | Coupon resets periodically based on a benchmark |
| Sukuk | Sharia-compliant fixed-income instrument |
| Gilt | UK government bond |
| Treasury | US government bond |
| Bund | German government bond |
Common exam traps
| Confusion | The fix |
|---|---|
| Macaulay vs Modified duration | Macaulay in years; Modified is % sensitivity. ModDur = Macaulay ÷ (1+y). |
| Current yield vs YTM | CY ignores capital gain/loss. YTM is the full IRR. |
| Price vs yield direction | Always opposite. Yields up → prices down. |
| Clean vs dirty price | Dirty = clean + accrued interest. Quotes are clean; settlement is dirty. |
| Coupon rate vs YTM | Coupon is fixed at issue. YTM moves with the market. Their relationship determines discount/par/premium. |
| Convexity sign | Standard bonds: positive (good). Callable: lower, can be negative near the call price. |
| Zero-coupon Macaulay | Always EQUAL to maturity (only one cash flow). Coupon bonds always have Macaulay duration LESS than maturity. |
Drill bonds in the ICWIM bank
icwim.com's ICWIM Ch 3 (Asset Classes) covers bond pricing in detail; Ch 5 (Analysis) tests the financial mathematics including duration and convexity. The Calculation Drill mode lets you focus on just the bond calcs.
Full ICWIM prep £49 — or the Cat 5 Pack (ICWIM + UAE FRR) for £79.
Related guides
- Derivatives Payoffs in 8 Diagrams — companion piece on options
- ICWIM Calculation Formulas Cheat Sheet — every formula tested
- ICWIM Chapter 5 Deep Dive: Macro Indicators — the macro side of Chapter 5
- ICWIM 8-week study plan — chapter-weighted prep