FNCE30001 Investments
Bond Pricing and Fixed Income
FNCE30001 is taught fixed-income first, and it starts here, in the heaviest examinable block (Weeks 1–2). A bond is a contract: the issuer pays a fixed coupon each period and returns the face (par) value at maturity, so its fair price today is just the present value of every promised cash flow, discounted at the yield. You split the cash flows into the level coupon stream (an annuity) and the single face repayment, price each, and sum. Get this one calculation fluent — on annual and on semi-annual conventions — and half the fixed-income marks are already yours, because the bond price and its inverse, the yield to maturity, recur on the mid-semester test and the comprehensive final. The chapter then nails the rest of the fixed-income toolkit: the price–yield inverse (price and yield always move in opposite directions, along a convex curve), premium / par / discount read straight off coupon vs yield, YTM vs current yield, the Australian ex-interest settlement convention, and pricing a risky bond off its default-adjusted expected cash flow.
What this chapter covers
- 011.1 The coupon-bond price formula (annuity of coupons + PV of face)
- 021.2 The semi-annual convention (halve C and y, double n)
- 03The price–yield inverse and convexity
- 041.4 Yield to maturity as an internal rate of return
- 051.5 Current yield vs YTM, and the premium / par / discount ordering
- 061.6 The Australian 7-day ex-interest convention
- 071.7 Risky bonds: default, recovery, promised vs expected yield, and the credit spread
Worked example: semi-annual bond pricing (a discount bond)
- +1(a) Convert to the half-year basis first. The single biggest mechanical error is mixing bases, so do this before anything else: C = 4/2 = $2 per period, y = 5%/2 = 2.5% per period, n = 2×3 = 6 periods.
- +1(b) Annuity factor for the coupons: (1.025)6 = 1.15969, so (1/0.025)(1 − 1/1.15969) = 5.50813.
- +1PV of coupons: 2 × 5.50813 = $11.02. PV of face: 100 / 1.15969 = $86.23.
- +1Price: 11.02 + 86.23 = $97.25.
- +1(c) Decide: the coupon rate (4%) is below the yield (5%), so the bond trades at a discount (P < Par). The capital gain pulling the price up to par at maturity compensates the buyer for the below-market coupon.
Key terms
- Coupon-bond price
- The present value of a bond's cash flows: P = (C/y)[1 − (1+y)−n] + Par/(1+y)n — the annuity value of the level coupon stream plus the PV of the single face repayment. C, y and n must all be on the same period basis.
- Semi-annual convention
- For a bond paying twice a year, convert everything to the half-year basis before pricing: halve the coupon, halve the yield, and double the number of periods. Plugging an annual coupon with a semi-annual n (or vice versa) is the most common bond mistake.
- Yield to maturity (YTM)
- The single rate that makes the PV of all cash flows equal the price — the bond's internal rate of return if held to maturity with coupons reinvested at that rate. For a coupon bond it has no closed form and is solved iteratively; for a zero it inverts directly. YTM equals the coupon rate only at par.
- Ex-interest (Australian convention)
- Australian Treasury bonds trade ex-interest when settlement falls 7 or fewer calendar days before a coupon date. Ex-interest means the seller keeps the imminent coupon, so the buyer drops it from the pricing equation. Settlement is counted in business days; the 7-day test is counted in calendar days.
- Credit spread
- The extra promised yield on a risky bond over a comparable risk-free bond of the same maturity: spread = z − z_rf. It is the market's price for bearing default risk; a wider spread signals a higher perceived chance of default (or more compensation demanded for it).
Bond Pricing and Fixed Income FAQ
When is a bond at a premium, par or discount?
Read it straight off the coupon rate versus the yield, before you compute. Coupon rate > YTM → premium (P > Par): you pay above par to collect above-market coupons. Coupon rate = YTM → par. Coupon rate < YTM → discount (P < Par): the capital gain to par compensates for the low coupon. Calling the direction first is a free sanity check that your number landed on the right side of par.
Why isn't YTM the same as the coupon rate?
The coupon rate is fixed at issue; the YTM moves with the market price every day. They are equal only when the bond trades at par. Two bonds with the same maturity but different coupons can have different YTMs and neither is mispriced. Never quote the coupon rate when the question asks for the yield, and remember the ordering: at a discount, coupon < current yield < YTM.
How does the Australian ex-interest rule change the price?
First reach settlement by counting business days from the trade date (skip weekends and AU holidays). Then apply the 7-day test in calendar days: if settlement is within 7 calendar days before the next coupon, the bond trades ex-interest, the seller keeps that coupon, and you drop it from the cash flows. Dropping a coupon lowers the price the buyer pays, because they are not paying for cash they will not receive. The two date counts use different rules — that is the bait.
How do I price a bond that might default?
Price the probability-weighted expected cash flow, not the promised par. With default probability b and recovery rate r_w, the expected redemption is Par[1 − b(1 − r_w)]; discount that to get the expected yield z*. The promised yield z (which discounts full par) is always higher than z* for a risky bond, and the credit spread z − z_rf is what the market charges for the default risk.
Exam move
Drill bond pricing until it is automatic on the four-step answer shape, because it is the single most-tested calculation and it appears on both the mid-semester test and the comprehensive final. Decide the period basis FIRST (annual or semi-annual) and never mix it; write the formula before the numbers; show the annuity factor and the face PV separately, then sum (a bare price scores zero). Call premium / par / discount off coupon-versus-yield as a sanity check. Practise the two date-count rules in the ex-interest convention — business days to settlement, calendar days for the 7-day test — and the risky-bond chain (expected cash flow → expected yield → spread). The price–yield inverse is the fixed-income reflex: a yield rise always means a price fall, every time.
