Bond traders watch one chart more than any other, and when its line bends the wrong way, people start saying "recession". This week you learn to price a bond and read that chart.
Learning objectives. After this week you can (1) price a bond as the present value of its payments; (2) explain why long bonds swing more when rates move; and (3) read a normal and an inverted yield curve.
Remember this: a bond's price is the present value of its payments at today's market rate.
1) A bond's price is a present value
Price = Σ Coupont ÷ (1 + r)t + Face ÷ (1 + r)n
A one-year $1,000 bond with a 5% coupon pays $1,050 in one year (the $50 coupon plus the face):
| Market rate | Arithmetic | Price | Trades at |
|---|---|---|---|
| 8% | 1,050 ÷ 1.08 | $972.22 | Discount |
| 5% | 1,050 ÷ 1.05 | $1,000.00 | Par |
| 3% | 1,050 ÷ 1.03 | $1,019.42 | Premium |
That is Week 6's inverse rule, worked out: a higher rate shrinks the PV of the same fixed payments. When the coupon rate equals the market rate, the bond prices at par.
2) Longer bonds swing harder
A two-year $1,000 bond, 5% coupon, 8% market rate: 50 ÷ 1.08 + 1,050 ÷ 1.082 = 46.30 + 900.21 = $946.50. That is further below par than the one-year bond's $972.22, because more of its money arrives later. Duration measures this sensitivity:
% price change ≈ − duration × change in rate
A duration-8 bond loses about 8% of its price when rates rise one point; a duration-1 bond loses about 1%. Two yields to keep straight: current yield is the coupon over today's price ($60 ÷ $960 = 6.25%); yield to maturity is the one rate that makes the PV of all payments equal the price.
3) The yield curve
Plot the yields of same-risk bonds against maturity and you get the yield curve.
- Normal: slopes upward. Lenders want extra yield to tie money up longer.
- Inverted: short yields above long ones. Investors expect the central bank to cut rates, usually because they expect a slowdown. Inversions have come before most recessions.
A riskier issuer pays the safe yield plus a credit spread. If a team's results collapse, lenders demand a bigger spread, and its bond price falls.
Quick self-check
(a) One-year $1,000 bond, 5% coupon, market rate 8%: price? (b) Rates rise one point. About how much does a duration-8 bond lose? (c) Short yields are above long yields. What is that curve called? (Answers: $972.22; about 8%; inverted.)
Common mistakes
- Mixing up coupon and yield. The coupon is fixed; the yield moves with the price.
- Calling a discount bond a bad bond. Buy at $950, hold to maturity, collect $1,000.
- Reading inversion as high long rates. It means short rates are high compared with long.
This week in the arena
Take the bond you sized in Week 6 (or a $9M face with a coupon you choose) and price it for one year at three market rates: 3%, your coupon rate, and 8%. Then ask what this week's race does to the rate lenders would want from your team.