Price, yield and the curve

Put £100 in an account paying 5% and in a year you have £105. Turn that round: £105 arriving in a year is worth £100 today, if 5% is what money earns. Working backwards from a future payment to its value today, by dividing by 1.05, is called discounting.

A one-year bond with a 5% coupon pays £105 at the end: the £100 back plus £5 interest. If investors want 5% for lending for a year, they pay 105 ÷ 1.05 = 100 for it. The rate that makes a bond’s payments add up to its price is its yield: the return you lock in by buying at that price.

Now investors want 7% instead of 5%. The bond still pays £105, so its price must fall until it offers 7%: 105 ÷ 1.07 = 98.13. A bond’s payments are fixed, so its price is the only thing that can move. When yields rise, prices fall; when yields fall, prices rise.

A bond whose coupon is higher than today’s yield pays more than the market rate, so buyers pay more than 100 for it: it trades at a premium. A coupon below the yield means a price below 100, a discount. When coupon and yield are equal, the price is exactly 100, called par.

A two-year bond pays a coupon in each year, so you discount each payment for the years you wait. A 3% two-year bond yielding 4.5%: 3 ÷ 1.045 for year one, plus 103 ÷ 1.045² for year two. That is 2.87 + 94.32 = 97.19. The later the payment, the more times it is divided.

Plot the yields on 2-year, 10-year and 30-year bonds and join them up: that line is the yield curve. It usually slopes upwards, because lenders want extra for tying money up longer. Traders measure gaps in basis points, hundredths of a percentage point: a 2-year at 4.0% and a 10-year at 4.6% are 60bp apart, a gap called 2s10s.

One-year money pays 4% and two-year money 4.5% a year. Lending for two years must pay about the same as lending for one year and then again for another. So the curve implies a one-year rate a year from now of about 5% (2 × 4.5 − 4). That is a forward rate: the future rate today’s yields lock in.

Traders describe a curve move in two words. The first says which way prices went: bull if yields fell, so bond prices rose; bear if yields rose, so prices fell. A day when 2-year and 10-year yields both rise is a bear move, whatever the size.

The second word says what happened to the gap between long and short yields: steepening if it widened, flattening if it narrowed. If the 2-year falls 20bp and the 10-year 5bp, yields fell and 2s10s widened by 15bp: a bull steepener, the usual reaction when markets start to expect rate cuts.