Bonds, yields and duration
Price a bond, then move interest rates and see why long bonds swing the hardest.
Interest the bond pays each year, as a share of its $1,000 face value.
The return investors demand today for lending this long.
A sudden move in yields. 100 basis points is one percentage point.
The curve bends upward: that bend is convexity. The red straight line is what duration alone predicts, and it always undershoots.
A 10-year bond paying 4.00% is worth $960.44 when the market demands 4.50%. If yields jump up by 200 basis points, its price moves -14.6%. Duration alone predicted -16.1%; adding convexity gets -14.5%.
Same shock, three maturities (+200 bp)
- 2 years-3.6%
- 10 years-14.6%
- 30 years-26.7%
Modified durations: 2y 1.9, 10y 8.0, 30y 16.7. Longer bonds wait longer for their money, so a change in rates is compounded over more years.
Cash flows, and what each is worth today
Pale bars are what you receive; solid bars are that payment discounted to today. Macaulay duration, 8.4 years, is the balance point of the solid bars.
This really happened
Long-term US Treasuries are often called the safest asset there is. When rates rose sharply in 2022, a fund of 20+ year Treasuries (TLT) fell 48% from Jul 2020 to Oct 2023, including interest paid. No default, no missed coupon: just duration.
Try this
- Set the coupon equal to the yield. The price lands on $1,000: the bond trades at par.
- Compare a 30-year bond at 1% yield with one at 8%. Which has the longer duration, and why?
- Push the rate shock to +300 bp, then −300 bp. Is the gain as big as the loss? That gap is convexity working for you.
The math
- percentage change in price
- modified duration, Macaulay duration ÷ (1 + y)
- change in yield
- convexity