Duration and Convexity

Why this matters

Duration and the Yield Curve gave duration as a rule of thumb: a bond with a duration of 6 falls about 6% when rates rise one point. That's enough to read a fund's fact sheet. It doesn't say where the 6 comes from, why a 30-year bond's duration is nowhere near 30, or why the rule goes wrong for large moves. This lesson builds the number from a bond's payments, then tests it against the largest yearly loss on Treasury bonds in the available record.

Duration as an average wait

A bond is a set of payments spread over time. A 3-year bond pays you in year one, in year two, and mostly in year three. One way to describe how long your money is tied up is to average those times, counting each payment by how much of the bond's value it represents. That average is the Macaulay duration.

A $1,000 bond with a 6% coupon paid once a year, priced at $1,000, so its yield is 6%.

Year	Payment	Value today at 6%	Share of the price	Year × share
1	$60	$56.60	5.66%	0.057
2	$60	$53.40	5.34%	0.107
3	$1,060	$890.00	89.00%	2.670
Total		$1,000.00	100%	2.833

Each payment is discounted to today, as in Present Value and Future Value Formulas. Eighty-nine percent of the bond's value arrives in year three, and a little arrives earlier, so the average wait is 2.83 years, slightly less than the bond's 3-year maturity.

Two things follow at once. A bond that pays coupons always has a Macaulay duration shorter than its maturity, because some of its value comes back early. And a zero-coupon bond, which pays everything on one date, has a duration equal to its maturity.

From years to sensitivity

The reason anyone computes that average is a piece of algebra: the further away a payment is, the more its present value changes when the discount rate changes, and the weighted average wait turns out to measure the sensitivity of the whole bond. One small adjustment is needed.

Modified duration = Macaulay duration ÷ (1 + yield per period)

Change in price ≈ −Modified duration × Change in yield

For the 3-year bond, modified duration is 2.833 ÷ 1.06 = 2.67. If the bond's yield rises from 6% to 7%, the estimate is a fall of 2.67%. Repricing the three payments at 7% gives $973.76, an actual fall of 2.62%. For a bond that pays twice a year, the yield per period is half the yearly yield, so a bond yielding 4% has its Macaulay duration divided by 1.02.

The rule of thumb that a bond moves by roughly its duration for each point of yield¹ uses this second kind. It isn't really a number of years any more. It's a percentage change per point of yield.

What makes duration longer or shorter

Bond	Macaulay duration	Modified duration
2-year, 4% coupon	1.94 years	1.90
5-year, 4% coupon	4.58 years	4.49
10-year, 4% coupon	8.34 years	8.18
30-year, 4% coupon	17.73 years	17.38
10-year, 8% coupon	7.50 years	7.35
10-year, 2% coupon	9.01 years	8.83
10-year, no coupon	10.00 years	9.80
30-year, no coupon	30.00 years	29.41

Coupons are paid twice a year. The first four rows change only the maturity, and the next three change only the coupon.

Three rules come out of the table and the arithmetic behind it. The first two are the ones FINRA gives investors: the longer the maturity, the higher the duration, and the higher the coupon, the lower the duration.¹

• Maturity raises duration, but less and less. Tripling the maturity from 10 years to 30 roughly doubles the duration. The 30-year bond pays sixty coupons on the way, and the final $1,000, discounted over 30 years, is worth only about $300 today.
• Coupons lower it. The 8% bond returns more of its value early than the 2% bond does, so its average wait is a year and a half shorter. With no coupons at all, the 30-year bond's duration is the full 30 years, which is why long zero-coupon bonds react so strongly to rates.
• Higher yields lower it too. When yields are high, distant payments are discounted heavily and count for less. The same 10-year 4% bond has a modified duration of 8.41 when it yields 2% and 7.68 when it yields 8%. Bonds are more sensitive to rate changes when rates are low.

The rest of this lesson is for members

Still to come in this lesson: How far the estimate can be trusted, Convexity, Try it, Testing it against a real year, The duration of a portfolio, What duration leaves out, the key takeaways, and 2 quiz questions with explanations.

Membership opens every lesson in the curriculum, the capstone projects, the interactive tools, and the members-only channels in the Hyperion Entities Discord. The first lesson of every module stays free.

Become a member Sign in