A portfolio manager holds $50 million face value of 10-year Treasuries and needs to know how many dollars are at risk if yields jump 25 basis points before tomorrow's Federal Open Market Committee (FOMC) announcement. Modified duration gives the answer in percent, money duration gives it in dollars, and the price value of a basis point gives it in dollars per basis point.
Start with a concrete bond. A 5-year bond with a 4% annual coupon pays you $4 a year for four years and $104 at the end. Most of its value arrives in year 5, but the coupon returns some sooner.
Macaulay duration (MacDur) is the weighted-average time until you receive the bond's cash flows. Each payment date is weighted by the share of today's full price that payment represents. The full price is the quoted (clean) price plus accrued interest. You can calculate modified duration from MacDur, or directly by repricing the bond (shown below).
Common mistakes
- Reversing the numerator in ApproxModDur. Using produces a negative duration. Trap value: on Example 2.
- Forgetting to divide MacDur by (1 + r). Reporting MacDur as ModDur skips the correction and overstates modified duration. For the 5-year bond at 5% YTM, reporting 4.62 instead of 4.40 is the classic error.
- Leaving semiannual duration in periods. The 20-year bond's MacDur of 29.9103 is in half-years. Dividing it by 1.017629 without halving gives a trap of 29.39. Dividing the 14.955 years by the annual 1.035258 instead of the semiannual 1.017629 gives a trap of 14.45. The right answer is 14.70.
Bottom line
- Macaulay duration is the PV-weighted average time to the bond's cash flows, measured in periods.
- Modified duration = Macaulay duration / (1 + periodic yield). It estimates the percentage price change for a 1% yield change.
- For semiannual bonds, annualize MacDur (divide by 2) and divide by one plus the semiannual yield.
- ApproxModDur = , with the down-yield price entered first.
Exam shortcut
For unit conversion: ModDur is the percent move per 100 bp, MoneyDur is dollars per 1.00 of yield change, and PVBP is dollars per 1 bp. Multiply by 0.0001 to step from MoneyDur to PVBP. If a question gives only a price and asks for duration, solve the YTM first, then reprice both ways. On a semiannual bond, check that your answer is in years, not half-years.
The full lesson (about 3,513 words, 23 min read) adds 4 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- yield-based duration measures
Browse all free CFA Level I lessons or jump into free CFA Level I practice questions.