Apple Numbers BONDMDURATION: Modified duration and local price sensitivity
Use BONDMDURATION in Apple Numbers with annual yield units, a verified relationship to Macaulay duration and carefully bounded sensitivity examples.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: BONDMDURATION is Apple’s modified Macaulay duration measure. Its published example returns approximately 4.95538808340513. For a small increase of 0.01 in annual yield, the local model predicts roughly a 4.955% price decrease, not a 4.955 currency-unit decrease.
Syntax and units
=BONDMDURATION(settle, maturity, annual-rate, annual-yield, frequency, days-basis)
Apple assumes par 100, requires maturity after settlement and documents strictly positive coupon rate and annual yield. Frequency is 1, 2 or 4. Optional days-basis defaults to 0; the modes are 0 NASD 30/360, 1 actual/actual, 2 actual/360, 3 actual/365 and 4 European 30/360. Apple BONDMDURATION help 15.4.
Keep annual-rate and annual-yield annual even when frequency is semiannual or quarterly. The result is a sensitivity coefficient, not a percentage that should be multiplied by 100 merely to match its printed numeric value.
Complete two-year example
Enter these values in a fresh table:
| Cell | Input |
|---|---|
| B2 | =DATE(2025,1,15) |
| B3 | =DATE(2027,1,15) |
| B4 | 0.06 |
| B5 | 0.08 |
| B6 | 1 |
| B7 | 0 |
Enter =BONDMDURATION(B2,B3,B4,B5,B6,B7) in B9. The independent reference is 1.79850903535114. Equivalently:
=BONDMDURATION(DATE(2025,1,15),DATE(2027,1,15),0.06,0.08,1,0)
For the same regular cash flows, independent BONDDURATION arithmetic gives Macaulay duration 1.94238975817923 years. Dividing by 1+0.08/1 yields 1.79850903535114. The modification accounts for the per-period yield factor.
This relation is independently derived for the stated coupon model and also reproduces Apple’s documented pair of examples. It is not a claim that the formulas were executed together in Numbers.
Reproduce Apple’s published value
Apple uses settlement April 2, 2010, maturity December 31, 2015, annual coupon 0.05, annual yield 0.05284, quarterly payments and actual/actual basis:
=BONDMDURATION(DATE(2010,4,2),DATE(2015,12,31),0.05,0.05284,4,1)
The documented result is 4.95538808340513. Independently, the 23 remaining quarterly coupons are 1.25 each, with final redemption 100. The first payment is 89/91 of a quarter away. Present-value-weighted time gives Macaulay duration 5.0208487599869117. Dividing by 1+0.05284/4 = 1.01321 gives 4.9553880834051300, matching the printed result.
The calculation uses the bond’s annual yield for discounting, not its coupon rate. Replacing 0.05284 with 0.05 in the divisor would answer a different calculation and would not reproduce the documented value.
Practical use: a small yield-change estimate
For Apple’s example, consider an illustrative increase of 0.25 percentage points in annual yield. The decimal change is 0.0025, so the first-order relative price estimate is:
relative price change ≈ -modified_duration * change_in_annual_yield
≈ -4.95538808340513 * 0.0025
≈ -0.0123884702085128
That is approximately a 1.238847% decrease. The relative-change denominator is the present value of all remaining cash flows: the full (dirty) price, including accrued interest at the fixed settlement date. A quoted clean price uses a different relative-change denominator when accrued interest is nonzero. For a chosen full-price baseline of 100, the linear estimate would be a decrease of about 1.238847 price units. A real model should use its actual full starting price rather than assume par.
The minus sign matters: positive-rate fixed cash flows decrease in present value when the discount yield increases. A decline in yield reverses the sign of the local estimate. A 0.25% relative change in the current yield is not the same as a 0.25-percentage-point change; use a decimal rate difference consistently.
Frequency and nonlinear limits
For the small two-year example with frequency 2, the independent modified duration is 1.83892532502679, obtained from Macaulay duration 1.91248233802787 divided by 1.04. Changing payment frequency changes the timing and discount factors; it is not just changing the displayed units.
The duration estimate is local and first order. Large yield changes introduce curvature, so multiplying duration by a large change does not produce an exact repriced value. Credit events, optional cash flows, fees and changing payment assumptions are outside this fixed-cash-flow calculation.
Apple’s DURATION constructs elapsed time and is not an alias. BONDDURATION provides the unmodified time-weighted measure; neither function guarantees a realized investment return.
Errors, availability and evidence scope
Check date order, strictly positive documented rates and the supported frequency/basis choices. The cited reference does not establish every exact error code or unsupported zero-input behavior. Use explicit dates through DATE and avoid guessing locale-specific text parsing.
Checked October 18, 2026 against Apple Formulas and Functions Help 15.4. The examples are independent calculations and identified published results; no native Numbers workbook or screenshot was produced. First introduction, older-release support and individual-client parity are not established by this shared iWork reference. These examples explain sensitivity mechanics, not investment advice.