Apple Numbers YIELD Function: Price-to-Yield Models and Rounding
Solve annual bond yields with complete cash flows, coupon-income comparisons and a reproduction of Apple’s rounded-price example.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
YIELD calculates an annual yield from a periodic-interest security’s quoted price and cash-flow terms. The coupon rate determines its scheduled interest payments; the yield is the rate consistent with the price paid. They need not be equal.
Availability and verification
The official YIELD reference was retrieved October 15, 2026. The original introduction release and last functioning version are unknown from the checked evidence. A documentation snapshot is not a last-working-build test or proof of every client’s behavior.
No native Apple Numbers session was executed. The worked values are independent calculations from the stated fixtures, with published source examples identified separately. The examples are instructional models, not financial advice or a claim about an actual security.
Syntax and quote units
YIELD(settle, maturity, annual-rate, price, redemption, frequency, days-basis)
Maturity must follow settlement. Apple’s documented coupon rate, price and redemption are positive. Price and redemption are per 100 of face value; normalize a cash purchase amount as purchase amount/face value*100. Frequency is 1, 2 or 4 payments per year.
The optional basis is 0 or omitted for US/NASD 30/360, 1 actual/actual, 2 actual/360, 3 actual/365, or 4 European 30/360. The examples avoid inventing a general algorithm for unusual month-end dates or multi-year actual/actual spans.
Apple calls the result an effective annual interest rate. The independent periodic-bond model below uses the annual yield divided by payment frequency for discounting; do not silently replace it with a separately compounded annual-percentage-yield calculation.
Complete three-year annual-coupon fixture
Use settlement January 1, 2026, maturity January 1, 2029, coupon rate 6%, price 95, redemption 100, annual frequency 1 and basis 0.
=YIELD(DATE(2026,1,1),DATE(2029,1,1),6%,95,100,1,0)
The independently solved target is 0.0793799735, approximately 7.9379973460%. The future cash flows per 100 are 6, 6 and 106. The yield solves 95 = 6/(1+y) + 6/(1+y)^2 + 106/(1+y)^3.
Substituting the full-precision independently solved yield back into that equation returns 95. This is a price/yield consistency check, not a claim about the native solver’s iteration method or tolerance.
Example: price sensitivity
Keep the same dates and cash flows:
| Price per 100 | Independent annual yield |
|---|---|
| 95 | 7.9379973460% |
| 100 | 6% |
| 105 | 4.1916981475% |
The yield falls as the price rises for this fixed positive-cash-flow model. A lower coupon rate is not being substituted; the coupon remains 6% in all three scenarios.
The price-to-yield relation is nonlinear. Averaging two prices and averaging their corresponding yields are different operations and need not produce a matching pair.
Advanced: coupon income versus total yield
At price 95, the annual coupon divided by price is 6/95, approximately 6.31578947%. That is a simple coupon-income ratio. It is below the solved 7.9379973460% yield because the bond also returns 100 at maturity after a purchase quote of 95.
At price 105, 6/105 is approximately 5.71428571%, while the solved yield is lower, 4.1916981475%, reflecting the redemption amount below the purchase quote. Neither coupon-income ratio alone solves the full discounted-cash-flow equation.
Advanced: verify Apple’s rounded-price example
Apple gives settlement May 1, 2009, maturity June 30, 2015, coupon 6.5%, price 106.50, redemption 100, semiannual frequency and basis 0. The published yield is approximately 5.25020473181683%.
Independently solving the conventional clean-price model for the 13 remaining coupons reproduces 5.2502047318%. The corresponding model uses 59/180 of a period to the next coupon and subtracts accrued interest of 3.25*121/180 from the discounted cash-flow total.
The companion 5.25% price control is approximately 106.5011078216, which rounds to 106.50. Therefore reversing the rounded price gives a yield slightly above 5.25%. That small difference is expected from input rounding, not evidence that the two formulas contradict each other.
Errors and troubleshooting
Confirm that price is per 100 rather than the whole transaction amount, and that frequency describes the actual payment schedule. A percentage display changes presentation, not the stored decimal result.
The official entry does not document a native iteration cap, guess parameter or complete error-code matrix. Do not import those details from an IRR function or another spreadsheet. Equal dates, invalid modes, zero rates and unusual negative-yield cases were not executed.
Yield calculations assume the stated cash flows occur. The model alone does not establish creditworthiness, reinvestment opportunities, fees or a guaranteed realized return. It is a reproducible worksheet calculation, not an investment recommendation.