Restate a yield at a different compounding frequency
The idea
A yield's periodicity is the number of compounding periods a year it assumes, and a yield can only be compared with another once you know it. To restate a yield from one periodicity to another, convert to a per-period rate, compound it to the new period length, and annualise. A bond yielding 6% on a semiannual basis is not the same as one yielding 6% annually — the semiannual figure compounds to 6.09% effective. Markets differ in convention, so comparing a semiannual-pay bond market with an annual-pay one requires a bond-equivalent restatement before any spread between them means anything.
Formulas
Converting a yield between periodicities
(1 + APR₂ ÷ 2)² = (1 + APR₄ ÷ 4)⁴
- APR₂
- the annual rate quoted with semiannual compounding
- APR₄
- the same yield quoted with quarterly compounding
Same bond, same cash flows, two quoting conventions. Equate the annual growth factors and solve for the one you want; the more frequent the compounding, the lower the quoted number for the same return.
A practice question
What does the periodicity of a stated yield refer to?
- The bond's remaining maturityWhy not: Maturity is when the cash flows end. Periodicity is how many times a year the rate is assumed to compound.
- The number of compounding periods assumed in a year — correct
- The frequency of credit rating reviewsWhy not: A ratings process, unrelated to how a yield is annualised.
A yield is a rate per period annualised by a stated convention. Without the periodicity, the annual figure does not identify a unique rate.
A harder one
A yield of 8% is quoted with semiannual compounding. What is the effective annual yield?
- 8.00%Why not: The stated rate. Two compounding periods lift the effective figure above it.
- 8.16% — correct
- 8.24%Why not: 1.02⁴ − 1 — quarterly compounding. Semiannual is two periods of 4%, which gives 8.16%.
4% compounded twice gives 1.04² − 1 = 8.16%, and the excess over 8% is interest on the first half-year's interest.
The hardest tier
Why must a yield be restated before comparing bonds from markets with different conventions?
- Because the spread would include the convention — correct
- Because different markets use different currenciesWhy not: A real complication, and a separate one. Two bonds in the same currency quoted on different periodicities are still not comparable.
- Because credit ratings are not comparable across marketsWhy not: Agency scales are broadly aligned across markets. The obstacle here is arithmetic rather than analytical.
A spread computed between yields on different periodicities measures the convention difference as well as the credit difference. An apparent spread of nine basis points can be entirely an artefact of annual versus semiannual quotation, and restating first is what makes the remaining difference economic.
In the app: 9 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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