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Lesson 1 of 6 · 14 min

Periodicity and converting annual yields

A quoted annual yield means nothing until you know how many times a year it compounds, so convert yields to the same periodicity before you compare them.

Video lesson · 7 min. Watch it first, then use the text below to go deeper.

In short

Periodicity = the number of compounding periods per year assumed in an annual yield. It normally matches the coupon frequency.
A semiannual bond basis yield (semiannual bond equivalent yield) is the yield per half-year × 2. Do not confuse it with the yield per semiannual period.
An effective annual rate (EAR) has a periodicity of 1: $(1 + APR_m/m)^m - 1$ .
Convert between periodicities with $(1 + APR_m/m)^m = (1 + APR_n/n)^n$ .
More frequent compounding ↔ lower stated annual rate for the same return. With negative yields, more frequent compounding gives a more negative rate.
A zero-coupon bond has no coupons, so the periodicity of its yield is a free choice; any one can be converted to any other.

Why yields need a periodicity

Investors want one number to compare bonds with different coupons, maturities and payment dates. For bonds maturing in more than a year, that number is an annualized, compounded yield-to-maturity (YTM).

But 'annualized' hides a choice: how many times a year does interest compound? That number is the yield's periodicity. A bond paying coupons twice a year is usually quoted with a periodicity of 2: the YTM per half-year times 2. A quarterly-pay bond is quoted with a periodicity of 4: the rate per quarter times 4.

Two bonds can show the same quoted yield yet give different returns, simply because their periodicities differ. Compare yields only after putting them on the same periodicity.

Money market rates (maturities of a year or less) are also annualized, but they are not compounded. Their conventions are covered in a separate module.

Effective annual rate: compounding inside the year

Each coupon received during the year is assumed to be reinvested at the same periodic rate until year-end. The more often that happens, the more interest-on-interest builds up, so the same stated rate earns more.

The annual return after all that compounding, expressed as a single annual figure, is the effective annual rate (EAR). It has a periodicity of 1, because it describes one compounding period in the year.

The table shows what a 6% stated annual rate earns over one year on a par bond priced at 100, for different periodicities.

6% stated annual rate, price 100, interest earned by year-end
Periodicity	Rate per period	Interest after one year	EAR
Annual (1)	6.00%	6.0000	6.0000%
Semiannual (2)	3.00%	6.0900	6.0900%
Quarterly (4)	1.50%	6.1364	6.1364%
Monthly (12)	0.50%	6.1678	6.1678%

The same 6% stated rate earns more the more often it compounds, but the gains shrink fast: going from annual to semiannual adds 9 bp, from quarterly to monthly only about 3 bp.

Read the table backwards: a quarterly-pay bond quoted at 6% delivers the same as an annual-pay bond quoted at 6.1364%. A higher number on one bond can just be a less frequent compounding convention.

Converting from one periodicity to another

Two annual rates are equivalent when they grow 1 unit of money to the same amount in one year. For a rate $APR_m$ with periodicity m and a rate $APR_n$ with periodicity n, set the growth factors equal: $(1 + APR_m/m)^m = (1 + APR_n/n)^n$ . Solve for the unknown rate.

The rule that checks every conversion: for given cash flows, the stated annual rate and the periodicity move in opposite directions. Compounding more often makes up for a lower rate, so converting a semiannual yield to a monthly one must give a smaller number, and converting it to an annual (EAR) basis must give a larger one.

The most common periodicity in the US, the UK and several other markets is 2, because most bonds there pay semiannual coupons. A yield with periodicity 2 is called a semiannual bond basis yield or semiannual bond equivalent yield. A bond yielding 1.8% per half-year has a semiannual bond basis yield of 3.6%.

Higher periodicity → lower stated annual rate (for positive yields).
EAR (periodicity 1) is the highest of all the equivalent stated rates when yields are positive.
With negative yields the same logic holds in signed terms: more frequent compounding gives a lower, i.e. more negative, rate.

One price, one return, many quoted yields. For the 7-year zero bought at 78.50, the stated yield falls as the periodicity rises, because more frequent compounding makes up for the lower number.

Zero-coupon bonds and negative yields

A zero-coupon bond (including a strip bond, a single coupon or principal payment separated from a coupon bond) has no coupons to fix its periodicity. You may compute its yield with any periodicity and convert it afterwards.

Two routes therefore give the same semiannual bond basis yield for a zero: (1) solve for an annual-compounding yield and convert it from periodicity 1 to 2, or (2) solve directly for the yield per half-year (twice as many periods) and multiply by 2.

Negative yields convert with the same formula. Because more frequent compounding of a negative rate shrinks the value faster, the equivalent monthly rate is slightly more negative than the annual one.

Formulas

Periodicity conversion: $\left(1 + \dfrac{APR_m}{m}\right)^{m} = \left(1 + \dfrac{APR_n}{n}\right)^{n}$
APR_m = annual rate with m compounding periods per year; APR_n = equivalent annual rate with n periods. Solve for the unknown APR.
Effective annual rate: $EAR = \left(1 + \dfrac{APR_m}{m}\right)^{m} - 1$
The special case n = 1 of the conversion formula.
Annualizing a periodic yield: $APR_m = m \times r_{\text{period}}$
The stated annual yield for periodicity m is the yield per period times m (e.g. semiannual bond basis = 2 × six-month yield).

Worked examples

Example 1 One zero-coupon price, four stated yields

A 7-year zero-coupon bond trades at 78.50 per 100 of par. What is its YTM with a periodicity of 1, 2, 4 and 12?

Example 2 Restating a semiannual YTM for comparison

A 4-year bond pays a 4.6% coupon semiannually and trades at 102.30. Find its YTM on a semiannual bond basis, then restate it with quarterly, annual and monthly periodicity.

Example 3 Negative yield on a zero

A 6-year government zero-coupon bond is issued at 104.50. What is its yield with annual, semiannual and monthly compounding?

On the exam

Expect a direct conversion (quarterly YTM to semiannual bond basis, monthly to EAR) with distractors that skip the conversion or report the per-period yield. Concept items ask which way a rate moves when the periodicity changes.

Reporting the yield per period (e.g. per quarter) as the annual yield, or multiplying a quarterly yield by 2.
Comparing two quoted yields with different periodicities without converting them first.
Thinking a zero-coupon bond's yield must use a periodicity of 1. Any periodicity works.
Assuming a negative yield becomes less negative with more frequent compounding. It becomes more negative.

Check yourself

Q1A bond's YTM is 4.40% stated with quarterly compounding. Its yield on a semiannual bond basis is closest to:
Q2A yield of 2.40% compounded monthly is equivalent to an effective annual rate closest to:
Q3A bond yields −0.80% with annual compounding. Restated with semiannual compounding, the yield will most likely be:
Q4A 5-year zero-coupon bond trades at 88.00 per 100 of par. Its yield on a semiannual bond basis is closest to:
Q5Which statement about periodicity is least accurate?

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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