Lesson 1 of 4 11–14% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Periodicity: converting yields between compounding frequencies

An annual yield only means something alongside its periodicity: the number of compounding periods per year it assumes. A semiannual bond basis yield is the half-yearly rate times two. To compare bonds, convert every yield to the same periodicity with (1 + APRm/m)^m = (1 + APRn/n)^n.

8 min cram, 32 min full read. Based on the 2027 CFA® Program Level I curriculum, Fixed Income, Learning Module 7. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Bond Valuation: Prices and Yields, Returns of Financial Assets

On the exam

Calculate annual yield on a bond for varying compounding periods in a year

Calculate annual yields for different compounding frequencies and convert between them. Expect to restate a yield as an effective annual rate or on a semiannual basis, to find the yield of a zero-coupon bond on either basis, and to compare two bonds after putting them on the same footing.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

What does periodicity mean for a yield?

A yield is a rate per period turned into an annual figure, and the annual figure only makes sense once you know how many periods went into it. That number is the periodicity , written $m$ . The stated annual rate is the periodic rate times $m$ .

Stated annual yield from a periodic yield Memorize

APR_m = m \times r_{\text{period}}

Periodicity usually matches the coupon frequency, which is 2 for most US and UK bonds. A semiannual bond basis yield is the yield per half-year times 2. You will also see it called the semiannual bond equivalent yield . “2.00% per semiannual period” and “4.00% on a semiannual bond basis” describe the same thing.

With a periodicity of 1 the figure is the effective annual rate : what a dollar actually earns over a year, with all compounding included.

One return, several stated yields

Annual yield of a bondalways quoted with a periodicity, spoken or assumed

Effective annual rate Memorize periodicity 1; the highest figure for the same return
Semiannual bond basis yield Memorize periodicity 2; the usual quote for coupon bonds
- Periodic yield × 2 a 2.00% half-year yield is a 4.00% yield
- Also called semiannual bond equivalent yield same thing
Quarterly basis Use it periodicity 4; lower than the semiannual figure
Monthly basis Use it periodicity 12; lower still

InteractiveOne stated rate, four compounding periods

Try this 0 of 2

Push the stated rate to 10% and read the monthly EAR
Set a negative rate: which compounding loses least?

Stated annual rate5.50%

EAR, semiannual compounding5.576%

Value of 100 after a year105.576

Annual → EAR 5.500%
Semiannual → EAR 5.576%
Quarterly → EAR 5.614%
Monthly → EAR 5.641%

Each step is one compounding period: the periodic rate (stated rate ÷ periods a year) earned on everything so far. Same stated rate, more steps, more growth.

Show working

Periodic rate = 5.50% ÷ 2 = 2.750% per period.

EAR = (1 + 0.0275)² − 1 = 5.576%.

Read the other way: a 5.576% annual return is quoted as 5.50% on a semiannual basis.

Effective annual rate on your calculator

1.0275y^x2=−1=×100=5.576%, the effective annual rate

y^x power

1.0275ENTER2y^x1−100×5.576%, the effective annual rate

ENTER enter (on the HP, puts the number on the stack) · y^x power

Before comparing two yields, put them on the same compounding basis.

Effective annual rate Memorize

EAR = \left(1 + \frac{APR_m}{m}\right)^{m} - 1

How do you convert a yield to another periodicity?

Two yields describe the same return when they grow money by the same amount over a year. Set the two growth factors equal and solve for the one you do not know.

Converting between periodicities Memorize

\left(1 + \frac{APR_m}{m}\right)^{m} = \left(1 + \frac{APR_n}{n}\right)^{n}

Solving for the new yield gives $APR_n = n\left[\left(1 + APR_m/m\right)^{m/n} - 1\right]$ . The exponent $m/n$ is the step people drop, and a quick sense check catches it: more compounding periods, lower stated rate.

Worked example One yield on four bases

A bond yields 5.5% on a semiannual basis. Restate it quarterly and monthly.

Quarterly: $4 \times [(1.0275)^{2/4} - 1] = 4 \times [1.0275^{0.5} - 1] =$ 5.463%.
Monthly: $12 \times [(1.0275)^{2/12} - 1] = 12 \times [1.0275^{1/6} - 1] =$ 5.438%.
Check the direction: 5.576% (EAR) > 5.5% > 5.463% > 5.438%. Each step to more compounding lowers the figure.

The quarterly basis on your calculator

1.0275y^x0.5=−1=×4= 0.054627

y^x power

1.0275ENTER0.5y^x1−4× 0.054627

ENTER enter (on the HP, puts the number on the stack) · y^x power

Rearrange itPeriodicity: restate a yield at another compounding frequency

The question asks for

A bond yields 5.50% on a basis of 2 compounding periods a year. Restated with 4 periods a year, the yield is closest to:

Stated yield, given basis (annual %)%Periods a year in the given yield, mPeriods a year wanted, n

A P R_{n} = n [(1 + \frac{A P R _{m}}{m})^{m / n} - 1]

4 × [(1 + 0.055 ÷ 2)^(2/4) − 1] = 5.463%

Stated yield, wanted basis (annual %)5.46%

On your calculator

1.0275y^x(2÷4)=−1=×4=0.054627

y^x power

1.0275ENTER2ENTER4÷y^x1−4×0.054627

ENTER enter (on the HP, puts the number on the stack) · y^x power

Exam tip On the TI, finish the power with [=] before you subtract 1. On the HP, the order is base, [ENTER], exponent, then [y^x].

How do you handle zero-coupon bonds?

A zero-coupon bond pays no coupons, so it has no natural periodicity. You choose one, and the bond’s return does not change. Zeros made by stripping the coupons and principal off a government bond are strip bonds . Analysts often restate a strip’s yield on a semiannual basis so it compares with coupon bonds that pay twice a year.

Yield of a zero-coupon bond Use it

EAR = \left(\frac{FV}{PV}\right)^{1/T} - 1, \qquad APR_2 = 2\left[\left(\frac{FV}{PV}\right)^{1/(2T)} - 1\right]

Worked example A six-year zero on two bases

A 6-year zero-coupon bond trades at 74.00 per 100 of par.

Effective annual: N = 6, so $(100 \div 74)^{1/6} - 1 =$ 5.146%.
Semiannual basis: N = 12 half-years, so $2 \times [(100 \div 74)^{1/12} - 1] = 2 \times 2.541\% =$ 5.082%.

Same bond, same return. The semiannual figure is lower because it compounds twice a year.

On your calculator

2NDCLR TVM with P/Y set to 1
6N74+|-PV100FV
CPTI/Y 5.1465, the EAR
12N
CPTI/Y 2.5410 per half-year
×2= 5.0819

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

fCLEAR FIN
6n74CHSPV100FV
i 5.1465, the EAR
12n
i 2.5410 per half-year
2× 5.0819

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Go deeper Why strip yields are restated, and where continuous compounding fits

A coupon bond’s quote already compounds twice a year, so a strip quoted as an annual effective yield is not directly comparable. Converting the strip to a semiannual basis, or the coupon bond to an EAR, puts both on one footing. Money market rates with maturities under a year are annualized but not compounded, which is the opposite extreme to the compounding here.

Continuous compounding is the limit as $m$ grows without bound: $EAR = e^{APR} - 1$ . It is the highest EAR a given stated rate can reach, and it belongs to the Quantitative Methods material on rates and returns. A bond question that states a periodicity of 2, 4 or 12 never wants it.

How do you compare two bonds with different periodicities?

Convert one of them so both share a basis, then compare. Do not compare the headline numbers.

The same rules work with negative yields. More compounding on a negative yield gives a more negative stated rate, because the loss compounds too.

Memorize this

MemorizePeriodicity and what it is called

Periods a year, m	Name	Rule
1	Effective annual rate	periodic yield × 1
2	Semiannual bond basis yield	periodic yield × 2
4	Quarterly basis	periodic yield × 4
12	Monthly basis	periodic yield × 12

Memory hook Same return, more compounding periods, lower stated number. The EAR is always the top of the ladder.

Know it cold $(1 + APR_m/m)^m = (1 + APR_n/n)^n$ . Only compare yields with the same periodicity.

MemorizeConverting without a formula sheet

Memory hook Grow one period, then take a root: raise (1 + periodic yield) to the power m ÷ n, subtract 1, multiply by n.

Know it cold $APR_n = n\left[(1 + APR_m/m)^{m/n} - 1\right]$ . Going to a higher n lowers the stated yield; to a lower n raises it.

MemorizeZero-coupon bond yields

Memory hook No coupons, so no natural periodicity: choose one. 2T half-years for a semiannual basis, T years for the EAR.

Know it cold With future value (FV) and present value (PV), $r = (FV/PV)^{1/N} - 1$ per period, with $N$ matching the periodicity you choose.

One return, several stated yields

Annual yield of a bondalways quoted with a periodicity, spoken or assumed

Effective annual rate Memorize periodicity 1; the highest figure for the same return
Semiannual bond basis yield Memorize periodicity 2; the usual quote for coupon bonds
- Periodic yield × 2 a 2.00% half-year yield is a 4.00% yield
- Also called semiannual bond equivalent yield same thing
Quarterly basis Use it periodicity 4; lower than the semiannual figure
Monthly basis Use it periodicity 12; lower still

Stated annual yield from a periodic yield

APR_m = m \times r_{\text{period}}

Quick example

A bond earns 2.10% every six months. Periodicity 2 gives a semiannual bond basis yield of 2 × 2.10% = 4.20%.

Converting between periodicities

\left(1 + \dfrac{APR_m}{m}\right)^{m} = \left(1 + \dfrac{APR_n}{n}\right)^{n}

Quick example

6.00% on a quarterly basis restated semiannually: 2 × [(1 + 0.06 ÷ 4)² − 1] = 2 × 0.030225 = 6.045%.

On your calculator

1.015y^x2=−1=×2= 0.06045

y^x power

1.015ENTER2y^x1−2× 0.06045

ENTER enter (on the HP, puts the number on the stack) · y^x power

Effective annual rate

EAR = \left(1 + \dfrac{APR_m}{m}\right)^{m} - 1

Quick example

4.00% on a semiannual basis: (1 + 0.04 ÷ 2)² − 1 = 4.04%.

On your calculator

1.02y^x2=−1= 0.0404

y^x power

1.02ENTER2y^x1− 0.0404

ENTER enter (on the HP, puts the number on the stack) · y^x power

Yield of a zero-coupon bond

EAR = \left(\dfrac{FV}{PV}\right)^{1/T} - 1, \qquad APR_2 = 2\left[\left(\dfrac{FV}{PV}\right)^{1/(2T)} - 1\right]

Quick example

A 10-year zero at 61.00 per 100. EAR = (100 ÷ 61)^(1/10) − 1 = 5.07%; semiannual basis = 2 × [(100 ÷ 61)^(1/20) − 1] = 5.01%.

On your calculator

2NDCLR TVM with P/Y set to 1
20N61+|-PV100FV
CPTI/Y 2.5023 per half-year
×2= 5.0046

fCLEAR FIN
20n61CHSPV100FV
i 2.5023 per half-year
2× 5.0046

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

How it is asked

Restate a quarterly yield on a semiannual basis

A bond yields 6.00% on a quarterly basis. Restated semiannually?

2 × [(1.015)² − 1] = 6.045%. Leaving it at 6.00%, or the EAR (6.136%), is the trap.
Compare two bonds on different bases

Bond X 4.20% monthly basis, bond Y 4.22% semiannual basis. Which yields more?

X on a semiannual basis = 2 × [(1.0035)⁶ − 1] = 4.237%, so X. Comparing 4.20 with 4.22 directly is the trap.
Zero-coupon bond on two bases

A 10-year zero at 61.00. Yield on a semiannual basis? As an EAR?

N = 20 half-years gives 2.5023% × 2 = 5.005%; N = 10 years gives 5.067% EAR.
Direction of the conversion

A yield is stated monthly and semiannually for the same price. Which is lower?

The monthly figure. More compounding periods, lower stated yield.

Exam traps

Comparing a quarterly-basis yield with a semiannual-basis yield directly. Put them on the same basis first.
Converting the wrong way. A monthly-basis yield is lower than the semiannual-basis yield for the same return, not higher.
Treating the yield per period as the annual yield. A 2.40% half-year yield is 4.80% on a semiannual basis, not 2.40%.
Using continuous compounding (e to the x) when the question gives a periodicity of 2, 4 or 12.

Checkpoint

4 exam-style questions. About 90 seconds each.

1A bond yields 4.80% on a semiannual bond basis. Its effective annual rate is closest to:
2A yield of 3.60% compounded monthly, restated on a semiannual bond basis, is closest to:
3For the same bond and price, a yield stated with monthly compounding will be:
4Bond M has a yield of 5.40% on a monthly basis. Bond N has a yield of 5.45% on a semiannual basis. Other features are the same. On a like-for-like basis, the bond with the higher yield is:

Key takeaways

Annual yield = periodic yield × periodicity. A yield without its periodicity is incomplete.
Convert with (1 + APRm/m)^m = (1 + APRn/n)^n.
More compounding periods, lower stated rate, for the same return.
A zero-coupon bond has no natural periodicity, so state the basis you used.
Always compare bonds on the same periodicity, usually semiannual.

Stated annual yield from a periodic yield

APR_m = m \times r_{\text{period}}

Quick example

A bond earns 2.10% every six months. Periodicity 2 gives a semiannual bond basis yield of 2 × 2.10% = 4.20%.

Converting between periodicities

\left(1 + \dfrac{APR_m}{m}\right)^{m} = \left(1 + \dfrac{APR_n}{n}\right)^{n}

Quick example

6.00% on a quarterly basis restated semiannually: 2 × [(1 + 0.06 ÷ 4)² − 1] = 2 × 0.030225 = 6.045%.

On your calculator

1.015y^x2=−1=×2= 0.06045

y^x power

1.015ENTER2y^x1−2× 0.06045

ENTER enter (on the HP, puts the number on the stack) · y^x power

Effective annual rate

EAR = \left(1 + \dfrac{APR_m}{m}\right)^{m} - 1

Quick example

4.00% on a semiannual basis: (1 + 0.04 ÷ 2)² − 1 = 4.04%.

On your calculator

1.02y^x2=−1= 0.0404

y^x power

1.02ENTER2y^x1− 0.0404

ENTER enter (on the HP, puts the number on the stack) · y^x power

Yield of a zero-coupon bond

EAR = \left(\dfrac{FV}{PV}\right)^{1/T} - 1, \qquad APR_2 = 2\left[\left(\dfrac{FV}{PV}\right)^{1/(2T)} - 1\right]

Quick example

A 10-year zero at 61.00 per 100. EAR = (100 ÷ 61)^(1/10) − 1 = 5.07%; semiannual basis = 2 × [(100 ÷ 61)^(1/20) − 1] = 5.01%.

On your calculator

2NDCLR TVM with P/Y set to 1
20N61+|-PV100FV
CPTI/Y 2.5023 per half-year
×2= 5.0046

fCLEAR FIN
20n61CHSPV100FV
i 2.5023 per half-year
2× 5.0046

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

All Fixed Income formulas, every lesson

Practice exam-style questions on Yield and Yield Spread Measures for Fixed-Rate Bonds

Next lesson Current yield, simple yield and yield conventions

Questions about this lesson?

Where this comes back

Yield and Yield Spread Measures for Floating-Rate Instruments Fixed Income
Term Structure: Spot, Par and Forward Curves Fixed Income

Questions candidates ask

What is a semiannual bond basis yield?

The yield per six-month period multiplied by two. It is the standard quote for bonds that pay coupons twice a year, so a 2.00% half-year yield is quoted as 4.00%.

How do you convert a quarterly yield to a semiannual yield?

Set (1 + quarterly yield ÷ 4)⁴ equal to (1 + semiannual yield ÷ 2)² and solve for the semiannual yield. In practice, raise 1 + yield ÷ 4 to the power 2, subtract 1 and multiply by 2.

Why does the periodicity of a yield matter?

The same return can be stated at different annual rates depending on how often it compounds. Two yields can only be compared once they share a compounding basis.

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