Calculate how different compounding frequencies change a bond's value and interpret the result.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.a

The idea

A quoted interest rate is meaningless until you know its compounding frequency. With m compounding periods a year and quoted rate R, one unit grows over T years to Periodic growth = (1 + R ÷ m)^(m × T). With continuous compounding it grows instead to Continuous growth = e^(R × T). For a given quoted rate, more frequent compounding means a higher effective annual rate. The discount factor is then smaller, so the present value of a fixed set of cash flows, and therefore a bond's price, is lower. The same price therefore corresponds to different quotes at different frequencies. The annual quote is highest, the quarterly quote is lower, and the continuous quote is lowest. To convert, equate growth factors, which gives Continuous rate = m × ln(1 + Rm ÷ m). The usual error is applying the number from a quote with one frequency to a discount formula built for another.

Formulas

Growth of one unit with periodic compounding

Growth = (1 + R ÷ m)^(m × T)

R: the quoted annual rate
m: compounding periods a year
T: the horizon in years

The discount factor is its reciprocal. The quote is only usable with the m it was quoted for; more frequent compounding at the same R gives more growth.

Growth of one unit with continuous compounding

Growth = e^(R × T)

R: the continuously compounded annual rate
T: the horizon in years

The limit of periodic compounding as m grows. For a given price the continuous quote is the lowest of all frequencies.

Continuously compounded rate from a periodic rate

Rc = m × ln(1 + Rm ÷ m)

m: the number of compounding periods a year
Rm: the annual rate compounded m times a year

Found by equating growth factors. The continuous rate is always below the periodic quote it matches.

A practice question

From the question bank, with the reason each wrong option is wrong.

A 5-year zero-coupon bond with face value 100 is priced using a 7% quoted yield compounded semi-annually. Its price is closest to:

71.30Why not: Compounds annually (1.07^5) instead of semi-annually, which understates the discounting.
70.47Why not: Uses continuous compounding, e^(-0.07 x 5), instead of semi-annual.
84.20Why not: Uses 5 semi-annual periods instead of 10: the 5-year bond has ten half-year periods.
70.89 - correct

Ten half-year periods at 3.5% each: 100 / 1.035^10 = 70.89. The semi-annual quote means the periodic rate is the quote divided by 2 and the number of periods doubles.

A harder one

An 8% continuously compounded rate is quoted to a counterparty that prices off semi-annual compounding. The equivalent semi-annually compounded rate is closest to:

8.162% - correct
8.000%Why not: Assumes the quoted rate is the same in any compounding convention.
7.844%Why not: Applies the formula for converting from semi-annual to continuous (2 x ln(1 + R/2)) in the wrong direction.
8.329%Why not: Gives the effective annual rate, e^0.08 - 1, rather than a semi-annual quote.

Equate growth factors: (1 + R/2)² = e^0.08, so R = 2 x (e^0.04 - 1) = 8.162%. A lower-frequency quote must be higher than the continuous quote for the same growth.

The hardest tier

A 5-year zero-coupon bond with face value 100 trades at 62.00. Its yield expressed with quarterly compounding is closest to:

10.03%Why not: Gives the annually compounded yield; quarterly compounding implies a lower quoted rate.
9.56%Why not: Gives the continuously compounded yield, which is below the quarterly quote.
40.13%Why not: Multiplies the annual yield by 4 instead of using 20 quarterly periods.
9.68% - correct

Solve 62 x (1 + R/4)^20 = 100: R = 4 x ((100/62)^(1/20) - 1) = 9.68%. The same price implies annual 10.03%, quarterly 9.68% and continuous 9.56%; the more frequent the compounding, the lower the quote.

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