Work out how much an investment is worth under various compounding frequencies.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.b

The idea

The value of an investment depends on how often interest is compounded, even at the same quoted rate. Invest A for T years at a rate R compounded m times a year. Terminal value = A × (1 + R/m)^(mT). With continuous compounding, interest is added at every instant. Terminal value = A × e^(RT). Two quotes are comparable only at the same frequency. Write Rm for a rate compounded m times a year, and Rc for its continuous equivalent. To turn a rate compounded m times a year into a continuous rate, take a log. Rc = m × ln(1 + Rm/m). To go the other way, from Rc back to m periods a year, use the exponential. Rm = m × (e^(Rc/m) − 1). The effective annual rate, EAR, is one year's growth as an annual rate. EAR = (1 + R/m)^m − 1. At a given positive quoted rate, compounding more often raises the terminal value, but by less each time. Common errors are counting periods wrongly, such as years instead of months, and comparing quotes at different frequencies, so convert to one frequency first.

Formulas

Terminal value with compounding m times a year

terminal value = A × (1 + R ÷ m)^(m × T)

A: the amount invested
R: the quoted annual rate
m: the number of compounding periods a year
T: the years invested

Count periods as m × T, not years. More frequent compounding at the same quoted rate raises the value, by less each time.

Terminal value with continuous compounding

terminal value = A × e^(R × T)

A: the amount invested
R: the quoted annual rate, continuously compounded
T: the years invested

The limit of ever more frequent compounding, and the convention used in derivative pricing.

Continuous rate equivalent to a rate compounded m times a year

Rc = m × ln(1 + Rm ÷ m)

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

Use it to put two quotes on one basis before comparing them. The continuous rate is always the lower number.

Rate compounded m times a year equivalent to a continuous rate

Rm = m × (e^(Rc ÷ m) − 1)

m: the number of compounding periods a year
Rc: the continuously compounded annual rate

The inverse of the log conversion. The result sits slightly above the continuous rate.

A practice question

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

USD 1,000 is invested at 5% per year, compounded continuously, for 3 years. The terminal value is closest to:

USD 1,161.83 - correct
USD 1,157.63Why not: Uses annual compounding (1.05³) instead of continuous compounding.
USD 1,150.00Why not: Uses simple interest with no compounding.
USD 1,051.27Why not: Applies the exponential for one year only and forgets to multiply the rate by the 3 years.

Terminal value = A x e^(R x T) = 1,000 x e^(0.15) = 1,161.83.

A harder one

A rate of 6% per year is quoted with continuous compounding. The equivalent rate with quarterly compounding is closest to:

6.05% - correct
6.18%Why not: Computes the equivalent annual-compounding rate, not the quarterly-compounded quoted rate.
5.96%Why not: Applies the conversion in the wrong direction: ln(1 + R/4) x 4 converts a quarterly rate into a continuous one.
6.00%Why not: Assumes the quoted rate does not change when the compounding frequency changes.

Rm = m x (e^(Rc/m) - 1) = 4 x (e^0.015 - 1) = 6.05%. The quarterly-compounded rate sits slightly above the continuous rate.

The hardest tier

Bank X offers 5.2% per year compounded continuously and Bank Y offers 5.3% per year compounded semiannually. For a one-year deposit of USD 1 million, which statement is correct?

Bank X pays more, by about USD 327Why not: Assumes continuous compounding always beats a higher stated rate; here the 0.1% higher nominal rate more than offsets the compounding gain.
Bank Y pays more, by about USD 327 - correct
Bank Y pays more, by about USD 1,000Why not: Takes the 0.1% nominal-rate gap on USD 1 million and ignores the different compounding frequencies.
The two pay the same, because both quotes compound more than once a yearWhy not: Treats any compounding frequency above annual as equivalent; the effective annual rates still differ.

Bank X effective annual rate = e^0.052 - 1 = 5.338%. Bank Y = (1 + 0.053/2)² - 1 = 5.370%. The difference on USD 1 million is about 327. Compare effective annual rates, never quotes with different frequencies.

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