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Find the nominal annual rate with periodic compounding.

When to use: Use when the compounding frequency differs from annual.

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Formula

k=m×[(FVPV)1/(n×m)−1]k = m \times \left[\left(\frac{FV}{PV}\right)^{1/(n \times m)} - 1\right]

Variables

SymbolNameDescriptionUnit
kNominal RateNominal annual rate%
FVFuture ValueEnding value$
PVPresent ValueStarting value$
nNumber of YearsTime periodyears
mPeriods/YearCompounding frequencyinteger

Real-Life Examples

Example 1: CD Rate

A CD grew from $10,000 to $13,000 in 5 years with monthly compounding. What was the rate?

Given

FV = 13,000PV = 10,000n = 5m = 12

Step-by-Step

1.k = 12 × [(13000/10000)^(1/60) - 1]
2.k = 12 × [(1.3)^(0.01667) - 1]
3.k = 12 × 0.004377
4.k = 0.0525 = 5.25%
Result:0.05

The CD rate was about 5.25% nominal with monthly compounding.

Example 2: Quarterly Growth

$50,000 grew to $72,000 in 8 years with quarterly compounding.

Given

FV = 72,000PV = 50,000n = 8m = 4

Step-by-Step

1.k = 4 × [(72000/50000)^(1/32) - 1]
2.k = 4 × [(1.44)^(0.03125) - 1]
3.k = 4 × 0.01141
4.k = 0.0456 = 4.56%
Result:0.05

The nominal quarterly-compounded rate was about 4.56%.

Frequently Asked Questions

Enter the ending value, starting value, years, and compounding frequency. The formula reverse-engineers the nominal annual rate that, with the specified compounding frequency, produced the observed growth.

The nominal rate is the stated annual rate before accounting for compounding. The effective rate is the actual annual return after compounding. A 12% nominal rate compounded monthly gives an effective rate of 12.68%.

When comparing investments that compound at different frequencies. Converting observed growth to a nominal rate at the correct compounding frequency allows apples-to-apples comparison with other similarly quoted rates.

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