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Rate from Lump Sum

Find the implied annual interest rate given PV, FV, and time.

When to use: Use to find the return earned on an investment or the rate implied by a price.

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Formula

k=(FVPV)1/n−1k = \left(\frac{FV}{PV}\right)^{1/n} - 1

Variables

SymbolNameDescriptionUnit
kInterest RateImplied annual rate%
FVFuture ValueEnding value$
PVPresent ValueStarting value$
nNumber of YearsTime periodyears

Real-Life Examples

Example 1: Investment Return

You invested $20,000 and it grew to $45,000 in 12 years. What was your annual return?

Given

FV = 45,000PV = 20,000n = 12

Step-by-Step

1.k = (45000/20000)^(1/12) - 1
2.k = (2.25)^(0.08333) - 1
3.k = 1.0694 - 1
4.k = 0.0694 = 6.94%
Result:0.07

Your investment earned about 6.94% per year.

Example 2: Real Estate Appreciation

A house bought for $150,000 is now worth $320,000 after 15 years.

Given

FV = 320,000PV = 150,000n = 15

Step-by-Step

1.k = (320000/150000)^(1/15) - 1
2.k = (2.1333)^(0.06667) - 1
3.k = 1.0516 - 1
4.k = 0.0516 = 5.16%
Result:0.05

The house appreciated at about 5.16% annually.

Frequently Asked Questions

Enter the ending value as FV, starting value as PV, and number of years as n. The formula gives the annualized rate of return, which is the constant annual rate that would have produced the same growth.

CAGR (Compound Annual Growth Rate) is the annualized rate at which an investment grows from its beginning to ending value, assuming profits are reinvested. It is identical to this formula's result: (FV/PV)^(1/n) - 1.

Annualized return accounts for compounding, while simple average return does not. An investment that gains 50% then loses 33% has a 0% annualized return but an 8.5% average return. Annualized return is always more accurate for measuring actual growth.

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