How to Solve the Continuous Compounding Formula for r

The continuous-compounding formula relates a starting balance to its value after interest has accumulated. If you know the initial principal, the final balance, and the elapsed time, you can rearrange the formula to find the continuously compounded rate $r$.

The formula solved for $r$

The standard model is

$$A = Pe^{rt},$$

where $P$ is the principal (starting amount), $A$ is the amount after time $t$, $r$ is the continuously compounded rate, and $e$ is the base of the natural logarithm. Rearrange it as follows:

$$\begin{aligned} \frac{A}{P} &= e^{rt} \\ \ln\left(\frac{A}{P}\right) &= rt \\ r &= \frac{\ln(A/P)}{t}. \end{aligned}$$

So the rate is the natural logarithm of the growth factor, divided by the elapsed time. Use the natural logarithm, written $\ln$, rather than treating the exponent as a base-10 logarithm.

For a positive principal and a positive elapsed time, the final amount must also be positive. If $A>P$, the calculated rate is positive; if $A=P$, it is zero; and if $0<A<P$, it is negative.

Worked example

Suppose \1{,}000$grows to$$1{,}250$in 3 years under continuous compounding. Here,$P=1000$,$A=1250$, and$t=3$ years:

$$\begin{aligned} r &= \frac{\ln(1250/1000)}{3} \\ &= \frac{\ln(1.25)}{3} \\ &\approx 0.07438. \end{aligned}$$

The continuously compounded annual rate is approximately $0.07438$, or 7.438% per year. To check, substitute the decimal rate into the original equation:

$$1000e^{(0.07438)(3)} \approx 1250.$$

The small difference from exactly $1250$ comes from rounding the rate. Keep more digits during your calculation if you need a more precise check.

Match the time and rate units

The units of $r$ are the reciprocal of the units used for $t$. If $t$ is measured in years, $r$ is a rate per year. If $t$ is measured in months, the result is a rate per month. To find an annual rate from a period measured in months, first express the elapsed time in years; for example, 18 months is 1.5 years.

Convert a decimal rate to a percent by multiplying by 100. For example, $r=0.07438$ corresponds to $7.438\%$, not $0.07438\%$.

The formula assumes the balance changes only through continuous compounding at a constant rate. If deposits or withdrawals occur during the period, the simple equation generally cannot recover the rate from just the initial and final balances; the timing and amounts of those cash flows also matter.

Common mistakes

• Using $\log_{10}$ instead of $\ln$. The exponential in the model has base $e$, so taking the natural logarithm directly gives $rt$.
• Reversing the ratio. Use $A/P$, final amount divided by principal. Using $P/A$ changes the sign of the result.
• Mixing time units. A rate found using time in months is monthly, not annual. Convert time or clearly label the result.
• Forgetting the percent conversion. The formula returns a decimal rate. Multiply by 100 only when reporting it as a percentage.
• Applying it to a different compounding convention. This formula returns the continuously compounded rate. A rate compounded annually, monthly, or daily may be quoted differently.

Continuous rate versus effective annual rate

If $t$ is in years and $r$ is the continuously compounded annual rate, the equivalent effective annual rate is $e^r-1$. For the example, this is approximately $e^{0.07438}-1=0.0772$, or 7.72%. This differs from 7.438% because the two percentages use different compounding conventions. Use $r=\ln(A/P)/t$ when the question asks specifically for the continuously compounded rate.