Example: A brokerage firm claims to be able to maintain a relative growth rate of \(3.25\%\) over the course of the year. What is the equivalent annual growth rate? Round your answer to the nearest hundredth.
Solution
Recall that "relative rate" always refers to the rate of the continuous exponential model, \(A=Pe^{rt}\). In order to convert the relative rate into the annual rate, we will set this equal to the annual rate formula, \(A=P(1+r)^t\) and then solve for the annual rate, \(r\): \[ \solve{ Pe^{0.0325t}&=&P(1+r)^t\\ e^{0.0325t}&=&(1+r)^t\\ \left(e^{0.0325t}\right)^\frac{1}{t}&=&1+r\\ e^{0.0325}&=&1+r\\ e^{0.0325}-1 &=&r\\ r&\approx&3.3\% } \] Note: In this example, I left \(t\) in the problem, but it is equally valid (both based on how the question was asked and the nature of these types of questions) to simply let \(t=1\) and solve it that way. Either method will yield the same answer.