Converting a rate between compounding frequencies
1. The idea
The same interest rate can be written in several ways. A rate is only complete with its compounding frequency: how often interest is added. Two rates are equivalent when they grow money by the same amount in a year. At a rate \(R_m\) with interest added \(m\) times a year, $1 grows in a year to \((1 + R_m/m)^m\); at a continuous rate \(R_c\), to \(e^{R_c}\). Setting the two equal gives
\[ R_c = m \ln\!\left(1 + \frac{R_m}{m}\right), \qquad R_m = m\left(e^{R_c/m} - 1\right), \]
where \(\ln\) is the natural logarithm, which undoes \(e^x\): \(\ln(e^x) = x\).
Between two frequencies \(m_1\) and \(m_2\), go through the growth in a year: \(R_2 = m_2\left[(1 + R_1/m_1)^{m_1/m_2} - 1\right]\).
An example. At 6% a year with quarterly compounding, $100 grows in a year to \(\$100 \times (1 + 0.06/4)^4 = \$106.14\). The continuous rate with the same growth is \(4 \ln(1.015) = 0.059554\), or 5.96%. The more often interest is added, the lower the rate needed for the same growth.
These conversions come before many other calculations. Before using a formula, match its compounding: most formulas of the site take rates with continuous compounding, so a rate quoted another way is turned into one first. Some payments later in this level are set with a simple rate for each period; those use the quoted rate as it is.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The two rates on a chart
First, a guess at the direction.
Now move the quoted rate. The equivalent rate stays a little below it, and the gap grows with the rate.
4. One worked example, then your turn
In the worked example, we turn a continuous rate into a semiannual one.
Now one between two frequencies, with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer to continuous compounding, from it, and between two other frequencies. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: