BM4.7 – Calculating Interest Rates

Chapter 4, Lesson 7

In this lesson students will:

• determine the periodic or nominal rate of compound interest given information about the future value, principle amount, term and frequency number of the investment

Rate of Compound Interest

Periodic Interest Rate

Recall that there are two types of interest we may refer to when discussing compound interest: the periodic rate $i$ and the nominal rate $r$. We discovered in Lesson 4.4 that these rates are connected by the formula $\displaystyle i = \frac{r}{f}$, where $f$ is the frequency number.

Suppose we were interested in solving for the periodic interest rate, given some information about the investment or debt. We could do this by substituting $i$ into the compound interest formula and isolating for $i$. We first divide both sides by $P$, but we run into a small problem.

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{t \times f}\\ FV &= P \left( 1+i \right)^{t \times f}\\ \frac{FV}{P} &= \left( 1+i \right)^{t \times f} \end{aligned}

How do we remove the power of $t \times f$? The simplest way to do this is to apply the power $\displaystyle \frac{1}{t \times f}$ to both sides of our equation. This will result in a $1$ as the power on the right side, and allows us to interpret the left side as some kind of root as we did in Lesson 4.2.

\begin{aligned} \frac{FV}{P} &= \left( 1+i \right)^{t \times f}\\ \left(\frac{FV}{P}\right)^{\frac{1}{t \times f}} &= \left( 1+i \right)^{\frac{t \times f}{t \times f}}\\ \sqrt[t \times f]{\frac{FV}{P}} &= 1+i\\ 1+i &= \sqrt[t \times f]{\frac{FV}{P}}\\ i &= \sqrt[t \times f]{\frac{FV}{P}} - 1 \end{aligned}

In practice, with the TI BAII, we will often use the formula

\begin{aligned} i &= \left( \frac{FV}{P}\right)^{\frac{1}{t \times f}} - 1 \end{aligned}

as it will be easier to program into the calculator.

Nominal Interest Rate

Recall that the periodic rate and nominal rates are connected by the formula $\displaystyle i = \frac{r}{f}$. Multiplying both sides by the frequency number, we obtain

\begin{aligned} r &= i \times f. \end{aligned}

This means if we would like to determine the nominal rate, we can use the formula given in the last section, then multiply the periodic rate $i$ by the frequency number $f$. In practice this will be the easiest way to determine the nominal rate. Let’s work through a few examples using the our formulas.