# 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.