# BM7.1 – Equivalent and Effective Interest Rates

Chapter 7, Lesson 1

In this lesson students will:

• understand why our current formulas for simple annuities will not work for a general annuity, and how we will make an adjustment to our current formulas to evaluate general annuities
• calculate the effective interest rate of a given non-continuous or continuous interest rate
• given a non-continuous or continuous interest rate, determine an equivalent interest rate that has a different compounding frequency

## The General Annuity Problem

Recall from Lesson 5.1 that a general annuity occurs when the frequency of compounds $f_{1}$ is different than the frequency of payments $f_{2}$. The formulas that we have developed for ordinary simple annuities and simple annuities due rely on the fact that $f_{1}=f_{2}$. Since $f_{1} \neq f_{2}$ for general annuities, we will need to develop a new tool to help us with these types of problems. The tool we will require is a formula to help us convert between between interest rates that have different frequencies. In order to develop this tool, we first look at the idea of effective interest rates.

## Effective Interest Rates

#### A Numerical Example

The first step in solving the general annuity problem is to do some algebraic work with interest rates. First, let’s consider a numerical problem: an interest rate of $6\%$ compounded monthly and an initial deposit of $\100$ (these numbers are chosen at random for convenience). We know by the future value formula for compound interest that after one year we would have:

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{tf} \\ &= 100 \left( 1 + \frac{0.06}{12} \right)^{12}\\ &= 106.17 \end{aligned}

What if we wanted to offer a nominal rate compounded annually so that we could obtain the same future value after one year for our initial $\100$? If we change $f$ to $1$ and set $FV = \106.17$, we should be able to solve for $r$:

\begin{aligned} FV &= P \left( 1 + \frac{r}{f} \right)^{tf} \\ 106.17&= 100 \left( 1 + r \right)\\ \frac{106.17}{100}&= 1+r\\ 1.0617 &= 1+r\\ r &= 0.0617 \end{aligned}

This means that $6.17\%$ compounded annually is effectively the same interest rate as $6\%$ compounded monthly since they give the same future value after one year. Given a nominal interest rate $r$ compounded $f$ times per year, the interest rate compounded annually that gives an equivalent future value after one year is called the effective interest rate of $r$ and is denoted as $r_{eff}$.

#### Formula for the Non-Continuous Case

Now that we have seen a numerical example, let’s develop a formula so we can calculate the effective interest for any given nominal rate. Recall that our given rate and our effective rate must have the same future value after one year ($t=1$). This means that $FV = P \left( 1 + \frac{r}{f} \right)^{tf}$ and $FV = P \left( 1 + r_{eff}\right)$ must be equal. After we divide both sides of the equation by $P$, it is fairly straightforward to isolate for $r_{eff}$.

\begin{aligned} P \left( 1 + r_{eff}\right) &= P \left( 1 + \frac{r}{f} \right)^{f}\\ 1 + r_{eff} &= \left( 1 + \frac{r}{f} \right)^{f}\\ r_{eff} &= \left( 1 + \frac{r}{f} \right)^{f} - 1 \end{aligned}

This yields a formula to calculate the effective interest rate, given information about a nominal rate $r$. Sometimes this formula is written as $r_{eff} = (1+i)^{f }-1$. A video outlining how we can use this formula is given below.

#### Formula for the Continuous Case

Developing a formula to deal with the case of continuous compounding is very similar. In this case, $FV = Pe^{r}$ and $FV = P \left( 1 + r_{eff}\right)$ must be equal. Dividing both sides of the equality by $P$ we obtain:

\begin{aligned} P \left( 1 + r_{eff}\right) &= Pe^{r}\\ 1 + r_{eff} &= e^{r}\\ r_{eff} &= e^{r} - 1 \end{aligned}

A video outlining how we can use this formula is given below.

## Equivalent Interest Rates

The next step in solving the general annuity problem is being able to convert a given rate $r_{1}$ with compound frequency $f_{1}$ into a new rate, called $r_{2}$, with compound frequency $f_{2}$. To ensure that they will give our customer the same future value after one year, we set their effective interest rates equal. That is, we say two nominal rates $r_{1}$ and $r_{2}$ are equivalent rates provided that their effective rates are equal.

#### Formula for the Non-Continuous Case

So given the compound interest rate $r_{1}$ with frequency $f_{1}$, how do we find the rate $r_{2}$ if we know the frequency $f_{2}$? Well first, we need to ensure that their effective interest rates are equal so that the future values are the same. To make answering this questions easier, we will also use $i_{1} = \frac{r_{1}}{f_{1}}$ and $i_{2} = \frac{r_{2}}{f_{2}}$ and try to solve for $i_{2}$. This gives us

\begin{aligned} r_{2_{eff}} &= r_{1_{eff}}\\ \left( 1 + \frac{r_{2}}{f_{2}} \right)^{f_{2}} - 1 &= \left( 1 + \frac{r_{1}}{f_{1}} \right)^{f_{1}} - 1\\ \left( 1 + i_{2} \right)^{f_{2}} - 1 &= \left( 1 + i_{1} \right)^{f_{1}} - 1\\ \left( 1 + i_{2} \right)^{f_{2}} &= \left( 1 + i_{1} \right)^{f_{1}} \end{aligned}

From here, we would like to remove the power of $f_{2}$, and we can do this by dividing this power on both sides of our equation (note that this is equivalent to taking an nth-root, but it will be easier to view it as a fraction for calculation purposes).

\begin{aligned} \left( 1 + i_{2} \right)^{f_{2}} &= \left( 1 + i_{1} \right)^{f_{1}} \\ (1 + i_{2})^{f_{2}/f_{2}} &= (1+i_{1})^{f_{1}/f_{2}}\\ 1 + i_{2} &= (1+i_{1})^{f_{1}/f_{2}}\\ i_{2} &= (1+i_{1})^{f_{1}/f_{2}} - 1 \end{aligned}

This means that the a formula to convert between different periodic rates is given by $i_{2} = (1+i_{1})^{f_{1}/f_{2}} - 1$. This is precisely the formula needed for us to solve the general annuity problem. A video outlining how we can use this formula is given below. Examples of how this will apply to annuities are in the lessons following this one.

#### Formula for the Continuous Case

Developing a formula to deal with the case of continuous compounding is very similar. In this case, let’s take $r_{1_{eff}}$ to be continuous and $r_{2_{eff}}$ to be non-continuous. We will have to remove the power of $f_{2}$ by dividing once again.

\begin{aligned} r_{2_{eff}} &= r_{1_{eff}}\\ \left( 1 + i_{2} \right)^{f_{2}} - 1 &= e^{r} - 1\\ \left( 1 + i_{2} \right)^{f_{2}} &= e^{r} \\ 1 + i_{2} &= e^{r/f_{2}}\\ i_{2} &= e^{r/f_{2}} - 1 \end{aligned}

The formula $i_{2} = e^{r/f_{2}} - 1$ allows us to convert a continuous rate to a nominal rate compounded $f_{2}$ times per year. A video outlining how we can use this formula is given below.