APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) — the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.

A 4% annual rate paid quarterly would be divided into four 1% payments.

Evaluating Exponents on the Calculator

When we need to calculate something like [latex]5^3[/latex], it is easy enough to just multiply [latex]5\cdot 5\cdot 5=125[/latex]. But when we need to calculate something like [latex]1.005^{240}[/latex] , it would be very tedious to calculate this by multiplying 1.005 by itself 240 times! So to make things easier, we can harness the power of our calculators.

Most scientific calculators have a button for exponents. It is typically either labeled like

[latex]\begin{array}{|c|}\hline \text{^}\\ \hline \end{array}\;[/latex], [latex]\begin{array}{|c|}\hline y^x\\ \hline \end{array}\;[/latex], or [latex]\begin{array}{|c|}\hline \ x^y\\ \hline \end{array}[/latex]

To evaluate [latex]1.005^{240}[/latex] we’d type 1.005 [latex]\begin{array}{|c|}\hline \text{^}\\ \hline \end{array}\;[/latex] 240, or 1.005 [latex]\begin{array}{|c|}\hline y^x\\ \hline \end{array}\;[/latex] 240. Try it out—you should get something around 3.3102044758.

Rounding

It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but keeping more digits is always better. If your calculator allows it, do all your calculations without rounding in the calculator and only round the final answer.

In many cases, you can avoid rounding completely by how you enter things in your calculator. For instance, in the example above, we needed to calculate

[latex]A=1000(1+\frac{0.05}{12})^{12 \cdot 30}[/latex]

We can quickly calculate [latex]12 \times 30 = 360[/latex], giving [latex]A=1000(1+\frac{0.05}{12})^{360}[/latex].

Now we can use the calculator.

[latex]\begin{array}{|c|c|} \hline \text{Type this}& \text{Calculator shows}\\ \hline 0.05 \; \begin{array}{|c|} \hline \div\\ \hline \end{array} \; 12 \; \begin{array}{|c|} \hline =\\ \hline \end{array} \;& 0.0041666666667\\ \hline \; \begin{array}{|c|} \hline +\\ \hline \end{array} \;1 \; \begin{array}{|c|} \hline = \\ \hline \end{array} \;&1.0041666666667\\ \hline \; \begin{array}{|c|} \hline y^x\\ \hline \end{array} \; 360 \; \begin{array}{|c|} \hline =\\ \hline \end{array} \;&4.46774431400613\\ \hline \; \begin{array}{|c|} \hline \times\\ \hline \end{array} \; 1000 \; \begin{array}{|c|} \hline =\\ \hline \end{array} \;&4467.74431400613\\ \hline \end{array}[/latex]

The previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter

[latex]1000 \; \begin{array}{|c|} \hline \times\\ \hline \end{array} \; (1 \; \begin{array}{|c|} \hline +\\ \hline \end{array} \;0.05 \; \begin{array}{|c|} \hline \div\\ \hline \end{array} \; 12) \; \begin{array}{|c|} \hline y^x\\ \hline \end{array} \;360 \; \begin{array}{|c|} \hline =\\ \hline \end{array} \;[/latex]

or

[latex]1000 \; \begin{array}{|c|} \hline \times\\ \hline \end{array} \; (1 \; \begin{array}{|c|} \hline +\\ \hline \end{array} \;0.05 \; \begin{array}{|c|} \hline \div\\ \hline \end{array} \; 12) \; \begin{array}{|c|} \hline \text{^}\\ \hline \end{array} \;360 \; \begin{array}{|c|} \hline =\\ \hline \end{array} \;[/latex]

Because of compounding throughout the year, with compound interest the actual increase in a year is more than the annual percentage rate. If $1,000 were invested at 10%, the table below shows the value after 1 year at different compounding frequencies:

[latex]\begin{array}{|c|c|} \hline \text{Frequency}& \text{Value after one year}\\ \hline \text{Annually, } n = 1 &$1100\\ \hline \text{Semiannually, } n = 2 &$1102.50\\ \hline \text{Quarterly, } n = 4 &$1103.81\\ \hline \text{Monthly, } n = 12 &$1104.71\\ \hline \text{Daily, } n = 365 &$1105.16\\ \hline \end{array}[/latex]

If we were to compute the actual percentage increase for the daily compounding, there was an increase of $105.16 from an original amount of $1,000, for a percentage increase of [latex]\frac{105.16}{1000}=0.10516=10.516\%[/latex] increase. This quantity is called the annual percentage yield (APY).

Notice that given any starting amount, the amount after 1 year would be [latex]A=P(1+\frac{r}{k})^k[/latex].

To find the total change, we would subtract the original amount, then to find the percentage change, we would divide that by the original amount:

[latex]\frac{P(1+\frac{r}{k})^k-P}{P}=(1-\frac{r}{k})^k-1[/latex]

Annual Percentage Yield

Notice this is equivalent to finding the value of $1 after 1 year and subtracting the original dollar.