Chapter 7.8, Problem 8HP

To determine

To Give: An example of two interest rates −one that is compounded monthly and one that is compounded annually-such that the effective monthly interest rates are approximately equal. Justify your answer.

Answer to Problem 8HP

0.25% compounded monthly and 3% compounded annually.

Explanation of Solution

Given information: Effective monthly interest rates are approximately equal.

Calculation:

If the interest is compounded annually, then n =1 so the compound interest formula simplifies to ,

  $A=P{(1+\frac{r}{n})}^{nt}=P{(1+\frac{r}{1})}^{1.t}=P{(1+r)}^t.$

Choose P =1 and any value of r and then rewrite the formula to have a power of 12t to find its effective monthly interest rate. I have choose to use $r=3\%=0.03:$

  $\begin{array}{}$ A=1{(1+.03)}^t.$$ A={(1.03)}^t.$$ A={({(1.03)}^{\frac{1}{12}.12})}^t.$$ A={({(1.03)}^{\frac{1}{12}})}^{12t}.$$ A={({1.03}^{\frac{1}{12}})}^{12t}.$$ A={({(1.03)}^{\frac{1}{12}})}^{12t}.$$ A\approx {1.0025}^{12t}.$\end{array}$

The effective monthly interest rate is then 1.0025-1=0.0025=0.25%.

If an interest rate is compounded monthly, then its effective monthly rate is equal to itself. Choosing an interest rate of 0.25% that compounded monthly and an interest rate of 3% that compounded annually will then give two interest rate that have approximately equal effective monthly interest rate.