Compounding with a 100% Interest Rate

Task

A man knows that money in an account where interest is compounded semi-annually will earn interest faster than money in an account where interest is compounded annually. He wonders how much interest can be earned by compounding it more and more often. In this problem we investigate his question.

For ease of computation, let's suppose the man invests \$1 at a 100% interest rate. If his interest is compounded annually, his year-end balance will be: $$\begin{align} \$1 + 100 \% \cdot \$1 &= \$1 + 1.00 \cdot \$1\\ & = \$1(1+1) \\ &= \$2. \end{align}$$

If his interest is compounded semi-annually, he earns half the annual interest at mid-year, and so his mid-year balance is: $$ \begin{align} \$1 + \frac{100 \%}{2} \cdot \$1 &= \$1 + \frac{1.00}{2} \cdot \$1 \\ &= \$1 \cdot\left(1+\frac{1}{2}\right) \\ &= \$ 1.5. \end{align}$$ At year-end he earns the other half of his annual interest giving him a year-end balance of:

IM Commentary

This task provides an approximation, and definition, of $e$, in the context of more and more frequent compounding of interest in a bank account. The approach is computational. Further progress toward finding how rapidly the approximations $\left(1 + \frac{1}{n}\right)^n$ approach $e$ could be studied by expanding and analyzing these longer and longer polynomials but going into these details is beyond the scope of the task.

This task is preliminary to F-LE Compounding Interest with a 5% Interest Rate which further develops the relationship between $e$ and compound interest.

Note that banks report account balances to the nearest cent. However, in this task we have kept several decimal places of accuracy in the account balance to clearly show changes in the account balance beyond the second decimal place.

Solution

1. At the end of March, the man will earn one quarter of the interest giving him an account balance of

   $$\begin{align} \$1 + \frac{100 \%}{4} \cdot \$1 &= \$1 + \frac{1.00}{4} \cdot \$1 \\ &= \$1\left(1+\frac{1}{4}\right) \\ &=\$1.25. \end{align} $$

   At the end of June, the man will earn the next quarter of the interest giving an account balance of

   $$\begin{align} \$ 1.25 + \frac{100 \%}{4} \cdot \$ 1.25 &= \$ 1.25+ \frac{1.00}{4} \cdot \$ 1.25 \\ &= \$ 1.25\left(1+\frac{1}{4}\right)\\ &= \$1\left(1+\frac{1}{4}\right)^2\\ &= \$1.5625.\end{align}$$

   At the end of September, the man will earn the next quarter of the interest giving an account balance of

   $$ \begin{align} \$1.5625 + \frac{100 \%}{4}\$1.5625 &= \$1.5625 + \frac{1.00}{4}\ \cdot \$1.5625 \\ &= \$1.5625\left(1 +\frac{1}{4}\right) \\ &= \$1 \cdot\left(1+\frac{1}{4}\right)^2 \left(1 +\frac{1}{4}\right) \\ &=\$1 \cdot\left(1+ \frac{1}{4}\right)^3\\ &= \$ 1.953125. \end{align}$$

   At the end of December, the man will earn the final quarter of the interest:

   $$ \begin{align} \$1.953125 + \frac{100 \%}{4}\$1.953125 &= \$1.953125 + \frac{1.00}{4}\ \cdot \$1.953125 \\ &= \$1.953125\left(1 +\frac{1}{4}\right) \\ &= \$1 \cdot\left(1+\frac{1}{4}\right)^3 \left(1 +\frac{1}{4}\right) \\ &=\$1 \cdot\left(1+ \frac{1}{4}\right)^4\\ &= \$ 2.44140625. \end{align}$$

2. Organizing and generalizing our previous work, we have, putting the number of times the interest is compounded in the first column and the expression showing the year end balance in the second column:

   $1$	$\$1(1+\frac{1}{1})^1$
   $2$	$\$1(1+\frac{1}{2})^2$
   $4$	$\$1(1+\frac{1}{4})^4$
   $n$	$\$1(1+\frac{1}{n})^n$

3. Based on the work in part (b) the account balance if interest is compounded $n$ times in the year will be, in dollars, $\left(1 + \frac{1}{n}\right)^n$. The table below shows the values of this expression for the given values of $n$:

   $n$	Year end account balance when interest is compounded $n$ times
   10	2.59374
   100	2.70481
   1000	2.71692
   10,000	2.71815
   100,000	2.71827

   Unlike the values calculated in part (a), these values have been rounded to five decimal places because the full decimal would be too long to record.

4. Based on the table, it does not appear that the man can make an unlimited amount of money off of his \$1 investment at the bank, if the bank compounds the interest more and more often. It appears that he will not make more than \$2.72. It is clear, in any case, that there are diminishing returns as the compounding frequency is increased. Going from $100$ times to $1000$ times increased the year-end balance by about one cent. Going from $1000$ to $10000$ times adds only about a tenth of a cent. And going from $10000$ times to $100000$ times yields only an extra hundredth of a cent. If this behavior continues, the account balance will not exceed $\$2.72$ no matter how often the interest is compounded.