Clever Carol went to her bank which was offering 12% interest on its savings account. She asked very nicely if instead of having 12% at the end of the year, if she could have 6% after the first 6 months and then another 6% at the end of the year. Carol and the bank talked it over and they realized that while the account would still seem like it was getting 12%, Carol would actually be earning a higher percentage. How much more will Carol earn this way?
Compound Interest Per Period of Time
Consider a bank that compounds and adds interest to accounts \begin{align*}k\end{align*}
 Once, with 12% at the end of the year \begin{align*}(k=1)\end{align*}
(k=1)  Twice (semiannually), with 6% after the first 6 months and 6% after the last six months \begin{align*}(k=2)\end{align*}
 Four times (quarterly), with 3% at the end of each 3 months \begin{align*}(k=4)\end{align*}
 Twelve times (monthly), with 1% at the end of each month \begin{align*}(k=12)\end{align*}
The intervals could even be days, hours or minutes. This is called the length of the compounding period. The number of compounding periods is how often interested is compounded. When intervals become small so does the amount of interest earned in that period, but since the intervals are small there are more of them. This effect means that there is a much greater opportunity for interest to compound.
Nominal interest is a number that resembles an interest rate, but it really is a sum of compound interest rates. A nominal rate of 12% compounded monthly is really 1% compounded 12 times.The formula for interest compounding \begin{align*}k\end{align*} times per year for \begin{align*}t\end{align*} years at a nominal interest rate \begin{align*}i\end{align*} with present value \begin{align*}PV\end{align*} and future value \begin{align*}FV\end{align*} is:
\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}\end{align*}
Just like simple interest and compound interest use the symbol \begin{align*}i\end{align*} to represent interest but they compound in very different ways, so does a nominal rate. A nominal rate of 12% may actually yield more than 12%.
Let's apply the formula above to an investment of $300 at a rate of 12% compounded monthly. If you wanted to know the amount of money the person would have after 4 years, you would take the following steps:
\begin{align*}FV=?, \ PV=300, \ t=4, \ k=12, \ i=0.12\end{align*}
\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}=300 \left(1+\frac{0.12}{12}\right)^{12 \cdot 4} \approx 483.67\end{align*}
Note: A very common mistake when typing the values into a calculator is using an exponent of 12 and then multiplying the whole quantity by 4 instead of using an exponent of \begin{align*}(12 \cdot 4)=48\end{align*}.
Examples
Example 1
Earlier, you were asked about Clever Carol and the difference in amount of money she would have if her interest was compounded once a year versus twice a year. If Clever Carol earned the 12% at the end of the year she would earn $12 in interest in the first year. If she compounds it \begin{align*}k=2\end{align*} times per year then she will end up earning:
\begin{align*}FV=PV \left(1+\frac{i}{k}\right)^{kt}=100 \left(1+\frac{.12}{2}\right)^{2 \cdot 1}=\$112.36\end{align*}
Example 2
How many years will Matt need to invest his money at 6% compounded daily \begin{align*}(k=365)\end{align*} if he wants his $3,000 to grow to $5,000?
\begin{align*}FV=5,000, \ PV=3,000, \ k=365, \ i=0.06, \ t=?\end{align*}
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 5,000 &= 3,000 \left(1+\frac{0.06}{365}\right)^{365t}\\ \frac{5}{3} &= \left(1+\frac{0.06}{365}\right)^{365t}\\ \ln \frac{5}{3} &= \ln \left(1+\frac{0.06}{365}\right)^{365t}\\ \ln \frac{5}{3} &= 365t \cdot \ln \left(1+\frac{0.06}{365}\right)\\ t &= \frac{\ln \frac{5}{3}}{365 \cdot \left(1 + \frac{0.06}{365}\right)}=8.514 \ years\end{align*}
Example 3
What nominal interest rate compounded quarterly doubles money in 5 years?
\begin{align*}FV=200, \ PV=100, \ k=4, \ i=?, \ t=5\end{align*}
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\ 200 &= 100 \left(1+\frac{i}{4}\right)^{4 \cdot 5}\\ [2]^{\frac{1}{20}} &= \left[ \left( 1+\frac{i}{4}\right)^{20}\right]^{\frac{1}{20}}\\ 2^{\frac{1}{20}} &= 1+\frac{i}{4}\\ i &= \left(2^{\frac{1}{20}}1\right)4 \approx 0.1411=14.11 \% \end{align*}
Example 4
How much will Steve have in 8 years if he invests $500 in a bank that offers 8% compounded quarterly?
\begin{align*}PV = 500, \ t=8, \ i=8 \%, \ FV=?, \ k=4 \end{align*}
\begin{align*}FV = PV \left(1+\frac{i}{k}\right)^{kt}=500 \left(1+\frac{0.08}{4}\right)^{4 \cdot 8}=\$942.27\end{align*}
Example 5
How many years will Mark need to invest his money at 3% compounded weekly \begin{align*}(k=52)\end{align*} if he wants his $100 to grow to $400?
\begin{align*}FV = 400, \ PV=100, \ k=52, \ i=0.03, \ t=?\end{align*}
\begin{align*}FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\
400 &= 100 \left(1+\frac{0.03}{52}\right)^{52 \cdot t}\\
t &= \frac{\ln 4}{52 \cdot \ln \left(1+\frac{0.03}{52}\right)}=46.22 \ years\end{align*}
Review
1. What is the length of a compounding period if \begin{align*}k=12\end{align*}?
2. What is the length of a compounding period if \begin{align*}k=365\end{align*}?
3. What would the value of \begin{align*}k\end{align*} be if interest was compounded every hour?
4. What would the value of \begin{align*}k\end{align*} be if interest was compounded every minute?
5. What would the value of \begin{align*}k\end{align*} be if interest was compounded every second?
For problems 615, find the missing value in each row using the compound interest formula.
Problem Number 
\begin{align*}PV\end{align*}  \begin{align*}FV\end{align*}  \begin{align*}t\end{align*}  \begin{align*}i\end{align*}  \begin{align*}k\end{align*} 
6. 
$1,000 

7 
1.5% 
12 
7. 
$1,575 
$2,250 
5 

2 
8. 
$4,000 
$5,375.67 

3% 
1 
9. 

$10,000 
12 
2% 
365 
10. 
$10,000 

50 
7% 
52 
11. 
$1,670 
$3,490 
10 

4 
12. 
$17,000 
$40,000 
25 

12 
13. 
$12,000 

3 
5% 
365 
14. 

$50,000 
30 
8% 
4 
15. 

$1,000,000 
40 
6% 
2 
Review (Answers)
To see the Review answers, open this PDF file and look for section 13.3.