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# Compound Interest per Period

## Interest that grows on the principal and on interest earned semi-annually, monthly, daily, etc.

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Compound Interest per Period

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*} times per year. If the original percent offered is 12% then in one year that interest can be compounded:

• Once, with 12% at the end of the year \begin{align*}(k=1)\end{align*}
• Twice (semi-annually), 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 earn12 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*}

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### Vocabulary Language: English

Compound interest

Compound interest refers to interest earned on the total amount at the time it is compounded, including previously earned interest.

future value

In the context of earning interest, future value stands for the amount in the account at some future time $t$.

nominal interest rate

A nominal interest rate or nominal rate is a number that resembles a regular interest rate, but it really is a sum of compound interest rates. For example, a nominal rate of 12% compounded monthly is really 1% compounded 12 times, and is equivalent to an effective annual interest rate of 12.86% because $\left(1+\frac{.12}{12}\right)^{12}=(1+0.01)^{12}=1.1286$.

nominal rate

A nominal interest rate or nominal rate is a number that resembles a regular interest rate, but it really is a sum of compound interest rates. For example, a nominal rate of 12% compounded monthly is really 1% compounded 12 times, and is equivalent to an effective annual interest rate of 12.86% because $\left(1+\frac{.12}{12}\right)^{12}=(1+0.01)^{12}=1.1286$.

number of compounding periods

The number of compounding periods is how often the interest will be accrued and added to the account balance, and is represented by the variable $k$.

present value

In the context of earning interest, present value stands for the amount in the account at time 0.

Simple Interest

Simple interest is interest calculated on the original principal only. It is calculated by finding the product of the the principal, the rate, and the time.