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# Compound Interest per Period

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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?

#### Guidance

Consider a bank that compounds and adds interest to accounts  $k$ 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 $(k=1)$
• Twice (semi-annually), with 6% after the first 6 months and 6% after the last six months $(k=2)$
• Four times (quarterly), with 3% at the end of each 3 months $(k=4)$
• Twelve times (monthly), with 1% at the end of each month $(k=12)$

The intervals could even be days, hours or minutes. 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.

The formula for interest compounding  $k$ times per year for  $t$ years at a nominal interest rate $i$ with present value  $PV$ and future value  $FV$ is:

$FV=PV \left(1+\frac{i}{k}\right)^{kt}$

Note: Just like simple interest and compound interest use the symbol  $i$ to represent interest but they compound in very different ways, so does a nominal rate. As you will see in the examples, a nominal rate of 12% may actually yield more than 12%.

Example A

How much will Felix have in 4 years if he invests $300 in a bank that offers 12% compounded monthly? Solution: $FV=?, \ PV=300, \ t=4, \ k=12, \ i=0.12$ $FV=PV \left(1+\frac{i}{k}\right)^{kt}=300 \left(1+\frac{0.12}{12}\right)^{12 \cdot 4} \approx 483.67$ 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 $(12 \cdot 4)=48$ . Example B How many years will Matt need to invest his money at 6% compounded daily $(k=365)$ if he wants his$3,000 to grow to $5,000? Solution: $FV=5,000, \ PV=3,000, \ k=365, \ i=0.06, \ t=?$ $FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\5,000 &= 3,000 \left(1+\frac{0.06}{365}\right)^{365t}\\\frac{5}{3} &= \left(1+\frac{0.06}{365}\right)^{365t}\\\ln \frac{5}{3} &= \ln \left(1+\frac{0.06}{365}\right)^{365t}\\\ln \frac{5}{3} &= 365t \cdot \ln \left(1+\frac{0.06}{365}\right)\\t &= \frac{\ln \frac{5}{3}}{365 \cdot \left(1 + \frac{0.06}{365}\right)}=8.514 \ years$ Example C What nominal interest rate compounded quarterly doubles money in 5 years? Solution: $FV=200, \ PV=100, \ k=4, \ i=?, \ t=5$ $FV &= PV \left(1+\frac{i}{k}\right)^{kt}\\200 &= 100 \left(1+\frac{i}{4}\right)^{4 \cdot 5}\\[2]^{\frac{1}{20}} &= \left[ \left( 1+\frac{i}{4}\right)^{20}\right]^{\frac{1}{20}}\\2^{\frac{1}{20}} &= 1+\frac{i}{4}\\i &= \left(2^{\frac{1}{20}}-1\right)4 \approx 0.1411=14.11 \%$ Concept Problem Revisited 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  $k=2$ times per year then she will end up earning:

$FV=PV \left(1+\frac{i}{k}\right)^{kt}=100 \left(1+\frac{.12}{2}\right)^{2 \cdot 1}=\112.36$

#### Vocabulary

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.

$\left(1+\frac{.12}{12}\right)^{12}=(1+0.01)^{12}=1.1286$

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  $k$ .