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# Compound Interest per Year

## Interest that grows on the principal and the interest earned in previous years.

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Compound Interest per Year

If a person invests $100 in a bank with 6% simple interest, they earn$6 in the first year and $6 again in the second year totaling$112.  If this was really how interest operated with most banks, then someone clever may think to withdraw the $106 after the first year and immediately reinvest it. That way they earn 6% on$106.  At the end of the second year, the clever person would have earned $6 like normal, plus an extra .36 cents totaling$112.36.  Thirty six cents may seem like not very much, but how much more would a person earn if they saved their 100 for 50 years at 6% compound interest rather than at just 6% simple interest? ### Compound Interest Per Year Compound interest allows interest to grow on interest. As with simple interest, \begin{align*}PV\end{align*} is defined as present value, \begin{align*}FV\end{align*} is defined as future value\begin{align*}i\end{align*} is the interest rate, and \begin{align*}t\end{align*} is time. The formulas for simple and compound interest look similar, so be careful when reading problems in determining whether the interest rate is simple or compound. The following table shows the amount of money in an account earning compound interest over time:  Year Amount Ending in Account 1 \begin{align*}FV=PV(1+i)\end{align*} 2 \begin{align*}FV=PV(1+i)^2\end{align*} 3 \begin{align*}FV=PV(1+i)^3\end{align*} 4 \begin{align*}FV=PV(1+i)^4\end{align*} … \begin{align*}t\end{align*} \begin{align*}FV=PV(1+i)^t\end{align*} For now you should assume that you are compounding the interest once a year or annually. An account with a present value of \begin{align*}PV\end{align*} that earns compound interest at \begin{align*}i\end{align*} percent annually for \begin{align*}t\end{align*} years has a future value of \begin{align*}FV\end{align*} shown below: \begin{align*}FV=PV(1+i)^t\end{align*} Applying this formula for years 1, 2, 3, and 4 for an initial deposit of100 at 3% compound interest, you would get the following results:

\begin{align*}PV=100, \ i=0.03, \ t=1,2,3 \ and \ 4, \ FV=?\end{align*}

 Year Amount ending in Account 1 \begin{align*}FV=100(1+0.03)=103.00\end{align*} 2 \begin{align*}FV=100(1+0.03)^2=106.09\end{align*} 3 \begin{align*}FV=100(1+0.03)^3\approx109.27\end{align*} 4 \begin{align*}FV=100(1+0.03)^4\approx112.55\end{align*}

Calculator shortcut:  When doing repeated calculations that are just 1.03 times the result of the previous calculation, use the <ANS> button to create an entry that looks like <Ans*1.03>.  Then, pressing enter repeatedly will rerun the previous entry producing the values on the right.

### Examples

Earlier you were introduced to a concept problem contrasting 100 for 50 years at 6% compound interest versus 6% simple. Now you can calculate how much more powerful compound interest is. \begin{align*}PV=100, \ t=50, \ i=6\%, \ FV=?\end{align*} Simple interest: \begin{align*}FV=PV(1+t \cdot i)=100(1+50 \cdot 0.06)=400\end{align*} Compound interest: \begin{align*}FV=PV(1+i)^t=100(1+0.06)^{50}\approx 1,842.02\end{align*} It is remarkable that simple interest grows the balance of the account to400 while compound interest grows it to about $1,842.02. The additional money comes from interest growing on interest repeatedly. #### Example 2 How much will Kyle have in a savings account if he saves$3,000 at 4% compound interest for 10 years?

\begin{align*}PV = 3,000, \ i=0.04, \ t=10 \ years, \ FV=?\end{align*}

\begin{align*}FV &= PV(1+i)^t\\ FV &= 3000(1+0.04)^{10} \approx \4,440.73\end{align*}

#### Example 3

How long will it take money to double if it is in an account earning 8% compound interest?

There are two ways you can solve this problem, through estimation or through computation.

Estimation Solution:  The rule of 72 is an informal means of estimating how long it takes money to double.  It is useful because it is a calculation that can be done mentally that can yield surprisingly accurate results.  This can be very helpful when doing complex problems to check and see if answers are reasonable.  The rule of 72 simply states \begin{align*}\frac{72}{\mathit{i}}\approx t\end{align*}  where \begin{align*}i\end{align*} is written as an integer (i.e. 8% would just be 8).

In this case \begin{align*}\frac{72}{8}=9\approx t\end{align*}, so it will take about 9 years.

Exact Solution:  Since there is no initial value you are just looking for some amount to double.  You can choose any amount for the present value and double it to get the future value even though specific numbers are not stated in the problem.  Here you should choose 100 for \begin{align*}PV\end{align*} and 200 for \begin{align*}FV\end{align*}

\begin{align*}PV = 100, \ FV=200, \ i=0.08, \ t=?\end{align*}

\begin{align*}FV &= PV (1+i)^t\\ 200 &=100(1+0.08)^t\\ 2 &= 1.08^t\\ \ln 2 &= \ln 1.08^t\\ \ln 2 &= t \cdot \ln 1.08\\ t &= \frac{\ln 2}{\ln 1.08}=9.00646\end{align*}

It will take just over 9 years for money (any amount) to double at 8%.  This is extraordinarily close to your estimation and demonstrates how powerful the Rule of 72 can be in estimation.

#### Example 4

How long will it take money to double at 6% compound interest?  Estimate using the rule of 72 and also find the exact answer.

Estimate: \begin{align*}\frac{72}{6}=12\approx\end{align*} years it will take to double

\begin{align*}PV = 100, \ FV=200, \ i=0.06, \ t=?\end{align*}

\begin{align*}200 &= 100 (1+0.06)^t\\ 2 &= (1.06)^t\\ \ln 2 &= \ln 1.06^t=t \ \ln 1.06\\ t &= \frac{\ln 2}{\ln 1.06} \approx 11.89 \ years\end{align*}

#### Example 5

What compound interest rate is needed to grow $100 to$120 in three years?

\begin{align*}PV=100, \ FV=120, \ t=3, \ i=?\end{align*}

\begin{align*}FV &= PV(1+i)^t\\ 120 &= 100(1+i)^3\\ [1.2]^{\frac{1}{3}} &= [(1+i)^3]^{\frac{1}{3}}\\ [1.2]^{\frac{1}{3}} &= 1+i\\ i &= 1.2^{\frac{1}{3}}-1 \approx 0.06266\end{align*}

### Review

For problems 1-10, 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*} 1. $1,000 7 1.5% 2.$1,575 $2,250 5 3.$4,500 $5,534.43 3% 4.$10,000 12 2% 5. $1,670$3,490 10 6. $17,000$40,000 25 7. $10,000$17,958.56 5% 8. $50,000 30 8% 9.$1,000,000 40 6% 10. $10,000 50 7% 11. How long will it take money to double at 4% compound interest? Estimate using the rule of 72 and also find the exact answer. 12. How long will it take money to double at 3% compound interest? Estimate using the rule of 72 and also find the exact answer. 13. Suppose you have$5,000 to invest for 10 years.  How much money would you have in 10 years if you earned 4% simple interest? How much money would you have in 10 years if you earned 4% compound interest?

14.  Suppose you invest $4,000 which earns 5% compound interest for the first 12 years and then 8% compound interest for the next 8 years. How much money will you have after 20 years? 15. Suppose you invest$10,000 which earns 2% compound interest for the first 8 years and then 5% compound interest for the next 7 years.  How much money will you have after 15 years?

To see the Review answers, open this PDF file and look for section 13.2.

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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 $t$.

present value

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

rule of 72

The Rule of 72 states that the approximate amount of time that it will take an account earning compound interest to double is $t \approx\frac{72}{i}$, where $i$ is written as an integer.

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.