5.2: Compound Interest
- Page ID
- 106254
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\dsum}{\displaystyle\sum\limits} \)
\( \newcommand{\dint}{\displaystyle\int\limits} \)
\( \newcommand{\dlim}{\displaystyle\lim\limits} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\(\newcommand{\longvect}{\overrightarrow}\)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Most banks, loans, and credit cards charge compound interest, not simple interest. Compound interest is both calculated on the original principal and on all interest that has been added to the original principal. You can think of this as "interest on your interest."
Compound interest is interest that is paid or earned on the principal as well as on the interest that has accrued.
Interest can be compounded at various time intervals. Interest on a mortgage or auto loan is usually compounded monthly. Interest on a savings account may be compounded quarterly (four times a year). Interest on a credit card may be compounded weekly or even daily. Here are some common compounding types that you may encounter in financial problems and in your everyday life.
| Compounding type | Number of compounding periods per year |
|---|---|
| Annually | 1 |
| Semiannually | 2 |
| Quarterly | 4 |
| Monthly | 12 |
| Daily | 365 |
Consider the following example that illustrates how interest is compounded.
We deposit $1,000 in a bank account offering 3% interest compounded monthly. Let's explore how our money will grow.
Solution
The \(3\%\) interest rate is an annual percentage rate (APR) – the total interest rate to be paid during the year. Because interest is being paid monthly, we will earn only \(\frac{3 \%}{12}=0.25 \% = 0.0025\) per month.
- In the first month, \(P=$1,000\) and \(r=0.0025\), so
\(I=$1,000(0.0025)=$2.50\)
\(F=$1,000+$2.50=$1,002.50\)
At the end of the first month, we will earn $2.50 in interest, raising our account balance to $1,002.50.
- In the second month, \(P=$1,002.50\) and \(r=0.0025\), so
\(I=$1,002.50(0.0025)=$2.51 \; \) (rounded)
\(F=$1,002.50+ $2.51=$1,005.51\)
Notice that at the end of the second month we will earn slightly more interest than in the first month. This is because we earned interest not only on the original $1,000 principal, but we also earned interest on the $2.50 of interest accumulated during the first month. This is the key advantage that compounding of interest gives us.
The table shows how the interest accumulates and is compounded for a complete year (12 months):
| Month | Starting Balance | Interest Earned | Ending Balance |
|---|---|---|---|
| 1 | $1,000.00 | $1,000.00 \(\times\) 0.0025 = $2.50 | $1,000.00 + $2.50 = $1,002.50 |
| 2 | $1,002.50 | $1,002.50 \(\times\) 0.0025 = $2.51 | $1,002.50 + $2.51 = $1,005.01 |
| 3 | $1,005.01 | $1,005.01 \(\times\) 0.0025 = $2.51 | $1,005.01 + $2.51 = $1,007.52 |
| 4 | $1,007.52 | $1,007.52 \(\times\) 0.0025 = $2.52 | $1,007.52 + $2.52 = $1,010.04 |
| 5 | $1,010.04 | $1,010.05 \(\times\) 0.0025 = $2.53 | $1,010.05 + $2.53 = $1,012.57 |
| 6 | $1,012.57 | $1,012.57 \(\times\) 0.0025 = $2.53 | $1,012.57 + $2.53 = $1,015.10 |
| 7 | $1,015.10 | $1,015.10 \(\times\) 0.0025 = $2.54 | $1,015.10 + $2.54 = $1,017.64 |
| 8 | $1,017.64 | $1,017.64 \(\times\) 0.0025 = $2.54 | $1,017.64 + $2.54 = $1,020.18 |
| 9 | $1,020.18 | $1,020.18 \(\times\) 0.0025 = $2.55 | $1,020.18 + $2.55 = $1,022.73 |
| 10 | $1,022.73 | $1,022.73 \(\times\) 0.0025 = $2.56 | $1,022.73 + $2.56 = $1,025.29 |
| 11 | $1,025.29 | $1,025.29 \(\times\) 0.0025 = $2.56 | $1,025.29 + $2.56 = $1,027.85 |
| 12 | $1,027.85 | $1,027.85\(\times\) 0.0025 = $2.57 | $1,027.85 + $2.57 = $1,030.42 |
So, after one year of compounding the interest monthly, the balance in the account is $1,030.42.
The calculations of interest and the new balance month-after-month can get tedious and long. To make these calculations more efficient, observe the pattern in the table: The future value after \(m\) months, \(F_{m}\), can be presented using the future value from the previous month, \(F_{m-1}\):
\(F_{m}=F_{m-1}\cdot(1 + \frac{0.03}{12}) \)
This observation leads to the standard compound interest formula.
\(F=P\left(1+\frac{r}{n}\right)^{nt}\)
where,
- F is future value
- P is present value, or principal
- r is the annual percentage rate (APR) written as a decimal
- n is the number of compounding periods per year
- t is number of years
Using the formula derived above, let's solve Example 1 again where \(P=$1,000\), \(r=0.03 \text{ annually}\), \(n=12 \text{ times}\), and \(t = 1 \text{ year}\). A screenshot of entering the calculation on a TI calculator is also shown.
\(F=1,000\left(1+\frac{0.03}{12}\right)^{12 \cdot 1}\)
The accumulated balance after 12 months is $1,030.42 and agrees with the result in the table.
So is there a significant difference between compounded interest and simple interest? The next example will investigate this question.
A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate than a savings or checking account, but you cannot access your investment for a specified length of time. Suppose you deposit $3,000 into a CD paying 6% interest compounded monthly. After 20 years, how will the value of the CD compare to the value of another type of account that pays 6% simple interest?
Solution
In this example,
\(\begin{array} {ll} P=\$ 3,000 & \text{the initial deposit} \\ r = 0.06 & 6\% \text{ annual rate} \\ n = 12 & \text{12 months in 1 year} \\ t = 20 & \text{since we’re looking for how much we’ll have after 20 years} \end{array}\)
The table and graph compares an account's balance using 6% simple interest to the CD balance using 6% interest compounded monthly every 5 years throughout 35 years.
\(\begin{array}{|c|c|c|}
\hline \text { Years } & \begin{array}{c}
\text { Value of Account Paying 6% Simple Interest } \\
\text { (\$15 per month) } \\
\end{array} & \begin{array}{c}
\text { Value of CD Paying } 6 \% \text { Interest Compounded Monthly } \\ (0.5 \% \text { per month})
\end{array} \\ \hline 5 & \$ 3,900 & \$ 4,046.55 \\
\hline 10 & \$ 4,800 & \$ 5,458.19 \\
\hline 15 & \$ 5,700 & \$ 7,362.28 \\
\hline 20 & \$ 6,600 & \$ 9,930.61 \\
\hline 25 & \$ 7,500 & \$ 13,394.91 \\
\hline 30 & \$ 8,400 & \$ 18,067.73 \\
\hline 35 & \$ 9,300 & \$ 24,370.65 \\
\hline
\end{array}\)
The graph shows there is not a great deal of difference between the two types of interest in the first few years. But as time increases, the value of the CD where interest is compounded increases at a faster and faster pace and at a much faster rate than the value of the account paying simple interest.
Using the formulas for simple interest and compound interest, we can find the future value \(F\) of the two different accounts in \(t=20 \text { years}\) with an initial deposit of \(P=$3,000\) and an interest rate of \(r=6 \%\).
Simple Interest: \(F=P(1+rt)= 3,000(1+0.06 \cdot 20) = $6,660\)
Compounded Interest: \(F=P\left(1+\frac{r}{n}\right)^{nt}=3,000\left(1+\frac{0.06}{12}\right)^{12 \cdot 20} = $9,930.61\)
When interest is compounded, the amount of money in the CD is \($9,930.61 - $6,600.00 = $3,330.61 \;\) more than if simple interest had accrued.
One can always use any scientific calculator to find value of \(F\) using the compound interest formula. However, computations become much more algebraically intense when solving for the principal \(P\), for the rate \(r\), or for the time \(t\). Therefore, it is recommended to use the Time Value Money Solver (TVM Solver) app available on a TI calculator to solve all problems related to compounded interest.
To access the TVM Solver, press the APPS key and then choose Finance....
Let us consider how to use the TVM Solver settings on following example.
Find the total amount accumulated and the total interest earned if $4,000 is invested for 2 years at a 6% compounded monthly.
Solution
When solving for a value using the TVM Solver app, you will need to enter all the given values, but skip over the value you are solving for (in this case \(FV\).) Once you have entered all given values, go back to the value you want to find and press the ALPHA button and then ENTER. A square will appear next to \(FV\) to show that this value was calculated using the given information.
According to the TVM Solver, the total amount in the account will accumulate to $4,508.64. This means that the account has earned \($4,508.64 - $4,000 = $508.64 \; \) in interest during the 2 years.
For the following examples, we will continue to make use of the TVM Solver.
Find the total amount accumulated and the total interest earned when $2,000 is invested for 4 years at 5% interest compounded quarterly (four times each year.)
Solution
For this scenario,
- \(N = 16\): The number of times the interest is compounded is \(4 \text { years } \times 4 \text { times per year } = 16\).
- \(I\% = 5\): Remember to leave this as a percentage. Do not change to a decimal.
- \(PV = -2000\): Recall that \('-'\) represents money leaving you and put into an account.
Completing the remaining values as in Example 3, we then solve for future value \(FV\) by moving the cursor back to the value for \(FV\) and pressing ALPHA and then ENTER. The result is shown on the calculator screen.
The accumulated amount in the account will be \($2,439.78. \) The total interest earned during the 2 years will be \($2,439.78 – $2,000.00 = $439.78\).
Use the TVM Solver app to solve:
Find the total amount accumulated and the total interest earned when $2,500 is invested for 3 years at 4% interest compounded semi-annually (twice a year).
- Answer
-
The accumulated amount in the account will be \($2,815.41. \) The total interest earned during the 3 years will be \($2,815.41 – $2,500.00 = $315.41.\)
How much do you need to invest into a 5-year CD (certificate of deposit) that pays 3% interest compounded monthly so that it will be worth $5,000 at the end of the term?
Solution
In this problem, the future value is given, \(FV = $5,000\), and we need to solve for the present value, \(PV\). We also are given that \(N = 5 \text{ years } \times 12 \text { times per year }= 60 \text { times}\) and \(I = 3 \%\).
As the TVM Solver calculates, you need to invest $4,304.35 in order to reach $5,000 in 5 years.
A rare violin has an estimated value of $12,000. Its value history indicates that it appreciates at 14% per year. How many years will it take to appreciate to $60,000?
Solution
Here we need to solve for \(N\). Set \(I = 14 \%\), \(PV=-12,000\) and \(FV = 60,000\).
The value \(N=12.28313558\) is in years so figuring the decimal 0.28 as part of a 52-week year gives \( 0.28 \times 52 = 14.56 \text { weeks}\). The final answer is about 12 years and 14 weeks.
Use the TVM Solver app to solve:
You investing $10,000 for 5 years into account where the interest is compounded monthly. What interest rate will you need to for the account to accumulate to $12,000 during that time?
- Answer
-
\(I \approx 3.65 \%\)
In summary, the TVM Solver app on the TI calculator is similar to the financial math functions in Microsoft Excel . Using the TVM Solver application for compound interest problems allows us to solve for any parameter (\(PV\), \(FV\), \(I\), \(N\), and so on) without having to solve a logarithmic or exponential equation. While the TVM Solver is useful for solving compound interest problems, it is not useful for solving simple interest problems.
The next two sections will use the TVM Solver app to solve annuity and loan repayment problems.