Compound Interest
Learning Outcomes
- Calculate one-time simple interest, and simple interest over time
- Determine APY given an interest scenario
- Calculate compound interest
Compounding
With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding. Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow? The 3% interest is an annual percentage rate (APR) – the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn [latex]\frac{3%}{12}[/latex]= 0.25% per month. In the first month,- P0 = $1000
- r = 0.0025 (0.25%)
- I = $1000 (0.0025) = $2.50
- A = $1000 + $2.50 = $1002.50
- P0 = $1002.50
- I = $1002.50 (0.0025) = $2.51 (rounded)
- A = $1002.50 + $2.51 = $1005.01
Month | Starting balance | Interest earned | Ending Balance |
1 | 1000.00 | 2.50 | 1002.50 |
2 | 1002.50 | 2.51 | 1005.01 |
3 | 1005.01 | 2.51 | 1007.52 |
4 | 1007.52 | 2.52 | 1010.04 |
5 | 1010.04 | 2.53 | 1012.57 |
6 | 1012.57 | 2.53 | 1015.10 |
7 | 1015.10 | 2.54 | 1017.64 |
8 | 1017.64 | 2.54 | 1020.18 |
9 | 1020.18 | 2.55 | 1022.73 |
10 | 1022.73 | 2.56 | 1025.29 |
11 | 1025.29 | 2.56 | 1027.85 |
12 | 1027.85 | 2.57 | 1030.42 |
Example
Build an explicit equation for the growth of $1000 deposited in a bank account offering 3% interest, compounded monthly.Answer:
- P0 = $1000
- P1 = 1.0025P0 = 1.0025 (1000)
- P2 = 1.0025P1 = 1.0025 (1.0025 (1000)) = 1.0025 2(1000)
- P3 = 1.0025P2 = 1.0025 (1.00252(1000)) = 1.00253(1000)
- P4 = 1.0025P3 = 1.0025 (1.00253(1000)) = 1.00254(1000)
- Pm = (1.0025)m($1000)
- m is the number of compounding periods (months in our example)
- r is the annual interest rate
- k is the number of compounds per year.
Compound Interest
[latex-display]P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}[/latex-display]- PN is the balance in the account after N years.
- P0 is the starting balance of the account (also called initial deposit, or principal)
- r is the annual interest rate in decimal form
- k is the number of compounding periods in one year
- If the compounding is done annually (once a year), k = 1.
- If the compounding is done quarterly, k = 4.
- If the compounding is done monthly, k = 12.
- If the compounding is done daily, k = 365.
Example
A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account after 20 years?Answer: In this example,
P0 = $3000 | the initial deposit |
r = 0.06 | 6% annual rate |
k = 12 | 12 months in 1 year |
N = 20 | since we’re looking for how much we’ll have after 20 years |
Years | Simple Interest ($15 per month) | 6% compounded monthly = 0.5% each month. |
5 | $3900 | $4046.55 |
10 | $4800 | $5458.19 |
15 | $5700 | $7362.28 |
20 | $6600 | $9930.61 |
25 | $7500 | $13394.91 |
30 | $8400 | $18067.73 |
35 | $9300 | $24370.65 |
Evaluating exponents on the calculator
When we need to calculate something like [latex]5^3[/latex] it is easy enough to just multiply [latex]5\cdot{5}\cdot{5}=125[/latex]. But when we need to calculate something like [latex]1.005^{240}[/latex], it would be very tedious to calculate this by multiplying [latex]1.005[/latex] by itself [latex]240[/latex] times! So to make things easier, we can harness the power of our scientific calculators. Most scientific calculators have a button for exponents. It is typically either labeled like:^ , [latex]y^x[/latex] , or [latex]x^y[/latex] .
To evaluate [latex]1.005^{240}[/latex] we'd type [latex]1.005[/latex] ^ [latex]240[/latex], or [latex]1.005 \space{y^{x}}\space 240[/latex]. Try it out - you should get something around 3.3102044758.Example
You know that you will need $40,000 for your child’s education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal?Answer: In this example, we’re looking for P0.
r = 0.04 | 4% |
k = 4 | 4 quarters in 1 year |
N = 18 | Since we know the balance in 18 years |
P18 = $40,000 | The amount we have in 18 years |
Rounding
It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but keeping more digits is always better.Example
To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.P0 = $1000 | the initial deposit |
r = 0.05 | 5% |
k = 12 | 12 months in 1 year |
N = 30 | since we’re looking for the amount after 30 years |
r/k rounded to: | Gives P30 to be: | Error |
0.004 | $4208.59 | $259.15 |
0.0042 | $4521.45 | $53.71 |
0.00417 | $4473.09 | $5.35 |
0.004167 | $4468.28 | $0.54 |
0.0041667 | $4467.80 | $0.06 |
no rounding | $4467.74 |
Using your calculator
In many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{12\times30}}[/latex] We can quickly calculate 12×30 = 360, giving [latex]{{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{360}}[/latex]. Now we can use the calculator.Type this | Calculator shows |
0.05 ÷ 12 = . | 0.00416666666667 |
+ 1 = . | 1.00416666666667 |
yx 360 = . | 4.46774431400613 |
× 1000 = . | 4467.74431400613 |
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter: 1000 × ( 1 + 0.05 ÷ 12 ) yx 360 =Solving For Time
Examples
If you invest $2000 at 6% compounded monthly, how long will it take the account to double in value?Answer: This is a compound interest problem, since we are depositing money once and allowing it to grow. In this problem,
P0 = $2000 | the initial deposit |
r = 0.06 | 6% annual rate |
k = 12 | 12 months in 1 year |
[latex]4000=2000{{\left(1+\frac{0.06}{12}\right)}^{N\times12}}[/latex] Divide both sides by 2000
[latex]2={{\left(1.005\right)}^{12N}}[/latex] To solve for the exponent, take the log of both sides
[latex]\log\left(2\right)=\log\left({{\left(1.005\right)}^{12N}}\right)[/latex] Use the exponent property of logs on the right side
[latex]\log\left(2\right)=12N\log\left(1.005\right)[/latex] Now we can divide both sides by 12log(1.005)
[latex]\frac{\log\left(2\right)}{12\log\left(1.005\right)}=N[/latex] Approximating this to a decimal
N = 11.581 It will take about 11.581 years for the account to double in value. Note that your answer may come out slightly differently if you had evaluated the logs to decimals and rounded during your calculations, but your answer should be close. For example if you rounded log(2) to 0.301 and log(1.005) to 0.00217, then your final answer would have been about 11.577 years. Get additional guidance for this example in the following: https://youtu.be/zHRTxtFiyxcLicenses & Attributions
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- Compound interest - the importance of rounding. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Question ID 6693. Authored by: Lippman,David. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.