Simple and Compound Interest
We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics.Learning Objectives
The leaning objectives for this section include:- Calculate one-time simple interest, and simple interest over time
- Determine APY given an interest scenario
- Calculate compound interest
Simple Interest
Discussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest.Simple One-time Interest
[latex-display]\begin{align}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align}[/latex-display]- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate (in decimal form. Example: 5% = 0.05)
Examples
A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?Answer:
[latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] = $300 | the principal |
r = 0.03 | 3% rate |
I = $300(0.03) = $9. | You will earn $9 interest. |
Exercises
Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn?Answer: Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course of five years, you would earn a total of $250 in interest. When the bond matures, you would receive back the $1,000 you originally paid, leaving you with a total of $1,250.
Further explanation about solving this example can be seen here. https://youtu.be/rNOEYPCnGwgSimple Interest over Time
[latex-display]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex-display]- I is the interest
- A is the end amount: principal plus interest
- [latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)
- r is the interest rate in decimal form
- t is time
APR – Annual Percentage Rate
Interest rates are usually given as an annual percentage rate (APR) – the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up. For example, a 6% APR paid monthly would be divided into twelve 0.5% payments. [latex-display]6\div{12}=0.5[/latex-display] A 4% annual rate paid quarterly would be divided into four 1% payments. [latex-display]4\div{4}=1[/latex-display]Example
Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?Answer: Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into two 2% payments.
[latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] = $1000 | the principal |
r = 0.02 | 2% rate per half-year |
t = 8 | 4 years = 8 half-years |
I = $1000(0.02)(8) = $160. | You will earn $160 interest total over the four years. |
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Compound Interest
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 |
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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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- money-grow-interest-save-invest-1604921. Authored by: TheDigitalWay. License: CC0: No Rights Reserved.
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- Simple interest over time. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
- Simple interest T-note example. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
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- Compound interest - the importance of rounding. Authored by: OCLPhase2's channel. License: CC BY: Attribution.
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