Ideas
Compound interest
Whereas simple interest is calculated on the principal (i.e. the initial) amount, meaning the amount of interest remains constant or fixed, compound interest is calculated on the current value of the investment or loan per period of time. Because of this, compound interest growth is another example of a geometric sequence.
For example, if you invest \$500 for 3 years at a rate of 6\% per annum, the investment would grow as follows:
| \text{Beginning of year } 1 | \$500 | ||
| \text{End of year } 1 | \$500 \times 1.06 | = | \$500 \times 1.06^1 |
| \text{End of year } 2 | (\$500 \times 1.06) \times 1.06 | = | \$500 \times 1.06^2 |
| \text{End of year } 3 | ((\$500 \times 1.06) \times 1.06) \times 1.06 | = | \$500 \times 1.06^3 |
| \text{End of year } n | \$500 \times 1.06^n |
After just a couple of calculations, a pattern can be seen. The size of the investment after n years can be written in the form \$500 \times 1.06^n. This pattern holds true for all forms of compound interest and is expressed with the following formula.
The compound interest formula is: V_n=V_0 \times \left(1+\dfrac{r}{100} \right)^{n} where
n is the number of periods (can be years, months, weeks),
V_n is the future value (final amount of our investment after n periods),
V_0 is the present value (the initial principal amount), and
r is the percentage interest rate per compounding period.
Note: This formula, is often written in the form A=P \left(1+r\right)^{n}. In this form, A refers to the future value of the investment after n periods, while P is the principal amount invested, r refers to the rate per period and n refers to the total number of periods. This formula also gives us the total amount (ie. the principal and interest together). If we just want to know the value of the interest, we can work it out by subtracting the principal from the total amount of the investment, I=A-P.
Examples
Example 1
Tina has \$900 in a savings account which earns compound interest at a rate of 2.4\% p.a. If interest is compounded monthly, how much interest does Tina earn in 17 months?
The compound interest formula:
Alternative formula:
Model compound interest using a recurrence relation
The recurrence relation used for modelling compound interest is the same as that of a reducing balance depreciation except that now, whatever is owed or invested, is increasing rather than decreasing over time.
Consider for example investing \$1000 at 10\% interest compounded annually. What will the value of the investment be after three years?
One approach is to repeatedly increasing each year by 10\%. In other words, the balance at the end of each year is 110\% of the previous year:
| \text{End of year } 1 | \$1000 \times 1.10 = \$1100 |
| \text{End of year } 2 | \$1100 \times 1.10 = \$1210 |
| \text{End of year } 3 | \$1210 \times 1.10 = \$1331 |
Another approach is to apply the compound interest formula:\begin{array}{c} &V_0 &= &\$1000 \\ &n &= &3 \\ &r &= &10 \\ \end{array}
We can now substitute these values into the formula.
| \displaystyle V_n | \displaystyle = | \displaystyle V_0 \times \left(r + \dfrac{1}{100}\right)^{n} |
| \displaystyle = | \displaystyle 1000 \times \left(1 + \dfrac{10}{100}\right)^{3} | |
| \displaystyle = | \displaystyle 1000 \times (1.1)^{3} | |
| \displaystyle = | \displaystyle \$1331 |
Both approaches give us the same result. The investment is worth \$1331 at the end of the three year period.
The following recurrence relation can be used: V_0=k, \, V_{n+1}=R \times V_n where
V_{n+1} is the value of the loan or investment after n periods,
R equals 1 + \dfrac{r}{100}, expressed as a decimal, where r\% is the ineterest rate
k is the initial amount, or principal amount
Unlike simple interest, compound interest is an example of geometric growth.
Compound interest rates are usually given as per annum, meaning the interest rate per year. This rate is called the nominal interest rate.
Although the annual interest rate is quoted, the compounding period may be different. For example, interest may be calculated monthly meaning that there are 12 interest periods within one year. The nominal interest rate therefore has to be converted to a compounding interest rate.
| \text{Term} | \text{Compounding periods per year} \left(n\right) | \text{Note} |
|---|---|---|
| \text{quarterly} | 4 | \text{there are } 3 \text{ months in each quarter} |
| \text{monthly} | 12 | \text{even though not all months} \\ \text{have the same amount of days} |
| \text{fortnightly} | 26 | \text{even though there are not exactly} \\ 26 \text{ fortnightly periods} |
| \text{weekly} | 52 | \text{even though there are not exactly } 52 \text{ weeks} |
| \text{daily} | 365 | \text{even though there are leap years} |
Exploration
Experiment with changing the values below to observe what happens when you change investments, interest rates, and periods.
The longer the duration of an investment, the higher will be its future value. The higher the interest rate of an investment, the higher will be its future value. The longer the compounding period, the more will be the effect on the future value.
Examples
Example 2
The balance of an investment, in dollars, at the end of each month where interest is compounded monthly is given by A_{n+1}=1.025A_{n},\, A_{0}=4000.
State the monthly interest rate.
Use the sequences facility on your calculator to determine the balance at the end of the first year.
Round your answer to the nearest cent.
Use the compound interest formula to determine the balance at the end of the first year and confirm the answer from the previous part.
Use the sequences facility on your calculator to determine at the end of which month and year the investment is first worth double the initial amount invested.
The following recurrence relation can be used:
Unlike simple interest, compound interest is an example of geometric growth.
Compound interest rates are usually given as per annum, meaning the interest rate per year. This rate is called the nominal interest rate.
Simple and compound interest graphs
Notice:
The simple interest graph is a straight line and the compound interest graph is a smooth curve.
Both graphs are increasing.
The simple interest line is increasing at a constant rate and the compound interest curve is increasing at an increasing rate.
Both graphs have the same y-intercept (present value or principal).
Knowing the basic shape of the curve that each type of investment makes will help us think about key points in the life cycle of an investment, and compare investment strategies.
The value of an investment earning simple interest is calculated using the formula A=P+Prn which is a linear equation in terms of n, the number of periods. Meanwhile, compound interest uses the formula A=P(1+r)^{n} which is non-linear equation in terms of n.
Examples
Example 3
When \$2250 is deposited a bank offers two types of savings accounts.
Complete the equation which describes the future value FV of the simple interest account after n years.
FV=⬚ + ⬚n
Complete the equation which describes the future value FV of the compound interest account after n years.
FV=⬚ (⬚)^n
Using the graphs provided, which account would have a greater balance in the 6th year?
Using the graphs provided, which account would have a greater balance in the 29th year?
Simple interest is calculated only on the principal (that is, the initial amount) so the amount of interest being added to a loan or investment remains constant or fixed. Compound interest is interest earned on the principal amount plus interest on the interest already earned. In general, we observed the following pattern.