The table below compares the effect of changing the number of compounding periods when $\$1000$$1000 is invested for one year at a nominal rate of $5%$5% p.a. The final amount is calculated using the compound interest formula $A=P\times(1+\frac{i}{n})^n$A=P×(1+in)n, where $i$i is the annual interest rate.
|
Number of compounding periods |
$1$1 |
$4$4 |
$365$365 |
|---|---|---|---|
| Calculation | $A=1000\times(1+\frac{0.05}{1})^1$A=1000×(1+0.051)1 $=1000\times(1.05)$=1000×(1.05) |
$A=1000\times(1+\frac{0.05}{4})^4$A=1000×(1+0.054)4 $=1000\times(1.05095)$=1000×(1.05095) |
$A=1000\times(1+\frac{0.05}{365})^{365}$A=1000×(1+0.05365)365 $=1000\times(1.05127)$=1000×(1.05127) |
| Final amount | $\$1050$$1050 | $\$1050.95$$1050.95 | $\$1051.27$$1051.27 |
| Amount of interest | $\$50$$50 | $\$50.95$$50.95 | $\$51.27$$51.27 |
| Effective annual interest rate | $\frac{50}{1000}=0.05=5%$501000=0.05=5% | $\frac{50.95}{1000}=0.05095=5.095%$50.951000=0.05095=5.095% | $\frac{51.27}{1000}=0.05127=5.127%$51.271000=0.05127=5.127% |
From the table we can see that the amount of interest earned increases when the number of compounding periods increases.
The effective annual interest rate has been calculated using the formula:
$\text{effective interest rate}=\frac{\text{amount of interest in first year}}{\text{loan amount}}\times100%$effective interest rate=amount of interest in first yearloan amount×100%
The published rate of $5%$5% per annum is called the nominal interest rate.
Note: If we only compound once per year then this nominal interest rate is the same as the effective interest rate.
$i_{effective}=(1+\frac{i}{n})^n-1$ieffective=(1+in)n−1
where $i_{effective}$ieffective is the effective interest rate per annum, expressed as a decimal
$i$i is the nominal (or published) interest rate per annum, expressed as a decimal
$n$n is the number of compounding periods per annum
Being able to calculate the effective interest rate can come in handy when we are choosing and comparing investments or loans. They allow us to more easily work out how much interest the investment or loan will actually earn and quickly compare rates that have different compounding periods.
When investing money, we want to have the highest possible effective interest rate.
When borrowing money, we want to have the lowest possible effective interest rate.
Worked examples
example 1
A bank advertises a nominal interest rate of $5.6%$5.6% per annum, compounded quarterly. Calculate the effective interest rate.
Think: Using the effective interest rate formula, we are solving for $i_{effective}$ieffective where $i=0.056$i=0.056 and $n=4$n=4.
Do:
| $i_{effective}$ieffective | $=$= | $\left(1+\frac{i}{n}\right)^n-1$(1+in)n−1 |
|
| $=$= | $(1+\frac{0.056}{4})^4-1$(1+0.0564)4−1 |
|
|
| $\approx$≈ | $1.057187-1$1.057187−1 |
Evaluate using a calculator |
|
| $=$= | $0.0572$0.0572 to $3$3 significant figures |
|
Therefore, the effective interest rate is $5.72%$5.72% p.a. to $2$2 decimal places.
example 2
The effective rate for an investment account which compounds monthly is $6.4%$6.4% p.a.. Calculate the nominal rate for this account correct to two decimal places.
Think: Using the effective interest rate formula, we want to solve for $i$i, where $n=12$n=12 and $i_{effective}=0.064$ieffective=0.064.
Do:
| $i_{effective}$ieffective | $=$= | $(1+\frac{i}{n})^n-1$(1+in)n−1 |
|
| $0.064$0.064 | $=$= | $(1+\frac{i}{12})^{12}-1$(1+i12)12−1 |
|
| $1.064$1.064 | $=$= | $(1+\frac{i}{12})^{12}$(1+i12)12 | |
| $\sqrt[12]{1.064}$12√1.064 | $=$= | $(1+\frac{i}{12})$(1+i12) | |
| $\sqrt[12]{1.064}-1$12√1.064−1 | $=$= | $\frac{i}{12}$i12 | |
| $\left(\sqrt[12]{1.064}-1\right)\times12$(12√1.064−1)×12 | $=$= | $i$i | |
| $\therefore\ i$∴ i | $=$= | $0.0622$0.0622 to $3$3 significant figures |
|
Therefore, the nominal rate of interest is $6.22%$6.22% p.a. to two decimal places.
Practice questions
Question 1
James invested $\$3000$$3000 at $4.6%$4.6% p.a. compounded daily.
Find the amount of interest earned in a year. You may assume that there are $365$365 days in a year (ignoring leap years).
Write your answer to the nearest cent.
Find the effective annual interest rate as a percentage to two decimal places.
Question 2
An investment earns interest at a rate of $7.2%$7.2% compounding semiannually.
What is the effective rate correct to two decimal places?