New Zealand
Level 6 - NCEA Level 1

# Compound Interest - Compounding Variations

Lesson

In Earning Interest on Interest, we were introduced to the compound interest, where new interest was calculated based on the amount of the principal and any previously earned interest. This was done using the formula:

Compound interest formula

$A=P\left(1+r\right)^n$A=P(1+r)n

However, the questions we have looked at already all compounded annually. How would we work out the total amount of an investment that was calculated weekly, monthly or semiannually (twice a year)?

Well, we still use the same formula but we just need to tweak a couple of things based on how the interest is calculated in each individual question. Basically, we need to find equivalent values so everything is expressed in the correct measurement unit. We need to change:

• the number of time periods, $n$n, e.g. if our investment compounded monthly for $3$3 years, our number of time periods would be $3\times12$3×12 or $36$36 months.
• the interest rate, $r$r, e.g. if our investment compounded weekly at a rate of $5.2%$5.2% p.a. for one year, our equivalent interest rate would be $5.2\div52$5.2÷​52 or $0.1%$0.1% per week.

Once everything is expressed in the correct units of measurement, we can use these values and apply the compound interest formula as normal. You'll notice that because the interest is being calculated more frequently, the total amount of the investment will increase faster than if interest is only calculated annually.

Alternatively, if you can easily work out the number of time periods, we can solve these kinds of questions using a slight variation of the compound interest formula:

Variation of the compound interest formula

$A=P\left(1+\frac{r}{k}\right)^{nk}$A=P(1+rk)nk

where:

$P$P is the principal amount

$r$r is the interest rate (this is usually expressed as an annual rate)

$n$n is the total duration of the investment (usually expressed in years)

$k$k is the number of times the interest accrues per time interval

### Outcomes

#### NA6-3

Apply everyday compounding rates

#### 91026

Apply numeric reasoning in solving problems