Write a recurrence relation for a loan or investment with a given compounding period practice questions | General Mathematics (Year 11, VCE Units 1 & 2)

Writing a compounding recurrence

For a stated nominal annual rate convertible at k equal compounding periods per year, divide the annual percentage by k. An effective annual rate cannot simply be divided by k. The multiplier and index must use the same period

r_p=\frac{r_a}{k},\quad R=1+\frac{r_p}{100},\quad V_{n+1}=R\times V_n

Count periods in a year For a stated nominal annual rate, count the equal compounding periods in one year $\displaystyle k=4\ \text{quarters per year}$
Convert the annual rate Divide the stated nominal annual percentage rate by k to obtain the rate for one compounding period. This rule does not apply to an effective annual rate $\displaystyle r_q=6\div4=1.5\%$
Form the multiplier Add 1 to the period rate written as a decimal $\displaystyle R=1+\frac{1.5}{100}=1.015$
Define the index The index counts compounding periods, so its unit changes when the frequency changes $\displaystyle n=2\times4=8\ \text{quarters}$

Worked example

An account holds $6,000 and earns nominal 4.8% p.a., compounded monthly. Write the recurrence relation for its balance

V_0=\$6{,}000,\quad r_a=4.8\%,\quad k=12

$\displaystyle k=12\ \text{months per year}$Monthly compounding has twelve periods in one year
$\displaystyle r_m=4.8\div12=0.4\%$Divide the stated nominal annual rate by twelve to find the rate for one month
$\displaystyle R=1+\frac{0.4}{100}=1.004$Add the monthly rate as a decimal to 1 to form the multiplier
$\displaystyle V_0=\$6{,}000,\quad V_{n+1}=1.004\times V_n,\quad n\ \text{counts months}$Use the starting balance and state that each index step represents one month

Common mistakes

Using the annual rate for one period

✕$\displaystyle V_{n+1}=1.048\times V_n$✓$\displaystyle V_{n+1}=1.004\times V_n$

The example uses a nominal annual rate compounded monthly, so divide it by twelve before forming the monthly multiplier

Leaving the index in years

✕$\displaystyle V_0=\$6{,}000,\quad V_{n+1}=1.004\times V_n,\quad n\ \text{counts years}$✓$\displaystyle V_0=\$6{,}000,\quad V_{n+1}=1.004\times V_n,\quad n\ \text{counts months}$

The monthly multiplier must be applied once for each month, so the recurrence index must count months as well

Missing the 1 in the multiplier

✕$\displaystyle R=\frac{0.4}{100}=0.004$✓$\displaystyle R=1+\frac{0.4}{100}=1.004$

The multiplier includes the original balance as 1 plus the growth rate for one period

Difficulty levels

introConvert an annual rate for yearly and one sub-annual period, then find each multiplier

standardWrite the rate and multiplier components for quarterly and monthly compounding under one annual rate

challengeChoose a relation for fortnightly, weekly or daily compounding and identify what its index counts

examChoose the correct relation or multiplier after checking rate conversion, index units and the added 1.

Questions about this topic

When is an annual rate divided for monthly or quarterly compounding?+

Divide a stated nominal annual rate by the number of equal compounding periods per year. An effective annual rate describes the full-year change and cannot simply be divided. These examples use nominal annual rates

What does n count in a compounding recurrence?+

n counts compounding periods. For monthly compounding, n counts months; for quarterly compounding, n counts quarters

Why is 1 added to the period rate?+

The 1 represents the whole balance already present, and the rate fraction represents the extra part added during the period

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 3: Sequences and finance

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