Align nominal rates, period rates and compounding counts practice questions | General Mathematics (Year 12, VCE Units 3 & 4)

Periodic compound models

A nominal annual rate is quoted for one year, but a periodic recurrence applies one rate at a time. Convert the rate and the time span to the same compounding period

r_p=\frac{r}{p},\quad R=1+\frac{r_p}{100},\quad n=p\times y,\quad V_n=V_0R^n

Match the rate to one period Divide the nominal annual rate by the number of compounding periods per year $\displaystyle r_p=\frac{12\%}{12}=1\%$
Form the multiplier Write one plus the period rate as a decimal so the existing balance is included $\displaystyle R=1+\frac{1}{100}=1.01$
Count compounding periods Multiply the number of years by the number of compounding periods in one year $\displaystyle n=12\times2=24$
Apply one multiplier per period Raise the period multiplier to the number of compounding periods $\displaystyle V_{24}=V_0\left(1.01\right)^{24}$

Worked example

An account starts with $1,500 and earns 6% p.a. compounded quarterly for 2 years. Find the balance after the 8 quarterly periods

V_0=\$1{,}500,\quad r=6\%,\quad p=4

$\displaystyle r_p=\frac{6\%}{4}=1.5\%,\quad R=1+\frac{1.5}{100}=1.015$Divide the nominal annual rate by four and form the quarterly multiplier
$\displaystyle n=4\times2=8$Two years of quarterly compounding gives eight compounding periods
$\displaystyle V_8=\$1{,}500\left(1.015\right)^8\approx\$1{,}689.74$Apply the quarterly multiplier eight times and round the final balance to the nearest cent

Common mistakes

Using the annual rate for each period

✕$\displaystyle V_{24}=\$1{,}000\left(1.12\right)^{24}$✓$\displaystyle V_{24}=\$1{,}000\left(1.01\right)^{24}$

A nominal annual rate must be divided by the number of compounding periods per year before it is used for one period

Using years as the exponent

✕$\displaystyle n=2$✓$\displaystyle n=12\times2=24$

The exponent counts monthly compounding periods, not calendar years

Replacing a power with a product

✕$\displaystyle V_8=V_0\left(8\times1.015\right)$✓$\displaystyle V_8=V_0\left(1.015\right)^8$

Repeated multiplication by the same multiplier is represented by a power, not by multiplying the multiplier by the period count

Difficulty levels

introConvert an annual rate to a quarterly or monthly period rate

standardConstruct a periodic recurrence and explicit rule for a shorter compounding period

challengeConvert in either direction or calculate a multi-year value after converting both rate and period count

examSelect the recurrence with a consistent nominal rate, multiplier and compounding-period index

Questions about this topic

Why is the nominal annual rate divided by the compounding frequency?+

The nominal rate is stated for one year. Dividing by the number of periods in that year gives the rate used in one compounding transition

Why is the exponent the number of compounding periods?+

The multiplier is applied once at each compounding period. The exponent therefore counts those applications rather than the number of calendar years

What changes between annual and monthly compounding?+

Monthly compounding uses a smaller rate in each period and applies the multiplier more times. The rate and exponent must both be changed to match the monthly 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 12, VCE Units 3 & 4, General Mathematics, chapter 7: Recursion and financial models

In this chapter

Iterate a first-order recurrence Link recurrence, explicit, tabular and graphical forms Model and analyse fixed-change finance Compare financial model assumptions and outcomes Model and analyse percentage-change finance Solve an inverse geometric finance model Align nominal rates, period rates and compounding counts Calculate and compare effective annual rates