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Model and analyse percentage-change finance

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An asset loses 7% of its current value each year. Which multiplier should be used?

Year 12, VCE Units 3 & 4, Chapter 7, 4 difficulty levels, Calculator allowedBuild a worksheet

Geometric financial models

A geometric financial model applies the same percentage change to the current value in every period. The multiplier is greater than 1 for growth and between 0 and 1 for reducing-balance depreciation

\[V_{n+1}=RV_n,\quad V_n=V_0R^n\]
  1. Identify the direction of change Compound interest increases the current value, while reducing-balance depreciation decreases the current value \(\displaystyle \text{growth}:R>1,\quad\text{decay}:0<R<1\)
  2. Convert the percentage to a multiplier Add the rate proportion for growth and subtract it for reducing-balance depreciation \(\displaystyle R=1+\frac{6}{100}=1.06,\quad R=1-\frac{7}{100}=0.93\)
  3. Apply the multiplier to the current value Each recurrence transition multiplies the value from the previous period by R \(\displaystyle V_{n+1}=RV_n\)
  4. Use the zero-indexed explicit rule The starting value has index zero, so the exponent counts the transitions \(\displaystyle V_0=7500,\quad V_4=7500(1.06)^4\)

Worked example

An investment starts at $7500 and earns compound interest of 6% per year. Find its value after four years, to the nearest cent

\[V_0=7500,\quad r=6\%\]
  1. \(\displaystyle R=1+\frac{6}{100}=1.06\)Compound interest increases the current value, so the rate proportion is added to 1
  2. \(\displaystyle V_{n+1}=1.06V_n\)Each year the previous balance is multiplied by 1.06
  3. \(\displaystyle V_4=7500(1.06)^4=9468.5772\)Four annual transitions give the exact balance before money rounding
  4. \(\displaystyle V_4\approx9468.58\)The investment value is $9,468.58 to the nearest cent

Common mistakes

Adding the depreciation rate
✕\(\displaystyle R=1+\frac{7}{100}=1.07\)✓\(\displaystyle R=1-\frac{7}{100}=0.93\)
Reducing-balance depreciation removes a percentage of the current value, so the multiplier is one minus the rate proportion
Multiplying by the index instead of using a power
✕\(\displaystyle V_n=V_0(nR)\)✓\(\displaystyle V_n=V_0R^n\)
A percentage change is applied repeatedly, so the multiplier is raised to the number of transitions
Using one fewer transition
✕\(\displaystyle V_n=V_0R^{n-1}\)✓\(\displaystyle V_n=V_0R^n\)
V_0 is the starting value, so the index n counts exactly n applications of the multiplier

Difficulty levels

introConvert a percentage rate to a multiplier and complete a geometric recurrence
standardCalculate a future value or total change from a geometric financial model
challengeFind a single-period change or the first strict threshold period
examRecognise the recurrence and explicit rule, calculate a value and interpret its meaning

Questions about this topic

Why is a percentage-change model geometric?+

Each period's change is calculated from the current value, so the same multiplier is applied repeatedly

Why is the depreciation multiplier less than 1?+

The asset retains the part left after the depreciation percentage is removed, so its multiplier is 1 minus the rate proportion

What does the indexed value represent?+

The indexed value is the value after that number of transitions from the starting value. It is the balance or asset value, rather than the change during the periods

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 recurrenceLink recurrence, explicit, tabular and graphical formsModel and analyse fixed-change financeCompare financial model assumptions and outcomesModel and analyse percentage-change financeSolve an inverse geometric finance modelAlign nominal rates, period rates and compounding countsCalculate and compare effective annual rates