Model reducing-balance depreciation with a recurrence relation, compare it with flat-rate depreciation, and find book value and scrap-value timing.

How to model reducing-balance depreciation with a recurrence relation, contrast it with flat-rate depreciation, find an asset's book value after n years, and work out when it reaches a scrap value.

Generated by Claude Opus 4.76 min answerUpdated 2026-05-29

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What this dot point is asking
Two ways an asset loses value
Book value and scrap value
Choosing the right model

What this dot point is asking

You must set up the depreciation recurrence, distinguish it from flat-rate depreciation, find book values, and determine when the value drops to a given scrap value.

Two ways an asset loses value

The course models depreciation two ways. Flat-rate (straight-line) depreciation removes the same dollar amount each year (arithmetic). Reducing-balance depreciation removes the same percentage of the current value each year (geometric), so the dollar loss shrinks over time.

The multiplier $(1 - i)$ is the fraction of value retained: a $20\%$ rate keeps $80\%$ , so $V_{n+1} = 0.8\,V_n$ .

Book value and scrap value

The book value is the modelled value after $n$ years. The scrap value is a chosen low value at which the asset is retired; questions often ask when the book value first falls to or below it.

Worked example

Book value and scrap-value timing

A \ $30\,000 van depreciates by$ 25% $reducing balance each year. It keeps$ 75% $each year, so$ r = 0.75$.

Recurrence: $V_{n+1} = 0.75\,V_n$ , $V_0 = 30\,000$ .

Book value after $3$ years: $V_3 = 30\,000 \times 0.75^3 = 30\,000 \times 0.421875 = 12\,656.25$ , about $12,656.

When does it first fall below a $5000 scrap value? Test successive years:

$V_5 = 30\,000 \times 0.75^5 = 30\,000 \times 0.2373 = 7119$ .
$V_6 = 30\,000 \times 0.75^6 = 30\,000 \times 0.17798 = 5339$ .
$V_7 = 30\,000 \times 0.75^7 = 30\,000 \times 0.13348 = 4004$ .

The book value first falls below \ $5000 during the$ 7 $th year, so the van reaches scrap value after$ 7$ years.

Choosing the right model

Read whether depreciation is "per year" as a percentage (reducing balance) or as a fixed dollar amount (flat rate). Reducing balance gives a curved decline that never quite reaches zero; flat rate gives a straight-line decline that eventually would reach zero.