Sum-of-years Digits

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Sum-of-years Digits

An asset often loses more of its value early in its lifetime. A method that exhibits this dynamic is desirable.

Assume an asset depreciates from price P to salvage value S in N years. First compute the value: sum-of-years = 1+2+ ... +N. The depreciation for the years after the asset's purchase is:

Table 11.1: Sum-of-years General Example

year number	annual depreciation
first	${N\over{\rm sum-of-years}}$ (P-S)
second	${{N-1}\over{\rm sum-of-years}}$ (P-S)
third	${{N-2}\over{\rm sum-of-years}}$ (P-S)
$\vdots$	$\vdots$
final	${1\over{\rm sum-of-years}}$ (P-S)

For the ith year of the asset's use this equation generalizes to

Annual Depreciation = ${{N+1-i}\over{\rm sum-of-years}}$ (P-S)

For our example, N=5 and the sum of years is 1+2+3+4+5=15. The depreciation during the first year is

($20,000-$5,000)(5/15) = $5,000

Table 11.2 describes how Declining Balance would depreciate the asset.

Table 11.2: Sum-of-years Example

Year	Depreciation	Year-end Value
1	($20,000-$5,000)(5/15) = $5,000	$15,000.00
2	($20,000-$5,000)(4/15) = $4,000	$11,000.00
3	($20,000-$5,000)(3/15) = $3,000	$8,000.00
4	($20,000-$5,000)(2/15) = $2,000	$6,000.00
5	($20,000-$5,000)(1/15) = $1,000	$5,000.00

And as expected, the value after N years is S.

Value after 5 years	=	P - (5 years' depreciation)
	=	P - $[t] l (\frac{5}{10}(P-S) + \frac{4}{10}(P-S) + \frac{3}{10}(P-S) +.\ .\frac{2}{10}(P-S) + \frac{1}{10}(P-S) )$
	=	P - (P-S)
	=	S

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