Derivation of Formula for Sum of Years Digit Method (SYD)

The depreciation charge and the total depreciation at any time m using the sum-of-the-years-digit method is given by the following formulas:
 

Depreciation Charge:

$d_m = (FC - SV) \dfrac{n - m + 1}{SYD}$

 

Total depreciation at any time m

$D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}$

 

Where:
FC = first cost
SV = salvage value
n = economic life (in years)
m = any time before n (in years)
SYD = sum of years digit = 1 + 2 + ... + n = n(1 + n)/2
 

Example of Depreciation Schedule using SYD
First cost, FC = \$500 000
Salvage value, SV = \$100 000
Economic life, n = 6 years
SYD = 1 + 2 + 3 + 4 + 5 + 6 = 21
Total depreciation, D = FC - SV = \$400 000
 

Depreciation schedule of the above given data using SYD:

Year Wearing Value Annual Depreciation, d Total Depreciation, D
1 6/21 (6/21)(400 000) = 114 285.71 114 285.71
2 5/21 (5/21)(400 000) = 95 238.10 114 285.71 + 95 238.10 = 209 523.81
3 4/21 (4/21)(400 000) = 76 190.48 209 523.81 + 76 190.48 = 285 714.29
4 3/21 (3/21)(400 000) = 57 142.86 285 714.29 + 57 142.86 = 342 857.14
5 2/21 (2/21)(400 000) = 38 095.24 342 857.14 + 38 095.24 = 380 952.38
6 1/21 (1/21)(400 000) = 19 047.62 380 952.38 + 19 047.62 = 400 000

 

Derivation of formulas
For economic life equal to n, the depreciation schedule can be tabulated as follows:

Year Annual depreciation, d
1 $\dfrac{n}{SYD} (FC - SV)$
2 $\dfrac{n - 1}{SYD} (FC - SV)$
3 $\dfrac{n - 2}{SYD} (FC - SV)$
4 $\dfrac{n - 3}{SYD} (FC - SV)$
... ...
m $\dfrac{n - (m - 1)}{SYD} (FC - SV)$
... ...
n - 1 $\dfrac{2}{SYD} (FC - SV)$
n $\dfrac{1}{SYD} (FC - SV)$

 

From the table above, the depreciation charge at any time m is   $\dfrac{n - (m - 1)}{SYD} (FC - SV)$.   Thus,
 

$d_m = (FC - SV) \dfrac{n - m + 1}{SYD}$

 

For the total depreciation Dm, take sum
$D_m = d_1 + d_2 + d_3 + \cdots + d_m$

$D_m = \dfrac{n}{SYD} (FC - SV) + \dfrac{n - 1}{SYD} (FC - SV) + \dfrac{n - 2}{SYD} (FC - SV) + \cdots + \dfrac{n - (m - 1)}{SYD} (FC - SV)$

$D_m = \dfrac{FC - SV}{SYD} \left\{ n + (n - 1) + (n - 2) + \cdots + [n - (m - 1)] \right\}$
 

The quantity   $n + (n - 1) + (n - 2) + \cdots + [n - (m - 1)]$   is a sum of Arithmetic Progression with common difference equal to -1 and number of terms equal to m.
 

Thus,
$\,n + (n - 1) + (n - 2) + \cdots + [n - (m - 1)] = \dfrac{m}{2}\left\{ \, n + [\,n - (m - 1)\,] \, \right\}$

$\,n + (n - 1) + (n - 2) + \cdots + [n - (m - 1)] = \dfrac{m}{2}\left\{ \, 2n - m + 1\, \right\}$

$\,n + (n - 1) + (n - 2) + \cdots + [n - (m - 1)] = \dfrac{m(2n - m + 1)}{2}$
 

Therefore,
$D_m = \dfrac{FC - SV}{SYD} \times \dfrac{m(2n - m + 1)}{2}$

$D_m = (FC - SV) \dfrac{m(2n - m + 1)}{2 \times SYD}$

 

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Derivation of Formula
Engineering Economy
depreciation
depreciation charge