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:
Total depreciation at any time m
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,
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,
$D_m = \dfrac{FC - SV}{SYD} \times \dfrac{m(2n - m + 1)}{2}$