# 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 D_{m}, 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,

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