# Depreciations

This chapter discusses the functions available in AIMMS for the depreciation of an asset. Depreciation can be performed in many ways, for example by a fixed amount in every period, or by depreciation amounts that decrease over time. An asset is characterized by its purchase (or initial) cost $$c$$ and its salvage value $$s$$ (the value at the end of the useful life of the asset).

## Useful Life

The accounting periods for depreciating the asset have a length of one year, but do not necessarily have to start at January 1. The useful life of the asset is either given as a fixed amount of $$L$$ years, or is computed dynamically on the basis of the characteristics of the depreciation. The first period is the period from the purchase date until the beginning of the next regular accounting period. If the purchase date does not coincide with the beginning of an accounting period, the depreciations take place in $$L+1$$ accounting periods.

## General Equations

The following system of equations are true for all types of depreciations supported by AIMMS, where $$d_i$$ is the actual depreciation in period $$i$$, $$\tilde{d_i}$$ is the generic depreciation computed in a method-dependent manner, and $$v_i$$ the value of the asset at the beginning of period $$i$$.

\begin{split}\begin{aligned} d_i &= \max(0, \min(\tilde{d_i}, v_i - s)) \\ v_i &= c - \sum_{j=1}^{i-1}d_j\end{aligned}\end{split}

The equations express that generic method-dependent depreciation method will be adapted to yield the actual depreciation value to make sure that the value of an asset $$v_i$$ can never drop below its salvage value $$s$$.

## Method-Dependent Equations

For each depreciation method available in AIMMS, the equations used to compute the generic method-dependent depreciation amount $$\tilde{d_i}$$ will be listed in the description of the depreciation function. In most occasions these equations use the fraction $$f_{PN}$$, which expresses the year fraction from the purchase date until the beginning of the next regular accounting period. Its value depends on the selected day-count basis method.

AIMMS supports the following linear depreciation by constant amounts functions:

AIMMS supports the following non-linear depreciation by linear declining amounts functions:

AIMMS supports the following non-linear depreciation by non-linear declining amounts functions: