(d) Arbitrary with Decreasing Amounts
(a) Fixed Percentage of Diminishing Value Method
The “Fixed Percentage of Diminishing Value” method estimates the periodic depreciation as a fixed percentage of the appraised or book value of the asset as at the time of the last appraisal. Thus, if the asset cost $1,000 and the fixed rate is 10%, the first depreciation estimate is $100 (10% of $1,000) giving an appraised value of $900; the second depreciation estimate is $90 (10% of $900), with a new appraised value of $810; the third estimate is $81 (10% of $810), with an appraisal of $729 for the asset; and so on. It is evident that a final zero valuation can never be reached (although it may be approximated) as the series becomes an indefinite or indeterminate series. If there is any scrap value, and there usually is, the series becomes determinate. From the standpoint of calculation the problem here is the determination of the fixed rate necessary to reduce the asset value to residual or scrap value in the given life-term. Using the standard notation, we may formulate the following equations:
| V₁ = | V(1 - d); | V₂ = | V₁(1 - d) = | V(1 - d)(1 - d); |
| V₃ = | V₂(1 - d) = | V(1 - d)(1 - d)(1 - d); whence |
| Vₙ = | V(1 - d)ⁿ, which solved for 1 - d gives |
| 1 - d = | ⁿ√ | Vₙ/V, | and, solving for d, we get |
While complex, the formula is readily solvable by means of logarithms. For an asset costing $150 with a service life of 5 years and a scrap value of $50, the rate is found by the above formula to be approximately 19.726%.
| d = | 1 - ⁵√ | 50/150, | = .19726 |