| Age in Periods | Periodic Depreciation Composed of: | Depreciated or Appraised Value | Total Accumulated Depreciation | ||
|---|---|---|---|---|---|
| Periodic Amount | Interest | Total | |||
| 0 | $ ..... | $ ..... | $ ..... | $150.00 | $ ..... |
| 1 | 18.10 | ..... | 18.10 | 131.90 | 18.10 |
| 2 | 18.10 | .90 | 19.00 | 112.90 | 37.10 |
| 3 | 18.10 | 1.85 | 19.95 | 92.95 | 57.05 |
| 4 | 18.10 | 2.85 | 20.95 | 72.00 | 78.00 |
| 5 | 18.10 | 3.90 | 22.00 | 50.00 | 100.00 |
| $100.00 | |||||
The following chart shows graphically the appraised values, the accumulating depreciation and the elements which compose it, the curve OD representing the fixed periodic amounts, and EF the theoretical interest accumulations. The curve OD is a straight line inasmuch as it represents fixed periodic amounts. The depreciation and interest curves, OC and EF, representing gradually increasing amounts, are both slightly concave and would become increasingly so the longer the period covered. The appraisal curve AB is slightly convex and its convexity is accelerated by lapse of time.
Graphic Chart—Sinking Fund Method
(b) Annuity Method
The “Annuity” method also makes use of the compound interest principle, but in addition to the method of the sinking fund it adds to the periodic depreciation charge as determined thereunder interest on the successive appraised values of the asset. The effect of this is to charge to the product, by way of Profit and Loss, interest on the capital invested in each asset used in manufacture. The appraised values of the asset are exactly the same as under the sinking fund method, but the expense charge to depreciation is larger under the annuity method by the interest on the appraised value of the asset. This charging of interest to the product under the title “depreciation” makes it necessary to capitalize the interest charge by adding it to the value of the asset. If the amount to be charged off, i.e., V-Vₙ is the same under both methods, for both to arrive at the same scrap value, Vₙ, the interest under the annuity method must be added to the value of the asset each time before deducting the depreciation charge, a part of which is this same interest. The annuity method thus makes a larger periodic charge than the sinking fund method.
The problem involved in the calculation of the periodic depreciation charge by the annuity method is sometimes stated as the method of finding a fixed or constantly equal periodic charge sufficient to charge off not only depreciation as such but also the interest which has been added to the value of the asset. The mathematical formula may be derived as follows, using the standard notation:
- VR = V(1 + r), or the asset with interest added to it
- VR - D₁ = V₁, appraised value at end of first period
- V₁R - D₁[31] = VR² - D₁R - D₁ = V₂,
- appraised value at end of second period
- V₂R - D₁[32] = VR³ - D₁R² - D₁R - D₁ = VR³ - D₁(R² + R + 1)
- = V₃, appraised value at end of third period, etc.
Generalizing, we have:
- Vₙ₋₁R - D₁ = VRⁿ - D₁(Rⁿ⁻¹ + Rⁿ⁻² ... + R² + R + 1) = Vₙ,
- scrap value. Whence