| (V - Vₙ)r | + | Vr(Rⁿ - 1) |
| Rⁿ - 1 | Rⁿ - 1 |
By reducing the second fraction, this may be written as:
| (5) | (V - Vₙ)r | + Vr |
| Rⁿ - 1 |
which is seen to be identical with formula (3) for the sinking fund except for the addition of Vr, which represents interest at r% on the investment V. The identity of the annuity formula (4) with the sinking fund formula (3) plus interest on investment can be established similarly for any of the periods.
It will be noted that by the annuity method the whole original value of the investment is always earning interest either in the sinking fund or in the diminishing appraised value. Thus, the portion of original value deducted each period earns interest in the sinking fund, while what is left as appraised value earns interest outside the fund. Thus, the annuity method of charging depreciation may be said to consist of two parts, viz., the fixed periodic amount and interest on the original investment. This will be seen from the appraisal schedule which follows. The same illustrative data are used as before, including interest at 5%.
| Age in Periods | Periodic Depreciation Charge Composed of: | ||||
|---|---|---|---|---|---|
| Fixed Amount | Interest on Fixed Amount | Interest on Investment | Total Charge | ||
| (a) | (b) | (c) | (d) | ||
| 0 | $ ..... | $ ..... | $ ..... | $ ..... | |
| 1 | 18.10 | ..... | 7.50 | 25.60 | |
| 2 | 18.10 | .90 | 6.60 | 25.60 | |
| 3 | 18.10 | 1.85 | 5.65 | 25.60 | |
| 4 | 18.10 | 2.85 | 4.65 | 25.60 | |
| 5 | 18.10 | 3.90 | 3.60 | 25.60 | |
| $90.50 | $9.50 | $28.00 | $128.00 | ||
| Age in Periods | Appraised Value Plus Interest | Depreciated or Appraised Value | Accumulated Depreciation | ||
|---|---|---|---|---|---|
| Including Interest | True Depreciation | ||||
| (e) | (f) | (g) | (h) | ||
| 0 | $ ..... | $150.00 | $ ..... | $ ..... | |
| 1 | 157.50 | 131.90 | 25.60 | 18.10 | |
| 2 | 138.50 | 112.90 | 51.20 | 37.10 | |
| 3 | 118.55 | 92.95 | 76.80 | 57.05 | |
| 4 | 97.60 | 72.00 | 102.40 | 78.00 | |
| 5 | 75.60 | 50.00 | 128.00 | 100.00 | |
It will be noted, as stated above, that the sum of the two interest items in columns (b) and (c) is the same for each period, viz., $7.50, interest on the original investment. The total of column (d), total charge for periodic depreciation, minus the total of column (c), interest on the diminishing appraised values, gives $100, the true depreciation as shown by column (h). True depreciation under the annuity method is the same as under the sinking fund method. The periodic depreciation charge differs, however.
A graphical illustration of the main items of the appraisal schedule is shown below:
