Question 3 of 6: Depreciation

Question 3: Depreciation (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

(a) Two methods of computing depreciation — pros and cons

Straight-line (SL) method. The depreciable value is written off in equal annual amounts, $d=(V_0-V_s)/n$, so the book value falls linearly. Advantages: simple, transparent, easy to audit, and the constant charge smooths reported earnings. Disadvantages: it ignores the time value of money, and it poorly matches reality for assets that lose most of their worth early or become obsolete before their nominal life ends.

Declining-balance (DB) method. A fixed fraction $f$ of the remaining book value is charged each year, $d_k=f\,V_{k-1}$, giving large early charges that taper off — an accelerated method. Advantages: better matches the true early loss of value of most process equipment, and, by front-loading the deduction, defers income tax and improves early cash flow — the basis of the Canadian Capital Cost Allowance (CCA) system, with its prescribed class rates and "half-year rule". Disadvantages: more complex, never reaches zero on its own (a salvage floor or a switch to straight-line must be imposed), and the heavy early write-off depresses early book profit.

(Two further recognised methods are the accelerated sum-of-the-years-digits scheme and the sinking-fund method used in part (b), which is the slowest write-off because the accumulating fund earns interest.)

(b) Sinking-fund annual charge to match the straight-line book value

Given. Original cost $V_0=\$20{,}000$; sinking-fund interest rate $i=4\%=0.04$; the target is that the book value at $n=8$ years equal the value it would have under straight-line depreciation at $\$2000$/yr.

Find. The uniform annual sinking-fund depreciation charge (deposit) $R$.

Approach. First fix the target: the straight-line book value at year 8 sets how much depreciation must have accumulated by then. In the sinking-fund method that accumulated depreciation is the future worth of a uniform annual deposit $R$ earning 4 %, so equate the two and solve for $R$.

1. Target accumulated depreciation from the straight-line reference. Straight-line at $\$2000$/yr for 8 years gives book value $$V_{8,\text{SL}} = V_0 - 8(\$2000) = \$20{,}000 - \$16{,}000 = \$4000$$ so the depreciation that must have accumulated by year 8 is $D_8 = V_0-V_{8,\text{SL}} = \boxed{\$16{,}000}$.
2. Equate to the sinking-fund accumulation. In the sinking-fund method the fund built by a level year-end deposit $R$ over 8 years at $i=4\%$ is the uniform-series compound-amount, and it must equal $D_8$: $$R\,\frac{(1+i)^{n}-1}{i} = D_8 \qquad\Rightarrow\qquad R = D_8\,\frac{i}{(1+i)^{n}-1}$$
3. Evaluate. With $(1.04)^{8}=1.36857$, the series factor is $\dfrac{1.36857-1}{0.04}=9.2142$, so $$R = \$16{,}000\times\frac{0.04}{(1.04)^{8}-1} = \frac{\$16{,}000}{9.2142} = \boxed{\$1736\text{/yr}}$$ The pump's annual sinking-fund depreciation charge is about $\$1736$.