Question 5 of 6: Depreciation

Question 5: 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.

(i) Three depreciation methods, with advantages and disadvantages

1. Straight-line method. The depreciable value (cost minus salvage) is written off in equal annual amounts, $d=(V-V_s)/n$. Advantage: simplest to compute and understand, and it spreads the cost evenly. Disadvantage: it ignores the time value of money and does not reflect that most equipment loses value fastest when new; the constant charge is unrealistic.

2. Declining-balance method (e.g. double declining balance). A fixed percentage is applied each year to the remaining book value, giving large early charges that taper off. Advantage: it is an accelerated method that matches the real pattern of rapid early value loss and front-loads the tax shield, improving the present value of after-tax cash flow (this is the basis of the Canadian Capital Cost Allowance system). Disadvantage: it never quite reaches a chosen salvage value without adjustment, and the arithmetic is less transparent than straight-line.

3. Sum-of-the-years-digits method. Also accelerated: the annual charge is the depreciable value times a declining fraction whose denominator is the sum $1+2+\dots+n$ and whose numerator counts the remaining years. Advantage: accelerated like declining balance but it writes the asset down exactly to salvage in $n$ years. Disadvantage: more arithmetic than straight-line and, like all book methods, the schedule is somewhat arbitrary.

A fourth method — the sinking-fund method — is the one the calculation in part (ii) uses: a uniform amount is deposited each year into a fund earning interest, so that the fund plus its interest exactly equals the depreciable value at end of life. It explicitly recognises the time value of money (its distinguishing advantage) but gives very small early charges, which is why tax authorities do not favour it.

(ii) Asset value at the end of the tenth year (sinking-fund method)

Given.

Find. The book (asset) value of the evaporator at the end of the tenth year.

Approach. Find the uniform annual deposit that grows to $(V-V_s)$ over 12 years at 8%, accumulate it for 10 years to get the depreciation charged to date, and subtract that from the original cost.

1. Annual sinking-fund deposit. The deposit $R$ must accumulate (with interest) to the depreciable value $V-V_s$ over the full life: $$R=(V-V_s)\,\frac{i}{(1+i)^{n}-1}=\$300{,}000\cdot\frac{0.08}{(1.08)^{12}-1}$$ With $(1.08)^{12}=2.518170$, $\;R=\dfrac{\$300{,}000(0.08)}{1.518170}=\$15{,}808.51\text{/yr}.$
2. Accumulated depreciation to the end of year 10. The fund (deposits plus their compound interest) after 10 years is the depreciation charged so far: $$D_{10}=R\,\frac{(1+i)^{a}-1}{i}=\$15{,}808.51\cdot\frac{(1.08)^{10}-1}{0.08}$$ With $(1.08)^{10}=2.158925$, $\;D_{10}=\$15{,}808.51(14.48656)=\$229{,}010.9.$
3. Asset (book) value at end of year 10. Subtract the accumulated depreciation from the original cost: $$V_{10}=V-D_{10}=\$320{,}000-\$229{,}010.9=\boxed{\$90{,}989}$$ Equivalently, in closed form $V_{10}=V-(V-V_s)\dfrac{(1.08)^{10}-1}{(1.08)^{12}-1}=\$320{,}000-\$300{,}000(0.763370)=\$90{,}989.$ The evaporator's book value at the end of the tenth year is about $90,990.