The annuity factor is a sum; uneven flows are a sum you do by hand

Equal flows: the shortcut

Lakehead’s boat lift pays 270,000 at the end of each of five years. Lesson 13.2 could value that as five single sums — 270,000 × 0.9091, 270,000 × 0.8264, and so on — and add them. An ordinary annuity is exactly that: an equal amount at the end of each period for a fixed number of periods. Because the amount is the same every year it can be factored out, and what remains is the sum of the single-sum factors:

$${\text{Annuity factor}}_{r,n}=\sum _{t=1}^n\frac{1}{(1+r{)}^t}$$

At 10 % for four years: $0.9091+0.8264+0.7513+0.6830=3.1698$, which the annuity table prints as 3.1699 — the table sums the unrounded factors before it rounds, so the last digit can differ by one. The exam’s table gives the annuity factor directly; the point of building one by hand is knowing what it is, so that you never apply it to flows that are not equal.

For five years at 10 % the factor is 3.7908, and the five inflows are worth

$$270,000\times 3.7908=1,023,516$$

today. (Lesson 13.5 will value the first four years as an annuity and year 5 — which carries the salvage and the working capital — as a single sum, because year 5 is not 270,000.)

An annuity due pays at the start of each period; its factor is one period less discounted. The course names it so you recognise it in a lease question, but every worksheet here is an ordinary annuity — end of period — unless the prompt says otherwise.

Unequal flows: no shortcut

Cobalt Ridge Mining is weighing a three-year contract to process ore for a neighbour: 150,000 in year 1, 200,000 in year 2 and 260,000 in year 3 as the neighbour’s volume ramps up. The flows are unequal, so there is nothing to factor out. Each is a single sum at its own year, discounted at Cobalt Ridge’s 12 %:

Year   Cash flow   Factor at 12 %   Present value
 1      150,000        0.8929           133,935
 2      200,000        0.7972           159,440
 3      260,000        0.7118           185,068
                                        -------
Total PV of the contract                478,443

Applying the three-year 12 % annuity factor (2.4018) to any one of the flows would be wrong: to the average flow it gives 488,366, overstating the contract by nearly 10,000, because the annuity factor weights the early and late years equally while the real contract is back-loaded into the most heavily discounted year.

Which to use

Equal flows, end of period, fixed count → the annuity factor, one multiplication. Anything else → a single-sum factor per year, then add. A mixed stream (Lakehead’s four equal years and one different year) is split: annuity for the equal run, single sum for the rest. Excel’s =NPV(rate, range) does the uneven sum in one cell; =PV(rate, n, -payment) does the annuity.