ENGR 4760: Engineering Economics · Study Notes · Topic 3 · Park 2.6–2.7
Before this topic, go back and review every formula from Topic 2. Real-world cash flows are rarely a clean single annuity or a single gradient. They are composites: a stretch of \$200 payments, then a few years at \$400, then a one-time \$600, perhaps with a gradient buried inside. The whole skill here is recognizing the familiar patterns inside the mess and breaking the problem back down into the formulas you already know.
Two reminders that trip people up, both carried over from Topic 2. A linear gradient's effect starts at the end of year two, so it reads as $0G, 1G, 2G, \ldots$ A geometric series runs $A_1, A_1(1+g), A_1(1+g)^2, \ldots$ Keep those alignments straight and the decomposition falls into place.
Brute force. Move every single cash flow to the reference point on its own. Each payment gets discounted with a single-payment factor: the year-1 amount back one period, the year-2 amount back two, and so on. With a calculator beside you, it is just a long column of $(P/F)$ conversions that you sum at the end. It takes longer, but it is the approach least likely to confuse you, and in this course time is rarely the constraint.
Grouping. Spot the sub-patterns and convert each group with one annuity factor instead of payment-by-payment. Say years 1–3 hold \$200 each. Use $(P/A, i, 3)$ on that group, but remember the $(P/A)$ habit: it lands the value one period before the first payment, so here it gives the worth at time 0 directly. A group sitting later in the timeline (say \$400 across years 5–7) converts with $(P/A)$ to a value at the start of that block, and then you slide that single number back to time 0 with one $(P/F)$ step. A lone payment, like a \$600 at year 8, is just one $(P/F, i, 8)$. Convert each group, sum the handful of results, done.
The two methods give the same answer. Brute force trades effort for safety; grouping trades a little setup thinking for far fewer calculations. Pick whichever you trust more on the day.
A common variant gives you the deposits but asks for the withdrawals (or vice versa). The unlock is always the same: bring both sides to the same point in time and set them equal. Until they sit at one common date you are comparing apples with oranges; once they do, a single equation pops out and you solve for the unknown.
Example. You make 4 annual deposits of \$2,000 during college at 5%, then withdraw an equal amount $A$ for 4 years afterward. What is $A$? Convert the deposits to a value at the end of year 4 with $(F/A, 5\%, 4)$; convert the withdrawals back to that same date with $(P/A, 5\%, 4)$; set them equal and solve. You get about \$2,431 per year. (You could instead bring both streams to time 0 with present-worth factors. Same answer, your choice of meeting point.)
Sometimes a series is regular except for one gap: payments every year, but nothing in year 9. The clean move is to add the missing payment and subtract it at the same instant. Adding and removing equal amounts at the same point in time changes nothing about the cash flow, but it restores a complete pattern you can hit with one annuity factor, plus a single correction term.
Example. Outflows of \$500 every year for 12 years at 6%, except year 9 is skipped. Pretend the \$500 is there: $P = 500\,(P/A, 6\%, 12)$ treats it as a full 12-year annuity. Then subtract the phantom year-9 payment, discounted on its own: $-\,500\,(P/F, 6\%, 9)$. The two terms together give the true present worth. The same trick handles a missing gradient step: insert the value that completes the gradient, then subtract it back.
What if payments happen every other year? Rather than fight the gaps, change the clock. Moving across two ordinary periods multiplies value by $(1+i)^2$, so define an equivalent rate over the longer step:
$$i_{eff} = (1+i)^2 - 1$$
At 5% annual, $i_{eff} = (1.05)^2 - 1 = 0.1025$, or 10.25% per two-year period. Now the gaps vanish: with the new rate, the every-other-year payments are just a plain annuity. This idea generalizes to any regular skip, and it is the same reasoning behind converting a nominal annual rate into a monthly one, only run in the other direction.
Example. A \$20,000 loan repaid every other year at 5% annual. Re-express at 10.25% per two-year period and apply $(A/P, 10.25\%, N)$ over the number of actual payments. Treating the biennial schedule as a clean annuity under the effective rate gives a payment of roughly \$4,625.
Every factor in this course has a spreadsheet twin, and once a problem gets long, the spreadsheet is faster and less error-prone than table lookups. The core five:
| You want | Function | Replaces |
|---|---|---|
| Present worth | PV(rate, nper, pmt, [fv]) | $(P/A)$ and $(P/F)$ |
| Future worth | FV(rate, nper, pmt, [pv]) | $(F/A)$ and $(F/P)$ |
| Periodic payment | PMT(rate, nper, pv, [fv]) | $(A/P)$ and $(A/F)$ |
| Interest rate | RATE(nper, pmt, pv, [fv]) | solving for $i$ |
| Number of periods | NPER(rate, pmt, pv, [fv]) | solving for $N$ |
You never have to use these unless a problem asks, but they are worth knowing. Watch the sign convention: money out is negative, money in is positive, the same up-arrow / down-arrow rule from your cash flow diagrams, now enforced by the software.
Investing. Real portfolios are composite cash flows: a bond ladder pays staggered coupons, a rental throws off rising rent against a lumpy roof replacement, a startup stake is years of nothing then one exit. Valuing any of them is exactly this decomposition: group the regular streams, treat the one-offs as single payments, discount everything to today, and add. The brute-force discounted-cash-flow spreadsheet a lot of analysts live in is precisely the $(P/F)$-on-every-row method, just automated.
Engineering. A project's life is a composite: upfront capital, a maintenance gradient, periodic overhauls every few years, a salvage value at the end. The effective-rate trick is the clean way to fold in overhauls that recur on their own cycle, and the missing-payment trick handles a year the line is down for retooling. Reduce the whole messy timeline to one present worth and competing designs become directly comparable.