PV of what comes in, minus what goes out today

Putting the three lessons together

Lesson 13.1 gave Lakehead’s boat lift as cash by year: −940,000 at t0, 270,000 in each of years 1–4, and 360,000 in year 5. Lessons 13.2–13.3 gave the factors. Lesson 13.4 gave the rate — Lakehead’s WACC is 10 %. Net present value is the present value of everything that comes in, less what goes out today:

$$\text{NPV}=\sum _{t=1}^n\frac{C{F}_t}{(1+r{)}^t}-I_0$$

Years 1–4 are an annuity; year 5 is a single sum:

Years 1–4   270,000 × 3.1699 (annuity, 4 yr, 10 %)   =    855,873
Year 5      360,000 × 0.6209 (single sum, 5 yr, 10 %) =    223,524
PV of inflows                                          1,079,397
Less: investment at t0 (undiscounted)                   (940,000)
NPV                                                     +139,397

The decision rule: NPV ≥ 0 → accept; NPV < 0 → reject. At 10 % the lift returns the 940,000, pays every provider of capital its required return, and creates 139,397 of value on top. Accept.

Two details carry marks. The t0 investment is not discounted — it is today’s dollars already; multiplying it by the one-year factor (giving 854,554 and an NPV of 224,843) is the most common error on the paper. And the factors come from the table the prompt gives — the first code block below rebuilds the schedule year by year with the single-sum factors and reaches 139,370, twenty-seven dollars short, because the annuity factor 3.1699 and the four single-sum factors it was rounded from do not agree to the last dollar. That is lesson 13.2’s rounding, not an error; the worksheet is graded on the factors the prompt supplies.

Payback: quick, and blind twice

The payback period asks how long the investment takes to come back in undiscounted cash:

$$\text{Payback}=\frac{940,000}{270,000}=3.48\text{ years}$$

It is easy, everyone understands it, and it says something about liquidity and risk — a project that pays back in a year is exposed to less that can go wrong. But it is blind twice: it ignores the time value of money (a dollar in year 3 counts the same as a dollar in year 1), and it ignores everything after payback (the 360,000 in year 5 — the salvage, the working capital, the best year — never enters the number). A project can pay back in two years and destroy value; another can pay back in six and be the best investment the company ever makes. Payback is a secondary check, never the decision.

The profitability index: NPV per dollar invested

NPV is an absolute amount, so it cannot compare a 940,000 project with a 200,000 one under capital rationing. The profitability index scales it:

$$\text{PI}=\frac{\text{PV of inflows}}{\text{Investment}}=\frac{1,079,397}{940,000}=1.148$$

Every dollar invested returns 1.148 in present value. PI > 1 says the same thing as NPV > 0; its use is ranking positive-NPV projects when the cash cannot fund all of them — highest PI first.

The IRR, at awareness level

The internal rate of return is the rate at which NPV is exactly zero — the project’s own yield. At 14 % (annuity factor 2.9137, single-sum 0.5194) the lift’s NPV is still +33,683, so its IRR is above 14 %; and since 14 % is well above the 10 % WACC, the project clears its hurdle with room. At 12 % (3.0373 and 0.5674) the NPV is +84,335. Finding the exact IRR is a trial-and-error or spreadsheet job (=IRR(range)); the course asks only that you know what it is and that NPV ≥ 0 at WACC and IRR ≥ WACC are the same test.