IGNITE EXTENDED / LESSON 21
How can cash received at different times be compared?
A cash-flow comparison with an NPV decision note.
A dollar received later cannot be compared directly with a dollar paid today when capital has an opportunity cost. Present value translates a future cash flow into an equivalent amount at the valuation date using a rate appropriate to the timing and risk.
The rate and period must agree. An annual rate applied to monthly cash flows requires a consistent monthly convention. Cash flows must also use consistent assumptions about inflation and currency. A precise formula does not correct inconsistent inputs.
Net present value adds the present values of the cash consequences of the decision, including the initial outlay. A positive NPV means the forecast exceeds the return represented by the discount rate under those assumptions. It does not establish that the forecast will occur.
Put the cash flows on a timeline before calculating. The timeline exposes a missed initial payment, a terminal cash flow counted in the wrong year or a comparison between different horizons. Explain the business assumptions behind the numbers alongside the result.
Illustrative investment: pay $1,000 now; receive $600 at each of the next two year ends. Discount rate: 10% per year.
$600 in year 1 has PV $545.45 at 10%.
$600/1.10 + $600/1.10² − $1,000 = $41.32.
An ordinary annuity contains equal payments at the ends of a finite number of periods. A perpetuity continues indefinitely. Their formulas summarize a sequence of discounted payments. Identify the first payment date before using either formula.
A growing perpetuity assumes the first future payment grows at a constant rate indefinitely, with the discount rate greater than the growth rate. Value becomes highly sensitive when the two rates approach one another. Long-run growth assumptions require a business explanation and a sensitivity analysis.
An annual percentage rate can describe a quoted rate with compounding within the year. The effective annual rate includes that compounding. Comparing products or projects requires a common rate convention.
If the quoted annual rate is 12% with monthly compounding, each month uses 1%. Over twelve months, one unit becomes 1.01¹². The effective increase is approximately 12.68%, not 12%. The chapter’s installment and mortgage examples apply the same principle to repeated payments.
At 10%, three annual payments of $100 have PV $248.69; a $100 annual perpetuity has PV $1,000.
$100/(10% − 3%) = $1,428.57. Small changes in r − g can have a large effect.
A 12% nominal rate compounded monthly gives (1.01)^12 − 1 ≈ 12.68% effective per year.
Selected reading: PDF pages 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Cash arriving at different times becomes comparable when each flow is valued at a common date using an appropriate opportunity cost of capital. Net present value compares those benefits with the required investment.
Future value moves money forward through time; present value reverses that process. The rate and number of periods must describe the same interval.
Place each receipt and payment at its date before calculating. Discount each flow separately, then add values measured at the common date.
A positive NPV indicates value above the investment under the stated cash flows and discount rate. The conclusion changes when those inputs change.
Annuity, perpetuity and growth formulas summarize specific cash-flow patterns. Beginning-versus-end timing and compounding frequency remain consequential.
Selected reading: PDF pages 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
The discount rate represents an opportunity with comparable risk. A rate chosen only because it produces the desired answer leaves the valuation unexplained.
Match currency, period and nominal or real assumptions. A quoted annual rate may differ from an effective annual rate when compounding occurs more often.
Distant or indefinitely growing flows can be sensitive to the rate and growth convention. Show that dependence rather than presenting the calculated value as an observed fact.
The illustrative two-year investment has NPV $41.32 at a 10% rate. That small positive result depends on receiving both $600 payments when expected. The calculation is a comparison under assumptions, not proof that the business will produce the forecast.
Which inputs deserve investigation before committing?
Illustrative alternatives: A costs $1,000 now and pays $600 at year-end 1 and year-end 2. B costs $1,000 now and pays $1,250 only at year-end 2. Use a 10% annual discount rate and assume comparable risk for this exercise.
Calculate both NPVs. Then name a reason the comparable-risk assumption might fail.
Relay can build a deep integration with a second EHR for $300,000 now. Daniel estimates incremental gross profit of $150,000 a year for three years from clinics on that EHR that cannot use Relay today (illustrative; at $319.20 of monthly gross profit per clinic, this equals about 39 clinics). The board uses a 20% required return for projects of this kind. Maya asks for a one-page decision note before the money is committed.
Write an NPV decision note: the cash-flow timeline, the discounted values at 20%, the NPV and IRR, the break-even annual gross profit, and a sensitivity table showing which assumptions could reverse the decision.
Clinics that would have joined Relay through the existing integration are not incremental. Including them turns a -$47k project into a +$16k one.
Gross profit from subscriptions arrives monthly. Treating it as a lump at each year end understates value (+$16.0k versus about +$44.0k here); state the convention and keep it consistent across projects being compared.
A point estimate hides that the decision flips with a modest change in uptake, life or rate. Report the break-even values alongside the base NPV.
An NPV decision note for Relay's $300k second-EHR integration: +$15.97k at 20%, IRR about 23.4%, break-even $142.4k a year, and a sensitivity table showing four ways the decision can flip.
Next lesson: Risk, diversification & required return
Check your understanding
Question 1
Relay's integration costs $300k now and returns $150k at the end of each of the next three years. What is the NPV at 20%?
125.00 + 104.17 + 86.81 = 315.97, less 300 = +15.97. The undiscounted +$150k ignores the opportunity cost of capital; -$27.58k is the result at 30%; +$42.48k is at 15%.
Question 2
Using the same integration cash flows, what is the NPV at 30%?
150/1.3 + 150/1.69 + 150/2.197 = 115.38 + 88.76 + 68.28 = 272.42, less 300 = -27.58. -$7.20k is the NPV at 25%, which shows the project's IRR sits between 20% and 25% (about 23.4%).
Question 3
At 20%, the three-year annuity factor is 2.1065. What is the minimum annual incremental gross profit for the $300k integration to break even?
300 / 2.1065 = $142.4k. $100k a year (300/3) ignores discounting; $165.2k is the break-even at 30%. The estimate of $150k is only about 5% above break-even, so the decision is sensitive to uptake.
Question 4
Daniel learns that about 20% of the clinics he counted would have adopted Relay anyway through the existing integration. Annual incremental gross profit falls to $120k. What happens at 20%?
120 x 2.1065 = 252.78, less 300 = -47.22. Only incremental cash flows belong in an NPV; revenue Relay would have earned without the project is not caused by it.
Question 5
Option A costs $1,000 and pays $600 at the end of years 1 and 2. Option B costs $1,000 and pays $1,250 at the end of year 2. At 10%, which is correct?
A: 545.45 + 495.87 - 1,000 = 41.32. B: 1,250/1.21 - 1,000 = 33.06. A larger undiscounted total can be worth less when it arrives later; A returns half its cash a year earlier.
Question 6
A founder argues the integration is clearly worth doing because 'the NPV is positive, so the project will make money.' What is the precise correction?
NPV is conditional on the forecast and the rate. Relay's +$15.97k disappears with a 5% shortfall in gross profit or a higher rate, so the note must show which inputs carry the decision.
Workbook
Module output: A linked financial analysis, operating forecast, valuation and staged-investment decision.
Sources
Original pages and full documents are in the Library edition.