Interest and growth
Valuing uneven cash flows
Value irregular cash-flow streams one flow at a time, and solve for the payment that closes a gap.
Learning objectives
- Calculate the present value of an uneven cash-flow stream
- Move a whole stream to a future date
- Use NPV() correctly when a year has no cash flow
- Solve for the missing payment that delivers a target value
Real cash flows are rarely level. A customer pays in stages, a new product ramps up, a settlement is paid in uneven installments. There is no shortcut formula for a stream like that — but there doesn't need to be.
Because present values add up, you value an uneven stream by discounting each cash flow by its own number of periods and summing the results. The same idea runs every NPV and DCF model.
PV = CF₁ ÷ (1 + r) + CF₂ ÷ (1 + r)² + … + CFₙ ÷ (1 + r)ⁿ FVₙ = PV × (1 + r)ⁿ
- CFₜ
- cash flow at the end of period t (zero if nothing arrives)
- r
- discount rate per period
- FVₙ
- value of the whole stream at the end of period n
Applying it: staged payments on a contract. A fictional builder, Bina Contracting, agrees a fit-out job for which the client pays SAR 150,000, SAR 250,000, SAR 400,000 at the ends of years 1 to 3, SAR 800,000 in total. At Bina's 10% discount rate the stream is worth SAR 643,501 today. If the client instead offered a single payment today, Bina should accept anything above that figure; if Bina's costs to do the work, in today's money, exceed it, the job destroys value however large the headline total looks.
One stream, any date. Because the value at one date converts to the value at any other by compounding or discounting, you can choose the date that suits the question. A lender asking how much the payments are worth on the day the job is finished compounds the PV forward; an investor asking what to pay today discounts. The ranking of two streams never changes with the date chosen, as long as the same rate is used for both.
Solving backwards. If you know the value the stream must reach but one payment is missing, (1) value the known payments on their own, (2) find the shortfall against the target, and (3) move that shortfall to the date of the missing payment. The last exercise in this lesson uses exactly these three steps.
Calculate
A client will pay you 4,000 at the end of year 1, 6,000 at the end of year 2 and 9,000 at the end of year 3. At an 8% discount rate, what is the stream worth today?
Check your understanding
Cash flows of 1,000 (year 1), 2,000 (year 2) and 3,000 (year 4) are listed in B2:B4 of a worksheet, with nothing for year 3. The discount rate is in B1. Which formula gives the present value?
Solving for a missing payment
Sometimes you know the value a stream must have and need the payment that gets you there — a final balloon payment, or the last installment that settles a debt. Work in today's money:
- Value the payments you know.
- The shortfall is the target minus that value.
- The missing payment must be worth the shortfall today, so compound the shortfall forward to the date the payment is made.
Challenge: Sizing the final payment
A buyer will pay for equipment in four annual installments. The first three are fixed (B5:B7). The seller needs the installments to be worth 50,000 today at a 6% discount rate. Discount the known installments (columns C and D), total them in B10, find the shortfall in B11, and size the year 4 payment in B13. B14 is your check: the value of all four payments, which should equal the target.
| A | B | C | D | |
|---|---|---|---|---|
| 1 | Discount rate | 6.0% | ||
| 2 | Target value today | 50,000.00 | ||
| 3 | ||||
| 4 | Year | Payment | Discount factor | Present value |
| 5 | 1 | 10,000.00 | ||
| 6 | 2 | 12,000.00 | ||
| 7 | 3 | 15,000.00 | ||
| 8 | ||||
| 9 | ||||
| 10 | PV of known payments | |||
| 11 | Shortfall in today's money | |||
| 12 | Year of the final payment | 4 | ||
| 13 | Required final payment | |||
| 14 | Check: PV of all four payments |
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