FOUNDATION 02 / Money & mathematics
Cash flows & annuities
Move several cash flows to one date and distinguish payments at the start and end of a period.
By the end of this lesson
- Draw a timeline before calculating.
- Value ordinary annuities and annuities due.
- Distinguish present value from net present value.
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One date, many cash flows
For annual cash flows C₁, C₂, …, Cₙ paid at the ends of years, PV = Σ Cₜ/(1+r)^t. Each amount has its own exponent because each waits a different length of time. Keep inflows positive and outflows negative from one chosen perspective.
A payment today is not discounted. For 500 today and 550 in one year at 10%, total PV = 500 + 550/1.10 = 1,000. Adding 500 and 550 directly would ignore their dates.
Equal payments at the end of each period
An ordinary annuity is a finite series of equal payments A at period ends. PV = A[1−(1+r)^−n]/r and FV = A[(1+r)^n−1]/r. The PV date is one period before the first payment; the FV date is the date of the final payment.
With a zero rate, both values equal A×n. Do not divide by zero in the standard formulas. If payments grow or vary, discount each separately rather than forcing an equal-payment formula.
Beginning-of-period payments
An annuity due makes the same n payments one period earlier, at t=0 through t=n−1. At the same valuation dates, multiply the ordinary-annuity PV or FV by (1+r). For a positive rate, earlier receipts have a higher value.
For monthly payments, use a monthly rate and number of months. A nominal annual rate compounded monthly can be divided by 12; an effective annual rate requires an equivalent-rate conversion.
From PV to NPV
Net present value includes all signed cash flows, including the initial outflow. NPV = C₀ + Σ Cₜ/(1+r)^t. A positive NPV means value exceeds the assumed required return under the stated cash-flow and discount-rate assumptions.
This does not establish that forecasts are reliable or that a project is suitable. Test timing, missing costs and sensitivity to the discount rate. A longer forecast is not automatically better evidence.
Three annual payments of 1,000
- Assume 10% per year; payments arrive at t=1, 2 and 3.
- PV = 1,000/1.10 + 1,000/1.10² + 1,000/1.10³ = 2,486.85.
- FV at t=3 = 1,000×1.10² + 1,000×1.10 + 1,000 = 3,310.00.
- If paid at t=0, 1 and 2 instead, PV = 2,735.54 and FV at t=3 = 3,641.00.
Try before looking at the explanation
Change one input in the practice tool below. Predict the direction first, calculate second, then explain why it changed.
Work the numbers
Common mistakes to catch
- Using the same exponent for every payment.
- Moving payments earlier without changing the valuation date consistently.
- Calling PV of benefits “NPV” without subtracting the initial cost.
Knowledge check · 6 original questions
Answer all six. At least five correct marks the lesson complete for self-study. You can retry; this is not a credential.
Keep these ideas
- Draw dates, choose the rate per period, then discount or accumulate.
- An annuity formula is a shortcut for a sum, not a different economic principle.