WEBVTT

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Timing changes present value. R20,000 now; R30,000 in year 3; R50,000 in year 5; r = 5%.

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MATHEMATICS · define the relationship and the meaning of each symbol before calculating.

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Ct: payment at year t (rand); r = 0.05 per year. Payments: R20,000 at year 0; R30,000 at year 3; R50,000 at year 5.

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INPUT · undiscounted payments · read the axis units and the numerical values.

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R20,000 now; R30,000 in year 3; R50,000 in year 5; r = 5%.

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Year 0: R20,000 / 1.05^0 = R20,000.00. Running present value = R20,000.00.

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Year 3: R30,000 / 1.05^3 = R25,915.13. Running present value = R45,915.13.

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Year 5: R50,000 / 1.05^5 = R39,176.31. Running present value = R85,091.44.

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OUTPUT · discounted contributions · read the axis units and the numerical values.

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R20,000 + R25,915.13 + R39,176.31 = R85,091.44. Later costs receive smaller present-value factors.

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CALCULATION · substitute the defined inputs; keep units through each intermediate step.

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Year zero: 20,000 × 1 = 20,000 R present value

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Year three: 30,000 / 1.05³ = 25,915.127956 R present value

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Year five: 50,000 / 1.05⁵ = 39,176.308323 R present value

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Total: 20,000 + 25,915.127956 + 39,176.308323 = 85,091.436279 R present value

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SENSITIVITY · change one declared input · read the axis units and the numerical values.

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Rate 0%: 100000 R present value. Same dated cost stream; change the annual discount rate only.

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Rate 5%: 85091.4 R present value. Same dated cost stream; change the annual discount rate only.

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Rate 10%: 73585.5 R present value. Same dated cost stream; change the annual discount rate only.

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Check: At r = 0 the total must equal R100,000. Year-zero cost must remain R20,000 for every rate. Compare independent term-by-term calculations, retaining unrounded values until the final displayed total.

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ONE ROW · The same invoice can occupy very different places in a present-value comparison.

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PARAMETER MAP · 7381 evaluated cells. The colour limits stay fixed across every row.

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Every cell discounts a single R50,000 payment. At year zero all rates meet at R50,000. At zero discount rate every date has the same present value. Moving away from those

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boundaries reveals the interaction between timing and rate. A programme total adds the individual discounted payments; this surface is one term, not a full option appraisal.

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VALUE LANDSCAPE · Height and colour encode Present value (R). This is a graph over the two input axes, not automatically a physical pavement shape.

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Scenario rates and a constant stated payment, not recommended appraisal rates. Use a consistent real or nominal price basis before comparing alternatives.

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Consistent price basis and rate are assumed; this is not a treatment recommendation.