Lifecycle cost comparisons: make scenarios reproducible

The date is part of the price

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

Road asset-management and pavement-management manuals are catalogued by the Department of Transport. Select the applicable project appraisal method before a real comparison. (Department of Transport, n.d.)

Put each payment on its own date

Governing relationship: PV = Σ Ct / (1 + r)ᵗ.

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.

The film first plots rand, then follows the calculation and plots present value (rand). R20,000 + R25,915.13 + R39,176.31 = R85,091.44. Later costs receive smaller present-value factors.

Discounted cash-flow scenario

Timing changes present value

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

Download example data (CSV)

Model, calculations and transcriptEquations, worked steps, source notes and a timed reading guide

Understand the model

Discounting expresses costs at different dates on one valuation date. It changes the comparison value, not the physical scope of work.

Work through the inputs and equations below, then compare the calculated output with the film. All numerical inputs in this study are synthetic.

Inputs and symbol key

Ct; t

Cost (rand) and year of occurrence relative to the valuation date.

r

Annual discount fraction (5% = 0.05).

PVt; PVtotal

Present value of one cost and their total (rand at the valuation date).

Values used in this film

• 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.

The mathematical relationships

1. Equation 1

   Source and scope
   $$PV_t=C_t(1+r)^{-t}$$

   Discount each cost using its own year exponent and a rate consistent with the price basis.

2. Equation 2

   Source and scope
   $$PV_{total}=\sum_t PV_t$$

   Add the separate discounted contributions. Retain full precision until rounding the reported total.

Work through the example

Use the relationships above with consistent units. These substitutions are our worked example, not values quoted from the source.

1. Year zero

   20,000 × 1 = 20,000

   R present value

2. Year three

   30,000 / 1.05³ = 25,915.127956

   R present value

3. Year five

   50,000 / 1.05⁵ = 39,176.308323

   R present value

4. Total

   20,000 + 25,915.127956 + 39,176.308323 = 85,091.436279

   R present value

Read the output

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

Check your understanding

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.

Assumptions and limits

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

Return to the full references and method notes to check the cited edition and its applicability.

Timestamped film transcript 27 short entries · select a time to revisit the film

1. 0:00

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

2. 0:01

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

3. 0:07

   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.

4. 0:14

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

5. 0:23

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

6. 0:27

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

7. 0:33

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

8. 0:40

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

9. 0:43

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

10. 0:52

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

11. 0:59

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

12. 1:05

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

13. 1:08

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

14. 1:11

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

15. 1:14

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

16. 1:17

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

17. 1:26

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

18. 1:29

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

19. 1:33

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

20. 1:36

    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.

21. 1:42

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

22. 2:00

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

23. 2:10

    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

24. 2:21

    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.

25. 2:31

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

26. 2:43

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

27. 2:51

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

Download the timed captions (VTT)

The transcript records the on-screen wording. Formula provenance and interpretation are explained in the source note above.

Use one comparison basis

Our recommended scenario record identifies price base, real or nominal values, discount-rate convention, timing, analysis period and residual value. State whether agency costs, user costs or both are included. Do not compare a broad-cost option against an agency-only option.

One future cost

A synthetic cost of R100,000 incurred at the end of year five has present value 100,000/(1.05^5) = R78,352.62 at an illustrative 5% annual discount rate. This is arithmetic, not a recommended South African discount rate.

Compare costs on a consistent basis

Treatment schedules and costs must come from an appropriate evidence base. A low present value does not establish technical suitability or justify a procurement decision.

Keep the cash-flow timing visible

The present-value example assumes one cost at the end of year five. If the same cost occurs today, its present value is R100,000 in this simple calculation. At an illustrative zero discount rate, the future amount also remains R100,000 before any other adjustments.

A real scenario usually contains multiple cash flows. Keep each amount and date inspectable before summing their discounted values. Distinguish a nominal scenario from one expressed in constant prices; document the convention used for the rate and costs.

Use sensitivity comparisons to show how assumptions affect the result. They do not determine a correct rate or make an unsuitable treatment acceptable. The interactive graph below shows the present value of a single R100,000 cost at different future dates, under the selected illustrative rate.

A complete small cash-flow example

Consider three invented agency cash flows: R20,000 now, R30,000 at the end of year three, and R50,000 at the end of year five. Use an illustrative 5% annual discount rate and one consistent price/rate convention. Discount each amount at its own date before summing.

Time	Cash flow	Present-value calculation	Present value
Year 0	R20,000	20,000 / 1.05⁰	R20,000.00
Year 3	R30,000	30,000 / 1.05³	R25,915.13
Year 5	R50,000	50,000 / 1.05⁵	R39,176.31
Total	R100,000 undiscounted	Sum unrounded present values	R85,091.44

Case review

The arithmetic is easy; the comparison needs discipline

Are both alternatives priced and timed on the same basis?

Year 0 · 20000 Present value (R)

The discriminating check

Align analysis period, real or nominal prices, service assumptions and any residual-value treatment.

Evidence supports the comparison

With a common basis, compare totals and inspect how ranking changes under plausible assumptions.

Evidence is missing or incompatible

With different horizons or price bases, reconcile the comparison before selecting the lower number.

What you can conclude. A lower present cost under one scenario is not a universal recommendation.

Original worked case. Read the source editions and method notes.

The undiscounted total is R100,000. The present-value total is R85,091.44 under these timing and rate assumptions. Applying the year-five factor to the entire R100,000 would be wrong because it would move the earlier costs to a later date.

A comparison between alternatives must use a consistent analysis period and cost scope. Do not compare agency-only costs for one option with agency and road-user costs for another. Record any residual value and explain its treatment. Keep real or nominal conventions explicit rather than mixing costs and rates drawn from incompatible bases.

At zero discount, this example's present-value total returns to R100,000. That boundary check is useful because it tests the arithmetic without claiming that zero is an appropriate appraisal rate. Likewise, the illustrative 5% is not a recommended South African project rate.

The final choice still requires technical suitability and an appropriate appraisal method. A smaller calculated present value cannot repair a treatment that does not meet the road's requirements. The official manual catalogue provides the referenced asset-management and pavement-management research context; the arithmetic here is original and synthetic. (Department of Transport, n.d.)

Calculation laboratory

Can discounting change how we compare alternatives?

Calculation step	Substitution or result	Units and interpretation
Year zero	20,000 × 1 = 20,000	R present value
Year three	30,000 / 1.05³ = 25,915.127956	R present value
Year five	50,000 / 1.05⁵ = 39,176.308323	R present value
Total	20,000 + 25,915.127956 + 39,176.308323 = 85,091.436279	R present value

Discounting puts costs at different dates onto one declared valuation date. It does not make the physical work cheaper, and a higher rate reducing present value is not an engineering improvement. The rate sensitivity isolates that valuation effect while leaving the work programme unchanged. Each cash flow retains its own exponent; discounting the entire undiscounted sum as though it occurs in year five would misplace the earlier expenditure.

For a future positive cost, the derivative is dPVt/dr = −tCt/(1+r)^(t+1). Later costs are more sensitive to the selected rate, which can affect rankings when alternatives have different expenditure timing. Compare options over a consistent analysis horizon and state how residual value and costs beyond that horizon are handled. A lower present cost alone does not establish equivalent service or acceptable risk.

This original arithmetic example assumes a consistent price basis. Mixing nominal cash flows with a real discount rate would confound inflation and time value. No official discount rate, forecast maintenance schedule or economic recommendation is asserted by the 5% scenario. Economic appraisal belongs alongside the technical comparison in the rehabilitation process; preserve the assumptions that allow a reviewer to recompute each contribution. (Department of Transport, 1997)

Controlled sensitivity experiment

Controlled case	Calculated output	What changes
Rate 0%	100,000.000000 R present value	Same dated cost stream; change the annual discount rate only.
Rate 5%	85,091.436279 R present value	Same dated cost stream; change the annual discount rate only.
Rate 10%	73,585.510180 R present value	Same dated cost stream; change the annual discount rate only.

CALCULATION LABORATORY

Can discounting change how we compare alternatives?

Compare the cases on a fixed scale. Select a case to inspect its input change and calculated output.

Sensitivity case1. Rate 0%2. Rate 5%3. Rate 10%

Rate 0%: 100,000 R present value. Same dated cost stream; change the annual discount rate only.

Reproduce and challenge the result

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.

These are original synthetic calculations. The sensitivity cases vary the stated input while retaining the other assumptions; they are not field observations or treatment thresholds. The film’s calculation and sensitivity chapters use the same numerical cases as this table.

Explore the timing–rate trade-off

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 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.

Reproduce the experiment. PV(t,r) = 50000/(1+r)^t. The downloadable table contains 7,381 evaluated cells, with the two input coordinates and output value retained on every row. These are parameter combinations, not that many independent field observations. The equation extends the worked example above; it is original teaching arithmetic, not a new standard or calibrated prediction.

Computed parameter atlas · 7,381 cells

The date is part of the price

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

Download every calculated cell (CSV) · Open full-size explorer · Equation source and tests (ZIP)

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

References and method notes

1. Department of Transport (n.d.). Roads: Engineering Standard Guidelines and Manuals. Official catalogue; accessed 5 September 2026. Original source (Accessed: 5 September 2026).TMH22 V8b Draft; TRH12; TRH15; TRH24 catalogue entries
2. Department of Transport (1997). TRH12: Flexible Pavement Rehabilitation Investigation and Design. Draft TRH12, Pretoria: Department of Transport for COLTO. Original source (Accessed: 5 September 2026).Chapter 1: rehabilitation design process; investigation, evaluation, analysis and economic appraisal

Use the source edition and project requirements that apply to your work. Educational examples are not site-specific design advice.