The choice of a discount rate is very important in calculating a cost benefit ratio. In a project where the costs are incurred in the first 5 years and the revenues grow steadily over the remaining 25 years, what will be the effect of a high discount rate (say 15%) on the stream of benefits and costs over a 30-year period, compared to a low discount rate (such as 5%)?

Select a single answer.

Question

The choice of a discount rate is very important in calculating a cost benefit ratio. In a project where the costs are incurred in the first 5 years and the revenues grow steadily over the remaining 25 years, what will be the effect of a high discount rate (say 15%) on the stream of benefits and costs over a 30-year period, compared to a low discount rate (such as 5%)?

Select a single answer.

BlackTom AI · Accepted Answer

Future cash flows are converted to present value by PV = CF / (1 + r)^t, so a higher discount rate disproportionately shrinks the present value of cash flows that occur far in the future.

1) Relevant concepts
- Present value (PV) and discounting: PV = CF/(1+r)^t.
- Time sensitivity: later-year cash flows are divided by (1+r)^t with larger t, so they fall faster as r increases.
- Cost timing: costs in years 1–5 are near-term; benefits in years 6–30 are long-term.

2) Step-by-step reasoning
- Early costs (years 1–5) are discounted only a few periods; later benefits (years 6–30) are discounted many periods. Increasing r reduces both, but the reduction is much larger for the distant benefits.

3) Calculations (intermediate steps shown) — illustrate with unit cash flows of 1 per year:
- PV of benefits as an annuity of 1 from t=6 to 30 equals PV of a 25-year annuity deferred 5 years:
  - At r = 5%: PV25 = (1 - 1/1.05^25)/0.05 ≈ 14.09; deflate to t=0 by 1/1.05^5 (≈0.7835) → PV ≈ 14.09 × 0.7835 ≈ 11.04.
  - At r = 15%: PV25 = (1 - 1/1.15^25)/0.15 ≈ 6.465; deflate by 1/1.15^5 (≈0.4972) → PV ≈ 6.465 × 0.4972 ≈ 3.21.
  - Comparison: benefits PV falls from ≈11.04 (5%) to ≈3.21 (15%) — a reduction to about 29% (a very large drop).
- PV of costs as annual 1 from t=1 to 5:
  - At r = 5%: PV5 = (1 - 1/1.05^5)/0.05 ≈ 4.33.
  - At r = 15%: PV5 = (1 - 1/1.15^5)/0.15 ≈ 3.35.
  - Comparison: costs PV falls from ≈4.33 to ≈3.35 — a modest reduction (~23%).

4) Why the correct answer is correct
Because benefits occur far into the future, a high discount rate (15%) heavily reduces their present value compared to a low rate (5%). The numerical example shows benefits are far more “penalized” than early costs.

5) Why other options are incorrect
- “Significantly reduce the value of the costs” / “reduce the value of the costs”: costs occur early and are much less reduced by a higher r.
- “There is negligible effect”: incorrect because long horizons amplify the effect on distant benefits.
- “It will reduce the value of the benefits”: true but understates the magnitude; the correct choice emphasizes the strong (significant) reduction.

6) Conclusion
The high discount rate will significantly reduce the value of the benefits — confirming the provided answer.

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