La Bourse / Volume III / Nº III.05 / Quiz, reflection, essays

A watercolour of people working at a long wooden table inside a glasshouse full of autumn trees. — Plate III.05 · Quiz, reflection, essaysThe Long Table.The table is the same width at both ends. It is the light that falls away. A discount rate is a claim about how fast it falls, and almost nobody who uses one has ever said out loud what theirs is made of.

ASSESSMENT · Chapter III.05 — Interest, Time, and the Discount Rate

Three instruments: a ten-point quiz, eight reflection questions, five essay prompts. The quiz checks comprehension rather than recall. The reflections are private and first-person. The essays are arguable from more than one side.

THE QUIZ — ten points

Four on recall.

1. Write the Ramsey equation, name each term with its unit, and say which terms are empirical.

rho = delta + eta · g. rho is the consumption discount rate, percent per year; delta the pure rate of time preference, percent per year; eta the elasticity of marginal utility of consumption, dimensionless; g the growth of consumption per head, percent per year. Only g is empirical. delta is an ethical position about whether a future person counts less for being later; eta is a position about how much less a pound matters to someone richer. One mark for the equation, one for identifying that two of the three terms are choices.

2. State the Green Book's declining schedule and how its headline rate is constructed.

3.5 percent for years 0–30; 3.0 percent to year 75; 2.5 percent to 125; 2.0 percent to 200; 1.5 percent to 300; 1.0 percent beyond. Constructed as a pure time preference of 0.5 percent, plus 1.0 percentage point for catastrophic risk, plus an elasticity of 1.0 applied to assumed per-head growth of 2.0 percent: 0.5 + 1.0 + 2.0 = 3.5. A lower schedule beginning at 1.5 percent applies to pure health effects.

3. Give the two parameter values on which the Stern–Nordhaus dispute turns, and each side's figure.

delta — Stern 0.1 percent, Nordhaus 1.5 percent. eta — Stern 1, Nordhaus 2. Holding growth common at 1.3 percent, the gap is 2.70 percentage points, of which 1.40 comes from delta and 1.30 from eta: almost exactly half each.

4. What is the certainty equivalent of a reduction in variance, for a decision-maker with constant absolute risk aversion a, and at what rate is it discounted?

(a / 2) · ΔVar, where 1/a is risk tolerance in the same units as the loss. It is a certain amount by construction, so it is discounted at the risk-free rate — never at a hurdle rate.

Four on application.

5. A colleague argues that because people discount hyperbolically, the government should adopt a declining social discount rate. Where is the error, and what is the sound argument for the same conclusion?

The error is treating a descriptive finding about individual behaviour as a normative rule for a planner. Hyperbolic preferences are time-inconsistent: a planner using them revises every plan the moment it becomes the present (Strotz, 1955). The sound argument is Weitzman's: if the true rate is uncertain and persistent, the expected discount factor is dominated at long horizons by the lowest possible rate, so the certainty-equivalent rate declines. Full marks require both halves — the rejection and the replacement.

6. Your treasury applies one hurdle rate of 11 percent to every project. A flood-defence asset is proposed that pays only in the years the business is already having its worst results. What is wrong, in the language of finance?

The single hurdle rate assumes a positive consumption beta. This asset's payoff is negatively correlated with the rest of the business, so its beta is negative and its correct rate is r_f + beta · premium, which sits below the risk-free rate. Applying 11 percent systematically underprices the entire class of protective assets. Credit an answer that notes the firm is effectively refusing to pay for insurance while buying insurance elsewhere.

7. A regeneration proposal is rejected because "the benefit is not a cash flow." Diagnose the appraisal, and say what you would add.

The benefit is a change in the distribution of a future cost: a lower expected loss and a lower variance. The appraisal has valued the second at zero because it has no line. Add three lines: the change in expected cost, the change in its variance, and the certainty equivalent of that change at the firm's stated risk tolerance. The stronger answer notes the opposite error is equally common — including the risk benefit and discounting it at the equity hurdle rate.

8. A team argues for a 1.4 percent discount rate on its restoration project. What must you require of them, and why?

That the same rate be applied to the organisation's long-dated liabilities in the same paper. A one billion pound decommissioning duty in year 100 requires a provision of £1.2m at 7 percent and £249.0m at 1.4 — 216 times more. Symmetry is what stops the rate becoming a negotiating position rather than a parameter. Credit any answer that identifies the underlying failure mode: arguing the rate down is easier than arguing the benefit up.

Two that require the arithmetic to be done.

9. A department proposes delta = 0.3 percent, eta = 1.5, and per-head growth of 1.4 percent. Compute the rate, the present value of £1,000,000 arriving in year 100, and the half-life of value. Compare with a flat 3.5 percent.

rho = 0.003 + 1.5 × 0.014 = 0.024 — 2.4 percent. PV = 1,000,000 / 1.024¹⁰⁰ = 1,000,000 × 0.093326 = £93,326. Half-life = ln 2 / ln 1.024 = 0.6931 / 0.023717 = 29.2 years. At a flat 3.5 percent the same benefit is £32,060, so the proposed rate values it 2.91 times higher — on a difference of 1.1 percentage points. The point of the question is that a rounding-sized change in the rate is a tripling of the answer.

10. A plant loses £5.0m with probability 0.20 in any year. A restoration costing £6.0m of capital and £0.15m a year takes the probability to 0.10 and the loss to £2.0m. Horizon 20 years. The board's risk tolerance 1/a is £10m, the risk-free rate is 2 percent, the asset's consumption beta is −0.30 against a market risk premium of 5 points, and the standing hurdle rate is 12 percent. Appraise it both ways.

Expected loss falls from 0.20 × 5.0 = £1.00m to 0.10 × 2.0 = £0.20m; mean saving £0.80m, net of opex £0.65m. Variance p(1−p)L² falls from 0.20 × 0.80 × 25 = 4.00 to 0.10 × 0.90 × 4 = 0.36; 3.64 (£m)² removed. Certainty equivalent = (a/2) · ΔVar = 0.05 × 3.64 = £0.182m a year. Hedge-adjusted rate = 0.02 + (−0.30 × 0.05) = 0.50 percent. Mean only at the 12 percent hurdle: 0.65 × 7.4694 = £4.855m — NPV −£1.145m, rejected. Priced properly: 0.65 × 18.9874 = £12.34m plus 0.182 × 16.3514 = £2.976m = £15.32m — NPV +£9.32m, accepted. The stronger answer states the honest caveat: the reversal is driven by the rate and the horizon, both of which are assumptions, and a paper that does not publish them alongside the answer is advocacy.

REFLECTION — eight questions, for one person and a pen

These are not for a room. Write the answers by hand if you can; the slowness is the point.

What is your own delta? Not the organisation's — yours. How much less does a person alive in a hundred years matter to you than a person alive now, and would you be willing to say the number out loud?

Recall a decision you made for someone who did not yet exist — a child, a successor, a reader. What rate did you use? What does the answer tell you about the rate you use at work?

Where in your own life do you hold something that pays only in a bad year? What does it cost you to hold it, and have you ever priced what it returns?

Think of a time you argued for a project by moving an assumption rather than by improving the evidence. What made that feel permissible at the time?

What is the longest horizon anyone has ever asked you to be accountable for, and what is the longest horizon you actually think about? Write both numbers down before you read the gap.

Which of your commitments exist because you did not trust your future self? List three. Then ask what that tells you about designing for other people.

What have you left out of a calculation because it had no line — not dishonestly, simply because the form had no field for it?

If you had to publish one number as your own risk tolerance — the amount of variability you are genuinely willing to carry — what would it be, and what would change if the people you work with knew it?

ESSAY PROMPTS — five

Each is arguable from more than one side. Each requires at least one source the chapter cites and at least one it does not.

1. Is delta an ethical parameter or an empirical one? Stern set pure time preference at 0.1 percent on the classical utilitarian ground that discounting a future person's welfare is indefensible; Nordhaus set it at 1.5 percent on the ground that a model must reproduce observed behaviour. Argue whether a descriptive calibration of a normative parameter is legitimate. Use Stern (2007) and Nordhaus (2007) directly, and at least one philosophical treatment of intergenerational obligation that the chapter does not cite.

2. The parameter with three jobs. Under the standard time-separable specification, eta governs intergenerational inequality aversion, intragenerational inequality aversion, and risk aversion simultaneously. Argue either that this conflation is a tolerable simplification whose costs are small in practice, or that it invalidates the use of a single Ramsey rate for policy. Engage Epstein and Zin (1989) and Gollier (2012), and find at least one empirical estimate of eta from a source the chapter does not cite — noting which of the three jobs that estimate actually measured.

3. Does the declining schedule solve the problem or relocate it? The UK, France and Norway have adopted declining rates, and the theoretical warrant is Weitzman's uncertainty argument rather than behavioural impatience. Argue whether declining schedules genuinely correct the treatment of the distant future, or whether they simply move the arbitrary choice from what rate to what schedule — and assess whether a policy maker could game the breakpoints. Use Weitzman (2001) and HM Treasury (2022), plus one critique of declining discounting that the chapter does not cite.

4. Pricing what is not a cash flow. The chapter argues that a regenerative asset's return is often a reduction in the variance of a future cost, and that this should be converted to a certainty equivalent and discounted at or below the risk-free rate. Write the strongest case against this method — that it introduces two unobservable parameters (a and the consumption beta) into an appraisal that previously had one, and that the resulting freedom is more dangerous than the omission it corrects. Then write the strongest rebuttal. Use the Quintana Roo reef policy or CCRIF as evidence that markets already price this, and at least one source on model risk or parameter uncertainty in valuation that the chapter does not cite.

5. The rate as a political instrument. Between 1992 and 2023, official US guidance moved from 7 percent to a pair of 3 and 7 percent to a single 2 percent, and the social cost of carbon moved with it from tens of dollars to hundreds. Argue whether the discount rate is a technical parameter that has been politicised, or a political parameter that has always been presented as technical — and what follows either way for how it should be governed. Use OMB (2003), EPA (2023) and the chapter's symmetry rule, and one account of regulatory cost-benefit practice that the chapter does not cite.