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Commerce · III.05 · MMXXVI · daylight

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TEN CONCEPT BRIEFS · Chapter III.05 — Interest, Time, and the Discount Rate

One page each. A reader who reads only these ten pages has the chapter.

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BRIEF 1 — The Ramsey Equation

The idea. The rate at which a society should discount future consumption is not an observed price. It is assembled from three terms, and two of them are ethical positions rather than measurements.

        rho  =  delta  +  eta · g

  rho    the consumption discount rate       % per year
  delta  pure rate of time preference        % per year
  eta    elasticity of marginal utility      dimensionless
  g      growth of consumption per head      % per year

Frank Ramsey wrote this in 1928 and called the discounting of future utilities "ethically indefensible" in the same paper — then used it anyway, because the mathematics needed it. Ninety-odd years later the profession is still having his argument.

Worked example. Take the United Kingdom's official rate. delta is 0.5 percent, plus one percentage point for the risk that the future does not arrive as expected; eta is 1.0; assumed growth is 2.0 percent. So 0.5 + 1.0 + (1.0 × 2.0) = 3.5 percent, which is exactly the Green Book's headline figure. The whole construction fits on a line.

Why it matters. Once a rate is decomposed it can be argued about in the right language. Somebody who wants a lower rate must now say which term they are moving and why, and "I would like a bigger number" stops being an available move. The Green Book's real authority comes from publishing the decomposition, not from the number.

You already know this because you have been in a meeting where a hurdle rate was quoted and nobody could say where it came from, and you noticed that the conversation had nowhere to go.

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BRIEF 2 — Pure Time Preference

The idea. delta is the rate at which you discount a future person's wellbeing purely because they are in the future. Not because they will be richer — that is eta · g, a separate term. Because they are later.

The two positions. The classical utilitarian answer, from Ramsey, Pigou and Sidgwick, is that the only defensible non-zero value is the probability that there is no future person there at all: extinction risk, on the order of a tenth of a percent. The Stern Review took that position. The descriptive answer is that markets reveal impatience, and a model that ignores it will not reproduce observed savings and returns; William Nordhaus took that position, at 1.5 percent.

Worked example — what delta alone does. Weight a person alive in 2300, two hundred and seventy-five years out, against a person alive today:

  at delta 0.1%    1 / 1.001^275  =  0.7597   ->  76.0%
  at delta 1.5%    1 / 1.015^275  =  0.0167   ->   1.67%
  ratio                                           45.6 x

Why it matters. No amount of data settles this. It is a choice about standing — about who counts — and it is made, usually silently, every time a long-horizon appraisal is run. A number that carries a factor of forty-five and has no empirical content deserves to be said out loud.

You already know this because you have made decisions for a child's benefit without discounting them at all, and you did not experience that as an error.

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BRIEF 3 — Eta, and the Three Jobs It Is Given

The idea. eta measures how fast the value of an extra pound falls as someone gets richer. In the standard model it is asked to do three different jobs at once, and it cannot do them independently.

1. Inequality between generations. If our descendants are richer, a pound given to them matters less. Higher eta, more discounting of the future.
2. Inequality within a generation. The same parameter governs how much you would sacrifice to move a pound from a rich person to a poor one today.
3. Aversion to risk. Under time-separable constant-relative-risk-aversion preferences — what almost every appraisal uses — eta is also the coefficient of relative risk aversion.

The problem. Those three are not the same question, and the answers people give to them are not the same number. Epstein and Zin (1989) built recursive preferences precisely to separate risk aversion from the willingness to substitute across time. Almost no practical appraisal uses them.

Worked example. With growth at 1.3 percent, moving eta from 1 to 2 adds 1.30 percentage points to the discount rate — which is 48 percent of the whole Stern–Nordhaus gap, the other 52 percent coming from delta. Two parameters, neither empirical, and the entire dispute.

Why it matters. When somebody argues for a higher eta on grounds of risk aversion, they have — under the standard model — also just argued for caring less about poor people today. Naming which job the parameter is doing is the whole of the discipline here.

You already know this because you already believe a hundred pounds means more to someone with nothing than to someone with everything. That belief is eta.

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BRIEF 4 — The Discount Factor and the Half-Life

The idea. A rate is abstract. A factor is arithmetic, and a half-life is memorable.

  factor(r, t) = 1 / (1 + r)^t          half-life = ln 2 / ln(1 + r)

Worked example, at the Green Book's 3.5 percent. One million pounds of benefit arriving in the future is worth, today:

Half-life: ln 2 / ln 1.035 = 0.6931 / 0.034401 = 20.1 years. Value halves every twenty years, so a benefit at year 100 arrives at about one thirty-first of its size.

The same number at other rates. At Stern's 1.4 percent the year-100 million is worth £249,003; at Nordhaus's 5.5 percent, £4,729. A ratio of 52.7 on the identical benefit.

Why it matters. Almost every argument about long-lived assets is really this table, unstated. Put the table in the paper and the argument becomes short.

You already know this because you have seen a thirty-year maintenance saving disappear from a business case and could not say exactly where it went. It went here.

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BRIEF 5 — Declining Discount Rates

The idea. If you do not know the right rate, the effective rate over long horizons falls toward the lowest rate you consider possible. This is arithmetic, not sentiment.

Why. You must average discount factors, never rates. A high-rate branch contributes almost nothing at long horizons — it has already shrunk to nothing — so the average is dominated by the low branch.

Worked example. The rate is 1 percent or 7 percent, equally likely, and persistent.

  horizon    E[factor]    implied rate     average of the rates
       1      0.962339         3.91%                    4.00%
      50      0.320993         2.30%                    4.00%
     100      0.185432         1.70%                    4.00%
     300      0.025267         1.23%                    4.00%

The certainty-equivalent rate slides from 4.0 toward 1.0. Martin Weitzman proved the general result in 1998; his 2001 survey of 2,160 economists produced the same shape empirically — 4 percent for years one to five, 3 to year twenty-five, 2 to seventy-five, 1 to three hundred, zero beyond.

Why it matters. It gives a rigorous reason for a declining schedule that has nothing to do with behavioural anomalies, and it is the reason three finance ministries have adopted one.

You already know this because you have watched a forecast's error bars widen with distance and understood, without being told, that the far end of the forecast was worth less as evidence and more as a range.

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BRIEF 6 — The Schedules Actually in Force

The idea. This is not a live controversy awaiting settlement. Governments have settled it, differently, and published.

Worked example — the same million at year 100. Flat UK 3.5 percent: £32,060. UK declining schedule: £50,818, or 1.59 times. Lebègue: £77,088. Norway: £45,119. US at 7 percent: £1,152; at 2 percent: £138,033.

Why it matters. Never construct an authority where you can cite one. A published ministry schedule survives an audit committee in a way that an analyst's reasoning does not, however good the reasoning.

You already know this because you have watched a proposal win on the strength of a standard everyone recognised rather than on the strength of its argument.

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BRIEF 7 — Hyperbolic Discounting, and What It Does Not License

The idea. People do not discount exponentially. Measured discount rates fall as the horizon lengthens, which produces preference reversals.

Worked example. Thaler's 1981 subjects, offered $15 today, were indifferent to $20 in a month, $50 in a year, and $100 in ten years. The implied annual rates are 345 percent, 120 percent and 19 percent. One exponential rate cannot produce three answers.

David Laibson's quasi-hyperbolic form adds one parameter: everything after now is scaled by beta. With beta = 0.7, $100 today beats $110 tomorrow (100 against 76.98), while $110 in 366 days beats $100 in 365 (68.99 against 62.74). The preference reverses with no new information.

What the evidence does not support. Andreoni and Sprenger (2012), putting a front-end delay on the sooner payment, recover a beta near one for money. Augenblick, Niederle and Sprenger (2015) find robust present bias over real effort — beta near 0.9 — in the same subjects. Money is fungible with a bank account, so a laboratory monetary discount rate is partly a measurement of credit access. Effort cannot be arbitraged.

And the normative point. A time-inconsistent planner is a defect to be bound, not a schedule to be adopted — Strotz proved that in 1955. The good case for declining social rates is Weitzman's uncertainty, not human impatience.

Why it matters. Commitment devices work because of this. Policy should not be designed by it.

You already know this because you have set an alarm across the room.

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BRIEF 8 — The Certainty Equivalent of Variance Removed

The idea. A regenerative asset often returns no cash flow at all. It returns a change in the shape of a future cost. To value it, first convert the variance reduction into a certain amount; only then discount it.

  certain-equivalent cost  =  mu  +  (a / 2) · sigma^2

  a       the decision-maker's constant absolute risk aversion
  1/a     risk tolerance, in the same units as the loss

Removing variance is therefore worth (a/2) · ΔVar per period, as a certain amount — so it is discounted at the risk-free rate, not at a hurdle rate.

Worked example. A plant loses £8.0m in a drought year, probability 0.15. Expected loss £1.20m; variance p(1−p)L² = 8.1600 (£m)². A watershed restoration takes the probability to 0.12 and the loss to £3.0m: expected loss £0.36m; variance 0.9504. Variance removed: 7.2096 (£m)². At a risk tolerance of £6.67m — that is a = 0.15 — the certainty equivalent is 0.075 × 7.2096 = £0.5407m a year, against a mean saving net of operating cost of £0.64m.

The risk term is worth 84 percent as much again as the entire expected saving, and it appears on no line of any management account.

Why it matters. Two failures are common and equally expensive: leaving it out because it has no line, and putting it in at the equity hurdle rate. Set 1/a once, as a board parameter, the way an insurance retention is set.

You already know this because you buy insurance at a premium above expected loss, and you do not regard that as irrational.

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BRIEF 9 — Consumption Beta and the Counter-Cyclical Asset

The idea. A discount rate is not a property of time. It is a property of when the payoff arrives relative to everyone else's fortunes.

  r_i  =  r_f  +  beta_i · (risk premium)

An asset whose payoff is largest exactly when everything else is worst has a negative beta, so its rate sits below the risk-free rate — and can go below zero.

Worked example. Risk-free 2.0 percent, market risk premium 5 points.

At a beta of −0.8, a pound in year 50 is worth more than a pound today. That is not a trick. It is what insurance has always meant, written as a rate.

Where the theory understates. The consumption-CAPM gets the sign right and the size badly wrong: with eta 1.5 and consumption volatility 2 percent, the model's premium per unit of beta is 0.060 percentage points against an observed equity premium near six — Mehra and Prescott's puzzle of 1985. Take the sign from the theory and the size from a market that already prices the risk: the reinsurance quote, the catastrophe bond spread, the parametric premium.

Why it matters. It is the formal reason a regenerative protective asset is worth more than a revenue-bearing asset of the same expected value — and the reason a single corporate hurdle rate systematically underprices the entire class.

You already know this because you value a spare part most on the day the machine fails.

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BRIEF 10 — The Symmetry Rule

The idea. A low discount rate justifies almost any project. That is a property of the instrument, not a licence to use one.

The arithmetic. A perpetual benefit of £1 a year justifies a capital cost of:

  at 10.0%   £10.00       at 3.5%   £28.57
  at  7.0%   £14.29       at 1.4%   £71.43

Moving from 7 percent to 1.4 multiplies the admissible cost by five. At that rate nearly any project with a permanent positive flow clears nearly any finite cost — so the rate has stopped discriminating and all the real work has moved, silently, to the benefit estimate, which is far less well identified.

The rule that keeps it honest. The same rate applies to liabilities as to benefits, in the same paper, always. A one billion pound decommissioning duty falling in year 100 requires a provision of £1.2m at 7 percent, £32.1m at 3.5, and £249.0m at 1.4 — two hundred and sixteen times more.

What it prevents. The failure mode of rate policy is that people notice arguing the rate down is easier than arguing the benefit up, and the rate quietly becomes the place every disagreement is settled. Symmetry removes the free lunch: whoever wants 1.4 percent for their restoration must take 1.4 percent on their clean-up.

Why it matters. An instrument that approves everything has stopped measuring. Symmetry is what turns the rate back into an instrument.

You already know this because you have watched an assumption become a negotiating position, and you knew the moment it happened.

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All figures in these briefs are computed in lib/verify/III_05.py, printed with their inputs and units, and sourced in the chapter's Works Cited.