Haute Lumière
Commerce · VII.08 · MMXXVI · daylight
TEN CONCEPT BRIEFS · Chapter VII.08 — The Long View and the Discount Rate
One page each. A reader who reads only these ten pages has the chapter.
BRIEF 1 — Average the Factors, Never the Rates
The idea. When you do not know the discount rate, you must average what a pound is worth under each possibility, not average the rates themselves. The rate is only the logarithm of the thing you care about, and averaging a logarithm is not the logarithm of an average.
correct: E[e^-rt] then read the implied rate back out
wrong: e^-E[r]t the mean rate applied as if it were certainWorked example. Seven equally likely rates, one percent to seven, mean four percent. At year 300, the wrong method gives a discount factor of e^-12 and a present value of £6.14 on a million. The right method gives 0.007485 and £7,485 — because the one-percent branch, which is still worth e^-3 = 0.0498, dominates an average in which the seven-percent branch contributes e^-21.
The number to carry. The certainty-equivalent spot rate at year 300 is 1.63 percent, not four. At year 1,000 it is 1.19, and the forward rate is 1.00 — the lowest rate in the support, exactly.
Why it matters. Every long-horizon appraisal that applies a single central rate has used the wrong operation. Not a conservative one. The wrong one.
You already know this because you know that the average of a set of survival times is not the survival time of the average, and that a portfolio's expected value is not the value of its expected holding.
BRIEF 2 — The Declining Certainty-Equivalent Rate
The idea. With a fixed but unknown rate, the certainty-equivalent rate falls with the horizon toward the lowest rate you consider possible. This is Weitzman's 1998 result, and it is a theorem rather than an ethical preference.
The mechanism, which is the whole thing. The forward rate at horizon t is the average of the candidate rates weighted by p_i · e^(-r_i t) — that is, by which branches still have any weight left. High rates annihilate themselves.
horizon forward rate weight on the 1% branch
0 4.0000% 14.29%
100 1.5756% 63.27%
400 1.0187% 98.17%
1000 1.0000% 100.00%What it does NOT say. It does not say what belongs in the support. The limit is the lowest rate you admit as possible, and choosing that floor is a judgement, not arithmetic. A schedule is exactly as defensible as its floor.
Why it matters. It gives a declining schedule a foundation that survives an auditor. Hyperbolic discounting does not: individual preference reversal is a defect to be bound, and a planner who adopts it will revise every plan the moment it becomes the present.
BRIEF 3 — Uncertainty Raises the Far Future's Weight
The idea. Hold the mean discount rate fixed and increase only the spread. The far future becomes worth more, and dramatically so.
distribution mean sd R(300) PV of GBP 1m at year 300
certain, 4% 4.00% 0.00% 4.0000% 6.14
narrow, 3-4-5% 4.00% 0.82% 3.3492% 43.29
wide, 2-4-6% 4.00% 1.63% 2.3654% 828.30
wider, 1-4-7% 4.00% 2.45% 1.3662% 16,597.74
widest, 0.5-4-7.5% 4.00% 2.86% 0.8662% 74,378.77The factor: 12,106 times, from certainty to the widest row, with the mean unchanged.
The inversion this performs. The common objection to long-horizon planning is we cannot know anything about the year 2300. The arithmetic answers: the less you know, the more the year 2300 is worth in today's money. Ignorance is an argument for the long view, not against it — and it has a factor attached.
Why it matters. It converts the strongest rhetorical objection into the strongest quantitative support, which is the rarest thing a piece of arithmetic can do.
BRIEF 4 — Where the Declining Schedule Actually Comes From
The idea. Two different uncertainties can make a rate decline, and they are not the same size.
Growth risk. Gollier's extended Ramsey rule subtracts a precautionary term: r(t) = delta + eta·ĝ - (eta²/2)·V(t)/t. With delta 0.5 percent, eta 2, growth 1.5 and a two percent standard deviation, the no-risk rate is 3.50.
shock persistence rho precautionary term long-run rate
0.0 0.080 pp 3.4200%
0.7 0.453 pp 3.0467%
0.9 1.520 pp 1.9800%Rate risk. The Weitzman mechanism moves the same rate from 4.00 to 1.00 — a full 3.00 percentage points.
The ratio: 6.6 to one. The decline in the schedule is overwhelmingly a fact about our ignorance of the rate, not about the riskiness of growth.
Why it matters. People defend declining schedules on precautionary grounds because it sounds prudent. It is the weaker of the two arguments by a factor of nearly seven, and defending a right practice with a weak reason is how the practice gets overturned.
BRIEF 5 — Time Inconsistency, Measured
The idea. A schedule indexed by years-from-now re-bases with every government, so the plan and its successor disagree — and neither is cheating.
Worked example. A million pounds arriving in year 200, on the Green Book schedule, is worth £6,207 today.
standing at year 30, 170 years still to run
the plan's own continuation value GBP 17,423
the successor, re-applying the same table GBP 11,244
the successor's share 64.5%
the plan over-states by 1.550x
standing at year 75 41.6% / 2.404xWhy the successor is right. It applied the published table from its own present, which is what the table instructs. The theorem's own schedule is indexed to calendar date and to what has been learned — by year 30 you genuinely know more about which branch you are on — and that schedule is time-consistent. No finance ministry publishes it.
Why it matters. It is a standing incentive for every successor to reprice your commitment down by a third and be correct. The defence is not a lower rate. It is a commitment device or a change of instrument.
BRIEF 6 — Past a Few Thousand Years, Present Value Is Not an Answer
The idea. There is a horizon beyond which no positive rate carries information, and it is nearer than people assume.
GBP 1 arriving in year 10,000 at 1.0% = 10^-43.2 pounds
GBP 1 arriving in year 300,000 at 1.0% = 10^-1,296 pounds
GBP 1 arriving in year 1,000,000 at 1.0% = 10^-4,321 poundsOne percent is the floor of the best public schedule in the world — the Green Book's rate for everything past year 300.
What the regulator did instead. EPA's Yucca Mountain standard, 40 CFR Part 197, originally ran to 10,000 years. The D.C. Circuit vacated that in NRDC v. EPA (2004) because the National Academies had found no scientific basis for the cut-off. The 2008 final rule runs to one million years: 15 millirem a year to year 10,000, 100 millirem median thereafter. No rate. A dose ceiling.
Why it matters. At that horizon the correct instrument is a constraint, and finance's job becomes meeting the constraint at least cost — which is a job it is extremely good at, and a much clearer brief than pricing a million years.
BRIEF 7 — Parfit's Non-Identity Problem
The idea. Any policy that changes the timing of anybody's life changes who gets conceived. So a person in the far future cannot be worse off than they would have been: under the other policy they would not exist.
Worked example. A person exists only if every ancestral conception happened when it did. k generations back there are 2^k - 1 of those. With a per-generation, per-lineage shift probability q:
generations ancestors q = 0.001 q = 0.010 q = 0.050
3 7 0.9930 0.9321 0.6983
7 127 0.8807 0.2790 0.0015
12 4,095 0.0166 1.34e-18 6.00e-92At a one-percent perturbation the seventh generation is 27.9 percent likely to contain the same people; the twelfth, one part in a million million million.
What it licenses, precisely. It defeats a claim of harm to an individual. It leaves standing every claim about the quality of the world — Parfit's same-number quality claim: if the same number will live either way, it is worse if those who live are worse off than those who would have lived.
Why it matters. A net present value of individual harms is exactly the object non-identity dissolves. A dose ceiling or a stock level is exactly the object it leaves untouched. The ethics and the arithmetic pick the same instrument.
BRIEF 8 — Ramsey's Own View, and the Problem It Creates
The idea. The man who wrote the discounting equation thought discounting future utilities was indefensible, and setting the term to zero creates a different, technical problem.
The quotations, because they are load-bearing. Ramsey (1928): "ethically indefensible and arises merely from the weakness of the imagination." Sidgwick (1874): the time at which a person exists cannot affect the value of their happiness. Pigou (1920): a defective telescopic faculty. Harrod (1948): "a polite expression for rapacity."
The problem. With delta = 0 an infinite horizon has no finite total.
a flow of 1 per year for 100 years: at delta = 0 100 at 0.5% 78.54
a flow of 1 per year for 10,000 years: at delta = 0 10,000 at 0.5% 200.00The second column converges to 1/0.005 = 200. The first diverges. Ramsey's own answer was the Bliss point — maximise the integral of (B - u). Koopmans (1960) and Diamond (1965) later proved that no complete, continuous, anonymous and Pareto ordering of infinite streams exists at all.
Why it matters. The defensible value of pure time preference is the hazard of extinction. Getting a well-defined optimum is a separate technical job, and conflating the two is how a good ethical argument acquires a bad reputation.
BRIEF 9 — The Population Term Every Calculation Contains
The idea. Under a total criterion the planner sums over people, so the discount rate on per-head consumption becomes delta - n + eta·g. Population growth is a discount rate adjustment, and nobody declares it.
2100 population n per year r = 3.5% - n PV of GBP 1m at year 76
low ~7.0 bn -0.2082% 3.7082% 62,836
average criterion 0.0000% 3.5000% 73,204
medium 10.2 bn 0.2872% 3.2128% 90,415
high 12.0 bn 0.5010% 2.9990% 105,851The sign flip. The medium variant raises the future's weight by 1.24 times. The low variant lowers it by 0.86 — because n is negative and a total criterion then discounts the future more heavily. On UN medium projections world population peaks in the 2080s, which means the criterion most often used to argue for the future changes sign inside this century.
Why it matters. It is not an argument against the total view. It is an argument for saying on the page which view you are using, beside the rate, the way you state a tax rate.
BRIEF 10 — A Funding Ratio Is a Discount Rate Wearing a Percentage Sign
The idea. "We are forty percent funded" is not a fact. It is a discount rate in disguise, and the honest form is the real return the pot is implicitly asserting.
Worked example. A duty of £130bn undiscounted across 120 years — £1.083bn a year level. A pot of £15bn:
rate present value of the duty apparent funding ratio
0.0% GBP 130.00 bn 11.5%
2.0% GBP 49.14 bn 30.5%
3.5% GBP 30.45 bn 49.3%A 4.27-times swing with nothing physical changed. So invert it.
assets, GBP bn implied real return, forever
15.00 7.22%
26.84 4.00%
40.00 2.58%£15bn is not "twelve percent funded". It asserts 7.22 percent real, forever, against a fund realising four. At four percent the pot required is £26.84bn.
Why it matters. The implied required return can be compared to something — the fund's own trailing realised return. A funding ratio can be compared to nothing, which is why it survives in board papers.
All figures in these briefs are computed in lib/verify/VII_08.py, printed there with their inputs and intermediate terms, and sourced in the chapter's Works Cited.