Interest, Time, and the Discount Rate

Volume III — Money, Energy, Information

Nine movements, one rate.

THE PLATE

A watercolour of people working at a long wooden table inside a glasshouse full of autumn trees. — Plate III.05The 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.

THE LETTER

There is one number in your organisation that decides more than any other, and it is usually the only one nobody can source.

It appears as the hurdle rate in a capital paper. It appears as the weighted average cost of capital in a valuation. It appears, in the public sector, as three and a half percent in a Green Book appraisal, or as seven percent in a US regulatory analysis, or as whatever the treasury function has been using since somebody set it. Everything long-lived in your portfolio lives or dies by it — the forest, the aquifer, the training programme, the repairable product line, the sea wall, the pension promise — and it is applied with less scrutiny than a travel expense.

This chapter is about that number. Not about whether it should be lower. About what it is made of, which terms are empirical and which are ethical, what happens to a benefit when it is applied for a century, and what the best public guidance in the world has actually done about it — because three governments have already done the thing most firms have not, and their schedules are published.

Then the part this edition needs and most treatments of discounting skip entirely. A regenerative asset very often does not return a cash flow. It returns a reduction in the variance of a future cost — the drought that costs less, the flood that does not arrive, the harvest that varies by eight percent instead of thirty. That is a real return and it is discounted differently from a cash flow. Done properly it can carry a rate below the risk-free rate. Done carelessly it is worth nothing at all, because it never appears on a line.

One warning, and it is the reason the honest negative in this chapter is longer than usual. A low discount rate will justify almost any project. That is not an argument for using one. It is a property of the instrument, and an instrument that approves everything has stopped measuring.

— The Editors

DISCOVERY

What is already working

The interesting fact about long-horizon discounting is that it is not a theoretical debate awaiting resolution. It has already been decided, in public, by finance ministries, and the schedules are in force.

The United Kingdom. The Green Book has carried a declining schedule of social time preference rates since 2003, and the 2022 edition sets it out in a single table: 3.5 percent for years zero to thirty, 3.0 percent for years thirty-one to seventy-five, 2.5 percent to year one hundred and twenty-five, 2.0 percent to year two hundred, 1.5 percent to year three hundred, and 1.0 percent beyond that. A second, lower schedule beginning at 1.5 percent applies to pure health effects. What makes this document remarkable is not the declining part. It is that the Treasury publishes the construction: the rate is a rate of pure time preference of 0.5 percent, plus a one percent allowance for catastrophic risk, plus an elasticity of one applied to assumed per-head growth of two percent. Nought point five, plus one, plus two, is three and a half. Any official in the United Kingdom can be asked where their number comes from and can answer in one sentence.

France. The Commissariat général du Plan, under Daniel Lebègue, reported in 2005 and cut the public discount rate from eight percent to four percent for the first thirty years, declining to two percent thereafter. Émile Quinet's 2013 review reset the structure again, separating a risk-free rate — 2.5 percent to 2070, then 1.5 percent — from a systemic risk premium of around two points for a project of average risk. That separation is the move this chapter is built on, and France made it official a decade ago.

Norway. The Hagen committee, NOU 2012:16, set four percent to year forty, three percent to year seventy-five, and two percent beyond.

The United States. OMB Circular A-94 fixed seven percent real in 1992; Circular A-4 in 2003 asked for both three and seven percent, and offered rates of one to three percent for effects reaching across generations. The 2023 revision of A-4 replaced the pair with a single consumption rate of 2.0 percent, derived from a thirty-year average of real returns on ten-year Treasury securities, and directed declining rates over long horizons. Both conventions are in live use in the United States today, which is itself instructive: the discount rate is the one parameter in cost-benefit analysis that changes with an administration.

The effect of these choices is not marginal. Take the United States Environmental Protection Agency's 2023 estimates of the social cost of carbon — one damage module, one set of climate models, three near-term Ramsey rates. At 2.5 percent, $120 a tonne. At 2.0 percent, $190. At 1.5 percent, $340. The same physical damage, priced nearly three times apart, by the rate alone.

Now the second family of cases, because the ministries are only half the story.

New York City, 1997. Facing a federal requirement to filter its drinking water, the city compared a filtration plant estimated at six to eight billion dollars of capital plus around three hundred million a year to run, against a watershed protection programme in the Catskill and Delaware basins costed at roughly one and a half billion over a decade. It bought the watershed. The arithmetic is four to five times on capital alone, before the avoided operating cost, which at a three and a half percent perpetuity is worth another eight and a half billion. What the city actually purchased was not a filtration service. It was a reduction in the probability and severity of a contamination event, and it is the largest regenerative infrastructure decision in the modern record.

Quintana Roo, Mexico, 2019. A stretch of the Mesoamerican Reef and the beach behind it became, for the first time anywhere, the named subject of a parametric insurance policy — arranged with Swiss Re and The Nature Conservancy, triggered by wind speeds at or above one hundred knots within a defined polygon, paying into a trust that funds reef repair. It paid after Hurricane Delta in 2020. Consider what that contract did: it took a living system whose economic contribution is entirely the reduction of variance in coastal loss, and gave that contribution a price, a trigger, a counterparty and a settlement date.

The Netherlands. Room for the River — some 2.3 billion euros across thirty-four sites, completed in 2019 — widened floodplains rather than raising dykes. The return is a distribution with a thinner tail.

The Caribbean. CCRIF SPC, formed in 2007, pays member governments parametrically within fourteen days of a qualifying event. Its product is liquidity at the exact moment liquidity is most expensive, which is the purest commercial expression of what variance reduction is worth.

Four continents, one pattern, and it is the pattern the rest of this chapter formalises: in every case the asset's return was a change in the shape of a distribution, and in every case somebody had to invent a way to book it.

THE ARITHMETIC

What works, what does not, and where the line sits

First, the equation. It has three terms and two of them are not empirical.

Frank Ramsey wrote it in 1928, in a paper whose first move was to declare the discounting of future utilities "ethically indefensible" and to do it anyway for tractability. The modern statement:

        rho  =  delta  +  eta · g

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

Read the three terms as three different kinds of claim, because they are.

g is a forecast. It is empirical, arguable, and the least controversial of the three. delta is the rate at which you discount a future person's wellbeing purely because they are in the future. There is no measurement that settles it; it is an ethical position, and the classical utilitarian answer — Ramsey's, Pigou's, Sidgwick's — is that the only defensible non-zero value is the probability that there is no future person there at all. eta is the one people misread. It governs how much less an extra pound is worth to a richer person than to a poorer one, so it is simultaneously a statement about inequality between generations, a statement about inequality within a generation, and — in the standard time-separable formulation — a statement about aversion to risk. Those three should not be forced to share one parameter, and under Epstein and Zin's (1989) recursive preferences they are not; under the specification almost every appraisal actually uses, they are.

Second, the dispute, reduced to the two numbers it turns on.

The Stern Review used delta of 0.1 percent — extinction risk only — and eta of 1, with growth of 1.3 percent. That yields 1.4 percent. William Nordhaus's DICE calibration used delta of 1.5 percent and eta of 2, with growth nearer 2 percent, yielding about 5.5 percent, chosen so that the model reproduced observed market returns on capital.

Hold g common at 1.3 percent, and the whole gap is 2.70 percentage points:

  from delta   1.40 pp     52% of the gap
  from eta     1.30 pp     48% of the gap

The split is almost exactly even, and neither half is empirical. One half is a claim about whether future people count as much as present people. The other is a claim about how much less a pound matters to someone richer. The famous disagreement over what to spend on climate is, arithmetically, those two sentences and nothing else.

What it does to a number is not subtle:

  a benefit of GBP 1,000,000 arriving in year 100
     at Stern's     1.4%    ->  GBP 249,003
     at Nordhaus's  5.5%    ->  GBP   4,729
     ratio                       52.7x

And delta alone, applied to a person alive in 2300 — two hundred and seventy-five years out — weights them at 76.0 percent of a person alive today under Stern, and at 1.67 percent under Nordhaus. A factor of 45.6, from one ethical parameter.

Third, a worked sensitivity, because a single rate is a false report.

Hold g at 1.5 percent and vary the two contested terms across ranges that serious people defend. Present value of one million pounds arriving in year 100:

  delta    eta      rho      PV year 100     half-life
   0.1%   1.00    1.60%        204,470        43.7 yr
   0.1%   2.00    3.10%         47,221        22.7 yr
   0.5%   1.00    2.00%        138,033        35.0 yr
   0.5%   2.00    3.50%         32,060        20.1 yr
   1.0%   2.00    4.00%         19,800        17.7 yr
   1.5%   2.00    4.50%         12,257        15.7 yr

Seventeen-fold, across parameters no one could call unreasonable. Nothing else in a business case has that much leverage and that little evidence behind it, and a paper that reports one rate without this table has not reported its result. It has reported one cell of it.

Fourth: what three and a half percent actually does.

  GBP 1,000,000 at 3.5%, flat
     year  10      GBP 708,919
     year  30      GBP 356,278
     year  50      GBP 179,053
     year 100      GBP  32,060
     year 200      GBP   1,028

The half-life is ln 2 / ln 1.035 = 20.1 years. A benefit halves in present value roughly every twenty years, so a benefit in year 100 is worth about one thirty-first of itself. A million pounds of avoided harm delivered to your great-grandchildren enters today's appraisal as thirty-two thousand pounds — and at Nordhaus's 5.5 percent, as four thousand seven hundred.

Apply the Green Book's declining schedule instead and the same year-100 million is worth £50,818, 1.59 times the flat rate. At year 200 the schedule gives £6,207 against £1,028 flat — 6.04 times. The declining schedule is not a rounding. It is the difference between a century being worth considering and not.

Fifth: why the schedule declines — and it is not impatience.

This is where most readers acquire a wrong reason for a right practice. The sound argument is Martin Weitzman's and it is arithmetic, not psychology. Suppose you do not know the true rate: it is one percent or seven percent, equally likely and persistent. You must average the discount factors, never the rates.

  horizon     E[factor]     implied rate     mean 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 four percent toward one percent — toward the lowest branch — because by year 100 the high-rate branch has annihilated itself and contributes nothing left to average. Uncertainty about the rate, entirely on its own, produces a declining schedule. Weitzman's 2001 survey of 2,160 economists produced the same shape empirically: 4 percent for years one to five, 3 percent to year twenty-five, 2 percent to seventy-five, 1 percent to three hundred, zero beyond.

Sixth: hyperbolic discounting — what the evidence carries, and what it does not.

Richard Thaler's 1981 subjects, offered fifteen dollars today, were indifferent to twenty dollars in a month, fifty dollars in a year, and a hundred dollars in ten years. Those imply annual rates of 345 percent, 120 percent and 19 percent respectively. One exponential rate cannot produce three answers. The measured rate falls as the horizon lengthens, which is the hyperbolic finding, and David Laibson's quasi-hyperbolic form captures it with one extra parameter: everything beyond the present moment is scaled by beta, and then discounted normally. With beta at 0.7, a hundred dollars today beats a hundred and ten tomorrow — 76.98 against 100 — while a hundred and ten in 366 days beats a hundred in 365, 68.99 against 62.74. The preference reverses with no new information. That is a real and robust phenomenon, and it is why commitment devices work.

Now the part usually left out. Monetary present bias largely disappears when the experiment is run properly. Andreoni and Sprenger (2012), putting a front-end delay on the sooner payment and using convex budgets, recover a beta close to one. Augenblick, Niederle and Sprenger (2015) find substantial present bias over real effort — beta near 0.9 — in the same subjects who show almost none over money. The explanation is not mysterious: money is fungible with a bank account, so a laboratory monetary discount rate is partly a measurement of the subject's access to credit. Effort cannot be arbitraged.

And the normative conclusion, which matters more than either result: none of this licenses a declining social discount rate. An individual whose preferences reverse is time-inconsistent, which is a defect to be bound — that is Strotz's 1955 finding and it is why we have mortgages, pension lock-ins and constitutions. A planner who adopts hyperbolic discounting as policy will revise every plan the moment it becomes the present. The defensible case for a declining schedule is Weitzman's and Gollier's uncertainty argument above, and the Ramsey terms themselves. Behavioural evidence describes what people do. It does not tell a treasury what to do.

Seventh: variance reduction is not a cash flow, and must not be discounted as one.

Here is the movement this edition needs. A regenerative asset frequently returns a change in the distribution of a future cost, not an addition to a future revenue. Two errors follow, and organisations commit them in roughly equal numbers: they omit it, valuing it at zero because it has no line; or they include it and discount it at the equity hurdle rate, which is wrong in the opposite direction.

Do it properly in two steps.

Step one — convert the variance reduction into a certainty equivalent. For a decision-maker with constant absolute risk aversion a facing a cost with mean mu and variance sigma², the certain-equivalent cost is mu + (a/2)·sigma². Removing variance is therefore worth (a/2)·ΔVar as a certain amount, and 1/a is the firm's risk tolerance in the same units.

Step two — discount each piece at its own rate. The certainty equivalent is certain by construction, so it takes the risk-free rate. The change in the mean is uncertain and — this is the whole point — it is largest exactly when things are worst, so its consumption beta is negative and its rate is r_f + beta · premium, which sits below the risk-free rate.

Work it. A processing plant carries a water-related production loss: with probability 0.15 a drought year costs £8.0m, otherwise nothing. Expected loss £1.20m a year; variance p(1−p)L² = 8.1600 (£m)², standard deviation £2.857m. A watershed restoration costing £6.0m of capital and £0.20m a year cuts the probability to 0.12 and the drought-year loss to £3.0m. Expected loss £0.36m; variance 0.9504; standard deviation £0.975m.

  mean saving, net of opex            GBP 0.64m per year
  variance removed                    7.2096 (GBP m)^2
  risk tolerance 1/a, at a = 0.15     GBP 6.67m
  certainty equivalent (a/2)·dVar     GBP 0.5407m per year

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 in the country.

Now the three treatments, over twenty-five years:

  treatment                                              PV      NPV
  mean only, at the 10% hurdle                        5.81    -0.19
  mean only, at 3.5%                                 10.55    +4.55
  mean at 0.5% risk-adjusted, plus variance at 2.0%  25.56   +19.56

Same project, same physics, three answers, and only the third is correct in theory.

And now the honest negative, which is the same machinery pointed the other way. Shorten the horizon to ten years — the plant's lease expires — and price the variance: NPV +£5.08m. Shorten it to ten years and value the mean only at 3.5 percent: NPV −£0.68m. Two competent analysts, one project, opposite recommendations, and neither has done anything dishonest. The rate and the horizon decide the outcome, and both are assumptions. Any use of this method that does not publish both alongside the answer is advocacy wearing a spreadsheet.

There is a second negative, and it is structural. The consumption-CAPM that justifies the low rate for a hedging asset gets the sign right and the size badly wrong. With delta 0.5 percent, eta 1.5, growth 1.8 percent and consumption volatility 2 percent, the model's risk premium per unit of beta is eta·sigma² = 0.060 percentage points, against an observed equity premium near six points — a hundredfold understatement, which is Mehra and Prescott's puzzle of 1985, and the same parameters produce a risk-free rate of 3.155 percent against an observed rate near one, which is Weil's. Take the sign from the theory and the size from a market that already prices this risk: the reinsurance quote, the catastrophe bond spread, the parametric premium. Those are observed numbers with counterparties attached.

Eighth, and last: a low rate justifies almost anything.

  a perpetual benefit of GBP 1 per year justifies a capital cost of
     at 10.0%     GBP  10.00
     at  7.0%     GBP  14.29
     at  3.5%     GBP  28.57
     at  1.4%     GBP  71.43

Moving from seven percent to Stern's 1.4 multiplies the admissible cost by five. At that rate very nearly any project with a permanent positive flow clears very nearly any finite cost — which means the rate has stopped discriminating between projects and all the real work has silently moved to the benefit estimate, which is far less well identified than the rate ever was. That is a failure of the instrument, and it is not a licence.

And a low rate is symmetric, which is the part its advocates rarely price. A one billion pound decommissioning duty falling in year 100 requires a provision of £1.2m at seven percent, £32.1m at three and a half, and £249.0m at 1.4 — two hundred and sixteen times more. Whoever argues the low rate for their project has argued it for their clean-up, their nuclear custody, their tailings dam and their pension promise in the same breath. Consistency here is not a nicety; it is the only thing that makes the argument survive an auditor.

DREAM

What becomes ordinary

In the organisation that has absorbed this, the capital paper carries three rates and says why.

The first page of every long-lived proposal shows a central case, a low case and a high case, and the range is drawn from the Ramsey terms rather than from sentiment — a stated delta, a stated eta, a stated growth assumption, each with a name beside it. The committee argues about eta explicitly, in the language of who is richer than whom, and the argument takes eleven minutes and is minuted. Nobody experiences this as philosophy. It is simply the part of the paper where the assumptions are declared, in the same spirit as the tax rate.

The treasury function maintains two discount rates, not one, and they are used for different things. A hurdle rate applies to cyclical, revenue-bearing projects that do well when the business does well. A second, lower rate applies to counter-cyclical assets — the ones that pay when everything else is failing — and the finance director can explain in a sentence why an insurance-shaped asset is worth more than a revenue-shaped asset of the same expected value.

Risk registers carry distributions instead of ratings. Where a register once said high against water availability, it now carries a probability, a loss given event, and a variance, all updated annually against the actual weather; and the regenerative options appear in the capital plan with their variance reduction costed and their certainty equivalent stated. The insurer is in the room for that conversation, because the insurer is the counterparty whose quote turns the estimate into a price.

And the long-lived things in the portfolio stop being orphans. The soil programme, the aquifer recharge, the apprenticeship, the repairable line, the sea wall — none of them any longer arrive at a committee asking to be believed in. They arrive with a rate, a horizon, a sensitivity table and a counterparty, which is to say they arrive as finance.

DESIGN

The structure that gets there

Four moves, in order, none of which requires anyone's permission to begin.

One: write down the rate you are actually using, and decompose it. Most organisations cannot do this. Find the hurdle rate, find who set it and when, and write it as delta + eta·g even if the original construction was nothing of the kind. The exercise is not academic — it converts a folk number into a declared assumption, and a declared assumption can be discussed. The Green Book's whole authority comes from having done this one thing in public.

Two: adopt a declining schedule for anything beyond thirty years, and take it off the shelf rather than inventing it. The Green Book's table is published, free, and defensible in front of any auditor precisely because a finance ministry stands behind it. The design principle is the one from Volume I: never construct an authority where you can cite one.

Three: split the rate by beta, not by department. Two rates, published: a hurdle rate for pro-cyclical revenue projects, and a lower rate for counter-cyclical protective assets, with a written rule for deciding which is which. The rule is a single question — does this pay most in the years the rest of the business does worst? — and it is answered with historical data, not with judgement, wherever three cycles of data exist.

Four: price variance explicitly, in three lines. Every long-lived proposal carries, on one page: the change in the expected cost; the change in its variance; and the certainty equivalent of that change at the firm's own stated risk tolerance. Publish a, or equivalently 1/a, once, as a policy parameter set by the board — the way an insurance retention is set — rather than letting each analyst choose one. That single act removes the largest degree of freedom in the whole method, which is the same act that makes the method trustworthy.

Governance. The rate, the schedule, the split and the risk tolerance are four numbers, and all four belong in one page of treasury policy, reviewed annually, signed. They must not live in a spreadsheet template, because a number that lives in a template is owned by whoever last edited it.

Sequence. Decomposition first, because it costs nothing and creates the vocabulary. The declining schedule second, because it is a citation. The beta split third, because it needs the vocabulary. Variance pricing last, because it needs the board to set a, and boards set parameters they have seen used.

DESTINY

How it holds when nobody is pushing

A rate policy survives for the same three reasons any practice survives, and fails for one that is specific to this chapter.

It holds when the sensitivity table is mandatory in the template. Not encouraged: in the form, unfillable-around. A single rate cannot then be reported, because the form has three cells.

It holds when the risk tolerance is a board parameter with a date on it. A number the board owns is a number the board revisits; a number an analyst owns is a number that drifts to whatever makes this quarter's paper pass.

It holds when the insurer, or internal audit, countersigns the variance estimate. An external mark converts an assumption into an observation, and the whole method depends on the difference.

And here is how it fails, honestly. It fails when the low rate becomes a tactic — when people notice that arguing the rate down is easier than arguing the benefit up, and the rate becomes the place where every disagreement is secretly settled. Within two cycles the rate is no longer a parameter; it is a negotiating position, and everyone in the room knows it. The defence is the symmetry rule and it must be written into the policy: the same rate applies to liabilities as to benefits, in the same paper, always. A team that wants 1.4 percent for its restoration project must accept 1.4 percent on its decommissioning provision, and the two hundred and sixteen-fold increase that brings. Symmetry is what makes the rate an instrument again rather than a lever.

It also fails, more quietly, when the horizon is set by a lease, a tenure or a budget cycle rather than by the asset. The rate gets all the attention and the horizon does all the damage. Put the horizon on the front page next to the rate, and let the two be argued about together.

DELIGHT

What it feels like

There is a specific pleasure in the meeting where the rate is finally said out loud. Somebody asks where the twelve percent came from, and instead of the usual shrug there is an answer: it is a pure time preference of half a percent, an elasticity of one, growth of two, and a risk premium of eight and a half that belongs to this kind of project and not to the one we are discussing. The room changes temperature. A number that had been weather becomes a decision, and a decision can be made differently.

Then there is the quieter pleasure of the counter-cyclical asset finally pricing. You have known for years that the thing which pays in the bad year is worth more than its expected value suggests — everyone who has run an operation through a drought or a recall or a flood knows it in their body — and one afternoon there is a line of arithmetic that says so, with a sign on it, and an insurer willing to quote against it. The intuition was right. It was simply homeless.

And there is something companionable about a long table. Working at a horizon of a century is not sombre work. It is the opposite of anxious: it is the one part of finance where being careful and being generous point in the same direction, and where the arithmetic, done honestly, mostly tells you to protect things.

OPERATIONALIZE THIS

At the level of finance

The instrument: a resilience facility repaid from the risk-transfer premium.

Not a green bond, which prices off the issuer's credit and pays for a label. This is narrower, harder and more bankable: a facility that funds regenerative capital expenditure and is repaid out of the measured reduction in the cost of carrying the risk — the insurance premium, the retained-loss provision, and the expected loss itself. It is the structure of an energy performance contract with an underwriter as the verifier.

The mechanics.

Structure. Internal first: treasury lends to the operating unit against a defined risk. Size it at verified capital cost plus fifteen percent, inside one signature, exactly as in Chapter I.01.
The baseline. Three quantities, agreed and signed before deployment: the annual probability of the loss event, the loss given event, and the current risk-transfer premium for that layer. The insurer's own submission is the cleanest baseline available, because it was written by a counterparty with money at stake and no interest in flattering you.
The trigger. Parametric where possible — a river gauge, a wind speed, a soil-moisture index, a rainfall deficit — because a parametric trigger settles in days rather than years and cannot be argued about. Quintana Roo and CCRIF are the working precedents.
Repayment. The measured reduction in premium plus an agreed share of the reduction in expected retained loss, until the facility clears; then full reversion to the operating unit.
Term. Set past the crossover and inside the asset's life, and put the horizon on the front page beside the rate.
Security. The premium reduction, which is contractual, rather than the avoided loss, which is statistical. This single choice is what makes the structure financeable.

The balance-sheet treatment. Capitalise the restored asset and depreciate over its regenerated life. The return arrives in three distinct places and they must not be double-counted: a lower insurance premium in operating expense; a smaller retained-loss provision, which under IAS 37 releases through the profit and loss account; and the certainty-equivalent value of the variance reduction, which is not booked anywhere and belongs in the appraisal only. Say so explicitly in the paper. Under IFRS S2 the resilience narrative and its quantitative effects are now disclosable, which means this arithmetic has an external audience for the first time — that is an opportunity rather than a burden, because a number that must be disclosed is a number that must be built.

The counterparty. The insurer or reinsurer that already writes the layer. They hold the loss model, they reprice annually, and they have every incentive to verify a genuine reduction. Brokers will structure a multi-year agreement with a premium adjustment clause tied to the intervention; that clause is the repayment stream and it is the whole transaction.

The number that decides it. One line, on the front page:

   premium saved  +  expected loss avoided  +  (a/2)·dVar
   ---------------------------------------------------------  >  r_f + beta·premium
                  capital cost  +  M&V cost

For the worked case: (0.32 + 0.64 + 0.5407) / (6.0 + 0.15) = 24.4 percent, against a hedge-adjusted rate of 0.5 percent and a conventional hurdle of 10 percent. It clears both. Put the premium term first, because it is the term an underwriter will countersign and therefore the term a credit committee will believe.

The first ninety days.

Day	Action	Artifact
1–15	Decompose the rate you currently use into `delta`, `eta`, `g`	One page, signed by treasury
16–30	Adopt the Green Book schedule for anything beyond thirty years	Treasury policy amendment
31–45	Pull the insurer's submission; extract probability, severity, premium	The signed baseline
46–60	Board sets the risk tolerance `1/a` as a policy parameter	Board minute
61–75	Price one regenerative candidate in three lines: mean, variance, CE	The three-line appraisal
76–90	Take the premium-adjustment clause to the broker	Term sheet with a trigger

APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

When has this organisation knowingly paid for something that only mattered in a bad year — and what made that decision possible at the time?
Where do we already use a rate other than the standard hurdle, and who authorised the exception? What did they see?
Which of our assets has quietly reduced how much our results move about, and how would we prove it from our own history?

Dream — what becomes possible

If every long-lived proposal arrived with three rates and a stated horizon, what would we approve next year that we would not approve this year?
Imagine our insurer as a partner in capital planning rather than a supplier of cover. What is the first conversation we would have?
If the board set the firm's risk tolerance as one published number, what would become arguable that is currently only assertable?

Design — what we build

What is the smallest long-lived asset we could appraise properly — mean, variance, certainty equivalent — inside this quarter?
Whose signature converts our discount rate from a habit into a policy, and what would they need to see first?
What data do we already hold that would tell us whether a given asset pays most in our worst years?

Destiny — how it holds

What would have to be true for the sensitivity table still to be in the template when everyone in this room has moved on?
If someone tried to win an argument by moving the rate rather than the evidence, who would notice, and what would they say?
Which of our liabilities would we be willing to discount at the same low rate we want for our best project — and what does our answer tell us?

WORKS CITED

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Samuelson, P. A. (1937). "A Note on Measurement of Utility." The Review of Economic Studies, 4(2), 155–161.

Strotz, R. H. (1955–56). "Myopia and Inconsistency in Dynamic Utility Maximization." The Review of Economic Studies, 23(3), 165–180.

Koopmans, T. C. (1960). "Stationary Ordinal Utility and Impatience." Econometrica, 28(2), 287–309.

Ainslie, G. (1975). "Specious Reward: A Behavioral Theory of Impulsiveness and Impulse Control." Psychological Bulletin, 82(4), 463–496.

Lucas, R. E. (1978). "Asset Prices in an Exchange Economy." Econometrica, 46(6), 1429–1445.

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Note on figures. Every figure in this chapter and its apparatus is computed in lib/verify/III_05.py and printed there with its inputs, units and source. The Green Book, Lebègue, Quinet, Hagen, OMB and EPA rates are quoted from the documents named above; the Ramsey decompositions, discount factors, annuity factors, certainty equivalents and sensitivity tables are computed.