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Commerce · VII.08 · MMXXVI · daylight

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WORKBOOK — THE STUDENT

Chapter VII.08 · The Long View and the Discount Rate

For the person studying this alone, or in a seminar, with no fund to manage and no board to persuade. You have something the executive does not: the longest horizon in the room is yours, and you can practise on it before anyone is depending on the answer.

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WHY THIS WORKBOOK IS DIFFERENT

The chapter is written for someone who signs a trust deed. You will probably not sign one for fifteen years. It would be reasonable to conclude that the material has to wait.

It does not, and the reason is the most useful sentence in this workbook: the long view is not a technique for managing endowments. It is a way of deciding under ignorance, and the ignorance is what you have most of. The chapter's central result — that widening uncertainty raises the weight on the far future — is a result about how to act when you cannot forecast. You cannot forecast. That is not a deficit at your stage; it is the exact condition the theorem was written for.

So you will do what the treasurer does. You will do it on the system you actually control, which for the next decade is a life with a very long remaining horizon and almost no committed capital. That is the strongest position in this chapter and almost nobody occupies it twice.

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PART ONE — DISCOVERY

Weeks 1–4: find the commitments that already held

Exercise 1.1 — The inherited ledger (90 minutes)

List ten things you use regularly that somebody paid for and never saw used. A bridge. A vaccine schedule. A body of law. A tree. A library. A dataset. A language's orthography. A building you study in. A sewer.

For each, write three columns: roughly when it was paid for, roughly what it cost, and — the hard one — what institution held the commitment between the payment and the use. Not the person. The institution. A charter, a trust, a statute, a guild, a church, a family. The answer to that third column is the whole subject of this chapter, and you will find that in about seven of the ten cases you do not know it.

Go and find out for two of them. That is the assignment.

Exercise 1.2 — Your own longest horizon (45 minutes)

Write the year you expect to be alive in that is furthest away and still plausible. Then write:

• what you will have done between now and then that will still exist,
• what decision you are making this year that reaches that far,
• and the discount rate you are implicitly applying to it.

Most people discover the third line is somewhere north of thirty percent a year, because they are behaving as though anything past five years is noise. Write the number. Do not defend it yet.

Exercise 1.3 — The appreciative interview (60 minutes, with another person)

Find somebody who has held a commitment for more than a decade — a researcher, a grower, a builder, a trustee, a parent, a craftsperson — and ask exactly this:

"Tell me about a commitment you kept for longer than was comfortable. Not the proudest one, the one that surprised you by lasting. What held it? What almost undid it, and what happened instead?"

Then stay quiet. Take notes on the mechanism, not the virtue. You are collecting commitment devices, and the person will usually name one without realising it: a contract, a witness, a public statement, a physical object, a second owner.

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PART TWO — THE ARITHMETIC

Weeks 5–9: compute it before you argue it

Exercise 2.1 — Reproduce the chapter (3 hours)

Open lib/verify/VII_08.py, read it, and then close it and rebuild these four results from the definitions alone, in a spreadsheet or in twenty lines of code:

1. The certainty-equivalent spot rate at horizons 0 to 1,000 for seven equally likely rates from one to seven percent. You should get 2.49 percent at year 100 and 1.63 at year 300.
2. The forward rate at the same horizons, Σp_i r_i e^(-r_i t) / Σp_i e^(-r_i t), and the weight on the lowest branch. You should get 63.3 percent at year 100.
3. The same-mean widening table. Certainty gives £6.14 on a million at year 300; a 0.5–4–7.5 spread gives £74,378.77.
4. The Green Book factor to year 200, which is 0.006207379.

If your numbers differ, yours is the one to trust until you find the discrepancy. That is not politeness. A figure you recomputed is a figure you own, and this book is built to be checked.

Exercise 2.2 — Break the model on purpose (60 minutes)

Run the normal-distribution closed form, R(t) = mu - sigma²t/2, with a mean of four percent and a standard deviation of two points. Find the year the rate reaches zero. It is 200, and 2mu/sigma² says so exactly.

Now answer in writing: why is that a fact about the model and not about the world? The answer is one sentence and it is the most transferable thing in the chapter — a distribution with no floor produces a limit with no floor, and the theorem's limit is the lowest rate in the support.

Exercise 2.3 — The non-identity calculation (90 minutes)

Compute (1 - q)^(2^k - 1) for q of 0.001, 0.01 and 0.05 and k from one to twelve. Confirm 27.9 percent at the seventh generation with q of one percent.

Then write six hundred words on what that licenses. Be strict with yourself. There is a strong temptation to conclude either therefore we owe them nothing or therefore the problem is a sophism. Both are wrong. It defeats a claim of harm to an individual and leaves every claim about the quality of the world standing. Defend that sentence or refute it; do not go around it.

Exercise 2.4 — The population term (60 minutes)

Compute n for the UN medium and low variants and produce the four-row table. Then answer: which criterion were you tacitly using before you read this chapter, and had you ever declared it?

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PART THREE — DREAM AND DESIGN

Weeks 10–14: build the apparatus

Exercise 3.1 — Your own commitment device (2 hours)

Take one commitment you actually want to hold for more than ten years. Now design it against the chapter's six devices, and write the cost of each in your case:

Choose one. Write which one and why the cost is one you are willing to pay. That last clause is the exercise; a device chosen without naming its cost is a resolution, and resolutions evaporate.

Exercise 3.2 — The calendar form (60 minutes)

Take any plan of yours with a horizon past five years. Write it twice: once as years from now, once as calendar dates. Then put the two side by side and find where they disagree — they will, because the years-from-now version quietly re-bases every time you revisit it. This is the 64.5 percent from the chapter, happening in your own notebook.

Exercise 3.3 — Read the primary source (3 hours)

Read Weitzman (1998) — it is eight pages — and then Gollier and Weitzman (2010), which is four. Then read the Green Book's Table 6.1 and its surrounding paragraphs. Write one page on where the published table departs from the theorem and why a ministry might have chosen the departure.

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PART FOUR — DESTINY AND DELIGHT

Weeks 15–18: make it hold, and enjoy it

Exercise 4.1 — The failure audit, run forward (60 minutes)

The chapter names the Nuclear Waste Fund failure: the money was collected faithfully for thirty years and the repository was not built. Funding a duty and discharging it are two different commitments, and only one was instrumented.

Look at your own commitment from 3.1 and answer: is there a version of it where you do the funding — the saving, the studying, the preparing — and never do the discharging? Almost always there is. Write the delivery covenant: the date, the observable act, and the person who would notice if it did not happen.

Exercise 4.2 — Sit with the markers (30 minutes, no writing)

Look up the WIPP marker designs and the EPA's million-year standard. Read the draft warning texts. Then do nothing with it for half an hour.

This is not a reflective flourish. Working at a horizon you will not see is a capacity that develops with exposure, the way a long-distance runner's sense of pace does, and it does not develop by being argued about. The people who designed those markers were engineers and lawyers doing an ordinary job, and the fact that the job existed at all is the most hopeful object in the chapter.

Exercise 4.3 — The two Ramseys (2 hours)

Read the first two pages and the last two pages of Ramsey (1928). Notice that the man who gave the world the discounting equation opened by calling the practice indefensible and then used it anyway for tractability, and that his own preferred device — the Bliss point — exists precisely because a zero rate over an infinite horizon has no maximum.

Then compute the two columns yourself: a flow of one per year for 100, 1,000 and 10,000 years, undiscounted and at half a percent. The second column converges to 200.00, which is 1/0.005. The first does not converge at all.

Write four hundred words answering one question: is the divergence a reason to discount, or a reason to change what you are maximising? Ramsey thought the second. Koopmans in 1960 argued the first. Diamond in 1965 showed the choice is narrower than either of them hoped. This is the cleanest example you will meet of an ethical position that is right and technically incomplete, and learning to hold both halves of that at once is worth more than the answer.

Exercise 4.4 — Teach one result (90 minutes)

Explain to somebody outside your field why widening uncertainty raises the value of the far future. You have succeeded when they can restate the mechanism — the high-rate branches annihilate themselves out of the average — without using the word discount.

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THE TERM PROJECT

One piece of work, carried the whole way

Build an implied-return calculator for a real long-horizon obligation, and publish it.

Pick a real one with public documents: a national decommissioning programme, a university endowment with a stated spending rule, a sovereign wealth fund, a municipal pension scheme, a conservation trust.

1. The liability schedule, undiscounted, by calendar year. This is 70 percent of the work and nobody does it. Most published numbers are already discounted at a rate buried in a note.
2. The asset pot, as reported, with its date.
3. The implied required real return — invert the funding ratio by solving for the rate that makes the pot equal the schedule's present value.
4. The trailing realised real return, as long a window as the fund publishes.
5. The restatement gap: (3) minus (4).
6. A one-page write-up, opening with the implied return and the gap, not with the funding ratio.

Ten to fifteen pages, plus your code. State your denominator: which liabilities you could not find, which years you interpolated, which numbers you could not verify. A survey that hides its gaps is an advertisement.

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SELF-ASSESSMENT

Score honestly, one to five.

1. I can explain why you average discount factors and not rates, to someone who has never seen the equation.
2. I can derive the declining certainty-equivalent rate from a distribution without looking it up.
3. I can state what the non-identity problem does and does not license, in two sentences, without overclaiming in either direction.
4. I can name the population criterion I am using and say what happens to it when population peaks.
5. I can take a funding ratio and turn it into an implied required return.
6. I have named one commitment device for one of my own commitments and named its cost.
7. I know the horizon past which I would stop pricing and start setting a standard, and I can say why.
8. I recomputed at least four of the chapter's figures myself and found where I disagreed.

Under three in the average, work the arithmetic exercises again. Over four, you are ready to do somebody else's fund, which is the term project.

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CARRYING IT FORWARD

Three things from this chapter will still be useful to you in thirty years, when every rate in it is out of date.

The operation. Average the thing you care about, not its logarithm. This generalises far past discounting and it will save you from a whole class of error in statistics, in risk, and in reading other people's summaries.

The instrument question. At this horizon, does a price still carry information? When the answer is no, reach for a constraint and optimise cost subject to it. Knowing which of the two you are holding is most of good judgement about long things.

The inversion. Ignorance is an argument for the long view. You will meet the opposite claim — we cannot know, therefore we should not weight it — perhaps fifty times in your career, from people who are otherwise careful. You now have an arithmetic answer and a factor of twelve thousand.

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APPRECIATIVE QUESTIONS FOR YOUR SEMINAR

1. When has a commitment made before any of us were born turned out well for us, and what specifically held it in the gap?
2. Where in our own field does somebody already use a standard rather than a price, and what did that make possible?
3. If our department published the real return it assumes on every long-lived thing it owns, what would become arguable that is currently only assertable?
4. What is the longest-horizon decision any of us will make in the next five years, and what would it take to make it well?
5. Which of us has already built a commitment device without calling it one?
6. What would we want a seminar in this room in a hundred years to have inherited from us — and what is the first instalment?