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Commerce · III.05 · MMXXVI · daylight

La Bourse  /  Volume III  /  Nº III.05  /  Workbook — the student

WORKBOOK — THE STUDENT

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

For the person studying this alone, or in a seminar, with no balance sheet to apply it to yet. The discount rate is the most portable idea in this edition: you are already using one, several times a day, and this term you are going to find out what it is.

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WHY THIS WORKBOOK IS DIFFERENT

The chapter was written for someone who signs capital papers. You may not sign anything yet. It would be easy to file discounting under things I will need later.

It is not later. You are running a portfolio right now — of hours, of sleep, of skill, of relationships, of money you do not have much of — and every allocation you make across time is a discount rate revealed. The person who reads this chapter and does nothing has a discount rate. They simply cannot say what it is, which means they cannot notice when it changes, and it changes most in exactly the situations where it matters most.

So the work of this term is threefold and none of it requires an institution. Learn to compute. A person who can put a date on "it pays off eventually" changes what a room can decide, and almost nobody in any room can do it on request. Learn to decompose. Every rate you meet — a hurdle rate, an interest rate, a government appraisal, your own impatience — can be pulled apart into terms and argued about term by term. Learn to price what has no line. That is the part that is genuinely new, and it is the part that will still be rare in ten years.

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PART ONE — DISCOVERY

Days 1–30: find the rates already in use

Exercise 1.1 — The rate inventory (90 minutes)

Find and write down, with its source, every discount rate you can locate in documents you already have access to. Aim for eight.

For each, write one sentence: who set this, when, and can I find the construction? You will find that almost none publish the construction, and that the ones that do — the Green Book, the French and Norwegian schedules — are the ones nobody argues with. That relationship is not a coincidence and it is the main lesson of the exercise.

Exercise 1.2 — Your own revealed rate (60 minutes)

Answer these honestly and quickly, without calculating:

1. What would someone have to give you in a month to make you give up £100 today?
2. In a year?
3. In ten years?

Now compute the implied annual rate for each, using r = ln(amount / 100) / t. Thaler's 1981 subjects produced 345 percent, 120 percent and 19 percent from exactly this instrument, and the fact that the three numbers differ is the whole of the hyperbolic finding.

Then ask the sharper question: how much of your one-month answer is impatience, and how much is that you have no savings? Someone with £10,000 in the bank would simply lend themselves the money and take the later payment. A monetary discount rate measured on a person without reserves is partly a measurement of their lack of reserves — which is why Andreoni and Sprenger's front-end delay matters, and why present bias over effort survives when present bias over money largely does not.

Exercise 1.3 — The appreciative interview (45 minutes, with another person)

Find someone who has held something for a long time — an instrument, a garden, a craft, a friendship, a piece of land — and ask exactly this:

"Tell me about something you have kept for much longer than was convenient. What did you have to give up to keep it? When did it start paying you back, and did you know in advance that it would?"

Take notes on the conditions, not the outcome. What you are collecting is a set of real crossover stories, told by people who lived through the years when it was not yet obvious. You will need them when you start putting numbers on horizons, because the arithmetic is easy and the holding is not.

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PART TWO — THE ARITHMETIC

Days 31–50: compute before you argue

Exercise 2.1 — Reproduce the chapter's figures (3 hours)

Do not take the numbers on trust. Open lib/verify/III_05.py, read it, then compute these independently — by hand, in a spreadsheet, or in whatever language you use.

1. Stern's rate: 0.001 + 1 × 0.013. Confirm 1.4 percent.
2. Nordhaus's: 0.015 + 2 × 0.020. Confirm 5.5 percent.
3. The present value of £1,000,000 at year 100 under each. Confirm £249,003 and £4,729, and the ratio of 52.7.
4. The half-life at 3.5 percent: ln 2 / ln 1.035. Confirm 20.1 years.
5. The Green Book declining schedule at year 100: 1.035⁻³⁰ × 1.03⁻⁴⁵ × 1.025⁻²⁵. Confirm £50,818, or 1.59 times the flat rate.
6. Weitzman's certainty-equivalent rate at year 100 from a 1 percent / 7 percent coin flip. Confirm 1.70 percent, and satisfy yourself about why it is not 4 percent.

Then do the thing that matters most: find one figure in this chapter you can check against an outside source, and check it. Bring any discrepancy to your seminar. This edition wants to be checked; that is the difference between a text and a doctrine.

Exercise 2.2 — Build the sensitivity table yourself (90 minutes)

Reproduce the chapter's table: hold g at 1.5 percent, vary delta across 0.1, 0.5, 1.0 and 1.5 percent, vary eta across 1.0, 1.5 and 2.0, and compute the present value of £1,000,000 at year 100 in every cell. Twelve cells.

Then answer three questions in writing.

• Which single cell would you defend, and on what grounds — empirical or ethical?
• The spread from the highest cell (£204,470) to the lowest (£12,257) is seventeen-fold. Name one other assumption in any appraisal you have ever seen with that much leverage.
• If you had to hand a decision-maker three cells instead of twelve, which three, and why those?

That last question is the professional skill. Anyone can produce twelve numbers. The job is choosing the three that make the decision legible.

Exercise 2.3 — Your own crossover, with a variance term (90 minutes)

Choose a real decision where the cheap option is volatile and the good option is steady. A bicycle against unpredictable fares. Cooking against a food budget that spikes. A fixed-rate contract against a variable one. Health insurance. A better lock.

Work it in four lines.

1. The mean. Expected annual cost each way. Most people stop here.
2. The variance. p(1−p)L² for a simple two-state case, or the variance of your last twelve months of actual spending if you have records.
3. Your risk tolerance. Name 1/a explicitly — the amount of variability in a year that you can genuinely absorb without it damaging something else. Then the certainty equivalent is (a/2) · ΔVar.
4. The rates. Discount the mean at the rate you would actually borrow at. Discount the certainty equivalent at your safest available rate.

Write the answer, then write one sentence on what changed between line one and line four. For most students the honest answer is the decision reversed, and that is the exercise working.

Exercise 2.4 — Find the honest negative (45 minutes)

The chapter contains at least three places where the method loses. Find them. Write in your own words why each is included and what it protects the reader from.

Then practise the move yourself. Take the strongest argument you personally believe about how long-lived things should be valued, and write the version that troubles you most. Not a straw version — the one you cannot answer.

If you cannot write it, you do not yet understand your own position well enough to defend it. That is not a criticism. It is the assignment.

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PART THREE — DREAM AND DESIGN

Days 51–75: build the apparatus

Exercise 3.1 — The decomposition habit (ongoing, 15 minutes each)

For the rest of the term, every time you meet a rate, decompose it in a notebook: what is the pure time preference, what is the growth assumption, what is the risk premium, and what is simply convention nobody has revisited. You will be wrong often. Being wrong in a notebook costs nothing and the habit is the asset.

By the end of the term you will have twenty decompositions, and you will find that you can no longer hear a hurdle rate without asking where it came from. That reflex is worth more than the chapter.

Exercise 3.2 — The present-tense description (60 minutes)

Write five hundred words describing your working life twenty years from now, in the present tense, as description rather than aspiration.

Constraints, and they are the exercise:

• Name one thing you are still holding that you started this year.
• Name the moment it became obviously worth holding, and the year.
• Include the years in between, honestly, including the ones where it was not obvious.
• No adjectives of achievement.

The middle constraint is the one that defeats people. A crossover is not a moment; it is a stretch of years during which you have to keep paying and cannot yet show why. Describing those years accurately is the same skill as sizing a horizon.

Exercise 3.3 — The three-line appraisal (60 minutes)

Take any proposal you can find — a student union motion, a departmental proposal, a family decision, a published business case — and rewrite its economics in exactly three lines: the change in expected cost, the change in its variance, and the certainty equivalent of that change at a stated risk tolerance.

Then note what the original document did with line two and line three. In almost every case the answer is nothing, and you have just found a real omission in a real document, using arithmetic that fits on a postcard.

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PART FOUR — DESTINY AND DELIGHT

Days 76–90: make it hold, and enjoy it

Exercise 4.1 — The symmetry test (30 minutes)

Find an argument — in the press, in a journal, in a seminar — that advocates a low discount rate for a project somebody wants. Then ask the chapter's question: would the same advocate accept that rate on the liabilities?

Write the answer. Where the answer is no, you have found the point at which an analysis became advocacy, and you found it with one question rather than with a critique. Learn this move. It is the cheapest rigour available to you.

Exercise 4.2 — The commitment device, designed (this week)

You now know why commitment devices work: beta is less than one for effort, and your future self is a different negotiator from your present one. Design one for something you actually want to hold across this term.

Make it small, specific and hard to reverse quietly. The test is not whether it is impressive. The test is whether reversing it would require you to do something rather than merely to omit something. Reversal is rarely a decision; it is an absence of renewal.

Exercise 4.3 — Delight, on purpose (ongoing)

Choose the hour, the place and the instrument for the computing part of this work, and make them genuinely pleasant. This is not indulgence. Delight is the adoption mechanism, and it is what determines whether you are still doing the decomposition habit in March.

Write one sentence: the part of this I look forward to is ___. If you cannot complete it, redesign the practice until you can.

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THE TERM PROJECT

One valuation, carried the whole way

Choose one long-lived thing you have genuine access to — a building on your campus, a woodland, a local flood scheme, a piece of public transport, a library's collection, a community asset — and appraise it properly.

Deliverables.

1. The rate memo (six hundred words). The rate you will use, decomposed into delta, eta and g, each defended. State whether you are adopting a published schedule and which one.
2. The sensitivity table (one page). Twelve cells minimum. Three cells chosen for the front page, with the reason.
3. The three-line appraisal (six hundred words plus workings). Expected cost change, variance change, certainty equivalent at a stated risk tolerance. State 1/a and defend it.
4. The beta argument (four hundred words). Does this asset pay most in the years its community does worst? Show the evidence, or say honestly that you could not find any.
5. The symmetry check (two hundred words). Apply your rate to the asset's own long-dated liabilities — decommissioning, maintenance, replacement — and report the result even where it is inconvenient.
6. The reflection (six hundred words). What you expected, what the arithmetic said, and what the gap tells you.

How it is assessed. Not on whether the asset turned out to be worth building. On whether a reader could reproduce every number, and on whether the symmetry check was reported honestly when it hurt.

A valuation that came out against your instinct and was computed cleanly is a first-class piece of work. A valuation that came out the way you hoped and cannot be reproduced is not.

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SELF-ASSESSMENT

The two that matter most are the fifth and the seventh. The rest is technique and can be learned in a fortnight. Those two are habits, and habits take a term.

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CARRYING IT FORWARD

What you will have, if you do this properly, is:

• Twenty decompositions in a notebook, and the reflex that produced them.
• A sensitivity table you built yourself, which almost nobody entering any profession has ever done.
• One completed valuation with a variance term in it — a document you can put in front of anyone.
• The symmetry question, which costs one sentence and reliably finds the weak point of a long-horizon argument.

That last one is worth more than it sounds. Most disagreements about long-lived things are settled, quietly, by moving a rate that nobody looks at. You will be able to see it happen. You can be that person before you have any authority at all, and it is the fastest route to being given some.

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APPRECIATIVE QUESTIONS FOR YOUR SEMINAR

1. When has one of us held something through the years before it paid, and what made holding possible?
2. Which published rate in the world do we find most defensible, and what exactly makes it so?
3. What is already working in how this institution treats its longest-lived assets — and what makes it work?
4. If every proposal in this department arrived with a sensitivity table, what would we approve next year that we would not approve this year?