01Six lectures, six puzzles
Each lecture opens on three items that a surface figure ranks one way and its model ranks another, answers four questions in four parts, and closes on a checkpoint of six questions with fresh numbers. The slides hold 30 frames; the sheet holds the formulas, the firms' inputs, an exercise for pairs with its check values, and a glossary.
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01How a carbon price enters a firm's cash flow, how a scenario sets a price path, what a stream of carbon costs is worth today, and which path costs a firm most once timing and discounting count.Puzzle: The Delayed path prices carbon at USD 700 a tonne in 2050 against the Orderly path's 400, yet at 5 per cent real the Orderly prices from 2026 to 2050 are worth USD 3,689 m per million tonnes emitted a year against Delayed's 3,326.
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02How often a hazard strikes, how large the loss of one event is, what a firm's yearly loss looks like, and how warming changes the losses.Puzzle: Port's largest loss on record is USD 300 m and Rice's USD 25 m, yet Rice's average annual loss is USD 5.03 m against Port's 3.31 m.
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03What a firm's Climate value-at-risk measures, how an earnings cap, cost pass-through and the horizon change it, how a portfolio's figure adds up, and how a scenario figure compares with a quantile.Puzzle: Steel emits 6 million tonnes a year and Cement 3, yet on the Orderly path Cement's Climate value-at-risk to 2040 is −62.4 per cent against Steel's −32.7.
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04How the Merton model turns assets, debt and volatility into a probability of default, how a shock to asset value moves the distance to default, how asset value and volatility follow from the equity price, and why one kind of shock moves three probabilities so differently.Puzzle: Carbon costs from 2026 to 2030 take 17.3 per cent of Cement's asset value and 9.3 per cent of Steel's, yet Steel's one-year probability of default rises 2.412 points against Cement's 0.618.
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05How one common factor makes defaults move together, how the loss of a large book is distributed, how the Basel internal ratings-based formula sets capital, and how a climate factor raises a concentrated book's capital.Puzzle: The Spread book has the highest probability of default, 2.0 per cent, yet at each book's own correlation the Coastal book, at 1.0 per cent, needs the most capital: 9.97 per cent of its lending against Spread's 4.33.
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06How a portfolio's financed emissions and carbon intensity are counted, how a portfolio halves its weighted average carbon intensity, which weights do so at the least tracking error, and which holdings pay as a Paris-aligned path compounds.Puzzle: Cement emits 6,000 tonnes per USD m of sales and Coal 5,000, yet halving the portfolio's weighted average carbon intensity at least tracking error cuts Coal's weight by 11.3 points and Cement's by 9.5.
One 90-minute lecture follows a single hazard through a book of three loans to the bank's capital: Climate risk in a loan book, from hazard to capital.
02Six firms of one economy
The six firms are stylised. Money is in millions of US dollars of 2025, the base year of every discount factor; earnings are a year's earnings before interest, taxes, depreciation and amortisation.
| Firm | Activity | Emissions, million tonnes a year | Earnings, USD m | Asset value, USD m | Debt, USD m | Carbon cost passed on, per cent |
|---|---|---|---|---|---|---|
| Coal | coal power plant | 4.0 | 320 | 2,600 | 1,430 | 80 |
| Cement | cement works | 3.0 | 125 | 1,000 | 350 | 90 |
| Steel | steel mill | 6.0 | 450 | 3,600 | 2,340 | 90 |
| Port | container port on a delta | 0.15 | 240 | 1,500 | 750 | 50 |
| Rice | rice mill on a river plain | 0.03 | 30 | 250 | 100 | 50 |
| Solar | solar farm | 0.004 | 60 | 400 | 280 | 50 |
Three carbon price paths follow the shapes of three scenario families of the Network for Greening the Financial System: Orderly rises from USD 100 a tonne in 2026 to 400 in 2050, Delayed stays at 10 to 2030 and reaches 700 in 2050, and Current rises from 10 to 40. The figures pass from lecture to lecture. The carbon costs of 2026 to 2030 on the Orderly path are the shock to asset value in lecture 4; the equity volatility lecture 4 calibrates, 42.9 per cent for Coal against 53.0 for Cement, sets the weight cuts of lecture 6; the three loan books of lecture 5 hold loans like all six firms, like Coal, Cement and Steel, and like Port and Rice.
03Licence and files
The six lectures are developed by the University of Twente and released under a Creative Commons Attribution 4.0 International licence (CC BY 4.0). They may be reused, adapted and redistributed, including commercially, with the attribution Mathematics of Green Finance Risk Management, University of Twente, GREEN FINANCE (Erasmus+ CBHE, Grant 101237817), CC BY 4.0. A question or a correction goes to Joerg Osterrieder, University of Twente, joerg.osterrieder@utwente.nl. The module's other teaching materials are on the Module 4 teaching page.
| File | Format | Length | Size | Licence |
|---|---|---|---|---|
M4_Mathematics_L1_Carbon_Price_Paths.pdf | 34 pages | 393 kB | CC BY 4.0 | |
M4_Mathematics_L1_Sheet.pdf | 2 pages | 322 kB | CC BY 4.0 | |
M4_Mathematics_L2_Physical_Losses.pdf | 34 pages | 482 kB | CC BY 4.0 | |
M4_Mathematics_L2_Sheet.pdf | 2 pages | 305 kB | CC BY 4.0 | |
M4_Mathematics_L3_Climate_VaR.pdf | 34 pages | 404 kB | CC BY 4.0 | |
M4_Mathematics_L3_Sheet.pdf | 2 pages | 346 kB | CC BY 4.0 | |
M4_Mathematics_L4_Merton_Default.pdf | 34 pages | 434 kB | CC BY 4.0 | |
M4_Mathematics_L4_Sheet.pdf | 2 pages | 358 kB | CC BY 4.0 | |
M4_Mathematics_L5_Vasicek_Capital.pdf | 34 pages | 471 kB | CC BY 4.0 | |
M4_Mathematics_L5_Sheet.pdf | 2 pages | 373 kB | CC BY 4.0 | |
M4_Mathematics_L6_Portfolio_Decarbonisation.pdf | 34 pages | 446 kB | CC BY 4.0 | |
M4_Mathematics_L6_Sheet.pdf | 2 pages | 299 kB | CC BY 4.0 |