Net Present Value: Discounting, Investment Appraisal and the Rate That Decides Everything
A pound today is worth more than a pound next year. Everyone nods at that. Almost nobody appreciates that the exact size of “more” — a single number chosen before any calculation begins — quietly determines whether a hospital gets built, whether a railway is worth it, and how much we should spend on climate change.
Suppose someone offers you a choice. £1,000 handed to you now, or £1,000 handed to you in five years. Guaranteed either way — no risk, no catch.
You take the money now. Obviously. But pause on why, because the reason is less obvious than the choice.
Now make it harder. £1,000 now, or £1,300 in five years. Now you have to actually think. And thinking about it properly — turning that vague preference for “sooner” into a number you can compare — is the entire subject of this article. It is also, without much exaggeration, the single most-used calculation in applied economics and finance.
Why is future money worth less?
Students often assume the answer is inflation. It is not — or rather, not only. Even in an economy with perfectly zero inflation, future money would still be worth less. There are three separate reasons, and good exam answers separate them.
1. Opportunity cost. Money you have now can be invested. £1,000 today at 5% becomes £1,276 in five years. So £1,000 in five years is genuinely inferior to £1,000 today — it has been robbed of five years of earning power. This reason matters most, and note that it is just compound interest running in reverse.
2. Risk and uncertainty. A promise of future money is a promise. The payer might default, the contract might fail, the firm might not exist. Certainty now beats probability later.
3. Time preference. People simply prefer consumption sooner rather than later — a psychological fact independent of both interest rates and risk. Economists call this the pure rate of time preference, and it is why you would still prefer the money now even if there were no investments available and no risk of default.
Inflation is a fourth consideration, handled separately by working consistently in either real or nominal terms.
The core idea. If compounding asks “what does £100 today become in the future?”, discounting asks “what must I have today to end up with £100 in the future?” It is the same equation solved for a different variable. If you understood A = P(1 + r)t, you already understand present value — you just have not rearranged it yet.
Present value
Start from compound interest and rearrange. If P grows to A over t years at rate r, then A = P(1 + r)t, so P = A ÷ (1 + r)t. That P is the present value of a future amount. In standard notation:
PV = FV ÷ (1 + r)t
PV = present value · FV = future cash flow · r = discount rate · t = periods until received
The term 1/(1 + r)t is the discount factor — the price today of £1 delivered at time t.
The term 1/(1 + r)t is the discount factor — the price today of £1 delivered at time t.
Back to our puzzle. Is £1,300 in five years better than £1,000 now, at a 5% discount rate?
PV = 1300 ÷ (1.05)5 = 1300 ÷ 1.27628 = £1,018.60. Yes — but only just. The future £1,300 is worth £1,018.60 in today’s money, so you should take it, by a margin of about £19.
Now notice something uncomfortable. Raise the discount rate to 6% and PV = £971.44 — and the answer flips. The decision was never really about the £1,300. It was about the rate.
How brutally discount factors fall away
| Year | @ 3% | @ 5% | @ 10% | @ 15% |
|---|---|---|---|---|
| 1 | 0.971 | 0.952 | 0.909 | 0.870 |
| 5 | 0.863 | 0.784 | 0.621 | 0.497 |
| 10 | 0.744 | 0.614 | 0.386 | 0.247 |
| 20 | 0.554 | 0.377 | 0.149 | 0.061 |
| 50 | 0.228 | 0.087 | 0.009 | 0.001 |
| 100 | 0.052 | 0.008 | 0.0001 | ~0 |
Read the bottom row slowly. At a 10% discount rate, £1,000,000 arriving in 100 years has a present value of about £72. At 15% it is worth roughly seven pence.
This is not a mathematical curiosity. It is why the climate discount rate is one of the most consequential arguments in modern economics — we will come back to it.
Net Present Value: the decision rule
Real projects are not single payments. You spend money now and receive a stream of cash flows over years. NPV handles that by discounting every cash flow back to today and adding them all up.
NPV = Σ [ Ct ÷ (1 + r)t ] − C0
Ct = net cash flow in period t · C0 = initial investment · summation from t = 1 to t = n
And the decision rule is refreshingly blunt:
| Result | Interpretation | Decision |
|---|---|---|
| NPV > 0 | Returns more than the discount rate; adds value | Accept |
| NPV = 0 | Exactly meets the required return | Indifferent |
| NPV < 0 | Returns less than alternatives; destroys value | Reject |
If projects are mutually exclusive — you can only build one factory — choose the highest positive NPV, not merely any positive one.
Worked example: appraising a machine
A firm is considering a £50,000 machine, using a 10% cost of capital.
| Year | Cash flow | Discount factor | Present value |
|---|---|---|---|
| 0 | −£50,000 | 1.0000 | −£50,000.00 |
| 1 | £15,000 | 0.9091 | £13,636.36 |
| 2 | £18,000 | 0.8264 | £14,876.03 |
| 3 | £20,000 | 0.7513 | £15,026.30 |
| 4 | £12,000 | 0.6830 | £8,196.16 |
| 5 | £8,000 | 0.6209 | £4,967.37 |
| NPV | £6,702.22 | ||
Decision: accept. The machine generates £6,702 more than the firm requires, in today’s money.
But look at the undiscounted sum first: £73,000 against a £50,000 outlay. Naively that looks like a £23,000 profit. Discounting reveals that nearly three-quarters of that apparent gain is an illusion created by ignoring timing.
Common error — discounting the Year 0 outlay. The initial investment happens now. Its discount factor is 1/(1 + r)0 = 1. Do not divide it by (1 + r). Similarly, a cash flow “at the start of year 3” is the end of year 2 — t = 2, not t = 3. Timing conventions cost more marks than the arithmetic does.
How sensitive is this to the rate?
| Discount rate | NPV | Decision |
|---|---|---|
| 5% | +£13,894 | Accept comfortably |
| 10% | +£6,702 | Accept |
| 15% | +£948 | Accept marginally |
| 16% | −£216 | Reject |
| 20% | −£4,447 | Reject clearly |
A one percentage point move — 15% to 16% — reverses the decision. The cash flow forecasts did not change at all. This is why sensitivity analysis is not optional in real appraisal work, and why examiners reward candidates who mention it.
Shortcuts: perpetuities and annuities
Some cash flow patterns are regular enough to collapse into a single formula. These come up constantly and are worth knowing cold.
Perpetuity — a payment forever
PV = C ÷ r
C = constant payment per period, received forever, starting next period
A bond paying £50 per year forever, discounted at 4%: PV = 50 ÷ 0.04 = £1,250.
It surprises students that an infinite stream has a finite value. But it follows directly from the discount factor table — payments far enough out contribute essentially nothing, so the infinite sum converges. Britain’s Consols, issued in 1751 and finally redeemed in 2015, were real perpetuities priced exactly this way.
Growing perpetuity — payments rising at rate g
PV = C ÷ (r − g)
Valid only when r > g. If g ≥ r the sum diverges and the formula returns nonsense.
This is the Gordon growth model, the backbone of equity valuation. A share paying a £2 dividend next year, growing at 3% forever, with investors requiring 8%: PV = 2 ÷ (0.08 − 0.03) = £40.
Common error — g approaching r. As g creeps toward r, the denominator shrinks toward zero and the valuation explodes toward infinity. If your dividend model returns a share price of £8 million, you have set g too close to r. No company grows faster than the discount rate forever — that would eventually make it larger than the world economy. Real valuations handle this with multi-stage models.
Annuity — a payment for a fixed number of periods
PV = C × [ 1 − (1 + r)−n ] ÷ r
n = number of payments. The bracketed term is the annuity factor.
Mortgages, car loans, pensions and lease payments are all annuities. £500 per month for 25 years at 0.4% monthly: annuity factor = 173.6, so PV = 500 × 173.6 = £86,800.
A useful sanity check: the annuity formula is just a perpetuity minus another perpetuity that starts at year n+1. That is where the derivation comes from if you ever need to reconstruct it.
Internal Rate of Return — and why it lies
The IRR is the discount rate at which a project’s NPV equals exactly zero — the project’s implied percentage return. In our machine example, NPV crossed from positive to negative between 15% and 16%, so the IRR is roughly 15.8%. The rule looks intuitive: accept if IRR exceeds the cost of capital.
IRR is popular with managers because a percentage feels more comparable than a pound figure. It is also, in three important situations, wrong.
Problem 1 — scale blindness. Project A: invest £100, get £150 back next year. IRR = 50%, NPV at 10% = £36. Project B: invest £10,000, get £13,000 back. IRR = 30%, NPV at 10% = £1,818. IRR prefers A. But A adds £36 of value and B adds £1,818. You cannot spend a percentage.
Problem 2 — multiple IRRs. If cash flows change sign more than once — a mine that costs money to open, earns for years, then costs money to decommission — the NPV polynomial can have several roots. A project can have an IRR of both 8% and 42%, and neither means anything.
Problem 3 — the reinvestment assumption. IRR implicitly assumes interim cash flows are reinvested at the IRR itself. If a project has a 40% IRR, that assumes you can redeploy every cash inflow at 40%. You cannot, or you would already be doing so.
When they disagree, trust NPV. NPV measures value added in currency, which is what shareholders and citizens actually consume. IRR measures a rate, which is dimensionless and therefore blind to scale. Graham and Harvey’s survey of US CFOs found roughly three-quarters use IRR routinely — which tells you about the psychology of decision-makers, not the mathematics.
Payback period: crude, popular, and not entirely stupid
Payback simply asks: how long until cumulative cash flows repay the initial outlay? Our machine: after year 1, £15,000 recovered; after year 2, £33,000; after year 3, £53,000 — so payback occurs during year 3, roughly at 2 + (17,000/20,000) = 2.85 years.
Its flaws are obvious and severe:
- It ignores the time value of money entirely (unless you use discounted payback)
- It ignores every cash flow after the payback point — a project could pay back in two years and then lose money for a decade
- The cut-off period is arbitrary
And yet firms use it constantly, which is worth taking seriously rather than dismissing. Payback is a crude proxy for liquidity risk and forecast reliability. Year-eight cash flow forecasts are close to fiction; payback refuses to rely on them. In a credit-constrained firm, a project that returns cash fast has genuine option value that NPV does not capture. For an evaluation question, “payback is theoretically inferior but responds to real constraints NPV assumes away” is a stronger answer than simply listing its flaws.
Case Study — HS2 and the Fragility of a Benefit–Cost Ratio
Britain’s HS2 high-speed rail project offers an unusually public demonstration of how much discounting assumptions do the work in appraisal.
The UK Treasury’s Green Book mandates a 3.5% social discount rate for public projects, falling to 3% after 30 years and declining further thereafter — a schedule designed specifically to stop long-horizon benefits from vanishing.
HS2’s benefit–cost ratio was estimated at around 2.3 in 2013, meaning £2.30 of benefit per £1 of cost. By 2020, after cost escalation and revised assumptions, the Oakervee Review put it between 1.0 and 2.7 depending on scenario — a range spanning “clearly worthwhile” to “barely breaks even.” In 2023 the northern leg was cancelled outright.
Three lessons worth carrying into an exam. First, the discount rate chose the answer: rail benefits accrue over 60+ years, so at a commercial 10% rate HS2’s NPV would be deeply negative, while at 3.5% it is defensible — and neither rate is objectively correct. Second, appraisal optimism is systematic: Flyvbjerg’s research on hundreds of infrastructure projects found rail cost overruns averaging 45%, and these errors are directional, not random. Third, NPV is only as good as its inputs — the mathematics is exact, the forecasts are not, and precision in the arithmetic can disguise enormous uncertainty in the assumptions.
The discount rate debate: Stern vs Nordhaus
Everything above has treated r as given. It is not. Choosing r is the argument, and nowhere more visibly than in climate economics.
Climate damages arrive in 2100 and beyond. Mitigation costs arrive now. So the entire policy question reduces to: how much is a life improved in 2150 worth to us today? The framework is the Ramsey equation:
r = δ + η·g
δ = pure rate of time preference · η = elasticity of marginal utility of consumption · g = consumption growth rate
Stern (2006) set δ = 0.1%, arguing on ethical grounds that discounting future people’s welfare simply because they are born later is indefensible — the only justification for any δ above zero being the small probability humanity ceases to exist. With η = 1 and g = 1.3%, Stern arrived at r ≈ 1.4%, concluding that aggressive immediate mitigation is worth roughly 1% of global GDP.
Nordhaus set δ ≈ 1.5%, arguing appraisal should use rates observed in actual markets, where people demonstrably do discount the future. His DICE model produced r ≈ 4.5% and a policy of gradual carbon pricing ramping up over decades.
Both won Nobel-level recognition for this work — Nordhaus the Prize itself in 2018 — and they disagree by a factor of roughly ten on how much to spend. They do not disagree about climate science. They do not really disagree about the economics. They disagree about δ, a single parameter that is fundamentally an ethical judgement wearing mathematical clothing.
Run the numbers: £1 trillion of damage in 2120 has a present value of £252 billion at Stern’s 1.4%, and £12 billion at Nordhaus’s 4.5%. A twenty-fold difference, produced entirely by a parameter neither economist can derive from data.
Research Spotlight — What Discount Rate Do Experts Actually Choose?
The question: if δ is an ethical judgement rather than an empirical fact, is there any consensus at all?
Drupp, Freeman, Groom and Nesje (2018) surveyed 197 economists who had published on discounting — the people best placed to have an informed view.
The findings were revealing. The median recommended social discount rate was 2%, with an interquartile range of 1% to 3%. But the full spread ran from 0% to over 10%. On δ specifically — the parameter at the heart of the Stern–Nordhaus dispute — the median was 0.5%, but nearly a third of respondents chose exactly zero, endorsing Stern’s ethical position.
Critically, three-quarters of respondents said they could accept a rate of 2% as a reasonable compromise for policy, even where it differed from their own preferred figure. So there is more agreement on a workable number than the public argument suggests — but the disagreement about why remains total.
A related strand: Weitzman (1998, 2001) and Gollier showed that when the correct rate is genuinely uncertain, the certainty-equivalent discount rate declines over time — because the lowest possible rate dominates the far future, contributing the largest present values. This is why the UK Green Book uses a declining schedule rather than a flat rate. Uncertainty about the discount rate is itself an argument for discounting the distant future less.
Why this matters for you: when an exam question hands you “assume a discount rate of 6%,” the mathematics is settled but the economics has already been decided for you. Noticing that is worth an evaluation mark.
Real vs nominal: keep them consistent
One rule, easy to state, easy to violate under exam pressure: discount nominal cash flows at nominal rates, and real cash flows at real rates. Never mix.
If cash flows are stated in today’s prices (real), use a real rate. If they include expected inflation (nominal), use a nominal rate. Convert between them with Fisher: (1 + i) = (1 + r)(1 + π). Discounting real cash flows at a nominal rate double-counts inflation and will systematically reject good projects — one of the most common errors in practice as well as in exams.
Summary of formulas
| Concept | Formula | Use when |
|---|---|---|
| Present value | PV = FV ÷ (1 + r)t | Single future cash flow |
| Net present value | NPV = Σ Ct/(1+r)t − C0 | Full project appraisal |
| Perpetuity | PV = C ÷ r | Level payment forever |
| Growing perpetuity | PV = C ÷ (r − g) | Dividends, r > g required |
| Annuity | PV = C[1 − (1+r)−n] ÷ r | Mortgages, pensions, leases |
| Ramsey equation | r = δ + η·g | Choosing a social discount rate |
| Fisher equation | (1 + i) = (1 + r)(1 + π) | Converting real ↔ nominal |
AP & Cambridge A-Level Exam Technique
1. Always build a cash flow table. Columns: Year, Cash flow, Discount factor, Present value. Examiners follow the structure and award method marks even if one row’s arithmetic slips. An unstructured wall of calculation loses marks it did not need to.
2. Year 0 is not discounted. The discount factor for t = 0 is 1. Write it in the table explicitly so the marker can see you knew.
3. Show discount factors to four decimal places. Rounding 0.9091 to 0.91 introduces visible error across five years. Carry precision, round only the final NPV.
4. State the decision, do not just compute the number. “NPV = +£6,702, therefore accept the project as it adds value at the firm’s cost of capital.” The concluding sentence is usually its own mark.
5. For mutually exclusive projects, highest NPV wins — not highest IRR. If a question sets IRR and NPV against each other, it is testing exactly this.
6. Evaluation marks live in the assumptions, not the arithmetic. Reliable points: the discount rate is itself an assumption and results are highly sensitive to it; cash flow forecasts beyond a few years are unreliable; NPV ignores non-monetary and strategic factors; real options are not captured. Two developed points beat five listed ones.
7. Mention sensitivity analysis. Saying “at 15% the project is marginal at +£948, so the decision is not robust to a small rise in the cost of capital” demonstrates genuine understanding rather than formula recall.
Practice Questions
Question 1 — Present value (4 marks)
Calculate the present value of £25,000 to be received in 7 years, using a discount rate of 8%. Explain what your answer means.
Answer.
PV = FV ÷ (1 + r)t [1]
PV = 25000 ÷ (1.08)7 = 25000 ÷ 1.71382 [1]
PV = £14,586.60 [1]
PV = FV ÷ (1 + r)t [1]
PV = 25000 ÷ (1.08)7 = 25000 ÷ 1.71382 [1]
PV = £14,586.60 [1]
Interpretation: £14,586.60 invested today at 8% would grow to exactly £25,000 in 7 years. An investor with access to an 8% return should therefore be indifferent between £14,586.60 now and £25,000 in 7 years — they are the same thing in different time clothing. [1]
Question 2 — NPV appraisal (8 marks)
A firm can invest £120,000 in equipment generating net cash flows of £35,000 (Yr 1), £45,000 (Yr 2), £50,000 (Yr 3) and £30,000 (Yr 4). Cost of capital is 12%. (a) Calculate the NPV. (b) Advise the firm.
(a) [6 marks]
Year 0: −£120,000 × 1.0000 = −£120,000.00
Year 1: £35,000 × 0.8929 = £31,250.00
Year 2: £45,000 × 0.7972 = £35,873.72
Year 3: £50,000 × 0.7118 = £35,589.40
Year 4: £30,000 × 0.6355 = £19,065.75
Year 0: −£120,000 × 1.0000 = −£120,000.00
Year 1: £35,000 × 0.8929 = £31,250.00
Year 2: £45,000 × 0.7972 = £35,873.72
Year 3: £50,000 × 0.7118 = £35,589.40
Year 4: £30,000 × 0.6355 = £19,065.75
NPV = +£1,778.87
(b) Advice [2 marks]
NPV is positive, so the firm should accept — the project adds £1,778.87 of value in today’s money at the required return. [1]
However the margin is thin: £1,779 on a £120,000 investment is about 1.5%. A small rise in the cost of capital, or a modest shortfall in the Year 3 forecast, would push NPV negative. The firm should conduct sensitivity analysis before committing. [1]
Note: undiscounted cash flows total £160,000 against a £120,000 outlay, which looks like a comfortable £40,000 profit. Discounting reveals the project is genuinely marginal.
Question 3 — NPV vs IRR (7 marks)
Two mutually exclusive projects, cost of capital 10%. Project X: invest £20,000, receive £26,000 in one year; IRR = 30%. Project Y: invest £200,000, receive £250,000 in one year; IRR = 25%. (a) Calculate each NPV. (b) Which should the firm choose, and why do NPV and IRR disagree?
(a)
Project X: NPV = (26,000 ÷ 1.10) − 20,000 = +£3,636.36 [2]
Project Y: NPV = (250,000 ÷ 1.10) − 200,000 = +£27,272.73 [2]
Project X: NPV = (26,000 ÷ 1.10) − 20,000 = +£3,636.36 [2]
Project Y: NPV = (250,000 ÷ 1.10) − 200,000 = +£27,272.73 [2]
(b) Choose Project Y. [1]
Why they disagree: IRR is a percentage and therefore dimensionless — it is blind to the scale of investment. X earns a higher rate on a much smaller base. NPV measures the absolute value added in currency, which is what the firm’s owners actually receive. Y adds £27,273 versus X’s £3,636; the firm cannot spend a 30% return. [1]
The general rule: where NPV and IRR conflict on mutually exclusive projects, NPV is correct. IRR is reliable only for independent accept/reject decisions with conventional cash flows and no capital rationing. [1]
Question 4 — The discount rate and climate policy (10 marks)
A climate mitigation programme costs £500 billion today and avoids £3 trillion of damage in 80 years. (a) Calculate the NPV at Stern’s 1.4% and at Nordhaus’s 4.5%. (b) Evaluate the claim that the choice of discount rate is a technical question best left to economists.
(a) [4 marks]
At 1.4%: PV of benefit = 3,000bn ÷ (1.014)80 = 3,000 ÷ 3.0497 = £983.7bn
NPV = 983.7 − 500 = +£483.7 billion → accept [2]
At 1.4%: PV of benefit = 3,000bn ÷ (1.014)80 = 3,000 ÷ 3.0497 = £983.7bn
NPV = 983.7 − 500 = +£483.7 billion → accept [2]
At 4.5%: PV of benefit = 3,000bn ÷ (1.045)80 = 3,000 ÷ 33.83 = £88.7bn
NPV = 88.7 − 500 = −£411.3 billion → reject [2]
Identical physical facts. Identical damage estimate. Opposite policy conclusion. The rate did all of the work.
(b) Evaluation [6 marks]
Against the claim — the rate is fundamentally ethical:
• Via Ramsey (r = δ + η·g), the pure rate of time preference δ is a judgement about how much the welfare of future people counts. Stern set δ ≈ 0 arguing that discounting people by birth date has no ethical basis; Nordhaus set δ ≈ 1.5% arguing appraisal should reflect observed behaviour. Neither position is derivable from data.
• Drupp et al. (2018) surveyed 197 expert economists and found δ recommendations ranging from 0% upward, with roughly a third choosing exactly zero. Expertise does not resolve the question — evidence that it is not a technical question.
• The stakes are distributional and intergenerational: the affected parties cannot vote, negotiate, or be compensated. That is a domain where democratic and ethical input has a strong claim.
For the claim — economists bring necessary structure:
• η and g are genuinely empirical and require technical estimation.
• Weitzman and Gollier showed that uncertainty over the correct rate implies a declining term structure — a technical result with major policy consequences that no non-specialist would derive. The Green Book’s declining schedule follows directly from it.
• Without economists the alternative is not a neutral process but an implicit, unexamined rate — which is worse.
Conclusion: the claim is best judged partly true. The machinery (η, g, term structure, uncertainty) is technical and requires expertise. But δ is an ethical parameter that economics can clarify and cannot settle. The honest position is that economists should present results across a transparent range of rates and make the value judgement explicit, rather than embedding it in a single number and presenting the output as a technical finding.
Summary
Present value converts money at different dates into a common currency so it can be compared. NPV extends that to whole projects and delivers a decision rule that is theoretically correct: accept positive NPV, and where projects compete, take the largest. IRR is more intuitive and less reliable — scale-blind, occasionally multi-valued, and built on a reinvestment assumption that rarely holds. Payback ignores nearly everything but survives because it proxies for liquidity and forecast risk.
The mathematics of all of this is exact. The inputs are not. Cash flow forecasts are guesses dressed as data, and the discount rate — the number that decides the outcome — is chosen before the calculation starts and cannot be derived from evidence alone. Stern and Nordhaus agree on the climate science and the formula, and disagree by a factor of ten on the policy, because of a single ethical parameter.
The formula gives you an answer. It does not give you the assumptions. Knowing the difference is the actual skill.
References
- Brealey, R., Myers, S. and Allen, F. (2020) Principles of Corporate Finance. 13th edn. New York: McGraw-Hill.
- Drupp, M.A., Freeman, M.C., Groom, B. and Nesje, F. (2018) ‘Discounting disentangled’, American Economic Journal: Economic Policy, 10(4), pp. 109–134.
- Flyvbjerg, B., Holm, M.S. and Buhl, S. (2002) ‘Underestimating costs in public works projects: error or lie?’, Journal of the American Planning Association, 68(3), pp. 279–295.
- Gollier, C. (2013) Pricing the Planet’s Future: The Economics of Discounting in an Uncertain World. Princeton: Princeton University Press.
- Graham, J.R. and Harvey, C.R. (2001) ‘The theory and practice of corporate finance: evidence from the field’, Journal of Financial Economics, 60(2–3), pp. 187–243.
- HM Treasury (2022) The Green Book: Central Government Guidance on Appraisal and Evaluation. London: HM Treasury.
- Nordhaus, W.D. (2007) ‘A review of the Stern Review on the Economics of Climate Change’, Journal of Economic Literature, 45(3), pp. 686–702.
- Nordhaus, W.D. (2017) ‘Revisiting the social cost of carbon’, Proceedings of the National Academy of Sciences, 114(7), pp. 1518–1523.
- Oakervee, D. (2020) Oakervee Review of HS2. London: Department for Transport.
- Ramsey, F.P. (1928) ‘A mathematical theory of saving’, The Economic Journal, 38(152), pp. 543–559.
- Stern, N. (2007) The Economics of Climate Change: The Stern Review. Cambridge: Cambridge University Press.
- Weitzman, M.L. (1998) ‘Why the far-distant future should be discounted at its lowest possible rate’, Journal of Environmental Economics and Management, 36(3), pp. 201–208.
- Weitzman, M.L. (2001) ‘Gamma discounting’, American Economic Review, 91(1), pp. 260–271.