8.3Lesson 8.3 · Module 8 — Capital Budgeting

When Risk Changes Over Time

Different stages or cash flows within the same investment can carry different risks. Learn to separate probability weighting from systematic risk, and to match discount rates to each stage.

One project can pass through stages with different economic risks
Probability affects expected cash flow; the discount rate compensates for systematic risk
The MIT oil-exploration example, worked stage by stage
Zero beta is not zero uncertainty
When one rate is enough — and when it is not

Start

Central question

Can the risk of one investment change as it moves from development to approval, construction, launch, and mature operation?

8.3.1Section 1 · One project, one rate?

A drug from preclinical to mature sales

A pharmaceutical development project looks very different at the trial stage than at the commercial stage. Should every stage automatically use the same discount rate?

One project, connected timeline

Development stage · key uncertainties

Will clinical trials succeed?
Will the drug meet safety and efficacy standards?
Will regulators approve it?
How much additional development spending will be required?

The project is the same molecule from preclinical through mature sales. But the economic source of risk changes across stages.

Should every stage automatically use the same discount rate?

8.3.2Section 2 · Three concepts that must stay separate

Expected cash flow, systematic risk, and timing

The most common valuation errors come from blending these three concepts. Keep them distinct.

A · Expected cash flow

E [C F] = i \sum p_{i} \cdot C F_{i}

Possible outcomes weighted by their probabilities.

B · Systematic risk

Is the cash flow especially likely to be low when the overall market and investors' wealth are also low? This is what CAPM compensates.

C · Timing

P V = \frac{C F _{t}}{( 1 + r ) ^{t}}

A dollar expected later is worth less today, even before adding risk.

Worked example · expected cash flow

E [C F] = \frac{1}{3} ($60 M) + \frac{2}{3} ($0) = $20 M

The 2/3 probability of finding nothing reduces the expected value. This is handled by probability weighting, not by inflating the discount rate.

Sort each statement into its concept

Click the concept that each statement primarily describes.

A trial has a 50% probability of success, producing $60M if it works.

The project's cash flows are strongly correlated with recessions.

The payoff arrives in three years rather than tomorrow.

Construction will cost $40M in year 1 and $20M in year 2.

A dollar expected in ten years is worth less than a dollar expected in one year, even before risk.

Demand for the product falls sharply when the overall economy contracts.

8.3.3Section 3 · More uncertainty does not mean a higher discount rate

Total uncertainty vs. systematic risk

A 50% probability of failure does not automatically justify a very high CAPM discount rate. The key question is whether the outcome moves with the market.

Definition · The critical distinction

Probability of failure affects expected cash flow. The discount rate should reflect systematic risk — whether the resulting cash flow covaries with broad market conditions.

The clinical-trial question

A trial has a 50% probability of success. The scientific outcome may be largely unrelated to broad market movements. Does a 50% probability of failure automatically justify a very high CAPM discount rate?

No. The probability of success belongs in the expected cash-flow estimate. The discount rate should reflect whether the value of the resulting cash flow moves with systematic market conditions.

Scientific uncertainty

Will the molecule work?
May be largely idiosyncratic
Primarily affects probability-weighted cash flow

Commercial market exposure

Demand, reimbursement, competition, pricing
Cyclical conditions
May affect systematic risk and required return

Classify each uncertain event

Is the risk primarily idiosyncratic, systematic, both, or is there insufficient information to tell?

A Phase 2 clinical trial fails because the molecule did not reach statistical significance on its endpoint.

Demand for the company's products falls during a recession as consumers cut discretionary spending.

A factory construction project experiences cost overruns because steel prices spike during a global supply crisis.

An early-stage software startup's product fails to achieve product-market fit.

Oil production revenue falls because global oil prices collapse during an economic downturn.

A specific construction contractor goes bankrupt due to poor management.

Total uncertainty and systematic risk are not the same concept.

A project can carry enormous total uncertainty yet have low systematic risk — or vice versa. The CAPM discount rate compensates only for the portion that covaries with the market. (0/6 classified above.)

8.3.4Section 4 · Primary worked example — oil exploration

Stage-specific valuation, one step at a time

The MIT oil-exploration example separates diversifiable exploration uncertainty from market-sensitive production risk. Work backward from the production cash flow.

The stage-specific principle

V_{0} = \frac{E [ V _{1} ]}{1 + r _{exploration}} where E [V_{1}] = p \cdot V_{1} (success) + (1 - p) \cdot 0

The Year 1 expected value combines the probability of discovery with the conditional production value. That expected value is then discounted at a rate matching the exploration stage's own systematic risk.

$p$: Probability of discovering oil (incorporated via probability weighting, not the discount rate).
$V_{1} (success)$: Value at Year 1 if oil is found — the production cash flow discounted at the production-required return.
$r_{exploration}$: Discount rate for the exploration-stage expected value, reflecting its own systematic risk.

Different stages use different rates because they carry different risks. The probability of finding nothing enters through E[V₁], not through inflating the discount rate.

Simplified instructional model

Project timeline

Year 1 · Exploration

Drilling occurs. At year-end: oil found (33%) or nothing (67%).

Year 2 · Production

If oil is found, production generates $60M.

Assumptions (adjust to explore)

Discovery probability33%

Production payoff (Year 2)$60M

Production discount rate20%

Exploration-stage discount rate5%

1Value the production cash flow at Year 1 (conditional on success)

Once oil is discovered, the remaining cash flow comes from producing and selling oil. That cash flow is valued using the required return associated with oil-production risk.

V_{1} (success) = \frac{$60 M}{1 + 20%} = $50.00 M

The production cash flow is discounted one year at the production-required return because it arrives at the end of Year 2, and at Year 1 one year remains.

2Incorporate the exploration outcome

The probability of finding oil is incorporated through the expected value. This is probability weighting, not discounting.

E [V_{1}] = 33% \times $50.00 M + 67% \times $0 = $16.67 M

The 67% probability of finding nothing enters as a zero payoff. It lowers the expected value without inflating the discount rate.

3Discount the exploration-stage value to today

Because this simplified example assumes the exploration outcome has beta 0, the Year 1 expected value is discounted at the risk-free rate.

V_{0} = \frac{$16.67 M}{1 + 5%} = $15.87 M

Zero beta does not mean zero uncertainty. It means the uncertainty is not priced as systematic market risk in this example.

Value if oil found (Yr 1)

$50.00M

Probability-wtd Yr 1 value

$16.67M

Present value today

$15.87M

Interpretation

The two stages use different discount rates because they carry different risks. Production cash flows are valued at 20% — the return investors require for oil-production market exposure. The exploration outcome is valued at 5% because its uncertainty is assumed unrelated to the market in this simplified model.

Move the exploration-stage rate to match the production rate and watch the present value change. The next section compares this stage-specific approach with using one rate for everything.

8.3.5Section 5 · Compare with the one-rate approach

Where the two methods diverge

Apply one rate to both years and see how the result differs from the stage-specific calculation. The gap reveals exactly where the rate treatment matters.

Same cash flows, different rate treatment

Discovery probability33%

Production payoff$60M

Single-rate (both years)20%

The stage-specific approach uses 20% for production and 5% for exploration. The single-rate approach uses one rate for both years.

Stage-specific approach

V_{1} = \frac{$60 M}{1.20} = $50.00 M

E [V_{1}] = 33% \times $50.00 = $16.67 M

V_{0} = \frac{$16.67}{1.05} = $15.87 M

Production risk is priced at 20%. Exploration outcome is priced at 5% because it is assumed uncorrelated with the market.

Single-rate approach

E [C F_{2}] = 33% \times $60 = $20.00 M

V_{0} = \frac{$20.00}{( 1.20 ) ^{2}}

V_{0} = $13.89 M

The expected payoff is discounted two full periods at 20% — as if exploration uncertainty carries the same market-risk compensation as oil production.

Stage-specific V₀

$15.87M

Single-rate V₀

$13.89M

Difference

$1.98M

The single-rate approach treats the exploration uncertainty as though investors require the same market-risk compensation as they require for oil production. That understates value by $1.98M here, because it double-charges for risk that the stage-specific model already handles via probability weighting.

Important limitation

This does not mean exploration always has zero beta. It means the analyst should identify the actual source of risk rather than applying one industry label to every stage. If exploration outcomes do move with market conditions — for example, drilling becomes uneconomic when oil prices collapse — then a higher exploration-stage rate is justified.

8.3.6Section 6 · Avoid double-counting risk

Probabilities and discount rates answer different questions

Counting the same uncertainty twice — once through probability weighting and again through an unsupported discount-rate premium — is a subtle but common error.

The double-counting trap

An investor multiplies the commercial payoff by a low probability of success and then uses an extremely high discount rate because “the project may fail.” This may count the same uncertainty twice — unless the higher discount rate is independently justified by systematic risk.

Probability weighting

p \times C F_{success}

Discounting

\frac{1}{( 1 + r ) ^{t}}

Adjust for time and systematic risk only.

Probabilities answer which outcomes may occur. Discount rates answer how investors price the systematic risk of those outcomes.

Evaluate each analyst approach

An investor multiplies the commercial payoff by a 30% probability of approval, then discounts the result at 25% because 'the project may fail.'

An investor probability-weights a drug's value by approval probability, then discounts at a rate estimated from commercial-stage comparables (demand, pricing, competition).

An investor ignores the probability of construction delays entirely and discounts expected operating cash flows at the risk-free rate.

An investor applies a 15% discount rate to all cash flows of a mining project, citing 'high risk,' without separating geological discovery risk from commodity-price exposure.

An investor discounts construction costs at a low rate (financed by fixed-price contract) but discounts operating revenue at a higher rate reflecting cyclical demand.

Important nuance

Some failure risk may be systematic.
Construction failures may become more common during financing crises.
Commodity projects may fail because market prices collapse.
The analyst must investigate the source of uncertainty.
Do not mechanically assign all failure probability to either category.

8.3.7Section 7 · Pharmaceutical development in practice

How risk shifts across clinical milestones

Return to the opening drug example. As the project moves from preclinical through commercialization, the dominant source of priced risk changes.

Stage 1 · Clinical development

Trial success probabilities
Remaining research expense
Safety and efficacy results
Regulatory milestones
Time to the next decision point

Stage 2 · Commercialization

Addressable patient population
Expected price and reimbursement
Market share and patent life
Manufacturing and marketing costs
Competitive treatments

Pharmaceutical development timeline

Illustrative scenario. Probabilities and costs are teaching examples, not industry benchmarks.

PreclinicalPrimarily idiosyncratic

Remaining cost

$15M

Probability to commercialization

Very low

Conditional value

Very low — far from revenue; science-driven

Major uncertainty

Does the molecule show activity and acceptable safety in animal models?

Risk classification

Largely unrelated to broad market conditions. Affects probability-weighted cash flow more than the discount rate.

What an investor would seek

Preclinical data, mechanism of action, animal safety profile, competitive landscape.

A successful trial does not merely increase the probability of success. It can also change the project's remaining risk profile. As the project moves from scientific uncertainty toward commercial market exposure, the dominant source of priced risk shifts — and the appropriate valuation treatment may shift with it.

8.3.8Section 8 · Factory construction and operation

A compact manufacturing example

The same principle applies beyond pharma. Separate the probability of successful completion from the value of operating cash flows.

Construction stage

Permitting and engineering delays
Contractor performance
Cost overruns
Equipment installation
Financing availability

Operating stage

Product demand
Selling prices
Capacity utilization
Input costs
Industry cyclicality
Technological obsolescence

A factory can be completed successfully and still produce poor financial returns. Probability of successful completion is a separate question from value of operating cash flows after completion.

Construction risk is not automatically idiosyncratic. Financing crises, commodity-price inflation, labor shortages, and supply-chain stress may create systematic exposure. The classification depends on the actual project and economic environment — not on a default assumption.

8.3.9Section 9 · Two sources of time-varying discount rates

The time value of money and the risk premium can both differ

Discount rates can vary across time for two distinct reasons. Understanding both prevents confusion.

1 · Term structure of interest rates

A risk-free cash flow due in one year may have a different risk-free rate from a risk-free cash flow due in ten years. The time value of money itself can vary with horizon.

2 · Term structure of risk premia

Cash flows at different stages may carry different market exposure: development versus commercialization, construction versus operation, fixed contracts versus cyclical sales.

Note

Different discount rates can arise because both the time value of money and the risk premium can differ across horizons. This lesson focuses on the second source — stage-specific risk premia. A full treatment of the yield curve belongs to the fixed-income module.

8.3.10Section 10 · When one discount rate is acceptable

Representing meaningful risk differences, not maximizing rates

The professional objective is not to use as many discount rates as possible. Sometimes one rate is the right answer.

When does complexity help?

The professional objective is not to use the maximum possible number of discount rates. It is to represent economically meaningful risk differences. One rate may be a reasonable approximation when risks cannot be separated or disclosure is insufficient.

One rate may be acceptable when…

All cash flows arise from similar operations
Business risk is relatively stable
Stages cannot be separated reliably
Public disclosure is too limited
Comparable risk estimates are highly uncertain
Sensitivity testing shows the decision is not materially affected

For each investment, which approach is justified?

Mature store expansion

A successful retailer opens 100 new locations using its established format, with stable unit economics and no distinct risk stages.

Early-stage drug program

A biotech company is advancing a novel molecule through clinical trials toward potential commercialization.

Factory with distinct construction and operating stages

A manufacturer builds a new plant under a fixed-price contract, then operates it to serve cyclical industrial demand.

Stable subscription contract

A software company signs long-term subscription contracts with predictable renewal rates and minimal economic-cycle sensitivity.

Poorly disclosed internal software project

A retailer's filings mention 'continued investment in digital capabilities' without any breakdown of stages, costs, or expected returns.

8.3.11Section 11 · How outside investors apply the principle

A connected analytical process

Outside investors rarely observe management's exact stage-specific betas or internal probability models. This practical workflow adapts the principle to incomplete information.

Identify

Separate

Price

Test & update

The loop closes at step 9: monitoring a milestone feeds back into step 1 of the next round. A successful trial does not merely confirm the old model — it produces a new project with different remaining risks, and the valuation should reflect that.

8.3.12Section 12 · Misconception checks

Common mistakes about stage-specific risk

Each card corrects a frequent error. Expand any question to see the reasoning.

8.3.13Section 13 · Applied practice

Four applied questions

Test the framework on realistic situations. Explanatory feedback follows every answer.

Try itMastery check

Pass with 3 of 4 correct

Answer all questions, then check your work. You can retry any time — mastery is based on correctness, not speed.

01
A mining company has a 25% probability of discovering a viable deposit. The geological result appears unrelated to the market. If discovered, revenue will be highly exposed to commodity cycles. Which statement is correct?
02
A company probability-weights a drug's value by its approval probability and also uses a 25% discount rate solely because approval is uncertain. What potential error should the investor investigate?
03
A data center is built under a fixed-price contract, but future lease revenue depends heavily on technology demand. Should construction and operating cash flows automatically use the same rate?
04
A mature store-opening program has stable unit economics and no clearly distinct risk stages. Is one discount rate necessarily inappropriate?

8.3.14Section 14 · Final takeaway

Risk is not always constant within one investment

The analytical sequence that ties the lesson together.

Separate project stagesEstimate possible cash flowsProbability-weight outcomesIdentify systematic riskApply appropriate ratesUpdate as uncertainty resolves

An investment does not necessarily have one permanent level of risk. As uncertainty resolves and the source of cash flow changes, the relevant valuation treatment may also change.

A single investment may pass through stages with different economic risks — and may justify different discount rates.
Expected cash flow incorporates possible outcomes and their probabilities. The discount rate compensates for systematic risk.
A high probability of failure does not automatically imply a high CAPM discount rate.
Zero beta does not mean zero uncertainty — it means the uncertainty is not priced as systematic market risk.
Probability weighting and an unsupported discount-rate premium can double-count the same risk.
Multiple rates are useful only when stages carry economically meaningful and distinguishable risk exposures.
When risks cannot be separated or disclosure is insufficient, one reasonable rate remains a valid approximation.
Investors should update valuations as milestones resolve uncertainty.

Toward Lesson 8.4

We now know how to estimate possible cash flows and match discount rates to their risks. The next lesson combines those pieces into the central value-creation measure: net present value.

Lesson summary

1A single investment may pass through stages with different economic risks — and may justify different discount rates.
2Expected cash flow incorporates possible outcomes and their probabilities. The discount rate compensates for systematic risk.
3A high probability of failure does not automatically imply a high CAPM discount rate.
4Zero beta does not mean zero uncertainty — it means the uncertainty is not priced as systematic market risk.
5Probability weighting and an unsupported discount-rate premium can double-count the same risk.
6Multiple rates are useful only when stages carry economically meaningful and distinguishable risk exposures.
7When risks cannot be separated or disclosure is insufficient, one reasonable rate remains a valid approximation.
8Investors should update valuations as milestones resolve uncertainty.

Replay this lesson Continue to Lesson 8.4

Sources and Notes

MIT OpenCourseWare, 15.401 Finance Theory I, Fall 2008, Andrew W. Lo — capital-budgeting lectures: discount rates may differ across horizons, different cash flows can carry different risks, and the rate used should correspond to the risk of the cash flow being valued.
MIT 15.401 — the oil-exploration example: separating diversifiable exploration uncertainty from market-sensitive production risk, and the principle that idiosyncratic uncertainty is not automatically rewarded with a higher expected return.
Applied extension: pharmaceutical development milestones, factory construction versus operation, and the practical investor workflow for stage-specific valuation under incomplete public disclosure.

Lesson content is adapted from MIT 15.401 capital-budgeting material for educational use. All cash flows are described as expected, not guaranteed. Examples use illustrative figures, not current market estimates.