Preserving Dependencies in Scenario Analysis

Why Dependencies Matter in Project Scenarios

A project scenario is more than a list of favorable or unfavorable values. It describes a set of conditions that are meant to occur together. Demand can depend on price; unit costs can depend on the volume purchased; sales can be limited by capacity; and terminal proceeds can depend on whether an expansion was made. Combining assumptions without preserving these links can produce a cash-flow forecast that is mathematically calculable but economically incoherent.

For project evaluation, the scenario must be translated into cash flows on a common timeline. This tutorial uses a simplified pre-tax operating model. Unless stated otherwise, cash flows occur at year-end, the effective annual discount rate is 10%, operating cash flow is revenue less variable and fixed operating costs, and no financing effects are included. Any terminal proceeds stated are net proceeds received at the end of the project. These assumptions are modeling choices, not universal descriptions of every project.

Definition: A dependency-preserving scenario is a set of assumptions whose values can reasonably occur together under the scenario’s stated conditions. For each period, units sold are $Q_t=\min(D_t,K_t)$, where $D_t$ is demand at the assumed price and $K_t$ is available capacity. Revenue is $p_tQ_t$, and simplified operating cash flow is $(p_t-c_t)Q_t-F_t$, where $p_t$ is price per unit, $c_t$ is variable cost per unit, and $F_t$ is fixed operating cost.

The demand input in that definition must be demand at the scenario’s price, not a volume forecast detached from price. Likewise, a unit-cost assumption that is available only above a purchase threshold should not be used when the scenario’s volume falls below that threshold. The capacity limit applies to units sold, not to the demand forecast itself.

From Scenario Assumptions to Cash Flows

A consistent scenario assigns assumptions period by period, applies operating constraints, and then records cash flows at the times they occur. Working capital is a common source of timing errors. Suppose the required working-capital balance at the start of operations is $W_1$, and it changes to $W_2$ later. An increase in the required balance uses cash; a decrease releases cash. If all working capital is recovered when the project ends, the final-period cash flow includes that recovery as well as any change required to reach the final balance.

\text{Cash flow from working capital in the final period} = -(W_2-W_1)+W_2 = W_1

The equality assumes that $W_1$ was invested before operations began, the balance is increased to $W_2$ during the project, and the entire $W_2$ balance is recovered at termination. It does not mean that the intermediate increase is ignored: the final-period change and the full recovery must both be shown to obtain the net release. If working capital is not fully recovered, or if some amount remains tied up after the forecast horizon, this expression does not apply without adjustment.

Define the scenario.
State the market or operating conditions and any relationships they imply among demand, price, costs, capacity, and terminal value.

Calculate feasible sales.
Use demand at the specified price and cap units sold at available capacity.

Build period cash flows.
Calculate operating cash flow, investment and expansion outlays, changes in working capital, and terminal proceeds in their correct periods.

Discount the resulting cash flows.
Use the same scenario throughout the cash-flow forecast and discount each cash flow from its actual time of receipt or payment.

Worked Scenario With Capacity and Working Capital

The following example shows why dependencies should be checked before calculating NPV. A forecast can have coherent individual inputs and still be wrong if the links among them or their timing are mishandled.

Worked Example: Maintaining a Consistent Two-Year Case

A hypothetical two-year project requires equipment costing $60,000 at time 0 and has capacity for 1,000 units in each year. A market case supports demand of 800 units in year 1 at a price of $45 per unit, and demand of 1,200 units in year 2 at a price of $42 per unit. The lower year-2 price reflects a planned volume discount in the market. The supplier’s stated costs are $27 per unit in year 1 and $25 per unit in year 2 under those same conditions. Fixed operating costs are $6,000 per year. Working capital equals 5% of each year’s revenue, is invested initially based on year-1 revenue, and is fully recovered at the end of year 2. Net terminal proceeds are $10,000 at the end of year 2. At an effective annual discount rate of 10%, calculate NPV.

Method. First apply the capacity constraint, then calculate each year’s operating cash flow and working-capital requirement. The year-2 demand forecast exceeds capacity, so units sold in that year are 1,000, not 1,200.

Year 1. Units sold are $\min(800,1{,}000)=800$. Revenue is $800\times \$45=\$36{,}000$. Variable costs are $800\times \$27=\$21{,}600$, so operating cash flow is $\$36{,}000-\$21{,}600-\$6{,}000=\$8{,}400$. Required initial working capital is $0.05\times \$36{,}000=\$1{,}800$. Thus time-0 cash flow is $-\$60{,}000-\$1{,}800=-\$61{,}800$.

Year 2. Units sold are $\min(1{,}200,1{,}000)=1{,}000$. Revenue is $1{,}000\times \$42=\$42{,}000$. Variable costs are $1{,}000\times \$25=\$25{,}000$, so operating cash flow is $\$42{,}000-\$25{,}000-\$6{,}000=\$11{,}000$. Year-2 working capital is $0.05\times \$42{,}000=\$2{,}100$. The increase from $1,800 to $2,100 requires a $300 investment, and the full $2,100 is recovered at termination. The net year-2 working-capital cash flow is therefore $-\$300+\$2{,}100=\$1{,}800$. Including $10,000 of terminal proceeds, year-2 cash flow is $\$11{,}000+\$1{,}800+\$10{,}000=\$22{,}800$.

NPV. With year-end cash flows and a 10% effective annual rate,

NPV=-\$61{,}800+\frac{\$8{,}400}{1.10} +\frac{\$22{,}800}{(1.10)^2} =-\$35{,}320.66

The NPV is $-\$35{,}320.66$, rounded to the nearest cent. The calculation uses the scenario’s stated price and cost conditions together, sells no more than capacity, and includes both the year-2 working-capital increase and full recovery. The common mistake avoided is treating all 1,200 units of year-2 demand as sales, or recording the $2,100 recovery without the $300 increase in the required balance.

Why the Operating Formulas Hold

The operating formulas follow from counting only units that can actually be sold. If demand is $D_t$ and capacity is $K_t$, sales cannot exceed either quantity, so the feasible quantity is the smaller of the two. Each sold unit contributes price less variable cost, $p_t-c_t$. Multiplying that contribution by feasible units gives total contribution; subtracting fixed costs gives the simplified operating cash flow.

Q_t=\min(D_t,K_t), \qquad OCF_t=(p_t-c_t)Q_t-F_t

This model assumes that all feasible units are sold at one price, variable cost is constant per unit within the scenario, and fixed costs do not change with the sales quantity. If pricing varies by customer, costs have tiers, capacity can be expanded, or taxes and depreciation affect cash flow, the model must be extended. The formula is not a claim that those effects are absent in every real project.

Worked Example: Diagnosing an Incompatible Demand and Price Pair

A common scenario error is to take the highest value from each separate input range, even where the inputs describe different market conditions.

Worked Example: Correcting a Mixed Market Case

A hypothetical one-year project costs $30,000 and has capacity for 1,000 units. Fixed operating costs are $5,000. Market research indicates demand of 700 units at a price of $40, or 1,300 units at a price of $36. A supplier charges $25 per unit below 900 units and $22 per unit for orders of at least 900 units. Net terminal proceeds are $8,000 in the lower-demand market and $12,000 in the higher-demand market. Working capital is 10% of revenue, is invested at time 0, and is fully recovered at year-end. The effective annual discount rate is 10%. Calculate the NPV of each coherent case and diagnose the proposed combination of $40 price, 1,300-unit demand, and $22 unit cost.

Lower-demand case. At $40, demand is 700, below capacity, so $Q=\min(700,1{,}000)=700$. Revenue is $700\times \$40=\$28{,}000$, and working capital is $0.10\times \$28{,}000=\$2{,}800$. Because 700 units is below the supplier’s 900-unit threshold, variable cost is $25 per unit. Operating cash flow is $700(\$40-\$25)-\$5{,}000=\$5{,}500$. Year-end cash flow is $\$5{,}500+\$2{,}800+\$8{,}000=\$16{,}300$. Therefore,

NPV=-\$30{,}000-\$2{,}800+\frac{\$16{,}300}{1.10} =-\$17{,}981.82

Higher-demand case. At $36, demand is 1,300, but capacity limits sales to 1,000 units. Revenue is $1{,}000\times \$36=\$36{,}000$, so initial working capital is $3,600. The 1,000-unit order qualifies for the $22 unit cost. Operating cash flow is $1{,}000(\$36-\$22)-\$5{,}000=\$9{,}000$. Year-end cash flow is $\$9{,}000+\$3{,}600+\$12{,}000=\$24{,}600$, giving

NPV=-\$30{,}000-\$3{,}600+\frac{\$24{,}600}{1.10} =-\$11{,}236.36

Diagnose the proposed combination. The 1,300-unit demand forecast applies at $36, not $40. At $40, the stated demand is 700 units, so the proposed 1,300-unit demand cannot be used with that price. Further, the $22 unit cost requires an order of at least 900 units, which the 700-unit demand at $40 does not support. For illustration, an incorrect calculation might cap the incompatible 1,300-unit demand at 1,000, apply the $40 price and $22 cost, and use the higher-market terminal proceeds. That gives revenue of $40,000, working capital of $4,000, operating cash flow of $1{,}000(\$40-\$22)-\$5{,}000=\$13{,}000$, and an apparent NPV of

-\$30{,}000-\$4{,}000+ \frac{\$13{,}000+\$4{,}000+\$12{,}000}{1.10} =-\$7{,}636.36

That result is not a valid case under the supplied market evidence. The corrected $40-price case is the lower-demand case, with NPV $-\$17{,}981.82$. The incorrect calculation overstates value by combining demand, unit cost, and terminal proceeds supported by other conditions. The common mistake avoided is to select optimistic assumptions independently rather than check whether one scenario can support them all.

Expansion and Terminal Cash Flows Must Belong to the Same Case

Capacity changes and terminal proceeds are also dependent assumptions. If a scenario includes an expansion, the expansion cost belongs in the period when it is paid, the expanded capacity is available only after that point, and working capital may need to increase to support the additional activity. Terminal proceeds should reflect the asset and operating configuration that actually exists at the project’s end.

Worked Example: Linking Expansion, Working Capital, and Disposal

A hypothetical two-year project requires $50,000 at time 0 and initially supports capacity of 800 units. At the end of year 1, the project expands capacity to 1,200 units by paying $12,000. Under the case being evaluated, demand is 700 units in year 1 and 1,100 in year 2. Price is $30 in year 1 and $28 in year 2; variable cost is $16 per unit in both years. Fixed costs are $4,000 in year 1 and $5,000 in year 2. Working capital is 5% of revenue: the year-1 amount is invested at time 0, the required balance changes at the end of year 1 to support year 2, and all working capital is recovered at the end of year 2. Net proceeds from the expanded asset are $9,000 at the end of year 2. At an effective annual rate of 10%, calculate NPV.

Time 0. Year-1 units sold are $\min(700,800)=700$, producing revenue of $700\times \$30=\$21{,}000$. Initial working capital is $0.05\times \$21{,}000=\$1{,}050$. Time-0 cash flow is $-\$50{,}000-\$1{,}050=-\$51{,}050$.

End of year 1. Operating cash flow is $700(\$30-\$16)-\$4{,}000=\$5{,}800$. The $12,000 expansion outlay is paid at this time. Year-2 sales are forecast at 1,100 units at $28, so year-2 revenue is $1{,}100\times \$28=\$30{,}800$, and the working-capital balance required for year 2 is $0.05\times \$30{,}800=\$1{,}540$. The increase from $1,050 to $1,540 requires an additional $490 at the end of year 1. Thus year-1 cash flow is $\$5{,}800-\$12{,}000-\$490=-\$6{,}690$.

End of year 2. Expanded capacity is 1,200 units, so all 1,100 units of demand can be served. Operating cash flow is $1{,}100(\$28-\$16)-\$5{,}000=\$8{,}200$. The full $1,540 working-capital balance is recovered, and the expanded asset produces $9,000 of net proceeds. Year-2 cash flow is $\$8{,}200+\$1{,}540+\$9{,}000=\$18{,}740$.

NPV. The expansion payment and additional working-capital investment occur at the end of year 1, not at time 0 or year-end 2.

NPV=-\$51{,}050+\frac{-\$6{,}690}{1.10} +\frac{\$18{,}740}{(1.10)^2} =-\$41{,}644.21

The NPV is $-\$41{,}644.21$, rounded to the nearest cent. The calculation is conditional on the stated demand case supporting expansion and on the $9,000 terminal proceeds being available for the expanded asset. The common mistake avoided is to use expanded capacity or expanded-asset proceeds without recording the expansion outlay and its associated timing.

Practical Checks Before Discounting

Before calculating an NPV, check whether the scenario’s assumptions describe one coherent operating path. These checks are especially useful when a table presents separate ranges for each input.

Demand and price: Confirm that demand is forecast at the price used in the case. If price changes demand, a demand value from one price cannot automatically be carried to another.
Demand and capacity: Use the lower of demand and available capacity for units sold. If an expansion changes capacity, apply it only from its effective date.
Volume and unit cost: Check whether rebates, supplier tiers, overtime, or production inefficiencies depend on the volume being modeled.
Revenue and working capital: Calculate required working capital from the scenario’s revenue, then record each investment and recovery at the stated time. Do not count full recovery while omitting an intervening increase in the required balance.
Operating plan and terminal proceeds: Use terminal proceeds appropriate to the asset and market condition that remain at the end of the scenario.

A scenario is not made coherent merely by labeling it “high,” “base,” or “low.” The assumptions must be mutually compatible, and the cash-flow timeline must reflect how the project would actually operate under those conditions. A scenario-specific NPV is meaningful only after both checks have been made.

Key takeaway: Build scenarios as linked operating cases, not as collections of independent best-case or worst-case inputs. Apply constraints first, derive the resulting revenues and costs, place investment and working-capital changes at the correct times, and use terminal proceeds consistent with the project configuration.

Check Your Understanding

For each question, preserve the stated links among operating assumptions and show the relevant cash-flow timing before checking the answer.

A project has demand of 950 units, capacity of 800 units, a price of $27 per unit, variable cost of $16 per unit, and fixed operating costs of $3,000. What are units sold, revenue, and simplified operating cash flow? (Answer: 800 units; $21,600 revenue; $5,800 operating cash flow.)
Working capital rises from $2,400 to $3,100 during a project’s final period and the full $3,100 is then recovered. What is the net working-capital cash flow in that period? (Answer: a $2,400 cash inflow, since $-\$700+\$3,100=\$2,400$.)
A one-year project costs $20,000, has capacity for 600 units, and faces demand of 500 units at $30 per unit. Variable cost is $18 per unit, fixed costs are $2,000, working capital is 5% of revenue and fully recovered at year-end, and net terminal proceeds are $4,000. At an effective annual rate of 10%, what is NPV? (Answer: $-\$12,795.45$.)
Market research supports demand of 900 units at a price of $32 and 1,200 units at a price of $28. Capacity is 1,000 units. What quantity is consistent with a $32 price, and why is it not appropriate to use the 1,200-unit demand forecast with that price? (Answer: 900 units; the 1,200-unit demand forecast applies to the $28 price condition.)
A two-year project has initial working capital of $1,200. At the end of year 1, the required balance increases by $500; at the end of year 2, the full year-2 balance is recovered. What is the net working-capital cash flow in year 2, assuming the year-2 balance is $1,700? (Answer: a $1,200 inflow, since the year-2 cash flow is $-\$500+\$1,700$.)

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