Why Use a Two-Input Sensitivity Table?
A project forecast depends on assumptions, and more than one assumption may change at a time. For a product project, sales volume and unit cost might both differ from their planning estimates. A two-input sensitivity table shows the project's NPV across selected combinations of those inputs. It helps a decision-maker see not only how each assumption matters, but also whether a combination creates or destroys value under the stated model.
A table is a structured set of calculations, not a probability model. Its rows and columns are chosen input values; each cell is the NPV that follows from that pair while the other assumptions remain fixed. The table does not say how likely any cell is.
Model and Cell Calculation
Consider a project that requires an initial investment \(I_0\) at time 0 and produces equal annual cash flows at each year-end for \(n\) years. Suppose each year it sells \(Q\) units at price \(p\), incurs unit cost \(c\), and pays fixed operating costs \(F\). Assume there are no taxes, working-capital changes, additional investment, or terminal cash flows. The effective annual discount rate is \(i\).
Under these assumptions, annual operating cash flow is the contribution per unit multiplied by volume, less fixed costs. Discounting that level cash flow as an ordinary annuity gives:
Here, \(a_{\overline{n}|}\) is the present value at time 0 of one dollar paid at the end of each year for \(n\) years. This formula applies to the stated level, year-end cash-flow model; it should not be used unchanged if cash flows vary by year, occur at different times, or include omitted items such as taxes or a terminal receipt.
The reason the annuity factor appears can be seen by writing the present value of the annual cash flows as a geometric sum. If annual cash flow is \(C=(p-c)Q-F\), the present value is \(C\sum_{t=1}^{n}(1+i)^{-t}\). Let \(v=(1+i)^{-1}\). Then the sum is \(v+v^2+\cdots+v^n\). Multiplying this sum by \(1-v\) cancels its intermediate terms and gives \(v-v^{n+1}\), so the sum equals \(v(1-v^n)/(1-v)\). Since \(1-v=i/(1+i)=iv\), this is \([1-v^n]/i=[1-(1+i)^{-n}]/i\), which is \(a_{\overline{n}|}\). Subtracting the time-0 investment gives the NPV formula.
To build the table, place one input's selected values in rows and the other input's values in columns. For every cell, substitute that specific pair into the cash-flow formula, find its present value using the same rate and horizon, and subtract the same initial investment. State the units in the headings: a volume of 10 may mean 10,000 units, while an NPV of 25 may mean $25,000.
State the cash-flow timing, project life, discount rate, initial investment, and assumptions that remain fixed.
Label each row and column with its input value and unit. Keep all other assumptions unchanged across cells.
Compute the annual cash flow for that pair, discount it, and subtract the initial investment.
Identify positive and negative cells and check whether the direction of changes is consistent with the model's economics.
Worked Table: Sales Volume and Unit Cost
The first example builds a complete table with volume and unit cost as the two changing assumptions. Amounts are shown in thousands of dollars, and volume is in thousands of units. This convention lets the arithmetic remain compact without changing the underlying dollar-per-unit calculations.
Worked Example: Volume-Cost Sensitivity Grid
A hypothetical project requires $400,000 at time 0. It sells each unit for $50, has annual fixed costs of $40,000, and produces equal year-end cash flows for three years. The effective annual discount rate is 10%. Ignore taxes and terminal cash flows. Construct an NPV table for annual volume \(Q=10,12,14\) thousand units and unit cost \(c=\$30,\$35,\$40\).
Method. Calculate the three-year annuity factor, then calculate annual cash flow and NPV for each volume-cost pair. The factor is:
For example, at \(Q=12\) thousand units and \(c=\$35\), annual contribution is \((\$50-\$35)(12{,}000)=\$180{,}000\). After $40,000 fixed costs, annual cash flow is $140,000, or $140 thousand. Thus NPV in thousands is \(-400+140(2.486851991)=-51.8407\), which rounds to \(-\$51.84\) thousand.
Applying the same steps to each cell gives the following table. The row labels are volumes in thousands of units; the column labels are unit costs in dollars per unit.
| Annual volume | Unit cost $30 | Unit cost $35 | Unit cost $40 |
|---|---|---|---|
| 10 thousand | −$2.10 thousand | −$126.45 thousand | −$250.79 thousand |
| 12 thousand | $97.37 thousand | −$51.84 thousand | −$201.05 thousand |
| 14 thousand | $196.84 thousand | $22.76 thousand | −$151.31 thousand |
For instance, the \(Q=10,c=30\) cell has annual cash flow \((50-30)(10)-40=160\) thousand. Its NPV is \(-400+160(2.486851991)=-2.1037\) thousand, or \(-\$2.10\) thousand to the nearest cent in the stated thousands convention. Positive cells occur at lower unit costs with sufficient volume; negative cells occur where the cash flows do not recover the initial investment at 10%. The table shows the result of each combination, not the chance of reaching it.
How to Read the Table Correctly
A table's pattern should follow from the cash-flow model, but the direction of change depends on the assumptions. In this model, increasing unit cost cannot increase NPV when volume is positive: each additional dollar of unit cost reduces annual cash flow by \(Q\) dollars. Increasing volume has a conditional effect. For a fixed price and cost, the change in NPV from a volume increase \(\Delta Q\) is:
Because \(a_{\overline{n}|}>0\) for a positive rate and positive project life, increasing volume raises NPV when \(p>c\), leaves it unchanged when \(p=c\), and lowers it when \(p<c\). This is why the direction of a table should be checked against the contribution margin, rather than assumed from the word “volume.”
A useful check is that adjacent cells differ in the direction implied by the formula, provided the other assumptions are held fixed. If a table says that raising unit cost improves NPV at positive volume, check for a sign error, a mislabeled row or column, or inconsistent units. But do not reject a decrease in NPV when volume rises unless you have first checked whether price exceeds unit cost.
Worked Example: When More Volume Reduces NPV
A hypothetical project costs $40,000 initially and sells a product for $10 per unit. Unit cost is $12, annual fixed costs are $5,000, and the project produces equal year-end cash flows for two years. The effective annual discount rate is 10%. Ignore taxes and terminal cash flows. Compare NPVs at annual volumes of 4,000 and 6,000 units.
Method. Use the same level-cash-flow model, while retaining the negative contribution margin. The two-year factor is \(a_{\overline{2}|}=[1-(1.10)^{-2}]/0.10=1.735537190\).
At 4,000 units, annual cash flow is \((10-12)(4{,}000)-5{,}000=-\$13{,}000\). Therefore, NPV is \(-40{,}000-13{,}000(1.735537190)=-\$62{,}561.98\).
At 6,000 units, annual cash flow is \((10-12)(6{,}000)-5{,}000=-\$17{,}000\). Therefore, NPV is \(-40{,}000-17{,}000(1.735537190)=-\$69{,}504.13\). The difference is \(-\$6{,}942.15\), which also equals the additional annual loss of $4,000 multiplied by 1.735537190. Both NPVs are negative, and the higher volume makes value worse because each additional unit sells below its unit cost. The check confirms the formula's direction.
Changing a Different Pair of Inputs
A sensitivity table can use other inputs, as long as the cash-flow model is explicit and the two varying assumptions are not mixed with hidden changes to other inputs. For example, price and volume can vary together while unit cost and fixed costs remain fixed. Each cell still represents one specified pair, and its NPV is calculated from that pair.
Worked Example: Sales Price and Volume
A hypothetical project requires $300,000 initially, has unit cost of $18 and fixed costs of $50,000 per year, and lasts four years with equal year-end cash flows. The effective annual discount rate is 8%. Ignore taxes and terminal cash flows. Construct the NPV table for prices of $30, $34, and $38 and volumes of 10,000, 12,000, and 14,000 units. Report NPVs in thousands of dollars to two decimal places.
Method. The four-year factor is \(a_{\overline{4}|}=[1-(1.08)^{-4}]/0.08=3.31212684\). For the price of $30 and volume of 10,000, annual cash flow in thousands is \((30-18)(10)-50=70\). The NPV is \(-300+70(3.31212684)=-68.1511\) thousand, or \(-\$68.15\) thousand.
Repeating the calculation for each cell gives:
| Price per unit | 10 thousand units | 12 thousand units | 14 thousand units |
|---|---|---|---|
| $30 | −$68.15 thousand | $11.34 thousand | $90.83 thousand |
| $34 | $64.33 thousand | $170.32 thousand | $276.31 thousand |
| $38 | $196.82 thousand | $329.30 thousand | $461.79 thousand |
For the $34, 12,000-unit cell, annual cash flow is \((34-18)(12)-50=142\) thousand; its NPV is \(-300+142(3.31212684)=170.3220\) thousand, which rounds to $170.32 thousand. The largest NPV in the grid is $461.79 thousand at $38 and 14,000 units; the only negative cell is at $30 and 10,000 units. In this grid, every contribution margin is positive, so increasing either price or volume increases NPV when the other input is fixed. That direction is consistent with the model.
Common Mistakes and Limits
- Assuming that more volume always helps. It helps only when the contribution margin \(p-c\) is positive; if \(p<c\), more units increase the operating loss.
- Changing more than the two labeled inputs. If one cell also uses a different discount rate or project life, the table no longer isolates the two stated assumptions.
- Mixing units. Multiplying a dollar-per-unit margin by volume in thousands produces thousands of dollars. Keep that scale consistent when subtracting investment and fixed costs.
- Reading signs as probabilities. A positive NPV cell is not necessarily likely, and a negative cell is not necessarily unlikely. The grid assigns no probabilities unless separate probability information is supplied.
- Over-interpreting the selected grid values. A table reports the combinations actually calculated. It does not establish what happens at every value between grid points or outside the selected range.
The central use of a two-input table is to make joint sensitivity visible while preserving a consistent NPV calculation. A defensible interpretation names the assumptions, checks units and cash-flow timing, and explains why the pattern of cells follows from the model.
Check Your Understanding
For each problem, show the cash-flow calculation before interpreting the NPV or table.
- A hypothetical project costs $120,000 initially, sells for $26 per unit, has unit cost $18 and annual fixed costs of $20,000, and operates for three years with equal year-end cash flows. The effective annual discount rate is 10%; ignore taxes and terminal cash flows. Find NPV at 7,000 units per year. (Answer: \(-\$30,473.33\))
- A project costs $100,000, sells for $32 per unit, and has annual fixed costs of $15,000. Its equal year-end cash flows last two years; the effective annual discount rate is 10%. Unit cost is either $20 or $24, and annual volume is either 5,000 or 8,000 units. Calculate all four NPVs in thousands of dollars, to two decimals. (Answer: At \(c=\$20\), NPVs are \(-\$21.90\) thousand at 5,000 units and $40.58 thousand at 8,000 units; at \(c=\$24\), they are \(-\$56.61\) thousand and \(-\$14.96\) thousand, respectively.)
- A project sells for $15 per unit, has unit cost $18 and annual fixed costs of $8,000. Its initial cost is $50,000; it produces equal year-end cash flows for three years, and the effective annual discount rate is 8%. By how much does NPV change when annual volume increases from 2,000 to 4,000 units? (Answer: NPV decreases by $15,462.58.)
- A project costs $250,000 initially, has unit cost $22 and annual fixed costs of $30,000, and lasts three years with equal year-end cash flows. The effective annual discount rate is 10%; ignore taxes and terminal cash flows. Find the NPV at each combination of price $30 or $34 and volume 8,000 or 10,000 units, and identify the highest cell. Report in thousands of dollars to two decimals. (Answer: At $30, NPVs are \(-\$165.45\) thousand and \(-\$125.66\) thousand; at $34, they are \(-\$85.87\) thousand and \(-\$26.18\) thousand. The highest cell is $34 and 10,000 units, but its NPV is still negative.)
- A project costs $300,000 initially, sells for $40 per unit, and has annual fixed costs of $80,000. It lasts four years with equal year-end cash flows; the effective annual discount rate is 8%. Unit cost is either $25 or $28, and annual volume is either 20,000 or 25,000 units. Which combination has the highest NPV, and what is that NPV in thousands of dollars to two decimals? (Answer: Unit cost $25 and volume 25,000 units; NPV is $677.08 thousand.)