Why Sensitivity Analysis Matters
A project discount rate is an estimate, not a directly observed constant. It depends on assumptions about systematic risk, financing, taxes, and market compensation for risk. Even when a base-case rate is calculated correctly, a small change in one of those assumptions can alter the present value of the project’s cash flows and, in some cases, change an accept-or-reject conclusion.
Sensitivity analysis makes that dependence visible. A one-way sensitivity changes one input while holding the others fixed. A scenario analysis changes several inputs together to represent a coherent alternative set of conditions. The first helps isolate drivers; the second shows how a project performs when assumptions move jointly. Neither method makes uncertain inputs certain, and neither substitutes for checking whether the chosen assumptions are reasonable.
Discount Rates and Project Value
For a project with initial investment \(I_0\), expected free cash flow \(FCF_t\) at the end of period \(t\), and a constant annual discount rate \(r\), net present value is
This form assumes the cash flows occur at the stated period ends, are measured consistently in nominal or real terms with the rate, and are discounted at a rate appropriate to their risk. If the risk or discount rate changes by period, use the corresponding period-specific rates rather than a single constant \(r\). In this tutorial, annual rates and annual period-end cash flows are used.
For a project discount rate based on a weighted average cost of capital, let \(E\) and \(D\) be the market values of equity and debt financing, \(V=E+D\), \(r_e\) the cost of equity, \(r_d\) the before-tax cost of debt, and \(T\) the corporate tax rate. The standard after-tax WACC is
This measure is appropriate when project cash flows are consistent with the financing assumptions represented by the weights and the project has risk comparable to the assets for which the rate was estimated. It is not a universal rate for every project: a project with different operating risk or financing policy needs a rate reflecting that difference. The cost of equity is commonly estimated using CAPM, \(r_e=r_f+\beta_E MRP\), where \(r_f\) is the risk-free rate, \(\beta_E\) is equity beta, and \(MRP\) is the market risk premium. The CAPM estimate assumes the stated beta and premium are appropriate for the market and period being analyzed.
Worked Example: Isolating the Effect of Equity Beta
A hypothetical project requires \(I_0=\$1{,}000\) and is expected to produce \(\$400\) at the end of each of the next three years. Use \(r_f=3\%\), \(MRP=5\%\), \(D/V=0.40\), \(E/V=0.60\), \(r_d=5\%\), and \(T=20\%\). The estimated equity beta is 1.20. Measure the effect on WACC and NPV if beta is instead 1.00 or 1.40, holding every other input fixed.
Method. Recalculate the CAPM cost of equity for each beta, then the WACC, and finally discount the same project cash flows at each WACC. This one-way comparison avoids attributing the effect of a beta change to a simultaneous change in leverage or debt cost.
Base case. CAPM gives \(r_e=3\%+1.20(5\%)=9.00\%\). The after-tax debt cost is \(5\%(1-0.20)=4.00\%\). Thus \(WACC=0.60(9.00\%)+0.40(4.00\%)=5.40\%+1.60\%=7.00\%\).
Lower beta. At \(\beta_E=1.00\), \(r_e=3\%+1.00(5\%)=8.00\%\), so \(WACC=0.60(8.00\%)+0.40(4.00\%)=6.40\%\).
Higher beta. At \(\beta_E=1.40\), \(r_e=3\%+1.40(5\%)=10.00\%\), so \(WACC=0.60(10.00\%)+0.40(4.00\%)=7.60\%\).
The present-value factors for the three annual payments are \(2.624316\) at 7.00%, \(2.653352\) at 6.40%, and \(2.595812\) at 7.60%, each calculated as \(\sum_{t=1}^{3}(1+r)^{-t}\). Therefore the base NPV is \(-\$1{,}000+\$400(2.624316)=\$49.73\). At 6.40%, NPV is \(-\$1{,}000+\$400(2.653352)=\$61.34\). At 7.60%, NPV is \(-\$1{,}000+\$400(2.595812)=\$38.32\). Dollar results are rounded to the nearest cent only at the final step.
Check. The higher beta produces a higher cost of equity and WACC, and the same positive future cash flows consequently have a lower present value. The NPV changes are consistent with that direction.
How the Main Assumptions Enter the Rate
Each input reaches WACC through a particular channel. Understanding that channel helps avoid changing the wrong quantity or counting the same risk twice. In practical estimates, a beta may be adjusted for target leverage before estimating the project’s cost of equity. One common approximation, when debt beta is treated as zero, is
where \(\beta_A\) is the asset beta and \(D/E\) is the target debt-to-equity ratio. This approximation assumes the tax-shield treatment embedded in the formula and negligible systematic risk in debt. It is not appropriate without adjustment when debt risk is material or a different leverage model is required. Once beta is estimated, CAPM translates beta and the market risk premium into a cost of equity; the WACC weights then combine that cost with the after-tax debt cost.
| Assumption changed | Primary effect | Important qualification |
|---|---|---|
| Equity beta | Changes \(r_e\) by the beta change times \(MRP\); WACC transmits this through \(E/V\). | Hold other inputs fixed for a clean one-way comparison. |
| Target leverage | Changes debt and equity weights and may change equity beta under a relevering model. | Recompute both weights and beta; changing only the weights misses an effect. |
| Tax rate | Changes the after-tax debt cost and may affect relevered beta. | Use a tax assumption consistent with the WACC model and project cash flows. |
| Debt cost | Changes the debt component of WACC, multiplied by \(D/V\) and \(1-T\). | Do not apply the tax adjustment a second time if the debt input is already after tax. |
| Market risk premium | Changes \(r_e\) by \(\beta_E\) times the change in \(MRP\). | It affects the equity component, not the debt component directly. |
Worked Example: Comparing One-Way WACC Sensitivities
Consider a hypothetical estimate with \(r_f=3\%\), \(MRP=5\%\), \(\beta_A=0.80\), \(D/E=0.50\), \(T=20\%\), and \(r_d=5\%\). Start with these assumptions and test five changes separately: equity beta rises by 0.10; \(D/E\) rises to 0.75; tax rises to 25%; debt cost rises to 6%; or \(MRP\) rises to 6%. For leverage and tax changes, recalculate beta using the stated zero-debt-beta approximation.
Base calculation. The equity beta is \(0.80[1+(1-0.20)(0.50)]=0.80(1.40)=1.12\). Thus \(r_e=3\%+1.12(5\%)=8.60\%\). Since \(D/E=0.50\), \(D/V=0.50/1.50=1/3\) and \(E/V=2/3\). The base WACC is \((2/3)(8.60\%)+(1/3)(5\%)(0.80)=5.7333\%+1.3333\%=7.0667\%\).
Equity beta rises by 0.10. With weights and other assumptions fixed, the cost of equity rises by \(0.10(5\%)=0.50\) percentage points. WACC therefore rises by \((2/3)(0.50)=0.3333\) percentage points, to 7.4000%.
Target leverage rises to \(D/E=0.75\). Relevered beta becomes \(0.80[1+0.80(0.75)]=1.28\), so \(r_e=3\%+1.28(5\%)=9.40\%\). The weights are \(D/V=0.75/1.75=3/7\) and \(E/V=4/7\). WACC is \((4/7)(9.40\%)+(3/7)(5\%)(0.80)=5.3714\%+1.7143\%=7.0857\%\), an increase of 0.0190 percentage points from the base.
Tax rises to 25%. Relevered beta becomes \(0.80[1+0.75(0.50)]=1.10\), giving \(r_e=8.50\%\). The after-tax debt cost is \(5\%(0.75)=3.75\%\). With the original weights, WACC is \((2/3)(8.50\%)+(1/3)(3.75\%)=5.6667\%+1.2500\%=6.9167\%\), a decrease of 0.1500 percentage points.
Debt cost rises to 6%. With beta, weights, and tax fixed, the after-tax debt cost rises from \(5\%(0.80)=4.00\%\) to \(6\%(0.80)=4.80\%\). WACC rises by \((1/3)(0.80)(1.00\%)=0.2667\) percentage points, to 7.3333%.
Market risk premium rises to 6%. The increase in cost of equity is \(1.12(1.00\%)=1.12\) percentage points. WACC rises by \((2/3)(1.12\%)=0.7467\) percentage points, to 7.8133%.
For these specified shocks, the market-risk-premium change has the largest absolute WACC effect, followed by the beta change, debt-cost change, tax change, and leverage change. This is a ranking of the particular changes specified, not a universal ranking: the input ranges matter, and a different plausible shock size could alter the order.
Why a Higher Rate Lowers NPV
For fixed future cash flows, each discounted term falls as the rate rises. The size of the NPV response depends on both the cash-flow amounts and their timing. Differentiating the constant-rate NPV expression with respect to \(r\) gives
To justify the result, write each present-value term as \(FCF_t(1+r)^{-t}\). Its derivative is \(FCF_t[-t(1+r)^{-t-1}]\); the initial investment is constant and has derivative zero. Summing the derivatives gives the displayed expression. If all future \(FCF_t\) are positive, the derivative is negative, so an increase in the discount rate reduces NPV. Larger or later cash flows generally contribute more to the magnitude of the rate sensitivity, although cash-flow timing and sign patterns must be considered in projects with interim outflows.
This derivative is a local measure: it describes the approximate NPV change for a small rate change near the starting rate. For a material change, recompute NPV at the changed rate rather than relying only on the derivative.
Worked Example: Combining Assumptions in Scenarios
A hypothetical project costs \(\$1{,}000\) initially and is expected to pay \(\$400\) at each year-end for three years. Compare a base case with a favorable and an adverse set of assumptions. In all cases use \(r_f=3\%\), the zero-debt-beta relevering approximation, and the after-tax WACC formula.
Base case. Let \(\beta_A=0.80\), \(D/E=0.50\), \(T=20\%\), \(r_d=5\%\), and \(MRP=5\%\). As calculated above, \(\beta_E=1.12\), \(r_e=8.60\%\), \(E/V=2/3\), and \(D/V=1/3\). WACC is 7.0667%.
Favorable case. Let \(\beta_A=0.75\), \(D/E=0.40\), \(T=25\%\), \(r_d=4.5\%\), and \(MRP=4.5\%\). Then \(\beta_E=0.75[1+0.75(0.40)]=0.975\), and \(r_e=3\%+0.975(4.5\%)=7.3875\%\). The weights are \(E/V=1/1.40=0.714286\) and \(D/V=0.40/1.40=0.285714\). The after-tax debt cost is \(4.5\%(0.75)=3.375\%\). Thus WACC is \((0.714286)(7.3875\%)+(0.285714)(3.375\%)=6.2411\%\).
Adverse case. Let \(\beta_A=0.90\), \(D/E=0.75\), \(T=15\%\), \(r_d=6\%\), and \(MRP=6\%\). Then \(\beta_E=0.90[1+0.85(0.75)]=1.47375\), and \(r_e=3\%+1.47375(6\%)=11.8425\%\). The weights are \(E/V=4/7=0.571429\) and \(D/V=3/7=0.428571\). The after-tax debt cost is \(6\%(0.85)=5.10\%\). Thus WACC is \((0.571429)(11.8425\%)+(0.428571)(5.10\%)=8.9529\%\).
Value comparison. The three-year annuity factors are 2.661137 at 6.2411%, 2.621122 at 7.0667%, and 2.533421 at 8.9529%. The corresponding NPVs are \(-\$1{,}000+\$400(2.661137)=\$64.45\), \(-\$1{,}000+\$400(2.621122)=\$48.45\), and \(-\$1{,}000+\$400(2.533421)=\$13.37\). Dollar values are rounded to the nearest cent only at the final step.
Check. The adverse assumptions produce the highest WACC and the lowest NPV; the favorable assumptions produce the lowest WACC and the highest NPV. Because several inputs moved at once, this comparison describes combined exposure but does not identify which single assumption caused the largest change. The one-way analysis is needed for that diagnosis.
Common Traps and Interpretation
- Changing a beta without tracing it through the rate. A change in equity beta first changes CAPM cost of equity; its WACC effect is then weighted by \(E/V\). Do not apply the beta change directly to the entire WACC.
- Changing leverage weights but not relevering beta. If the model uses target leverage to estimate equity beta, a leverage change affects both the weights and the cost of equity. Recompute both consistently.
- Applying the tax adjustment twice. The WACC formula multiplies the before-tax debt cost by \(1-T\). If a stated debt cost is already after tax, do not reduce it again.
- Comparing unlike shock sizes. A one-percentage-point change in debt cost and a 0.10 change in beta are not equivalent-sized changes. State the perturbation and, when comparing drivers, use defensible ranges or report sensitivity per unit of input.
- Using only the local derivative for a large rate shift. The derivative is a local approximation. Recalculate cash-flow present values for the full changed rate when the shift is substantial.
- Assuming a positive base NPV guarantees robustness. The base result is conditional on its inputs. A scenario can materially reduce value, so report the range and the assumptions behind it.
Check Your Understanding
For each problem, state the calculation path and check that the direction of the result is consistent with the assumptions.
- A hypothetical project has \(r_f=4\%\), \(MRP=5\%\), \(\beta_E=1.10\), \(E/V=0.70\), \(D/V=0.30\), \(r_d=6\%\), and \(T=25\%\). Calculate its CAPM cost of equity and WACC. (Answer: \(r_e=9.500\%\); \(WACC=8.000\%\).)
- A project’s asset beta is 0.80, target \(D/E=0.75\), and tax rate is 20%. Under the zero-debt-beta relevering approximation, calculate equity beta. (Answer: \(\beta_E=1.28\).)
- Equity beta is 1.25 and \(E/V=0.60\). If the market risk premium rises by 1.50 percentage points, with all other assumptions fixed, by how many percentage points does WACC rise? (Answer: 1.125 percentage points.)
- A hypothetical project costs \(\$900\) and pays \(\$350\), \(\$350\), and \(\$400\) at the ends of years 1, 2, and 3. At a constant annual discount rate of 10%, calculate NPV. (Answer: \(NPV=\$7.96\), rounded to the nearest cent.)
- A project has \(D/V=0.40\) and \(T=25\%\). If its before-tax debt cost rises by 0.75 percentage points, with equity cost and financing weights fixed, what is the WACC increase? (Answer: 0.225 percentage points.)