Why a Positive-NPV Project May Not Be Funded
A project can increase the value of a firm and still fail to attract its existing shareholders’ funding. The reason is that the project’s benefits do not necessarily go entirely to equity. If the firm has debt outstanding, project cash flows may first improve the amount creditors can recover in states where the firm would otherwise default. Shareholders then compare their own increase in equity value with the contribution they must provide.
This conflict is called debt overhang: shareholders may reject a project with positive net present value (NPV) because some of its value accrues to existing debtholders. The key is to keep two questions separate: does the project create value for the firm as a whole, and does the value it creates for equity justify the new equity investment?
Model and Valuation Method
The examples use a one-period, two-state model. State probabilities are risk-neutral probabilities, not estimates of real-world frequencies. For a risk-free rate \(r_f\), a maturity payoff \(Z_s\) has present value \(\sum_s q_s Z_s/(1+r_f)\), where \(q_s\) is the risk-neutral probability of state \(s\). This valuation assumes no arbitrage, a traded risk-free rate for discounting, and that the stated probabilities are risk-neutral for the payoffs being valued.
The firm has one fixed debt claim due at maturity, and equity has limited liability. There are no taxes, transaction costs, bankruptcy costs, additional financing claims, or changes to the debt contract unless stated. The project’s investment is paid today by existing shareholders as a new equity contribution; the project’s terminal benefits are included in the firm’s assets. These simplifying assumptions isolate the distribution of project value between equity and existing debt.
The equality holds in both payoff regions. If \(V_s\leq D\), debt receives \(V_s\) and equity receives zero. If \(V_s>D\), debt receives \(D\) and equity receives \(V_s-D\); together those payoffs equal \(V_s\). Therefore, after the project, the state-by-state increases in debt and equity payoffs must add to the project’s asset benefit \(G_s\).
The first equality defines the project’s NPV. For the second, add the state-by-state identity \(\Delta E_s+\Delta B_s=G_s\), multiply each side by its risk-neutral probability, sum over states, and discount at \(r_f\). This shows why a firm-level positive NPV does not guarantee a positive investment incentive for shareholders: their decision uses the equity-value increase, not the combined increase in equity and debt value.
A Numerical Debt-Overhang Case
The calculation has a consistent order: value the project’s total benefits, calculate the debt and equity payoffs in each state both with and without the project, and compare each claim’s value increase with the project cost. A state-by-state payoff table is especially useful near the debt face amount, where a small change in assets can change who receives the project’s benefits.
Worked Example: Positive Firm NPV but a Shareholder Loss
A firm has existing debt with a face amount of $110 due in one year. Without a proposed project, the firm’s assets at maturity will be $66 in the low state and $154 in the high state. A project costs $34 today and adds $22 to assets in the low state and $66 in the high state. Each state has risk-neutral probability \(0.5\), and the one-year risk-free rate is 10%. Determine the project NPV and whether existing shareholders will fund it.
Method. First value the project benefits, then apply the debt and equity payoff formulas in both states. This avoids the common mistake of treating the whole project benefit as a gain to shareholders.
Total project value and NPV. The risk-neutral expected terminal benefit is \(0.5(22)+0.5(66)=11+33=44\). Its present value is \(44/1.10=40\). Therefore, \(\operatorname{NPV}=40-34=\$6\). The project increases total firm value by $6 after accounting for its cost.
Low state, without the project. Assets are $66. Debt receives \(\min(66,110)=\$66\); equity receives \(\max(66-110,0)=\$0\).
Low state, with the project. Assets are \(66+22=\$88\). Debt receives \(\min(88,110)=\$88\); equity receives \(\max(88-110,0)=\$0\). The project adds $22 to debt’s payoff and nothing to equity’s payoff in this state.
High state, without the project. Assets are $154. Debt receives \(\min(154,110)=\$110\); equity receives \(154-110=\$44\).
High state, with the project. Assets are \(154+66=\$220\). Debt still receives \(\min(220,110)=\$110\); equity receives \(220-110=\$110\). The project adds $66 to equity’s payoff and nothing to debt’s payoff in this state.
Shareholder decision. The present value of the equity payoff increase is \([0.5(0)+0.5(66)]/1.10=33/1.10=\$30\). Shareholders would contribute $34, so their net gain from funding is \(30-34=-\$4\). They will reject the project under the stated assumptions.
Debtholder gain and check. The present value of the debt payoff increase is \([0.5(22)+0.5(0)]/1.10=11/1.10=\$10\). The firm’s NPV reconciles: \(-\$4+\$10=\$6\). Thus the project is positive-NPV, but shareholders bear a $4 net loss while existing debt gains $10. The state-by-state payoff increases also check: \(22=22+0\) in the low state and \(66=0+66\) in the high state.
Interpreting the Payoff Allocation
In the low state of the example, the firm’s assets remain below the debt face amount after investment. The project raises the amount available to creditors from $66 to $88, while equity remains out of the money. In the high state, debt is already paid in full without the project, so the project’s $66 benefit goes entirely to equity. Because only the high-state benefit reaches equity, the present value of equity’s gain is less than the $34 contribution.
This is the defining incentive problem. It is not enough to observe that the project improves asset values or that its NPV is positive. Shareholders need to evaluate the incremental value of their own claim under the existing capital structure. The fixed debt face amount creates a threshold: below it, additional asset value may improve debt recovery; above it, additional value generally accrues to equity.
| Question | Calculation | Decision implication |
|---|---|---|
| Does the project create value? | Present value of project benefits less investment | Positive NPV means the combined claims gain value. |
| How much value reaches equity? | Present value of state-by-state equity payoff increases | Compare this amount with the new equity contribution. |
| How much value reaches existing debt? | Present value of state-by-state debt payoff increases | This can explain why total NPV is positive even when equity declines. |
The result is distinct from risk shifting. In risk shifting, shareholders may prefer a riskier project because they receive more upside while some downside falls on creditors. In debt overhang, the central issue is that the project’s benefits may improve recovery on existing debt, leaving too little incremental value for shareholders to justify funding. The calculations use the same debt and equity payoff definitions, but the investment incentive differs.
How Debt Terms Can Change the Incentive
Reducing the debt face amount can move the equity payoff threshold. If less debt is due at maturity, more project value may accrue to equity in states where the firm previously could not repay its creditors in full. That may make shareholders more willing to fund the project. However, a debt reduction is not itself a creation of project value: it changes how value is divided between claimholders and may transfer value from creditors to shareholders. Whether creditors agree to renegotiate is a separate question.
Worked Example: Finding a Debt Face Amount That Removes the Disincentive
A firm’s assets at maturity will be $40 or $120, each with risk-neutral probability \(0.5\). Its existing debt has face amount $80. A project costs $32 today and adds $20 in the low state and $60 in the high state. The risk-free rate is 10%. Find the greatest debt face amount at which shareholders would be willing to fund the project, assuming the debt face amount can be renegotiated before the investment decision.
Method. Find the present value of the project’s equity payoff increase as a function of the renegotiated debt face amount \(D'\), and set that value equal to the $32 contribution. The common mistake is to set total project value equal to the investment cost; that tests firm NPV, not the shareholders’ incentive.
Check total NPV first. The risk-neutral expected project benefit at maturity is \(0.5(20)+0.5(60)=10+30=40\). Its present value is \(40/1.10=\$36.3636\), so project NPV is \(36.3636-32=\$4.3636\). The project creates value for the firm.
At the original debt face amount of $80, low-state assets rise from $40 to \(40+20=\$60\), still below debt, so equity receives zero with or without the project. In the high state, equity rises from \(120-80=\$40\) to \(180-80=\$100\), a $60 increase. The present value of the equity increase is \(0.5(60)/1.10=\$27.2727\), less than $32. Shareholders will not fund at the original debt face amount.
For a renegotiated face amount \(D'\) between $40 and $60, low-state equity is zero before the project and \(60-D'\) after it. High-state equity increases by $60: both the original and project asset values exceed \(D'\), and the project adds $60. Thus the present value of the equity increase is \([0.5(60-D')+0.5(60)]/1.10=(60-D'/2)/1.10\).
For shareholders to be just willing to contribute $32, set this present value equal to $32: \((60-D'/2)/1.10=32\). Multiplying by 1.10 gives \(60-D'/2=35.2\), so \(D'/2=24.8\) and \(D'=\$49.60\). At this amount, the low-state equity increase is \(60-49.60=\$10.40\), and the high-state increase is $60. Their expected increase at maturity is \(0.5(10.40)+0.5(60)=5.20+30=\$35.20\); discounted at 10%, it is \(35.20/1.10=\$32\), exactly the investment cost.
Conclusion and limitation. Under this model, the greatest debt face amount at which shareholders will fund is $49.60; at a lower face amount, their project-related equity gain is at least $32. This result establishes only the shareholders’ incentive conditional on renegotiation. It does not show that creditors will accept the reduction or that renegotiation is costless.
Common Exam Traps
- Calling positive firm NPV sufficient for investment. It is not sufficient when shareholders must fund the project. Calculate the present value of the equity payoff increase separately.
- Ignoring the debt threshold state by state. Apply \(\min(V_s,D)\) and \(\max(V_s-D,0)\) before taking expected values. A project benefit below the threshold may accrue to debt even if it increases the firm’s assets.
- Subtracting the investment twice. Project NPV is project-benefit value minus \(I\). Shareholders’ net gain is equity-value increase minus their contribution \(I\). Do not subtract \(I\) again when calculating the payoff increase to equity or debt.
- Mixing payoff dates or valuation probabilities. In these examples, \(I\) is paid today, terminal payoffs are one year away, and risk-neutral expected payoffs are discounted once at the stated risk-free rate.
- Treating debt renegotiation as project value creation. A face-amount reduction can shift value between claims. Keep that transfer separate from the project’s operating NPV.
Check Your Understanding
For each question, use the one-period limited-liability model and the stated risk-neutral probabilities and risk-free rate.
- A firm has debt of $90 due in one year and assets of $50 or $130 in two equally likely risk-neutral states. A project costs $20 today and adds $10 or $50 to assets, respectively. The risk-free rate is 0%. Calculate project NPV, the present value of the equity gain, and the shareholders’ net gain. (Answer: Project NPV is $10; equity gains $25 in present value, so shareholders’ net gain is $5.)
- A firm has debt of $100, assets of $60 or $140 in two equally likely states, and a project costing $24 that adds $20 or $60, respectively. The risk-free rate is 5%. Calculate total NPV and the shareholders’ net gain. (Answer: Total NPV is approximately $14.10; the present value of the equity gain is approximately $28.57, so shareholders’ net gain is approximately $4.57.)
- A firm has debt of $120 and assets of $80 or $180 in two equally likely states. A project costs $30 and adds $10 in the low state and $70 in the high state. The risk-free rate is 10%. Find total NPV and the net gain to shareholders after their contribution. (Answer: Total NPV is approximately $6.36; the present value of the equity gain is approximately $31.82, so shareholders’ net gain is approximately $1.82.)
- A firm has debt of $100 and assets of $70 or $150 in two equally likely states. A project costs $28 and adds $20 or $60, respectively. The risk-free rate is 0%. Calculate project NPV, the present value of the equity gain, and the present value of the debt gain. (Answer: Project NPV is $12; the equity gain is $30, the debt gain is $10, and shareholders’ net gain after investing $28 is $2.)
- A firm has debt of $100 and assets of $50 or $150 in two equally likely states. A project costs $25 and adds $10 or $60, respectively. The risk-free rate is 0%. Find project NPV and determine whether shareholders fund it. (Answer: Project NPV is $10; the equity payoff increase is \(0\) in the low state and $60 in the high state, giving a present value of $30. Shareholders’ net gain is $5, so they fund it.)