Risk Shifting and Asset Substitution

Why Risk Can Shift Value Between Shareholders and Debtholders

A company that has issued debt may face a choice between a relatively safe project and a riskier one. If the project’s risk changes after the debt is in place, shareholders and debtholders may not evaluate the choice in the same way. Shareholders have limited liability: they can lose their investment, but they do not have to pay firm debts from personal assets. This makes the equity payoff especially sensitive to high outcomes, while debtholders have a contractual claim that is capped at the amount owed.

The resulting conflict is called risk shifting or asset substitution: shareholders may favor replacing a safer asset or project with a riskier one because some downside falls on debtholders while shareholders retain much of the upside. The key is to calculate each claim’s payoff in every possible state before taking an expected value.

Definition: Suppose a firm owes debt with face value $D$ at a specified future date, and $V$ is the value of the assets available to satisfy claims at that date. With limited liability and no bankruptcy costs, the equity payoff is $\max(V-D,0)$, and the debt payoff is $\min(V,D)$. These payoffs add to $V$ in every state.

The debt payoff cannot exceed the amount owed, $D$. If assets are worth less than $D$, debtholders receive the available asset value and shareholders receive zero. If assets are worth more than $D$, debtholders receive $D$ and shareholders receive the surplus. This simple priority structure is enough to explain why the two groups can prefer different project risks.

The comparisons below are one-period illustrations. Probabilities are assumed to be risk-neutral probabilities, so expected terminal payoffs give values when discounted at the risk-free rate. Each example uses an annual risk-free rate of 0.0%, so no discounting changes the figures. There are no mortality-contingent cash flows; the stated state probabilities and payoffs fully specify each distribution. All scenarios and data are hypothetical.

How a Risk Increase Can Raise Equity Payoff

Consider two projects with the same expected terminal asset value. A safe project pays a fixed amount. A risky project has a spread of possible outcomes, including a low state in which the firm cannot repay its debt and a high state in which shareholders receive a substantial surplus. Even when the expected asset value is unchanged, the expected equity payoff can rise because the high-state surplus belongs to shareholders while the low-state equity payoff is already zero.

Worked Example: Equal Expected Assets, Different Claim Payoffs

A hypothetical firm owes $D=\$80$ at the end of one year. It can choose a safe project that pays $\$100$ with certainty or a risky project that pays $\$40$ with probability 0.5 and $\$160$ with probability 0.5. The risk-free rate is 0.0% per year, and the stated probabilities are risk-neutral. There are no mortality-contingent cash flows or bankruptcy costs. Compare expected equity and debt payoffs.

Method. First calculate the asset value in each state. Then apply the debt and equity payoff formulas separately in each state and take probability-weighted averages.

Safe project. The asset value is $\$100$. Equity receives $\max(100-80,0)=\$20$, and debt receives $\min(100,80)=\$80$. Expected project value is $\$100$.

Risky project: expected assets. The expected asset value is $0.5(\$40)+0.5(\$160)=\$20+\$80=\$100$, equal to the safe project.

Risky project: equity. In the low state, equity receives $\max(40-80,0)=\$0$. In the high state, it receives $\max(160-80,0)=\$80$. Thus expected equity is $0.5(\$0)+0.5(\$80)=\$40$.

Risky project: debt. In the low state, debt receives $\min(40,80)=\$40$. In the high state, it receives $\min(160,80)=\$80$. Thus expected debt is $0.5(\$40)+0.5(\$80)=\$60$.

Check and interpretation. Under the safe project, the claims total $\$20+\$80=\$100$. Under the risky project, they total $\$40+\$60=\$100$. Expected equity rises by $\$20$, exactly matching the $\$20$ reduction in expected debt payoff; the expected total asset value has not changed. The common mistake is to calculate equity from average assets, $\max(100-80,0)=\$20$. That is the safe-project equity payoff, not the risky project’s expected equity payoff: the risky project’s payoffs must be calculated state by state first.

Why the Payoff Comparison Works

The result follows from the shape of the equity payoff. As a function of firm asset value $V$, equity receives zero for $V\leq D$ and receives $V-D$ for $V>D$. The slope therefore changes from zero to one at $D$. This is a convex payoff: a rise in asset value above the debt threshold benefits shareholders, while a fall below that threshold cannot make their payoff negative.

Write equity’s state payoff.
$E(V)=\max(V-D,0)$. This function is convex in $V$: its slope never decreases as $V$ rises.

Compare the expected payoff with payoff at the mean.
For a random asset value with a finite expected value, Jensen’s inequality for the convex function $E$ gives $\mathbb{E}[\max(V-D,0)]\geq\max(\mathbb{E}[V]-D,0)$.

Relate equity to debt when there are no costs.
State by state, $\min(V,D)=V-\max(V-D,0)$. Taking expectations gives $\mathbb{E}[\min(V,D)]=\mathbb{E}[V]-\mathbb{E}[\max(V-D,0)]$.

Thus, for two projects with the same expected asset value and the same debt obligation, a mean-preserving increase in risk can increase expected equity payoff while reducing expected debt payoff. A mean-preserving increase in risk spreads out possible outcomes without changing the expected asset value. This comparison describes a value transfer between claims under the stated assumptions; it does not say that every riskier project is better for the firm.

More Than Two Outcomes

The same method applies when there are several possible outcomes. In each state, determine which side of the debt threshold the asset value falls on. This avoids treating the whole distribution as though it were a single average payoff.

Worked Example: A Three-State Risky Project

A hypothetical firm owes $D=\$100$ at the end of one year. A safe project pays $\$120$ with certainty. A risky project pays $\$40$ with probability 0.25, $\$100$ with probability 0.50, and $\$240$ with probability 0.25. The risk-free rate is 0.0% per year, and these are risk-neutral probabilities. There are no mortality-contingent cash flows or bankruptcy costs. Compare expected claim payoffs.

Method. Build a state-by-state payoff comparison, then weight each payoff by its state probability.

Check expected assets first. The risky project’s expected asset value is $0.25(\$40)+0.50(\$100)+0.25(\$240)=\$10+\$50+\$60=\$120$, the same as the safe project.

Safe project. Equity receives $\max(120-100,0)=\$20$; debt receives $\min(120,100)=\$100$.

Risky project, equity. The payoffs are $\max(40-100,0)=\$0$, $\max(100-100,0)=\$0$, and $\max(240-100,0)=\$140$. Expected equity is $0.25(\$0)+0.50(\$0)+0.25(\$140)=\$35$.

Risky project, debt. The payoffs are $\min(40,100)=\$40$, $\min(100,100)=\$100$, and $\min(240,100)=\$100$. Expected debt is $0.25(\$40)+0.50(\$100)+0.25(\$100)=\$10+\$50+\$25=\$85$.

Check and interpretation. The risky project’s expected claims total $\$35+\$85=\$120$, matching expected assets. Relative to the safe project, expected equity rises by $\$15$, while expected debt falls by $\$15$. The middle state matters: because asset value exactly equals the debt due, equity receives zero there and debtholders are repaid in full.

Distress Costs Can Reduce Total Value

So far, every dollar of asset value has been available to satisfy claims. In financial distress, legal, administrative, or operating costs may reduce what remains for investors. If a risky project makes distress more likely, a gain to shareholders can be accompanied by a loss to debtholders that is larger than the gain. In that case, risk shifting is not merely a transfer between claims: it also reduces total value.

Worked Example: Risk Shifting with a Distress Cost

A hypothetical firm owes $D=\$80$ at the end of one year. Its safe project pays $\$100$ with certainty. A risky project produces gross asset value $\$40$ with probability 0.5 and $\$160$ with probability 0.5. In the low state only, financial distress consumes $\$20$ before claims are paid. The risk-free rate is 0.0% per year, and the stated probabilities are risk-neutral. There are no mortality-contingent cash flows. Compare the claims after distress costs.

Method. Deduct the distress cost in the state where it occurs, then apply the debt and equity payoff formulas to the asset value remaining for investors.

Safe project. It creates no distress in this scenario. Equity receives $\max(100-80,0)=\$20$, and debt receives $\min(100,80)=\$80$. Total value available to investors is $\$100$.

Risky project, assets available. In the low state, $\$40-\$20=\$20$ remains after the distress cost. In the high state, $\$160$ remains. Expected value available to investors is $0.5(\$20)+0.5(\$160)=\$10+\$80=\$90$.

Risky project, equity. The low-state equity payoff is $\max(20-80,0)=\$0$, and the high-state payoff is $\max(160-80,0)=\$80$. Expected equity is $0.5(\$0)+0.5(\$80)=\$40$.

Risky project, debt. The low-state debt payoff is $\min(20,80)=\$20$, and the high-state payoff is $\min(160,80)=\$80$. Expected debt is $0.5(\$20)+0.5(\$80)=\$50$.

Check and interpretation. The claims total $\$40+\$50=\$90$, exactly the expected value available after costs. Shareholders gain $\$20$ compared with the safe project, but debtholders lose $\$30$, so total investor value falls by $\$10$. This is consistent with the expected distress cost of $0.5(\$20)=\$10$. The trap is to compare the risky project’s gross expected assets of $\$100$ with the safe project and ignore the cost in the distressed state.

Assumptions and Common Exam Traps

The payoff comparison is most direct when the debt face value is fixed, the project choice does not change the debt contract, and the stated probability distribution can be used to value terminal payoffs. Risk-neutral probabilities and a common discount rate make expected payoffs directly comparable as values; with a positive risk-free rate, a one-period expected payoff would be discounted by $1+i$. In real markets, different systematic risks may require different discount rates, so expected cash flows alone need not establish market values.

The no-cost identity $\min(V,D)=V-\max(V-D,0)$ assumes that all available assets are divided between debt and equity. If distress costs consume assets, replace $V$ with the amount remaining after those costs. Likewise, the conclusion about risk depends on holding expected asset value fixed. A riskier project with a sufficiently lower expected payoff may reduce expected equity as well as total value.

Do not apply the payoff formula to average assets. In general, $\mathbb{E}[\max(V-D,0)]$ is not equal to $\max(\mathbb{E}[V]-D,0)$. Calculate payoffs state by state.
Do not confuse a transfer with value creation. With no costs and equal expected assets, a higher expected equity payoff can be exactly offset by a lower expected debt payoff.
Do not ignore costs that occur before claims are paid. Deduct distress costs in the states where they occur, then allocate what remains.
Do not infer that any increase in risk benefits shareholders. The payoff comparison isolates risk while holding expected assets and debt terms fixed; other changes can reverse the result.

Key takeaway: Limited liability makes equity’s payoff convex in firm value. When a fixed-debt firm takes on more risk without changing expected assets, shareholders may gain at debtholders’ expense. If the added risk causes distress costs, the resulting shareholder gain can come with a larger loss in total value.

Check Your Understanding

For each hypothetical one-period case, calculate payoffs state by state. Unless a question says otherwise, use a risk-free rate of 0.0%, risk-neutral probabilities, no mortality-contingent cash flows, and no distress costs.

A firm owes $\$70$. A safe project pays $\$95$ with certainty. An alternative pays $\$20$ with probability 0.4 and $\$145$ with probability 0.6. Compare expected equity and debt payoffs under the two projects. (Answer: Safe: equity $\$25$, debt $\$70$; risky: expected equity $\$45$, expected debt $\$50$; equity gains $\$20$ and debt loses $\$20$.)
A firm owes $\$120$. The safe project pays $\$138$. A risky project pays $\$60$ with probability 0.3, $\$120$ with probability 0.4, and $\$240$ with probability 0.3. Verify equal expected asset values and compare expected claims. (Answer: Both projects have expected assets of $\$138$; safe equity and debt are $\$18$ and $\$120$, while risky expected equity and debt are $\$36$ and $\$102$.)
A firm owes $\$80$. A safe project pays $\$90$. A risky project pays $\$30$ or $\$150$, each with probability 0.5. Find expected equity and debt under each project and state the change in each claim. (Answer: Both have expected assets of $\$90$; safe equity and debt are $\$10$ and $\$80$, while risky expected equity and debt are $\$35$ and $\$55$. Equity gains $\$25$, and debt loses $\$25$.)
A firm owes $\$100$. A safe project pays $\$130$. A risky project pays gross assets of $\$40$ with probability 0.25 and $\$160$ with probability 0.75. A $\$24$ distress cost is incurred only in the low state. Calculate expected assets available to investors and expected claim payoffs. (Answer: Expected assets available are $\$124$; expected equity is $\$45$, and expected debt is $\$79$. Compared with the safe project’s $\$30$ equity and $\$100$ debt, equity gains $\$15$, debt loses $\$21$, and total value falls by $\$6$.)
A firm owes $\$80$. Its safe project pays $\$100$. A risky project pays $\$40$ with probability 0.25 and an unknown amount $H$ with probability 0.75. Find $H$ so that expected assets equal $\$100$, then compare expected equity and debt with the safe project. (Answer: $0.25(40)+0.75H=100$, so $H=\$120$; risky expected equity is $\$30$ and debt is $\$70$, compared with safe equity of $\$20$ and debt of $\$80$.)

What You'll Learn