Deriving Debt As Risk-Free Debt Minus A Put

Why Risky Debt Contains a Put

A zero-coupon corporate bond promises a fixed amount at maturity, but the firm may not have enough assets to pay it. When the firm is solvent, debtholders receive the promised amount. When it is insolvent, they receive only the assets available to them. This limited recovery makes the debt worth less than an otherwise comparable default-free claim.

The shortfall has the payoff of a put option on the firm’s assets. A put with a strike equal to the debt due at maturity pays the difference between that promised amount and the assets’ value, when the assets fall short. The debt claim can therefore be understood as a default-free payment with this shortfall removed.

Definition: Consider a firm with asset value $V_T$ at time $T$ and a single zero-coupon debt claim with face value $D$ due then. Assume the debt has priority over equity, equityholders have limited liability, and all firm assets are available to satisfy the debt. The debt payoff is $\min(V_T,D)$. A European put on the firm’s assets with strike $D$ has payoff $(D-V_T)^+$, where $x^+=\max(x,0)$.

For the basic identity, assume there are no interim coupons or asset distributions, no bankruptcy costs, no other senior claims, and no early default or settlement before $T$. These assumptions make the terminal claim depend only on $V_T$. With different debt terms or recovery rules, the payoff must be specified anew; the simple decomposition may not apply unchanged.

The Payoff Identity

At maturity, there are two cases. If $V_T\geq D$, debtholders receive $D$, and the put expires worthless. If $V_T<D$, debtholders receive $V_T$, while the put pays $D-V_T$. In either case, the debt payoff equals the promised payment less the put payoff:

\min(V_T,D)=D-(D-V_T)^+

This identity explains the economic interpretation. The put is the amount by which the firm’s assets fail to cover the promised debt payment. Subtracting that shortfall from the full promised payment leaves the actual payment to debtholders.

To convert the maturity identity into a value today, use a no-arbitrage model in which the debt and the put are valued consistently. Let $B_0$ be the value of the risky debt today, $P_0$ the value of the put today, and $i$ the effective annual risk-free rate. For a single maturity $T$ years away, the value of a default-free payment of $D$ at $T$ is $D/(1+i)^T$. Therefore:

Theorem: Under the stated assumptions and consistent no-arbitrage valuation, $B_0=D/(1+i)^T-P_0$. Risky debt equals the value of default-free debt with the same promised payment and maturity, less the value of a European put on the firm’s assets with strike $D$ and maturity $T$.

The identity is a valuation statement, not a claim that the firm literally issues a put option to debtholders. It describes how the limited-recovery payoff can be decomposed into two simpler payoffs. In a complete market, the same conclusion follows by valuing each payoff under the risk-neutral measure and discounting at the risk-free rate.

Deriving the Value in a One-Period Binomial Model

Suppose firm assets are worth $V_0$ today and, after one year, are either $uV_0$ or $dV_0$, where $u>d$. The risk-free gross return over the year is $1+i$. In the standard one-period binomial model, no-arbitrage requires $d<1+i<u$. The risk-neutral probability $q$ of the up state is chosen so that the expected asset value under that probability grows at the risk-free rate:

q=\frac{(1+i)-d}{u-d}

This is a pricing probability, not necessarily the real-world probability that assets rise. Under the model, a claim with terminal payoff $H$ has value $[qH_{\text{up}}+(1-q)H_{\text{down}}]/(1+i)$. Applying this rule to both the put and debt gives the same decomposition as the payoff identity.

Find the two asset values.
Compute $uV_0$ and $dV_0$ at maturity.

Calculate the risk-neutral up probability.
Use $q=((1+i)-d)/(u-d)$, checking that $0<q<1$.

Find each claim’s terminal payoff.
The debt pays $\min(V_T,D)$; the put pays $(D-V_T)^+$.

Discount the risk-neutral expected payoff.
Divide each expected payoff by $1+i$, then check that debt value equals default-free debt value less put value.

Worked Example: A One-Year Debt Claim and the Up-Probability Trap

A firm’s assets are worth $75 today. In one year, their value will be either 1.4 times or 0.8 times today’s value. The effective annual risk-free rate is 5%, and a zero-coupon debt claim with face value $90 is due in one year. Find the risk-neutral up probability, the put value, and the risky debt value. Use dollars to the cent for final values.

Method. Use the one-period binomial risk-neutral valuation rule. The up and down asset values are $1.4(75)=105$ and $0.8(75)=60$. The risk-free gross return is $1.05$, so:

q=\frac{1.05-0.8}{1.4-0.8} =\frac{0.25}{0.6} =\frac{5}{12} =0.4167

The put pays $\max(90-105,0)=0$ in the up state and $\max(90-60,0)=30$ in the down state. Its value is:

P_0=\frac{(5/12)(0)+(7/12)(30)}{1.05} =\frac{17.5}{1.05} =16.6667

The comparable default-free debt is worth $90/1.05=85.7143$. Subtracting the put gives risky debt value $85.7143-16.6667=69.0476$, or $69.05 to the cent.

Check by valuing debt directly. The debt pays $\min(105,90)=90$ in the up state and $\min(60,90)=60$ in the down state. Thus:

B_0=\frac{(5/12)(90)+(7/12)(60)}{1.05} =\frac{72.5}{1.05} =69.0476

Both methods give $69.05. Common mistake avoided: the expression $(u-(1+i))/(u-d)=0.5833$ is the risk-neutral down probability here, not the up probability. Using it as the up probability produces $[0.5833(90)+0.4167(60)]/1.05=73.81$, which reverses the state weights. The correct up probability is $((1+i)-d)/(u-d)=0.4167$.

Why the Decomposition Holds for Prices

The maturity payoff identity alone does not assign prices; a valuation model is also needed. In a one-period binomial model, the risk-neutral pricing rule provides that link. Applying the rule separately to the debt and the put yields:

\begin{aligned} B_0 &=\frac{q\min(uV_0,D)+(1-q)\min(dV_0,D)}{1+i},\\ P_0 &=\frac{q(D-uV_0)^+ +(1-q)(D-dV_0)^+}{1+i}. \end{aligned}

For either terminal state, the debt payoff plus the put payoff equals $D$: $\min(V_T,D)+(D-V_T)^+=D$. Therefore the risk-neutral expected sum of the two payoffs is also $D$. Discounting gives:

B_0+P_0=\frac{D}{1+i} \qquad\Longrightarrow\qquad B_0=\frac{D}{1+i}-P_0.

This derivation relies on using the same asset states, risk-neutral probabilities, and discount rate for both claims. The put’s value is not obtained by multiplying a real-world default probability by a loss and then discounting unless a separate model justifies that approach.

Worked Example: Two-Year Debt in a Recombining Asset Tree

A firm has assets worth $100 today. In each of the next two years, the assets move up by a factor of 1.2 or down by a factor of 0.9. The effective annual risk-free rate is 5%, and zero-coupon debt with face value $120 is due in two years. Assume the same binomial factors apply in each year. Find the debt and put values. Round final dollar values to the cent.

Method. First calculate the one-year risk-neutral up probability: $q=(1.05-0.9)/(1.2-0.9)=0.15/0.30=0.5$. The two-year asset values are $100(1.2)^2=144$, $100(1.2)(0.9)=108$, $100(0.9)(1.2)=108$, and $100(0.9)^2=81$. The corresponding risk-neutral probabilities are $0.25$, $0.50$, and $0.25$ for the up-up, one-up-one-down, and down-down outcomes.

The put payoffs at maturity are $(120-144)^+=0$, $(120-108)^+=12$, and $(120-81)^+=39$. Thus the put’s expected payoff under the risk-neutral probabilities is:

$$ 0.25(0)+0.50(12)+0.25(39)=15.75 $$

Discount for two years using $(1.05)^2=1.1025$: $P_0=15.75/1.1025=14.2857$, or $14.29. The default-free debt value is $120/1.1025=108.8435$, or $108.84. Therefore risky debt is $108.8435-14.2857=94.5578$, or $94.56.

Check directly. Debt pays $120, $108, or $81 in the three outcomes, so its risk-neutral expected payoff is $0.25(120)+0.50(108)+0.25(81)=104.25$. Its value is $104.25/1.1025=94.5578$, again $94.56. The check also confirms that the middle outcome has probability 0.50, because either order of one up move and one down move leads to the same asset value.

Common mistake avoided: discounting the two-year payoff by only 1.05 understates the discount period. The correct factor is $1.05^2$, since the claim is paid two years from today.

Interpreting the Put as Default Risk

The put value measures the market value of the shortfall embedded in the debt payoff under the model’s pricing assumptions. A larger put value means a larger reduction from the comparable default-free debt value. It is not generally equal to the expected dollar loss under actual probabilities, because the option price reflects the timing and risk of the shortfall as well as discounting.

The debt’s default state is the state in which $V_T<D$. In that state, the put finishes in the money and the debt pays less than face value. A higher face amount $D$, all else equal, raises the strike of the put and can increase its value. A change in asset value, asset volatility, maturity, or interest rates can also affect the put price; the precise effect depends on the model and its assumptions.

The decomposition is useful because it separates the promised payment from the shortfall claim. It also clarifies why debt is not simply valued as a default-free bond whenever the firm has limited liability. But the basic formula is not a complete model of every corporate bond. Coupons introduce payments at multiple dates; bankruptcy costs can reduce the assets available to creditors; and seniority, collateral, covenants, or early default can change payoffs. Those features require the relevant claims and timing to be represented explicitly.

Do not confuse face value with market value. $D$ is the contractual payment at maturity; $B_0$ is the value today.
Do not use actual default probabilities as risk-neutral probabilities. Risk-neutral probabilities are selected for pricing within the model.
Do not subtract the put payoff from today’s debt value without discounting. The decomposition subtracts the put’s value today, $P_0$, not its maturity payoff.
Do not apply the identity mechanically when recovery rules change. Define the debt payoff first, including any costs or priority claims.

Key takeaway: With one zero-coupon debt claim, limited liability, and no additional recovery complications, the maturity payoff is $\min(V_T,D)=D-(D-V_T)^+$. Consistent no-arbitrage valuation therefore gives risky debt value as comparable default-free debt value minus the value of a put on firm assets.

Check Your Understanding

For each problem, state the model assumptions needed and use risk-neutral valuation where a binomial tree is specified.

A firm has assets worth $80, with one-year up and down factors of 1.25 and 0.875. The effective annual risk-free rate is 5%, and debt face value is $90. Find the risk-neutral up probability, put value, and debt value. (Answer: up probability 0.4667; put value $10.16; debt value $75.56.)
Assets are worth $120; after one year they will be either 1.2 or 0.9 times their current value. The effective annual risk-free rate is 5%, and zero-coupon debt of $130 is due then. Find the put value and debt value, checking the debt value directly from its two payoffs. (Answer: put value $10.48; debt value $113.33.)
A firm’s assets are worth $90 and follow a two-year recombining binomial tree with annual up factor 1.2 and down factor 0.9. The effective annual risk-free rate is 5%, and debt face value is $110. Find the two-year put value and risky debt value. (Answer: put value $14.22; debt value $85.56.)
A one-year default-free payment of $100 has a present value of $95. A put on the firm’s assets, with strike equal to the debt face value and the same maturity, is priced at $14. What is the risky debt value under the decomposition? Explain what the $14 represents. (Answer: $81; it is the present value of the shortfall payoff, not the real-world expected loss.)
At maturity, firm assets are worth $82 and zero-coupon debt of $100 is due. Under the stated limited-liability assumptions, calculate the debt and put payoffs and verify the payoff identity. (Answer: debt payoff $82; put payoff $18; $100-$18=$82.)

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