Credit Spreads, Default Hazard, and Structural Debt Valuation

Summary

The document defines a credit spread from the prices of defaultable and risk-free zero-coupon bonds, then relates default risk to a hazard rate and survival probability. It also introduces a structural company valuation in which equity resembles a call option on firm value and debt resembles a capped claim on that value. The question asks why the defaultable bond price might equal the company’s debt value divided by face value.

The material gives definitions and formulas but does not answer that question or establish the proposed equality. In a standard structural model, debt value depends on promised repayment, recovery at default, discounting, and the distribution of firm value; matching a bond price ratio to debt value requires compatible assumptions and normalization. The displayed formulas also contain apparent notation inconsistencies, including a debt payoff expression and survival-probability definition, so they should not be treated as a complete derivation.

Key ideas

• A credit spread compares the yield implied by a defaultable bond with that of a risk-free bond.
• A hazard rate describes default likelihood conditional on survival up to a given time.
• Survival probabilities can be expressed through the cumulative hazard rate.
• The document models equity as a call-like claim and debt as a claim on firm value.
• The proposed link between bond price and company debt value is posed but not derived.

Tags

# Credit spread model


# Credit spread model












Let $c(t,T):=-\frac{1}{T-t}[\mathrm{ln}(P_1(t,T))-\mathrm{ln}(P_0(t,T))]$, with:

- $c$ measure of how a company is prone to fail;

- $P_0(t,T):=e^{-r(T-t)}$ price of no-defaultable bond.

- $P_1(t,T):=\mathbb{I}_{(\tau>t)}e^{-\int_{t}^{T}R_sds}$ price of defaultable bond, where $R_s:=r_s+\gamma_s$.

- $\gamma_s:=\frac{f(t)}{\bar{F}(t)}:=\lim_{h\rightarrow 0}\frac{P(\tau\leq t+h|\tau>t)}{h}$ hazard rate, where $\tau$ is a generical instant default of defaultable bond on $(\Omega, F, {F_{t}}_{(t\geq 0)},\mathbb{P})$ such that ${(\tau\leq t)}\in F_{t}$.

- $F(t):=P(\tau \leq t)$ the CDF of a generical instant default.

- $\bar{F}(t):=1-P(\tau>t):=1-F(t)=e^{-\int_{0}^{t}\gamma_sds}$ the complement of $F(t)$.

- $P(\tau>T|\mathfrak{H}_t):=\mathbb{I}_{(\tau\geq T)}e^{-\int_{t}^{T}\gamma_sds}$ the survivale rate of defaultable bond over the maturity $T$.

Then let:

- $V_t:=S_t+F_t$ the portfolio's company, evaluated in $t$, that invests in stocks and bonds.

- $F_t:=Fe^{-r(T-t)}\phi({d(t,2)})+V_t\phi({-d(t,1)})$ the debt quote value of portfolio's company that acts like a put option with payoff $F_T=F-(F-V_T)^{+}$

- $S_t:=V_t\phi({d(t,1)})-Fe^{-r(T-t)}\phi({d(t,1)})$the equity quote value of portfolio's company that acts like a call option with payoff $S_T=(V_T-F)^{+}$.

I ask you: why it's possibile to observe that $P_1(t,T)=\frac{F_t}{F}$?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.