Asset Volatility, Default Probability, And Credit Spreads

Why Asset Volatility Matters to Creditors

A firm with debt due at a future date may have assets worth more than its promised payment at that date—or less. In the structural view, shareholders receive what remains after debt is paid, while creditors receive no more than the amount owed. Because these payoffs depend on the firm’s asset value, uncertainty about that value matters to both parties. In particular, more volatile assets can make repayment less certain and reduce the market value of risky debt.

The connection is not simply “more volatility means a higher default probability” in every situation. The effect on default probability depends on the firm’s starting asset value relative to its debt. The effect on the value of risky debt is more direct under the model below: greater volatility increases the value of the shortfall protection creditors have effectively granted shareholders, reducing debt value. Keeping those two conclusions distinct is important when interpreting a credit spread.

Model and assumptions: At time 0, a firm has asset value $V_0$ and one zero-coupon debt obligation promising $F$ at time $T$. There are no interim payouts, taxes, or bankruptcy costs, and shareholders have limited liability. The firm’s assets follow a geometric Brownian motion with constant volatility $\sigma$. For valuation, assume frictionless markets, a constant continuously compounded risk-free rate $r$, and risk-neutral pricing. Under the risk-neutral probability measure, $V_T$ is lognormally distributed: $$ V_T=V_0\exp\left((r-\tfrac{1}{2}\sigma^2)T+\sigma\sqrt{T}Z\right),\qquad Z\sim N(0,1). $$ Default at maturity occurs when $V_T<F$; repayment in full occurs when $V_T\geq F$. These assumptions describe a simplified structural model, not every feature of an actual firm’s debt.

Risk-Neutral Probabilities of Default and Repayment

Define $$ d_2=\frac{\ln(V_0/F)+(r-\tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}. $$ Under the risk-neutral distribution, $\ln(V_T/F)$ is normally distributed. Standardizing the event $V_T\geq F$ therefore gives the repayment probability $P^Q(V_T\geq F)=\Phi(d_2)$, where $\Phi$ is the standard normal cumulative distribution function. The risk-neutral default probability is its complement, $P^Q(V_T<F)=\Phi(-d_2)$. The superscript $Q$ indicates that these probabilities come from the risk-neutral distribution used in pricing.

These are not automatically the probabilities an analyst should report as a forecast of actual default frequency. If the firm’s assets instead have physical expected continuously compounded return $\mu$, then under the physical measure, $$ P(V_T<F)=\Phi\left(-\frac{\ln(V_0/F)+(\mu-\tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}\right). $$ The physical measure uses $\mu$; the risk-neutral measure uses $r$. A risk-neutral probability is useful for valuation, while a physical probability is the model-based probability of the event under the assumed real-world asset return. They answer different questions.

Worked Example: Comparing Risk-Neutral Default Probabilities

A hypothetical firm has assets worth $120$ today and a single debt payment of $100$ due in one year. The continuously compounded risk-free rate is $4\%$, and there are no interim asset payouts. Calculate the risk-neutral default and repayment probabilities at asset volatilities of $20\%$ and $40\%$. Use the model’s risk-neutral distribution and report probabilities to four decimals.

Method and common mistake avoided. Compute $d_2$ separately for each volatility and then use $\Phi(-d_2)$ for default. Do not treat $1-\Phi(-d_2)$ as default; that is the repayment probability.

For $\sigma=0.20$, the inputs are $V_0/F=120/100=1.2$, $T=1$, $r=0.04$, and $\sigma^2/2=0.20^2/2=0.02$. Thus $$ d_2=\frac{\ln(1.2)+(0.04-0.02)(1)}{0.20\sqrt{1}} =\frac{0.182322+0.020000}{0.20} =1.01161. $$ The repayment probability is $\Phi(1.01161)\approx0.8442$, so the default probability is $\Phi(-1.01161)\approx0.1558$. They sum to $0.8442+0.1558=1.0000$.

For $\sigma=0.40$, $\sigma^2/2=0.40^2/2=0.08$. Therefore $$ d_2=\frac{\ln(1.2)+(0.04-0.08)(1)}{0.40\sqrt{1}} =\frac{0.182322-0.040000}{0.40} =0.35580. $$ The repayment probability is $\Phi(0.35580)\approx0.6389$, and the default probability is $\Phi(-0.35580)\approx0.3611$. Again, $0.6389+0.3611=1.0000$.

In this case, increasing volatility from $20\%$ to $40\%$ raises the risk-neutral default probability from $0.1558$ to $0.3611$. This result reflects the firm’s initial asset value being above the promised debt payment; it is not a universal rule for every starting asset value.

Risky Debt as a Risk-Free Bond Less a Put

At maturity, creditors receive $\min(V_T,F)$: the firm pays $F$ if it has enough assets and transfers its remaining assets if it does not. For any terminal asset value, $$ \min(V_T,F)=F-\max(F-V_T,0). $$ The term $\max(F-V_T,0)$ is the payoff of a put option on the firm’s assets with strike $F$. Thus risky debt is equivalent to a risk-free zero-coupon bond promising $F$, less this put option. The put represents the limited-liability shortfall borne by creditors.

Under the model assumptions, the Black–Scholes expressions use $$ d_1=d_2+\sigma\sqrt{T} =\frac{\ln(V_0/F)+(r+\tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}. $$ The time-0 risky debt value $D_0$ is $$ D_0=V_0\Phi(-d_1)+Fe^{-rT}\Phi(d_2). $$ Equivalently, the risk-free bond value less the put value is $$ D_0=Fe^{-rT}-\left(Fe^{-rT}\Phi(-d_2)-V_0\Phi(-d_1)\right). $$ The equivalence follows by expanding the right side and using $1-\Phi(-d_2)=\Phi(d_2)$. It also explains why, under these assumptions, risky debt cannot be worth more than the risk-free bond with the same promised payment: the put value subtracted from that bond is nonnegative.

Worked Example: Volatility, Debt Value, and Credit Spread

A hypothetical firm has $V_0=100$, owes $F=100$ in one year, and has no interim asset payouts. Take the continuously compounded risk-free rate to be $r=0$. Calculate the risky debt value and continuously compounded credit spread for asset volatilities of $20\%$ and $60\%$. Report dollar values to the cent and rates to four decimals.

Method and common mistake avoided. Value the debt as the discounted expected terminal payoff under the risk-neutral model, then calculate the yield from the price. Do not use the equity call value as though it were debt value; the debt value is the risk-free bond value less the asset put.

For $V_0=F=100$, $T=1$, and $r=0$, the formulas simplify to $d_2=-\sigma/2$, $d_1=\sigma/2$, and $D_0=100\Phi(-d_1)+100\Phi(d_2)=200\Phi(-\sigma/2)$.

At $\sigma=0.20$, $d_1=0.10$ and $d_2=-0.10$. Since $\Phi(-0.10)\approx0.460172$, $$ D_0=200(0.460172)=92.0344\approx \$92.03. $$ The continuously compounded yield $y$ on a one-year zero-coupon bond satisfies $D_0=Fe^{-yT}$. Hence $$ y=\frac{\ln(F/D_0)}{T} =\ln(100/92.0344) \approx0.0830. $$ The matched risk-free rate is zero, so the credit spread is $s=y-r\approx0.0830$, or $8.30\%$ per year.

At $\sigma=0.60$, $d_1=0.30$ and $d_2=-0.30$. Since $\Phi(-0.30)\approx0.382089$, $$ D_0=200(0.382089)=76.4178\approx \$76.42. $$ The yield and spread are $$ y=\frac{\ln(100/76.4178)}{1}\approx0.26895,\qquad s=y-r=0.26895-0=0.26895. $$ To four decimals, the yield and spread are both $0.2690$, or $26.90\%$ per year. The risk-free bond is worth $100e^0=\$100.00$, so both risky debt values are below that benchmark. The computed yields exceed the risk-free rate, as expected under the stated model.

Why Higher Volatility Lowers Debt Value

The debt payoff $\min(V_T,F)$ is capped at $F$: if assets rise far above the debt payment, creditors still receive only $F$. But if assets fall below $F$, creditors bear the shortfall. The put-option decomposition captures this asymmetry. In the standard option model, a put’s value increases with volatility when its other inputs are held fixed. Since the risky debt value equals the risk-free bond value less that put, greater asset volatility lowers debt value. With the promised payment and maturity unchanged, a lower price means a higher yield and credit spread.

The default probability has a more nuanced response. Let $A=\ln(V_0/F)+rT$, so $$ d_2=\frac{A-\tfrac{1}{2}\sigma^2T}{\sigma\sqrt{T}} =\frac{A}{\sigma\sqrt{T}}-\frac{\sigma\sqrt{T}}{2}. $$ When $A>0$, $d_2$ falls as $\sigma$ rises, so $\Phi(-d_2)$, the risk-neutral default probability, rises. When $A<0$, the first term can move $d_2$ upward as volatility initially increases, so default probability need not rise. Intuitively, if assets begin below the promised payment, greater volatility can increase the chance that they rise above it. Do not infer the direction of the probability change from the debt-value result alone.

Worked Example: Calculating a Credit Spread From a Bond Price

A hypothetical three-year zero-coupon risky bond with face amount $100$ trades for $94$. The matching continuously compounded risk-free rate is $0.015$ per year. Calculate the bond’s continuously compounded yield and credit spread, and check the price against the risk-free benchmark.

Method and common mistake avoided. For a continuously compounded yield, divide the log price ratio by the full maturity. Omitting the maturity in the denominator misstates both the yield and the spread.

The yield $y$ satisfies $94=100e^{-3y}$. Taking logarithms gives $$ y=\frac{\ln(100/94)}{3} =\frac{0.0618754}{3} =0.0206251. $$ The credit spread is the risky yield minus the matching risk-free rate: $$ s=y-r=0.0206251-0.0150000=0.0056251. $$ Thus, to four significant figures, the yield is $0.02063$, and the spread is $0.005625$, or about $0.5625\%$ per year.

The risk-free benchmark price is $100e^{-0.015(3)}=100e^{-0.045}\approx95.5997$. The quoted risky price, $94$, is lower than this benchmark, consistent with a positive spread. A bond price alone does not determine a default probability: that inference also requires a model for the distribution of asset values and the payoff to creditors in default.

Interpretation and Common Traps

Confusing physical and risk-neutral probabilities. The risk-neutral calculation uses $r$ and is part of pricing; a physical default estimate uses the assumed physical asset return $\mu$. They are not interchangeable.
Assuming default probability always rises with volatility. This need not hold when the firm starts with assets below the promised payment. Check the inputs to $d_2$ and the direction of the probability change.
Using a spread as a standalone default probability. A spread reflects the model’s pricing of credit risk and the bond’s terms. It does not identify a default probability without assumptions about recovery and other risks.
Forgetting maturity in a continuous-compounding yield calculation. For a zero-coupon bond, use $y=\ln(F/D_0)/T$, not just $\ln(F/D_0)$ when $T\ne1$.
Using a risk-free discount benchmark incorrectly. Under the stated model, the risky debt claim is a risk-free bond minus a nonnegative put, so it cannot exceed the matched risk-free bond value.

Key takeaway: In the structural model, risk-neutral repayment and default probabilities come from the terminal asset distribution, while risky debt is valued as a risk-free bond less a put on firm assets. Greater volatility increases that put’s value and lowers risky debt value, but its effect on default probability depends on the firm’s starting position relative to its obligation. The model and its assumptions are aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, state the measure or pricing convention being used and retain enough digits in intermediate calculations to avoid rounding errors.

A hypothetical firm has $V_0=125$, $F=100$, $T=1$, $r=0.02$, and $\sigma=0.30$, with no interim payouts. Find $d_2$, the risk-neutral default probability, and the repayment probability. (Answer: $d_2\approx0.6605$; default $0.2545$; repayment $0.7455$.)
A firm has $V_0=F=100$, $T=1$, $r=0$, and $\sigma=0.40$. Using the structural model, find risky debt value and the continuously compounded credit spread. Give the value to the cent and the spread to four decimals. (Answer: $D_0\approx\$84.15$; spread $\approx0.1726$, or $17.26\%$ per year.)
Under the physical measure, a hypothetical firm has $V_0=90$, $F=100$, $T=1$, $\mu=0.08$, and $\sigma=0.20$. Calculate the physical default and repayment probabilities. (Answer: default $0.5897$; repayment $0.4103$.)
A three-year zero-coupon risky bond with face amount $100$ is quoted at $94$; its matching continuously compounded risk-free rate is $0.015$. Calculate its yield and spread, and check whether the price is below the risk-free benchmark. (Answer: yield $\approx0.02063$; spread $\approx0.005625$, or $0.5625\%$ per year; risk-free benchmark $\approx\$95.60$, above the quoted price.)
A hypothetical firm has $V_0=105$, $F=100$, $T=2$, $\sigma=0.25$, $r=0.03$, and physical asset return $\mu=0.07$, with no interim payouts. Calculate risk-neutral and physical default probabilities separately. (Answer: risk-neutral default $\approx0.4479$; physical default $\approx0.3604$.)

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