Deriving Default Probability in the Merton Model

Introduction

In a structural credit model, default is tied to the value of a firm’s assets relative to its debt. The Merton model makes this connection precise: if a firm’s asset value at the debt horizon is below the amount due, the firm defaults. Because the model treats asset value as lognormal, the default probability can be derived from a standard normal probability.

The calculation depends on which probability measure is being used. A physical probability uses the asset’s expected growth rate under the real-world measure. A risk-neutral probability uses the risk-free rate and is used in pricing arguments. These are different quantities, even when they use the same current assets, debt threshold, horizon, and volatility.

Definition: Let $V_0$ be a firm’s asset value today, $D$ the promised debt payment due at time $T$, $\sigma$ the constant asset volatility, and $\mu$ the constant physical expected asset return. Under the basic Merton assumptions, distance-to-default is $$ \mathrm{DD}=\frac{\ln(V_0/D)+(\mu-\sigma^2/2)T}{\sigma\sqrt{T}}. $$ The physical probability of default by the debt horizon is $P(V_T<D)=\Phi(-\mathrm{DD})$, where $\Phi$ is the standard normal cumulative distribution function.

Model Assumptions and the Asset Distribution

The basic one-debt Merton model assumes a firm has traded assets whose value follows a geometric Brownian motion, with constant volatility and constant drift over the horizon. It also assumes no asset payouts, a single debt payment $D$ at time $T$, and default only at that horizon. The debt threshold is the amount due at $T$, not necessarily the face amount of debt stated at some other date.

Under the physical probability measure, write the asset process as

\frac{dV_t}{V_t}=\mu\,dt+\sigma\,dW_t,

where $W_t$ is a standard Brownian motion. Applying the standard solution for geometric Brownian motion gives

V_T=V_0\exp\!\left((\mu-\sigma^2/2)T+\sigma W_T\right).

Since $W_T/\sqrt{T}\sim N(0,1)$, the logarithm of the terminal asset-to-debt ratio is normally distributed:

\ln(V_T/D)=\ln(V_0/D)+(\mu-\sigma^2/2)T+\sigma\sqrt{T}\,Z, \qquad Z\sim N(0,1).

The term $-\sigma^2/2$ is essential. It is the adjustment in the drift of the logarithm, not an optional correction to the asset return. Omitting it changes the distribution used to calculate default probability.

Deriving Distance-to-Default and Physical Default Probability

In the model, default at the debt horizon occurs when $V_T<D$. Taking logarithms preserves the inequality because the logarithm is increasing:

V_T<D \quad\Longleftrightarrow\quad \ln(V_T/D)<0.

Substitute the distribution of $\ln(V_T/D)$ and divide by its positive standard deviation $\sigma\sqrt{T}$. The resulting standard normal threshold is $-\mathrm{DD}$:

P(V_T<D) = P\!\left( Z< -\frac{\ln(V_0/D)+(\mu-\sigma^2/2)T}{\sigma\sqrt{T}} \right) = \Phi(-\mathrm{DD}).

This is the derivation: distance-to-default is the mean of the log asset-to-debt ratio, measured in standard deviations from the default boundary at zero. A larger positive distance means the model places the boundary farther into the lower tail, so the default probability is smaller. With a continuous asset distribution, the probability that $V_T$ is exactly $D$ is zero, so using $<$ or $\leq$ gives the same probability.

Worked Example: Physical Default Probability at a One-Year Horizon

A hypothetical firm has debt of $D=\$100$ million due in one year. Its current asset value is $V_0=De^{0.09}$, its physical expected asset return is $\mu=0.05$, and its annual asset volatility is $\sigma=0.20$. Find the physical distance-to-default and probability of default. Report the probability to four decimals.

Method. Use the physical drift $\mu$, form the log asset-to-debt ratio, and standardize by $\sigma\sqrt{T}$. The stated ratio gives $\ln(V_0/D)=0.09$ exactly.

Step 1: Apply the volatility adjustment. Here, $\sigma^2/2=0.20^2/2=0.04/2=0.02$, so the drift of log assets is $\mu-\sigma^2/2=0.05-0.02=0.03$.

Step 2: Calculate distance-to-default. With $T=1$, the numerator is $0.09+(0.03)(1)=0.12$, and the denominator is $0.20\sqrt{1}=0.20$. Therefore, $\mathrm{DD}=0.12/0.20=0.6000$.

Step 3: Convert to probability. The standard normal table gives $\Phi(-0.6000)=0.2743$ to four decimals. The physical probability of default is therefore $27.43\%$.

Check and common mistake. A positive distance must correspond to a default probability below $0.5$, as it does here. Using $\Phi(\mathrm{DD})$ instead would give $0.7257$, the probability of finishing above the threshold, not below it.

Risk-Neutral Probability of Default

For a risk-neutral calculation, suppose the asset process has no payouts and the risk-free rate $r$ is constant and continuously compounded. Under the risk-neutral measure, the asset drift is $r$, so the same derivation applies with $\mu$ replaced by $r$. Define the risk-neutral distance parameter $d_2$ by

d_2=\frac{\ln(V_0/D)+(r-\sigma^2/2)T}{\sigma\sqrt{T}}.

Then the risk-neutral probability that terminal assets fall below the debt threshold is

Q(V_T<D)=\Phi(-d_2),

where $Q$ denotes the risk-neutral measure. This is a model-implied pricing probability, not automatically the firm’s real-world forecast probability of default. A physical probability uses $\mu$; a risk-neutral probability uses $r$. If the model includes continuous asset payouts at rate $q$, the respective log drifts become $\mu-q$ and $r-q$; the formulas above assume no payouts.

Worked Example: Risk-Neutral Probability With a Two-Year Debt Horizon

A hypothetical firm has $D=\$100$ million due in two years, asset volatility $\sigma=0.20$, and a continuously compounded risk-free rate $r=0.04$. Its current assets satisfy $\ln(V_0/D)=0.101421356$, corresponding to approximately $\$110.67$ million. Find $d_2$ and the risk-neutral probability of default. Report the probability to four decimals.

Method. Since this is a risk-neutral calculation, use $r$, not a physical expected asset return. The debt horizon is $T=2$, so both the drift contribution and the volatility scaling must use two years.

Step 1: Calculate the adjusted log drift. The volatility adjustment is $\sigma^2/2=0.20^2/2=0.02$. Thus $r-\sigma^2/2=0.04-0.02=0.02$.

Step 2: Calculate the numerator and denominator. The numerator is $0.101421356+(0.02)(2)=0.141421356$. The denominator is $0.20\sqrt{2}=0.282842712$.

Step 3: Standardize and find the probability. Dividing gives $d_2=0.141421356/0.282842712=0.5000$. Therefore $Q(V_T<D)=\Phi(-0.5000)=0.3085$ to four decimals.

Check and common mistake. The calculation uses the two-year total drift contribution $0.04$, not the one-year contribution $0.02$. Also, $0.3085$ is the risk-neutral probability under the stated model; it should not be presented as a physical default forecast without a physical drift assumption.

Interpreting and Using the Distance

Distance-to-default is expressed in standard deviation units, so it helps compare asset cushions across firms with different horizons and volatilities. It is not itself a probability. To obtain a probability, apply the appropriate normal tail function. Conversely, a target probability can be converted to a target distance using the inverse standard normal distribution.

For example, if a physical default probability target is $p$, then $\mathrm{DD}=\Phi^{-1}(1-p)$. Substituting this target into the distance formula allows the required asset value, debt threshold, or other input to be solved for. This is a direct algebraic use of the model and does not require estimating parameters from equity prices.

Worked Example: Solving for Assets at a Target Distance

A hypothetical firm has debt of $D=\$100$ million due in two years. Under the physical measure, its asset expected return is $\mu=0.05$ and asset volatility is $\sigma=0.20$. Find the current asset value required for a physical distance-to-default of $1.0000$. State the corresponding default probability.

Method. Rearrange the distance formula to isolate $\ln(V_0/D)$, then exponentiate to recover the asset-to-debt ratio. A distance of $1.0000$ corresponds to a standard normal lower-tail probability of $\Phi(-1)=0.1587$.

Step 1: Rearrange the equation.

\ln(V_0/D) = \mathrm{DD}\,\sigma\sqrt{T}-(\mu-\sigma^2/2)T.

Step 2: Substitute the assumptions. The volatility adjustment is $0.20^2/2=0.02$, so $\mu-\sigma^2/2=0.05-0.02=0.03$. The first term is $(1.0000)(0.20)\sqrt{2}=0.2828427$, and the second is $(0.03)(2)=0.0600000$. Thus $\ln(V_0/D)=0.2828427-0.0600000=0.2228427$.

Step 3: Exponentiate and calculate assets. The required ratio is $V_0/D=e^{0.2228427}\approx1.2496$. Hence $V_0\approx1.2496(\$100\text{ million})=\$124.96$ million, rounded to the nearest $\$0.01$ million.

Step 4: Check the implied probability. Substitution into the distance formula gives a distance of $1.0000$, so $P(V_T<D)=\Phi(-1.0000)=0.1587$. The positive asset cushion is consistent with a probability below one-half.

Common mistake. The log drift over two years is $0.03(2)=0.06$, not $0.03$. Using a one-year drift would solve a different horizon and produce an incorrect asset requirement.

Common Exam Traps and Model Limits

The formulas are compact, but a few distinctions determine whether the result is meaningful. In particular, the time horizon, drift measure, debt threshold, and direction of the normal tail must all match the question.

Using the wrong drift. Use $\mu$ for a physical probability and $r$ for a risk-neutral probability under the no-payout model. Do not label one as the other.
Dropping the volatility adjustment. The log drift is $\mu-\sigma^2/2$ or $r-\sigma^2/2$, not simply $\mu$ or $r$.
Using the wrong time units. If $T$ is in years, the annual drift is multiplied by $T$, and volatility is multiplied by $\sqrt{T}$. Do not multiply volatility by $T$.
Taking the wrong normal tail. Default is below the threshold, so the probability is $\Phi(-\mathrm{DD})$ or $\Phi(-d_2)$, not $\Phi(\mathrm{DD})$.
Misstating what the threshold represents. $D$ is the promised payment at the horizon in this one-period model. The model does not represent interim default before $T$, multiple debt maturities, or recovery mechanics by this formula alone.

The results rely on constant parameters and a lognormal asset value. They are useful for understanding how assets, leverage, volatility, and time combine in a structural default calculation, but actual default risk can also reflect changing business conditions, complex liabilities, and model uncertainty.

Key takeaway: Under the basic Merton assumptions, standardize the terminal log asset-to-debt ratio. The physical default probability is $\Phi(-\mathrm{DD})$, using $\mu$; the risk-neutral probability is $\Phi(-d_2)$, using $r$. The volatility correction, horizon, and probability measure must be applied consistently.

Check Your Understanding

For each problem, use the basic no-payout Merton model. Report probabilities to four decimals and distances to four decimals where applicable.

A hypothetical firm has $\ln(V_0/D)=0.04$, $T=1$, $\mu=0.05$, and $\sigma=0.20$. Calculate its physical distance-to-default and default probability. (Answer: $\mathrm{DD}=0.3500$; probability $0.3632$.)
For a risk-neutral calculation, let $\ln(V_0/D)=0.08$, $T=1.5$, $r=0.04$, and $\sigma=0.20$. Calculate $d_2$ and the probability of finishing below the debt threshold. (Answer: $d_2\approx0.4491$; probability $0.3267$.)
A firm has $\ln(V_0/D)=0.15$, $T=4$, physical asset return $\mu=0.07$, and volatility $\sigma=0.30$. Calculate its physical distance-to-default and probability of default. (Answer: $\mathrm{DD}=0.4167$; probability $0.3384$.)
A firm has $T=1$, $\mu=0.06$, and $\sigma=0.25$. What current asset-to-debt ratio gives physical distance-to-default $1.0000$? (Answer: $V_0/D=e^{0.22125}\approx1.2476$.)
For a firm with $\ln(V_0/D)=0.12$, $T=2$, $\mu=0.03$, and $\sigma=0.20$, calculate the physical distance and probability. Identify which drift is used and why. (Answer: $\mathrm{DD}\approx0.4950$; probability $0.3103$; use $\mu$ because the requested probability is physical.)

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