You are estimating the probability of default for a given firm using the options model of default risk where this probability is given by Prob Default = N(-DD), and DD is the distance-to-default, calculated using the formula we covered in class.

While implementing this model, you decided to consider the risk of default over a one-year horizon and set the face value of the debt in one year as the current book value of the debt. You also collected the following additional information on the firm:

Current book value of debt = $190M
Current market value of equity = $400M
Equity volatility = 0.81 (measured during the past year)
Value of D/A (measured during the past year) = 0.4

Moreover, you decided to use the simple approach described in class to compute the value of the firm’s assets and its asset volatility. Finally, you decided to set the value of
𝜇
equal to 5%. Following these steps, you first calculated DD and then determined the risk of default using the table below for the standard normal distribution.

Standard Normal Distribution (Mean = 0, Std Dev = 1).

P(Z ≤ z)          z

3.0%           -1.88

2.0%           -2.05

1.0%           -2.33

0.5%           -2.58

0.2%           -2.88

0.1%           -3.09

In this context, what can you say about the probability of default?

Question

You are estimating the probability of default for a given firm using the options model of default risk where this probability is given by Prob Default = N(-DD), and DD is the distance-to-default, calculated using the formula we covered in class.

While implementing this model, you decided to consider the risk of default over a one-year horizon and set the face value of the debt in one year as the current book value of the debt. You also collected the following additional information on the firm:

Current book value of debt = $190M
Current market value of equity = $400M
Equity volatility = 0.81 (measured during the past year)
Value of D/A (measured during the past year) = 0.4

Moreover, you decided to use the simple approach described in class to compute the value of the firm’s assets and its asset volatility. Finally, you decided to set the value of 
𝜇
 equal to 5%. Following these steps, you first calculated DD and then determined the risk of default using the table below for the standard normal distribution.

Standard Normal Distribution (Mean = 0, Std Dev = 1).

P(Z ≤ z)          z

3.0%           -1.88

2.0%           -2.05

1.0%           -2.33

0.5%           -2.58

0.2%           -2.88

0.1%           -3.09

In this context, what can you say about the probability of default?

BlackTom AI · Accepted Answer

Distance-to-default (DD) is computed as DD = [ln(V/D) + (μ − 0.5 σ_V^2)T] / (σ_V √T), and the default probability = N(−DD). Use the class “simple approach” where the historical D/A ratio gives the asset base: A = E / (1 − D/A), and asset volatility σ_A is scaled down from equity volatility by E/A.

1) Relevant knowledge points and concepts
- Merton/options model: equity is a call on firm assets; DD summarizes distance between asset value and debt face value in volatility units.
- Simple heuristics taught in class: A = market equity / (1 − D/A), σ_A = σ_E × (E/A), T = 1 year, μ = 5%, and default threshold D is the one-year face value (book debt = $190M).

2) Step-by-step reasoning and calculations
- Given: E = $400M, D/A = 0.4, σ_E = 0.81, D (face in 1 yr) = $190M, μ = 0.05, T = 1.
- Compute assets: A = E / (1 − D/A) = 400 / 0.6 = 666.6667 (M).
- Compute asset volatility: σ_A = σ_E × (E/A) = 0.81 × (400 / 666.6667) = 0.81 × 0.6 = 0.486.
- Compute ln(V/D): ln(666.6667 / 190) = ln(3.50877) ≈ 1.255.
- Compute 0.5 σ_A^2: σ_A^2 = 0.486^2 = 0.236196 ⇒ 0.5σ_A^2 = 0.118098.
- Numerator = ln(V/D) + (μ − 0.5σ_A^2)T = 1.255 + (0.05 − 0.118098) = 1.255 − 0.068098 = 1.186902.
- DD = 1.186902 / 0.486 ≈ 2.44.

3) Why this implies the correct probability range
- Prob(Default) = N(−DD) = P(Z ≤ −2.44). From the provided normal table: z = −2.33 corresponds to 1.0% and z = −2.58 corresponds to 0.5%. Since −2.44 lies between −2.33 and −2.58, the PD lies between 0.5% and 1.0% (excluding endpoints).

4) Why other answer choices are incorrect
- Options giving probabilities between 0.1–0.2%, 1.0–2.0%, or 2.0–3.0% correspond to z-values much more negative (DD much smaller) or less negative (DD much smaller/larger) than the computed DD ≈ 2.44; they do not match the computed DD and implied tail probability from the normal table.

Conclusion: Using the simple-class heuristics and the inputs, DD ≈ 2.44, so the probability of default lies between 0.5% and 1.0% (excluding endpoints), which matches the provided correct answer.

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