Debugging Merton Model Inputs for Default Probability and Recovery

Summary

The document presents a numerical question about applying the Merton structural credit model to estimate a firm's asset value, asset volatility, default probability, and recovery rate from equity value, debt, equity volatility, the risk-free rate, and time to maturity. The questioner follows a textbook procedure using a spreadsheet solver to infer asset value and volatility, but reports that different starting values produce markedly different outputs, including an implausible recovery rate above one in one run.

The examples show sensitivity to initialization and raise concerns about the solver setup, model equations, or interpretation of expected loss and recovery. However, the document contains no answer or diagnosis, so it does not establish which calculation is wrong or provide a corrected estimate. It is useful as a troubleshooting case: validate the equations and constraints, inspect solver convergence, and check how default probability, expected loss, and recovery are defined before relying on the output.

Key ideas

• The Merton model infers firm asset value and asset volatility from equity and debt inputs.
• The questioner uses a spreadsheet solver and reports outcomes that vary with starting values.
• One reported recovery estimate exceeds one, signaling a possible setup, convergence, or interpretation problem.
• The document provides no resolution, so the reported figures should not be treated as validated model results.

Tags

# Question about Merton model to estimate default probability and recovery rate of the company


# Question about Merton model to estimate default probability and recovery rate of the company












I recently come across Merton's model to estimate the default probability and recovery rate of the company. Here is the inputs

```
Market value of equity =  4,242,509,661
Debt to be paid =  3,397,334,000
equity Volatility = 34%
Risk-free rate = 0.38%
Time to maturity = 2.29
```

I simply follow the way that Chapter 20 in John Hull 6th edition suggested, in which excel "solver" is used to find the total market value of the asset and its volatility. However, the result is strange and I cannot get positive expected loss and recovery rate greater than 1.

For your information, I initialize

```
V =  7,500,000,000 
sig_V = 10%
```

But the solution becomes

```
V =  7,500,000,000 
sig_V = 19%
Expected loss = 0.0046807
Recovery rate = 8.2132 > 1
```

If I initialize

```
V =  7,619,759,237 
sig_V = 10%
```

Then, the solution becomes,

```
V =  7,619,759,237 
sig_V = 19%
Default probability = 0.00304
Recovery rate = 0.04853
```

Did anyone come across the Merton's model before? Can anyone explain what's wrong of the model or what I did? Thanks.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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