Estimating Merton Default Risk: Equity Volatility and Model Inputs
Summary
The document discusses estimating default probability with the Merton structural credit model from monthly equity returns, a risk-free rate, equity value, and debt face value. It outlines a workflow that converts prices to log returns, estimates return dispersion, annualizes equity volatility, and then attempts to solve for asset value and asset volatility. The response corrects the annualization approach: estimate the sample standard deviation of monthly log returns, then scale volatility by the square root of twelve. It also notes that the proposed asset-value formula is not addressed as a valid model equation, and points to Excel-based worked material elsewhere.
The answer emphasizes separating volatility from returns and warns that a long historical window may be unsuitable for forecasting a firm's near-term default risk because company conditions change. The document does not provide a complete derivation of the Merton equations, solver setup, or a calculated probability. Its guidance is therefore limited to return and volatility estimation, with model implementation still requiring further study.
Key ideas
- Monthly log returns are computed as the difference between consecutive log prices.
- Monthly return volatility should be estimated with a standard deviation, not a variance mislabeled as volatility.
- The response annualizes monthly volatility by multiplying it by the square root of twelve.
- The proposed formula for asset value is not validated or corrected in the document.
- A long historical sample may not represent a firm's current default risk because conditions change.
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Full text
# How to use Merton model to calculate default probability with monthly stock prices?
# How to use Merton model to calculate default probability with monthly stock prices?
I want to calculate the estimated default probability with only given data the monthly returns for the last 20 years, the risk-free rate ($R_f$), equity value (EV) and the face value of debt ($D$). My steps so far are:
- Find each month's lognormal returns: Ln(Month);
- Subtract from each result in step 1 the average of the lognormal returns and then raise them to the power of 2 and then sum it, in order to find the monthly equity volatility;
- Calculate the annualized equity volatility by doing $$\left(1 + \frac{\textrm{monthly equity volatility}}{12}\right)^{12 \times 20} - 1$$
- Calculate Asset Value (AV) using the formula: AV = EV * equity volatility + D (not sure if its correct)
- Attempt to solve the equations to derive the asset volatility but get stuck when using the Excel solver.
How to proceed?
## Answer by Dmitry Pavliv (score 2)
https://quant.stackexchange.com/a/9702
detailed description of the solution of this problem using Excel is in the second chapter of the book Credit Risk Modeling using Excel and VBA Gunter Löffler
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/10199
your steps are a bit too complicated to me.
in Step 1 and 2 you do two things: you caculate monthly log-returns and then their standard decviation.
Given prices $P_t$ indexed by time the log return is given by $$ r_t = \ln(P_t/P_{t-1}) = \ln(P_t) - \ln(P_{t-1}). $$ The formula for standard-deviation (the sample estimator of it) should be clear: $$ \sigma = \frac1{n-1} \sum_{t=1}^n (r_t-\bar{r})^2, $$ where $\bar{r}$ is the average return. Usually software packages have a function for standard-deviation. Then you annualize volatility $\sigma_a $ by the square-root of time rule: $$ \sigma_a = \sigma \sqrt{12}. $$ So much for the first three steps. Your formula 3 is a mixture of various ways to calculate yearly and monthly returns from one another (geometric returns not log returns). But for calculating a yearly vola you need the formula above.
Last comments: Be sure you understand the math. If you mix up volatility and returns then you need to study some more.
Second: for a text book example: ok, use the past $20$ years of data. For real life: don't estimate a default probability for the, say, comming year using data that is that old. The world changes and so do firms and I don't expect data from $20$ years ago to be relevant at the moment.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.