The formula worked step by step for a private borrower, the listed-company version, and why the EBITDA definition moves the answer most.
Distance to default in the Merton model is the number of standard deviations of asset value between the firm's assets and its debt: DD = [ln(V/D) + (μ − σ²/2)T] / (σ√T), and the probability of default is N(−DD). For an illustrative private borrower with 120.0m of debt, valued at 6.5x its lender-adjusted EBITDA of 28.46m with 25 per cent asset volatility, DD is 1.74 and the one-year PD 4.07 per cent. Value the same company on management's EBITDA and the PD is 0.21 per cent; on its audited EBITDA, 12.32 per cent.
Worked in full in Credit Analysis by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Merton treats equity as a call option on the firm's assets with a strike equal to the debt. The firm defaults if, at the horizon, its assets are worth less than what it owes. Two quantities decide how likely that is: the cushion between asset value and debt, measured in logarithms, and how volatile asset value is. Distance to default divides the first by the second. It is a structural measure, so unlike a spread or a rating it can be computed for any borrower whose asset value and volatility can be estimated, including one that has never issued a bond.
| Input | Value |
|---|---|
| Debt, the default point D | 120.0m |
| EBITDA, lender adjusted | 28.46m |
| Going-concern multiple, illustrative | 6.5x |
| Asset value V | 184.99m |
| Asset volatility σ | 25% |
| Drift μ, set to the risk-free rate | 3.40% |
DD = [ ln(V / D) + (μ − σ² / 2) × T ] / (σ × √T)
PD = N(−DD)
Excel: =(LN(V/D)+(Mu-Sigma^2/2)*T)/(Sigma*SQRT(T)), then =NORM.S.DIST(-DD,TRUE)
Step 1, the cushion. V/D is 184.99 / 120.0 = 1.5416, and its logarithm 0.4328. Assets can fall 35.1 per cent before they no longer cover the debt.
Step 2, the drift. 3.40 per cent less half of 25 per cent squared is 0.00275. Added to the cushion, 0.4356.
Step 3, scale by volatility. Over one year σ√T is 0.25, so DD = 0.4356 / 0.25 = 1.742. The assets sit 1.74 standard deviations above the default point.
Step 4, the probability. N(−1.742) = 4.07 per cent. With the drift set to the risk-free rate this is a risk-neutral probability; a higher expected asset return would lower it.
For a private borrower, V is a multiple times EBITDA. That makes the PD a function of whichever EBITDA is chosen, and borrowers present several.
| EBITDA basis | EBITDA | Leverage | Asset value | DD | PD |
|---|---|---|---|---|---|
| Management adjusted | 37.60 | 3.19x | 244.40 | 2.86 | 0.21% |
| Lender adjusted | 28.46 | 4.22x | 184.99 | 1.74 | 4.07% |
| Statutory | 24.60 | 4.88x | 159.90 | 1.16 | 12.32% |
| Downside case | 23.45 | 5.12x | 152.42 | 0.97 | 16.66% |
The statutory PD is 57 times management's. The model has done nothing wrong; it has faithfully converted an argument about add-backs into an argument about default probability. For the add-backs behind those four numbers, see what an add-back bridge is actually worth.
| Asset volatility | DD | PD | Multiple | DD | PD |
|---|---|---|---|---|---|
| 20% | 2.23 | 1.27% | 5.5x | 1.07 | 14.14% |
| 25% (base) | 1.74 | 4.07% | 6.0x | 1.42 | 7.75% |
| 30% | 1.41 | 7.99% | 6.5x (base) | 1.74 | 4.07% |
| 35% | 1.16 | 12.33% | 7.5x | 2.31 | 1.03% |
Both inputs are estimates for a private company, and either moves the PD by a factor of three or more across a plausible range. Report the PD with its multiple and volatility, never alone.
If the borrower is listed, V and σ are solved rather than assumed, from two equations: equity value E = V·N(d1) − D·e−rT·N(d2), and σE·E = N(d1)·σ·V. With a market value of equity of 70.0m and equity volatility of 55 per cent, iterating gives V = 185.86m and asset volatility of 20.91 per cent, so d2 = 2.151 and the PD is 1.57 per cent.
Using equity volatility as asset volatility. Equity is levered, so it is far more volatile than the assets. Put 55 per cent into the formula for the private borrower and DD collapses to 0.57, a PD of 28.31 per cent. Asset volatility is roughly equity volatility times E/V, after the option adjustment.
Reading N(−DD) as a calibrated default rate. The normal distribution has thin tails, and real asset values jump. Commercial implementations therefore map DD to an empirical default frequency observed for firms at that distance rather than reading the normal table. The same implementations usually set the default point at short-term debt plus half of long-term debt rather than all of it, which raises DD for a borrower with long-dated debt. Use the formula to rank borrowers and to see which input drives the answer, and calibrate before quoting it as a probability. Recovery, which Merton treats crudely, still has to be estimated separately; expected loss: PD x LGD x EAD puts the two together.
Credit Analysis follows one borrower through four defensible EBITDAs and shows what each does to leverage, cover and recovery; the free workbook for this case reproduces every one of them.
Higher is safer: it counts the standard deviations of asset value between the assets and the debt. In the illustrative case a DD of 1.74 corresponds to a 4.07 per cent one-year PD under the normal distribution, 2.86 to 0.21 per cent and 0.97 to 16.66 per cent. Because the normal tails are thin, practitioners map DD to observed default frequencies before using it as a probability.
Estimate them. Asset value is commonly enterprise value, a multiple times EBITDA, and volatility is taken from listed peers' asset volatility. At 6.5x lender EBITDA of 28.46m, V is 184.99m; moving asset volatility from 20 to 35 per cent moves the PD from 1.27 to 12.33 per cent, so both assumptions should be reported with the result.
Only to solve for asset volatility, never in its place. For a listed firm with equity of 70.0m at 55 per cent volatility and 120.0m of debt, solving the two Merton equations gives asset volatility of 20.91 per cent and a PD of 1.57 per cent. Using 55 per cent directly as asset volatility would give a PD of 28.31 per cent.
This article is one calculation from Credit Analysis. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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