Probability of Default Calculator
The formula
The intuition behind the Merton model
Merton (1974) reframed equity as a call option on the firm's assets with a strike equal to the face value of debt. If assets finish above the default point at maturity, equity holders repay the debt and keep the remainder; if assets finish below, they hand the keys to creditors and equity is worthless. Default therefore is not a mysterious event — it is the ordinary outcome of asset value falling past a known threshold, and the probability of that happening is measurable from the asset value, the threshold, the volatility and the time available. The distance-to-default statistic simply expresses "how many standard deviations of asset volatility away is the default point" in the same units a statistics student already understands.
Why market-based PD beats ratio-based PD
Credit-rating agency models lean on accounting ratios, which are reported quarterly and lag the market by weeks. The Merton approach uses equity prices, which update continuously and embed every piece of public information investors have. Empirically, market-based measures such as Moody's KMV Expected Default Frequency react faster — the classic example is that KMV flagged Enron's deterioration months before its rating was downgraded to junk. The trade-off is volatility: market PDs swing with sentiment and liquidity, so a spike during a broad market selloff may reflect panic rather than a genuine change in the firm. Best practice is to read the market-based PD alongside a ratio-based one and investigate when they diverge.
Frequently asked questions
What is distance to default?
How do I estimate asset volatility when only equity trades?
Is a 1% one-year PD high?
Does the model assume anything unrealistic?
Source & methodology
Merton, R. C. (1974), "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates", Journal of Finance 29(2); Crosbie & Bohn (2003), Moody's KMV "Modeling Default Risk".
Probability of Default explained: formula, worked example and honest limits
You are asking the only question that matters at a table or a sportsbook: does the arithmetic support this decision? Searchers land on a Probability of Default page for three reasons — to plan, to compare options, and to check that somebody else’s arithmetic is honest. This tool serves all three, and everything after the calculator explains the model instead of hiding it.
In plain terms, Probability of Default answers the question "Probability of default?". The measurement matters because it converts a situation that feels complicated — several inputs pulling in different directions — into a single number you can act on. Within Investing & Markets, Fixed Income is the context where this figure is quoted most often, and professionals treat it as a first-pass screen rather than a verdict.
In Investing & Markets the same quantity is often quoted under two or three different conventions. The formula printed above fixes one convention and applies it consistently, which is what makes results from this page comparable with each other.
Every figure in the prose is generated by the engine rather than written beside it — the worked example resolves to probability of default = 12.71% on the default inputs, and the sensitivity section ranks the inputs that move the result — Market value of assets first, at ≈75.0 % of the headline for a ten-percent error either way in that input. Read the sections in order and you will be able to reproduce every figure by hand.
The Probability of Default formula, explained
Here is the complete model this tool runs — DD = [ln(V/D) + (r − σ²/2)T] / (σ√T); PD = Φ(−DD) — with each input explained below. If a competitor’s page gives you a different answer from the same inputs, this is the text that settles which of you is right.
Market value of assets (example value $850,000,000.00) is a money amount; keep it in the currency the rest of your inputs use, or the comparison steps lose meaning. The example value is there to demonstrate the mechanics, not to describe your situation — overwrite it.
Default point (total debt / face of liabilities) (example value $620,000,000.00) is a currency figure — enter the total exactly as your document states it, fees included. If you are unsure what belongs in this field, look at the printed steps: the substitution shows exactly how the engine uses it.
Asset volatility (% per year) (example value 28) is a quantity rather than a category — precision here is cheap and worth having. Check the convention first: if the source says “6.5 % per year”, type 6.5 and let the formula do the rest.
Risk-free rate (% per year) (example value 4) is a quantity rather than a category — precision here is cheap and worth having. Rates are the most mis-transcribed input on any page like this; copy it from the document rather than retyping from memory.
Horizon (years) (example value 1) is a numeric input whose unit is fixed by the label; a wrong unit is a wrong answer wearing a plausible disguise. Be precise about the boundary (exact date, start of month); time inputs decide the exponent in most formulas.
A fully worked Probability of Default example
Rather than describe the calculation, here it is end to end — every substitution in order, starting from the inputs above. Starting figures: Market value of assets, Default point (total debt / face of liabilities), Asset volatility (% per year), Risk-free rate (% per year). Run to the end, the probability of default resolves to 12.71% — Elevated risk — Assets close to liabilities on the bands this tool uses.
- Distance to default DD = [ln(V/D) + (r − σ²/2)·T] ÷ (σ·√T).
- ln(V/D) = ln($850,000,000 ÷ $620,000,000) = ln(1.3710) = 0.3155.
- Drift = (r − σ²/2)·T = (4.3% − 3.92%) × 1 = 0.0038.
- Denominator = σ·√T = 0.2800 × 1.0000 = 0.2800.
- DD = (0.3155 + 0.0038) ÷ 0.2800 = 1.140.
- PD = Φ(−DD) = Φ(-1.140) = 12.7056%.
Distance to default — 1.14 std dev. Compare it against the main result of 12.71% before drawing conclusions.
Asset coverage of debt — 1.37 ×. Read next to the headline, this is the figure that shows the shape of the trade-off.
Implied 5-year PD — 29.66%. Compare it against the main result of 12.71% before drawing conclusions.
Annualised default risk — 12.71%. Read next to the headline, this is the figure that shows the shape of the trade-off.
Re-running with your own figures is instant; re-running by hand once is the best way to know the tool is telling you the truth.
The Probability of Default corridor: your result across a realistic band
The honest version of any estimate is a corridor, and this one is cheap to produce: Probability of Default was recomputed at 5 points along the dominant input, with every other field left untouched.
| Market value of assets setting | Probability of default |
|---|---|
| 80 % of the example value | 36.56% |
| 90 % of the example value | 22.24% |
| example inputs | 12.71% |
| 110 % of the example value | 6.93% |
| 120 % of the example value | 3.66% |
Across 5 settings of Market value of assets, the result spans 3.66% to 36.56%. A decision that only works at one row of this table is a decision waiting for an input to move. Reproduce it in the form above by editing Market value of assets alone; the substitution steps will follow your figures exactly.
The levers behind Probability of Default, ranked by effect
Nobody should have to guess which input matters. The engine was re-run once per field below — ten percent up, ten percent down, everything else frozen — and the results ranked by how far they moved the probability of default. For Probability of Default the ranking is measured rather than guessed, and it belongs to these defaults: shift the starting point and the order can shift with it.
- Nudge Market value of assets by ±10 % and Probability of Default lands between 6.93% and 22.24% (sensitivity ≈75.0 %)
- Nudge Default point (total debt / face of liabilities) by ±10 % and Probability of Default lands between 6.47% and 21.18% (sensitivity ≈66.7 %)
- Asset volatility: ten percent up gives 15.62%, ten percent down gives 9.74% — ≈23.4 % of the probability of default either way
- Horizon +10 % → Probability of Default = 13.82% · −10 % → 11.49% (the probability of default shifts ≈9.5 %)
- Risk-free rate +10 % → Probability of Default = 12.39% · −10 % → 13.03% (the probability of default shifts ≈2.5 %)
Market value of assets is the dominant lever here: ten percent either way moves the result by about 75.0 %, from 6.93% to 22.24%. Spend your verification effort there first, and treat Default point (total debt / face of liabilities) as the second priority (≈66.7 % per ten percent). The rest of the inputs earn only a rounding-error of attention, and that asymmetry is the practical payoff of measuring rather than asserting.
Is your Probability of Default good — reading the result bands
A number on its own is trivia; a number against a range is intelligence. On the example inputs the band reads “Elevated risk — Assets close to liabilities”; your own figures place you on a different rung of the same ladder. How much of that band is luck? On Market value of assets alone ten percent either way covers 6.93% to 22.24%. The practical reading rules: read the band as a starting hypothesis, then test it: change the two dominant inputs by ten percent and see whether the verdict survives. Verdicts that survive stress are the ones worth acting on.
Related measures answer related questions, and the distinctions matter: Probability of Default is the quantity, Probability of default is this tool’s reading of it, and the decision that follows belongs to neither of them separately.
Why Probability of Default results go wrong — and how to prevent it
Copying a single headline number without checking the companion outputs. The tool reports Distance to default, Asset coverage of debt and Implied 5-year PD alongside the main result precisely because Probability of Default in isolation misleads — the secondary columns are where outliers and trade-offs show up.
Copying a single headline number without checking the companion outputs. The tool reports Distance to default, Asset coverage of debt and Implied 5-year PD alongside the main result precisely because Probability of Default in isolation misleads — the secondary columns are where outliers and trade-offs show up.
One mistake is specific to this engine: assuming every input matters equally. The computation above shows Market value of assets dominating — a careful figure there beats false precision everywhere else.
Another is comparing across contexts: a Probability of Default computed with one set of conventions should never be lined up against a figure computed with another. Check the inputs behind both before ranking them.
Before you trust your Probability of Default: a short checklist
- Compare the output against one published benchmark or a colleague’s independent estimate.
- Record the date and the inputs alongside the number so future-you knows what was assumed.
- Re-verify Market value of assets against a source document — ten percent of error there moves the headline by ≈75.0 %.
- Check that every field the formula needs actually arrived from a document, not from recollection.
- Compare the result with one independent method or published benchmark.
- Run the ±10 % stress on the top two levers and check that your conclusion survives both.
Five steps from raw inputs to a figure you can defend
The tool does the arithmetic; the routine below is what makes the arithmetic trustworthy. Five steps, a minute total — and afterwards you will know not just what the number is, but which assumption produced it.
Step 1 — collect, then type. Everything this tool asks for is listed above (market value of assets, default point (total debt / face of liabilities), asset volatility (% per year), risk-free rate (% per year), horizon (years)). Pull each figure from something you can point at — paperwork, a dashboard, a label — before touching a field. Guesswork at the input stage cannot be repaired at the interpretation stage. When in doubt, prefer the boring input: the interesting one is usually the one you misread.
Step 2 — baseline before exploration. Get one defensible Probability of default on screen before you start nudging numbers. Without a baseline, every variation looks interesting and none of them means anything. Suspiciously tidy answers deserve the same scrutiny as untidy ones.
Step 3 — play with the inputs on purpose. Ten percent up, ten percent down, one field at a time. Within a few minutes you will know this calculation better than most people who quote its results. Numbers are cheap to produce and expensive to retract; the extra minute is the cheapest insurance available.
Step 4 — run the scenario trio. Expected, pessimistic, optimistic. The gap between pessimistic and expected is the number to plan against; the optimistic result is useful only as a ceiling. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
Step 5 — schedule the next run. In Investing & Markets the inputs drift quietly. Put a reminder a quarter out, re-enter the same fields, and read the change rather than the level. Repeating a calculation with the same numbers proves only that the arithmetic is reliable — which is worth knowing.
From calculation to decision: where Probability of Default earns its keep
Most Fixed Income disagreements are arithmetic disagreements in disguise. When Probability of default of 12.71% sits on the table with its working shown, the conversation stops being about who sounds more confident and starts being about which input is wrong. Compare the calculation to the rough guess you arrived with, and ask why they differ.
Used properly this is a five-minute filter that saves weeks. Compute Probability of Default, reject what fails, and only then bring in the expensive analysis for the survivors. Most surprises in numbers are surprises about inputs, not about formulas.
Nothing here is stored or transmitted, so the habit of checking is free: open the page, re-enter current figures, compare with the number you saved last time. If the verdict changes with a ten-percent nudge, report the nudge alongside the number.
Quick answers: what searchers ask about Probability of Default
What affects Probability of Default the most?
Measured on this engine, Market value of assets is the dominant lever: moving it ±10 % shifts the probability of default across 6.93% to 22.24% — about 75.0 % of the headline, with Default point (total debt / face of liabilities) next at ≈66.7 %. The ranked list in the sensitivity section is computed live, not asserted from a template.
Which matters more: default point (total debt / face of liabilities) or market value of assets?
For Probability of Default, Market value of assets wins the head-to-head (≈75.0 % against ≈66.7 % per ten-percent change). Knowing which input dominates is the difference between careful work and merely slow work.
What is a typical Probability of Default?
There is no honest universal average for Probability of Default, but there is a reference point: the defaults on this page produce 12.71% (Elevated risk — Assets close to liabilities). Replace the inputs with your own and compare — that comparison is the useful meaning of “typical”.
How do I calculate Probability of Default by hand?
Use the stated model — DD = [ln(V/D) + (r − σ²/2)T] / (σ√T); PD = Φ(−DD) — and substitute in the order the tool prints (it starts from Distance to default DD = [ln(V/D) + (r − σ²/2)·T] ÷ (σ·√T).). On the example inputs the hand route lands on 12.71%, exactly what the calculator shows. A spreadsheet reproduces it in two or three cells; doing it once by hand is what makes the automated answer trustworthy.
What happens to Probability of Default if market value of assets changes by 10 %?
Try it in the form above and the printed substitution will follow you; the pre-computed answer for the defaults is 6.93% to 22.24% across the ten-percent probe, i.e. ≈75.0 % sensitivity on the probability of default.
Is the Probability of Default result of 12.71% good or bad?
Neither, on its own — the label “Elevated risk — Assets close to liabilities” is the band this figure falls in for the example inputs, and your own inputs will place you elsewhere on the same ladder. Judge the number against your history and your alternatives, and remember that a ten-percent error in Market value of assets alone moves it by ≈75.0 %.
Does Probability of Default account for fees and taxes?
Only what you put in. Fees belong inside Market value of assets where they shrink returns before compounding touches them; taxes are outside the scope of a pre-tax Probability of Default and should be layered on top before decisions.
What affects Probability of Default the most?
Sensitivity depends on where you start, but the inputs Market value of assets and Default point (total debt / face of liabilities) typically move Probability of Default further than the rest because they enter the formula multiplicatively. If you can only measure two things well, measure those and hold the others at conservative defaults.
What is distance to default?
The number of asset-volatility standard deviations between today's asset value and the default point, adjusted for expected drift. A DD of 3 means assets would have to fall three standard deviations before creditors are impaired.
How do I estimate asset volatility when only equity trades?
Equity volatility is higher than asset volatility because equity is levered. The standard approach solves the two Merton equations simultaneously (equity value and equity volatility as functions of asset value and asset volatility) by iteration. A quick proxy for moderate leverage is asset vol ≈ equity vol × (E/V).
Can this tool see the numbers I type?
It cannot. There is no submission step: the calculation happens in your browser and the result disappears when you close the page. Treat the fields above as a scratchpad rather than a form.
Glossary for Probability of Default
- Distance to default
- 1.14 std dev for the Probability of Default worked case; it sits next to the main result because it carries context the headline leaves out.
- Asset coverage of debt
- 1.37 × for the Probability of Default worked case; it sits next to the main result because it carries context the headline leaves out.
- Implied 5-year PD
- 29.66% in the default Probability of Default calculation — one of the secondary figures an experienced reader checks before trusting the primary number.
- Probability of default
- The primary output of this tool: 12.71% for the default inputs, computed exactly from DD = [ln(V/D) + (r − σ²/2)T] / (σ√T); PD = Φ(−DD).
- Worked example
- The calculation written out in full so that nothing on this page has to be taken on trust.
- Verdict band
- The plain-language label — here “Elevated risk — Assets close to liabilities” — mapping the computed figure onto experience-based ranges for Investing & Markets.
- Sensitivity (±10 %)
- How far the Probability of Default moves when a single input is nudged ten percent either way; the ranked list on this page is computed live from this engine.
Where the Probability of Default model comes from
Everything above runs on one stated formula, applied without hidden coefficients: DD = [ln(V/D) + (r − σ²/2)T] / (σ√T); PD = Φ(−DD). Source of record: Merton, R. C. (1974), "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates", Journal of Finance 29(2); Crosbie & Bohn (2003), Moody's KMV "Modeling Default Risk". Specialist software layers extra coefficients on top of the same algebra; none are hidden here, and that transparency is what makes a disagreement traceable to a specific input instead of a black box.
Treat the output as a precise answer to the question you typed: exact arithmetic, transparent formula, published source, no hidden rounding. The sensitivity table above is part of that promise rather than a bonus: it lets you see how much of the result depends on how carefully you measured.
After the number: what to do with it
Compare the computed edge against the actual price being offered; a positive theoretical edge still loses to bad odds.
This page was written for players deciding whether a wager is mathematically worth taking, and the promise is that nothing above required trust. Bookmark the tool, re-run it whenever an input moves by more than a few percent, and keep the date with the number.