Probability of Default and Loss Given Default

Ask someone how risky a loan is and you usually get one number back. But there are always two, and they have almost nothing to do with each other. Think about insuring a car. One question is how likely it is to be stolen this year. A completely separate question is what you would actually be out of pocket if it were, after the police recover half the vehicles and your policy covers part of the rest. A car that is often stolen but always recovered is not the same risk as one that is rarely stolen but gone forever when it goes.

Credit uses the same split. Probability of default ($PD$) is the chance the borrower fails to pay over a stated period. Loss given default ($LGD$) is the share of your money that never comes back, conditional on that failure. Together with the exposure at default ($EAD$), the amount outstanding at the time, they give the expected loss:

$$EL = PD \times LGD \times EAD .$$

In words: how often, times how badly, times how much. Three independent estimates that get multiplied at the very end, never guessed at as a single blob.

Where each number comes from

$PD$ is estimated from three places, and they rarely agree. Historical default rates by rating bucket give you long-run averages published every year by the agencies. Statistical models score a borrower from its own financials, leverage, coverage and equity price. Market-implied figures are backed out of bond spreads or CDS, which always give higher numbers because a spread pays for risk premium as well as default. See Credit Spreads.

$LGD$ is simply one minus the recovery rate, and it is driven by structure rather than by how good the business is. Seniority comes first: senior secured lenders queue ahead of subordinated ones, so the same bankruptcy hands them very different outcomes. Then collateral quality, jurisdiction and how long the workout drags on. Historically, senior secured debt recovers around $70\%$ of face value and subordinated debt closer to $30\%$, though the range around those averages is enormous.

Worked example: the horizon changes everything

A $PD$ quoted without a horizon is meaningless. Suppose a borrower has a one-year $PD$ of $2\%$ and that rate stays flat. The chance of surviving one year is $0.98$. Surviving five years means surviving each of them in turn:

$$0.98^5 = 0.9039 .$$

So the cumulative five-year default probability is $1 - 0.9039 = 0.0961$, about $9.6\%$. A borrower who looks perfectly comfortable over one year has close to a one-in-ten chance of failing before a five-year bond matures. This is why credit desks distinguish the marginal yearly rate from the cumulative figure, and why lending long to a mediocre borrower is a very different trade from lending short.

Worked example: a small loan book

Now take a portfolio of 40 loans, $2.5m each, so $100m of exposure. Each borrower has $PD = 5\%$ and, being unsecured, $LGD = 45\%$.

Expected loss for the book is $0.05 \times 0.45 \times 100{,}000{,}000 = 2{,}250{,}000$, or $2.25\%$ of the book each year. The expected number of defaults is $40 \times 0.05 = 2$, and if defaults were independent the standard deviation of that count would be $\sqrt{40 \times 0.05 \times 0.95} = 1.38$.

Six defaults would cost $6 \times 2{,}500{,}000 \times 0.45 = 6{,}750{,}000$, three times the budgeted loss. Under independence that happens about $1.4\%$ of the time, roughly a once-in-seventy-year event. Real loan books saw exactly that in 2002 and again in 2009, which tells you the independence assumption is the weak link, not the arithmetic.

Drag the sliders below to see it. Set trials to 40 and the probability to 0.05, then watch how much of the mass sits well to the right of the mean. Push the probability up and the whole distribution slides and widens.

What this means in practice

Banks are required to estimate $PD$, $LGD$ and $EAD$ separately under the Basel internal-ratings framework, and the same triple drives loan pricing, provisioning under IFRS 9 and CECL, and the risk limits on a credit desk. Getting the split right matters commercially: a lender that improves $LGD$ by taking collateral has cut its expected loss without turning down a single borrower. And $EAD$ is the quiet one. On a drawn term loan it is obvious, but on a revolving credit line borrowers draw it down hard exactly when they are in trouble, so the exposure you model must be larger than the exposure you see today.

Key terms

• PD — probability of default over a stated horizon; meaningless without that horizon.
• Marginal vs cumulative PD — the chance of failing this year, versus at any point up to a date.
• LGD — the fraction of exposure lost when default occurs; one minus the recovery rate.
• EAD — the amount outstanding at the moment of default, which can exceed today's balance.
• Expected loss — the product of the three, budgeted as a cost rather than held as a risk.