Estimating Default Probability from a Loan Credit Spread
Summary
The document presents the credit triangle as a quick way to infer an annualized default probability from a loan’s credit spread and expected recovery rate. It defines the spread as the loan rate minus a comparable default-free rate, with known administration or other fees potentially removed. The approximation relates spread to default probability multiplied by loss given default; rearranging gives an implied probability when the recovery assumption is available.
An example uses a loan rate of 6%, a risk-free rate of 1%, and a recovery assumption of 40% to derive an annualized implied default probability of 8.33%. A portfolio illustration compares interest income with credit losses under different default counts. The response describes the rule as a back-of-the-envelope approximation to a full hazard-rate model, and its estimate depends on the quality of the risk-free benchmark, fees, recovery statistics, collateral, and the assumed similarity of loans. The calculation is not a complete pricing or portfolio-loss model.
Key ideas
- The credit spread is approximated by default probability multiplied by loss given default.
- Subtract a maturity-matched default-free yield from the loan rate to estimate the spread.
- Recovery assumptions can reflect loan security, historical recovery data, and pledged collateral.
- The inferred probability is approximate and omits the detail of a full hazard-rate model.
- Portfolio interest income offsets losses only up to a level determined by default frequency and recovery.
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Full text
# Interest rate implied probability of default # Interest rate implied probability of default Is there an equation or rule of thumb to determine the probility of default for a loan with a specific interest rate? Let's say, a bank offers a company a loan with an interest rate of 6%, by which they assume the change of default is X. ## Answer by Dom (score 10, accepted) https://quant.stackexchange.com/a/30232 You can use the "credit triangle" which states that the (annualised) credit spread $S$ equals the annualised probability of default $p$ times the loss given default LGD which equals par minus the expected recovery amount $R$, i.e. $S=p(1-R)$. This is a "back-of-the-envelope" approximation to a full hazard rate credit model - from experience I find that the percentage error is usually less than 5%. The credit spread $S$ can be obtained by subtracting the default-free rate from the loan interest rate. The default-free rate is the yield of a par government bond with a similar maturity to the loan. You may also subtract the impact of administration fees and other fees if you know what they are. The expected recovery rate can be obtained from rating agencies default and recovery statistics for loans. This will depend on whether the loan is secured or unsecured. For a specific loan you can take into account the value of any collateral pledged to the bank by the borrower. So if you have $S$ and $R$ you can solve for $p=S/(1-R)$. Let me give an example. Suppose loan interest rate is 6% and the risk free rate is 1%. In this case $S=0.05$. Assume that $R = 0.4$. In this case $p=0.05/(1-0.4)=0.05/0.6=8.33\%$. So the annualised implied probability of default is 8.33%. So suppose you had 100 loans similar to this one. You would be in profit if 8 or fewer defaulted. More than 8 defaults would result in a loss. Let us check this. Assume $1m per loan. 8 defaults: Loss is \$1m x 8 x 0.60 = \$4.8m. Interest received = \$100m x 5% = \$5m. 9 defaults: Loss is \$1m x 9 x 0.60 = \$5.4m. Interest received = \$100m x 5% = \$5m. With 9 defaults we lose more than we get in interest.....
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