# 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.

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 $1m per loan. 8 defaults: Loss is \$1m x 8 x 0.60 = \$4.8m. Interest received = \$100m x 5% = \$5m. 9 defaults: Loss is \$1m x 9 x 0.60 = \$5.4m. Interest received = \$100m x 5% = \$5m. With 9 defaults we lose more than we get in interest..... • Please keep in mind that this is an approximation also with regard to the input parameters. More precisely the credit spread$S\$ is the difference between loan rate and the bank's funding rate (that is default-free rate plus the bank's own credit spread) minus other fees for administration of the loan etc.. Moreover the calculated recovery rate for a specific loan might heavily depend on the collateral offered to the bank. – Dr_Be Sep 22 '16 at 8:26
• I agree with your points. I will amend my answer to make that clear. – Dom Sep 22 '16 at 14:36
• Could you give an example calculation? I'm having some difficulty understanding what value to give R – Matthias Oct 3 '16 at 10:45
• Let's say the loans interest rate is 6% and the risk free rate is 1% Assuming R = 0. Then p = (6%-1%)/1 = 5%, so the assumed risk of default is 1/20. In other words at 6% interest, 1/20 loans could default in order to break-even. Is this assessment correct? – Matthias Oct 3 '16 at 10:54