# annualized probability of default for loan including time component

i am struggling with this. say i am given an annual probabilty of default for a company going insolvent as 0.02. so 2%.

say this client then takes out a 100k , 150 day loan on jan 1st 2018, what is predicted default amount? would you be right to say ok 150 days is half the year so just halve the probability to 0.01 and then use this to predict default amount. the issue i have with this is that it treats probability as linear, so we are saying as time goes on, the probability of defaulting increases, is this right to say? eg. (150/365) * 0.02 * 100k = predicted default amount.

am i right in thinking you can model this as binomial ? i am not sure how to calculate this. do i need to find default rate per month first using p_annual = p_month^12 ?? i could rearrange this to find p_month then do (150/days in month)*p_month

i am confused how to calcaulte the predicted default amount at a loan level (i.e. 30 day, 80 day loan level given the annual PD).

Briefly stated, you have four alternative modelling strategies:

• Predict default at a certain timestamp in the future, which is indeed Binomially distributed;
• Predict the probability of default at each future timestamp - as a function of time, most often survival models (Cox, Kaplan Meier) are used for this;
• Predict the default amount by (non)linear regression;
• A combined approach where you first predict probability of default, and in a second model predict the most likely default amount at that timestamp

I would start with predicting default itself at a specific timestamp. Investigate whether this is reasonably predictable with the predictor variables you have at your disposal. You can use the debtor approach straight away for doing this. The combined approach is more involved and entails building conditional prediction models from stratified data sets (stratified sampling).

A final note: the probability of default is likely to somewhat depend on the amount itself. In that case, you cannot model default amount independently from the probability of default - as that approach would ignore this dependency, and hence be (slightly) biased.

• what about a hazart rate approach? could i use that here Jul 24 '20 at 10:58

To answer the question "calcaulte the predicted default amount at a loan level", the metric typically used is "Expected loss (EL)."

EL = Probability of default (PD) * Loss given default (LGD) * Exposure at time of default (EAD)

In the example you gave, you would need LGD and EAD in addition to PD to compute the EL. Keeping PD calculation aside for a while, LGD would depend on a variety of factors such as presence of any collateral (did the borrower mortgage any of his assets against the loan), efficiency of bankruptcy laws, etc. EAD is simply the amount that's due at the time default happened.

To answer your primary question, I will give a practical example followed by the theory behind it. Corporate default rates are usually measured over multiple time horizons - one year, two years, three years.. and so on (see table 25, pg 59 ,https://www.spratings.com/documents/20184/774196/2018AnnualGlobalCorporateDefaultAndRatingTransitionStudy.pdf). This table describes the Cumulative default rates (CDR), i.e. the probability of default BY a certain time period. Consider CDR1, CDR2 and CDR3:

• CDR1: probability of default by end of first year, say this is 1%
• CDR2: probability of default by end of second year, say this is 2.5%
• CDR3: probability of default by end of third year, say this is 3.5%

Using this CDR curve, you can easily compute the marginal default rate (MDR) curve, which signifies the probability of default DURING a certain time period. For instance, if I am interested in probability of default during year 2, i.e. MDR2, I can calculate it using CDR1 and CDR2.

CDR2 = P (default by Y2)

 = Probability that company either defaulted in year 1, or if it didn't default in year 1, it defaulted in year 2

= P (default in Y1) + P(no default in Y1) * P(default in Y2)

= CDR1 + (1 - CDR1) * MDR2


Therefore, MDR2 can be calculated as (CDR2 - CDR1)/(1 - CDR1), or in our example, (2.5% - 1%)/(1 - 1%) = 1.52%.

Similarly, MDR3 can be calculated as (CDR3 - CDR2)/(1 - CDR2) = (3.5% - 2.5%)/(1 - 2.5%) = 1.03%

Practically, if you observe CDR (in the table 25 referred earlier), while it's an increasing function, it tends to flatten as time horizon increases. This can be intuitively explained by the fact that companies that have already survived multiple business cycles, are less likely to default incrementally. Therefore, it is easy to appreciate that MDR is a declining function, i.e. if t2 > t1, MDR(t2) < MDR(t1).

If you can appreciate the flattening nature of CDR and the declining nature of MDR, it is easy to connect it to the Hazard rate model. As per hazard rate model,

CDR(t) = 1 - Exp(−λt); where λ represents the default intensity

MDR(t) = first differential of CDR(t) = λ*Exp(−λt)

Note that for a given λ, as time horizon increases, CDR(t) approaches 1, this indicates that all firms, no matter how creditworthy will eventually default.

Coming to the original question of computing the partial time default rates, you can see that for a given set of CDR values (which you will have if you have history of typical borrower behavior over multiple time horizons), you can compute λ (by answering the question which value of λ best fits the CDR curve). Once you have λ, you can easily compute default rates over partial times.

CDR(6 months) = 1 - exp(-0.5λ)

CDR(3 months) = 1 - exp(-0.25λ)