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)