Smoothing a table of Probabilities of Default

I have a table with empirical Cumulative Default Rates per tenor (time) and per rating level.
In both dimensions, the values should increase, but not linearly. For example, a rating BBB must have a higher Cumulative Default Rate than a rating A, and the Cumulative Default Rate over 2 years has to be more than over 1 year. I am looking for a smoothing function that makes the values always increase.
I found that by doing a polynomial regression in the second or third degree I get a fairly good smoothing, and it is fairly near to the original values. However, if I do it in the time dimension and then in the rating dimension, my rating regression breaks the time dimension. In the same way, if I start with the rating dimension and then do the regression on the time dimension, it breaks the rating dimension.
I am looking for something like a two-dimensional polynomial regression, or whichever other model that would fit my needs. I coded my problem in R and tried also with the GAM methods and with Loess, but the results are so far away from the original values that I must discard those results. I would appreciate very much any idea from the community. Thank you.


Active Member
Cumulative means accumulated. It's the total so far of some underlying measurement over a period of samples, in this case time in years.

For example, a daily apple harvest
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Well-Known Member
As I read your question you want to smooth a surface. It seems unlikely that you will find a function of x and y which will satisfy you and you will need to use some numerical method. You could look up Kriging which is a surface smoothing technique.
Thank you @katxt. From the first few informations I read about Kriging it looks promising and might match what I need. I will go more into detail and do some research on it. Thanks again for pointing me to it.


Well-Known Member
Fine. I'm still not sure how a two variable cumulative function works. For one variable cum x = sum of all <=x. For two variables are you using cum x,y = sum of all <= x AND <= y? kat
Indeed the sum is only in one dimension. Probabilities of Default show the probability that a company with a given rating (for example BBB) are unable to honour their duties within a certain time. The cumulation is that the Probability of Default in 2 years includes the Probability of Default in 1 year. PD[2y]=PD[1y]+PD[from 1yr to 2yr].
At the same time, on average a PD of a company with a rating BBB should be higher than a PD of a company with a rating A. However in some situations empirical observations might not be like that. That is why a smoothed table must have this 'increasing' relationship.