Combining probabilities and building term structures

Consider the following two term structures of PD, it represents two measures of default risk from two different perspectives and models.

By Bayes: the joint probability of default by either driver (or) is the sum of the probabilities less the probability of both drivers (and) simultaneously. Here is the climate related default probability and is the sector related default probability, and is the assumed default correlation.

The origin of the term in the square root derives from the definition of correlation as a normalised covariance. Since default events are true/false by the indicator function.

For example as a function:

const pdcor = function(pdc, pds, cor) { 
  var pdj = cor * Math.sqrt(pdc * (1-pdc) * pds * (1-pds));
  return pdc + pds - pdj - pdc*pds; 
}

If the default drivers are uncorrelated, then and the joint probability of default is simply the product of the two probabilities.

Select the default correlation:

Calculation of the joint pd using the default correlation:

const jpd = pd.map( d => ({...d, jointpd: pdcor(d.climpd , d.sectpd, cor) }));

Cumulative default probability term structure

Term structure of PD, cumulative probabilities requires iterating over the incremental PDs, by bootstrapping suriviving to the previous period and defaulting in the next. Etc ...

For example as a function:

const pdcum = function(pd) { 
  var n = pd.length; 
  var cumpd = [pd[0]];
  var margpd = [pd[0]];
  for (let i = 1; i < n; i++) { 
    margpd[i]= (1-cumpd[i-1])*pd[i];
    cumpd[i] = cumpd[i-1] + margpd[i]; 
  } 
  return [margpd, cumpd]; }

Calculation of the joint cumulative probability of default:

const cjpd = jpd.map((d,i) => ({ 
    ...d, 
    margpd: pdcum(jpd.map(d => d.jointpd))[0][i], 
cumjointpd: pdcum(jpd.map(d => d.jointpd))[1][i] }))

Results

Bivariate normal correlated PD solution

Using bivariate normal distribution, the estimate used here is a numerical integration, called Daley's T (reference below).

Daley, D. J. (1974). Computation of bi- and tri-variate normal integrals. Applied Statistics, 23(3), 435. doi:10.2307/2347136

Where through probit, the probabilities are quantiles of the normal distribution, the joint integration is the area represent by the joint integration to the limits of x and y:

Comparing the bivariate normal with the simple joint PD calculation

Comparing the bivariate normal with the simple joint PD calculation:

The percentage error ie proportional difference between the bivariate normal and simple joint PD calculation.