Journal of Risk Model Validation

Risk.net

Addendum to Rubtsov and Petrov (2016): “A point-in-time–through-the-cycle approach to rating assignment and probability of default calibration”

Torsten Pyttlik, Mark Rubtsov and Alexander Petrov

In June 2016, The Journal of Risk Model Validation published a paper by Rubtsov and Petrov (2016) called “A point-in time–through-the-cycle approach to rating assignment and probability of default calibration”. This paper included a system of equations that were solved numerically. Following publication, Torsten Pyttlik and Roland Wolff proposed an analytical solution, which the authors believe adds substantial value to their original work. In the attached PDF the authors have written a  short follow-up, presenting the details of that analytical solution.

In June 2016, The Journal of Risk Model Validation published a paper by Rubtsov and Petrov (2016) called “A point-in-time–through-the-cycle approach to rating assignment and probability of default calibration”. On p. 102 of the paper, the authors solved a system of equations (5.7)–(5.9) numerically; these equations are reproduced below as (1)–(3):

  𝔼[Φ-1(dr)] =μr-ρr𝔼(Z^)1-ρr,   (1)
  𝔼[(Φ-1(dr))2] =11-ρr[𝔼(Br2)-2ρr𝔼(BrZ^)+ρr𝔼(Z^2)]  
    =11-ρr[(γr+μr2)-2ρrμr𝔼(Z^)+ρr𝔼(Z^2)],   (2)
  𝔼[(Φ-1(dr))3] =(1-ρr)-3/2[𝔼(Br3)-3ρr𝔼(Br2Z^)+3ρr𝔼(BrZ^2)-ρr3/2𝔼(Z^3)]  
    =(1-ρr)-3/2[(3μrγr+μr3)-3ρr(μr2+γr)𝔼(Z^)+3ρrμr𝔼(Z^2)-ρr3/2𝔼(Z^3)].   (3)

Torsten Pyttlik has recently proposed an analytical solution to this system, and we present the details of that solution below. We believe it adds substantial extra value to the original material.

Let Yr:=Φ-1(dr) for brevity. The original equations (5.7)–(5.9) then become

  𝔼[Yr] =μr-ρr𝔼[Z^]1-ρr,   (4)
  𝔼[Yr2] =11-ρr[γr+μr2-2ρrμr𝔼[Z^]+ρr𝔼[Z^2]],   (5)
  𝔼[Yr3] =(1-ρr)-3/2[μr3+3[μrγr-ρr(μr2+γr)𝔼[Z^]+ρrμr𝔼[Z^2]]-ρr3/2𝔼[Z^3]].   (6)

Rearranging (4) gives

  μr=1-ρr𝔼[Yr]+ρr𝔼[Z^].   (7)

Taking the square of (4) and subtracting that from (5) and then rearranging gives us

  𝔼[Yr2]-𝔼[Yr]2 =11-ρr[γr+ρr(𝔼[Z^2]-𝔼[Z^]2)],   (8)
  γr =(1-ρr)𝕍[Yr]-ρr𝕍[Z^].   (9)

Here, we have introduced the variance, defined as

  𝕍[X]:=𝔼[X2]-𝔼[X]2.  

Note that (9) might result in γr<0 if ρr>0, which is undesirable since γr was defined as a variance when the original system of equations was set up. Negative values of ρr could therefore be considered, which would require an extensive modification of (1)–(3), using -ρr and changing signs in several places.

Taking the third power of (4) and subtracting this from (6) gives

  𝔼[Yr3]-𝔼[Yr]3=(1-ρr)-3/2[3[μrγr-ρrγr𝔼[Z^]+ρrμr𝕍[Z^]]-ρr3/2(𝔼[Z^3]-𝔼[Z^]3)].  

Inserting (7) and (9) into the inner square brackets on the right-hand side yields, after rearranging, an expression that is solvable for ρr alone:

  𝔼[Yr3]-3𝔼[Yr]𝕍[Yr]-𝔼[Yr]3 =-(ρr1-ρr)3/2[𝔼[Z^3]-3𝔼[Z^]𝕍[Z^]-𝔼[Z^]3],  
  ρr =1[1+(𝕊[Yr]/𝕊[Z^])-2/3].   (10)

Here, we have defined

  𝕊[X]:=𝔼[X3]-3𝔼[X]𝕍[X]-𝔼[X]3,  

which is the nonnormalized skewness (to obtain normalized skewness, multiply 𝕊[X] by 𝕍[X]-3/2).

After evaluating ρr from (10), use (9) and (7) to obtain values for γr and μr, respectively.

Note that if the distribution of Z^ is symmetrical, ie, 𝕊[Z^]=0, then (10) has no solution if 𝕊[Yr]0. There is no unique solution if both 𝕊[Z^]=0 and 𝕊[Yr]=0. For the limiting case ρr=1, the whole system of equations (4)–(6) would be invalid.

References

Rubtsov, M., and Petrov, A. (2016). A point-in-time–through-the-cycle approach to rating assignment and probability of default calibration. The Journal of Risk Model Validation 10(2), 83–112 (http://doi.org/bzcb).

 

You need to sign in to use this feature. If you don’t have a Risk.net account, please register for a trial.

Sign in
You are currently on corporate access.

To use this feature you will need an individual account. If you have one already please sign in.

Sign in.

Alternatively you can request an indvidual account here: