credit var calculation explaination (garp15-p2-11)

[NNath]

Hi David, This sample question from GARP, in-order to calculate the C-VaR uses the current value of the Bond



In your example you use exp. terminal value of the bond.




Here is the explanation from GARP. Do you think they are correct.

Rationale: The 95% credit VaR corresponds to the unexpected loss at the 95th percentile minus the expected loss,
or the expected future value at the 95% loss percentile minus the current value. Using the probabilities in the given
ratings transition matrix, the 95% percentile corresponds to a downgrade to BBB, at which the value of the bond
would be estimated at 101. Since cash flows for the bond are not provided, we cannot derive the precise expected
and unexpected losses, but the credit VaR (the difference) is easily derived by subtracting the estimated value
given a BBB rating from the current value. 95% credit VaR = 110 – 101 = 9.

 

[David Harper CFA FRM]

Hi @NNath

Thank you! I think it's a flawed but very instructive question. I sent my contacts at GARP a link to my response (this post here), as I think the issues are materially interesting.

First, there is an issue with the 0.05 (ie, 95%) quantile choice. This is a discrete distribution, so a good question should locate the quantile within a probability bin, not on the border. We have a case here where the first three probabilities sum to 95.0%. FRM authors support at least three choices for the 95% quantile:

• $101; i.e., the "conservative" choice, due to higher loss of $9 = $100 - 101, which is consistent with Jorion's approach
• $105; i.e., a valid choice consistent with assigned Dowd and elegant because it locates the VaR outside the discrete 5% so we can say "about 5% of time, we expect losses to EXCEED $5 = $110 -105"
• $103 is smart because it's technically accurate if the weights straddle the values; this is consistent with both Linda Allen and John Hull

This ambiguity can be avoided by, for example, simply changing the 7.00% Probability to 6.0% and the 4.00% to 5.0%. Then the first three probabilities sum to 94.0% and the 95% quantile falls unambiguously WITHIN the BBB bin. Okay, that said, let's assume $101 is the correct 0.050 quantile ...

There are two ways to produce a (C)VaR, absolute or relative. As I've written many times, the question really needs to specify which of these two VaRs because, while there are tendencies and predominant use-cases, either is valid. In the case of CVaR, the Basel committee has previously defined CVaR as UL + EL. However, Malz does not. Malz defines CVaR as UL(α) or "net of expected loss." Here is Malz (emphasis mine):

Unexpected loss (UL) is a quantile of the credit loss in excess of the expected loss. It is sometimes defined as the standard deviation, and sometimes as the 99th or 99.9th percentile of the loss in excess of the expected loss. The standard definition of credit Value-at-Risk is cast in terms of UL: It is the worst case loss on a portfolio with a specific confidence level over a specific holding period, minus the expected loss.

Further, my example diagram above (which you copied) is just a copy of Malz Figure 6.5. This diagram illustrates CVaR as UL, net of EL. More specifically, it shows that CVaR = expected terminal value (i.e., net of expected loss) - 0.0010 quantile (in the case of a 99.9% CVaR). As Malz writes, "This is quite different from the standard definition of VaR for market risk. The market risk VaR is defined in terms of P&L. It therefore compares a future value with a current value. The credit risk VaR is defined in terms of differences from EL. It therefore compares two future values."

So, if you are still with me, Malz (who is assigned) defines his CVaR as a relative VaR, which can be interpreted as the unexpected loss relative to the expected future value. This relative 95.0% CVaR is given by:

• Expected future value of $108.16 - 0.050 quantile of bond value = $7.16.
  • This is exactly consistent with Malz example 6.5.
    
  • I retrieved the $108.16 as the mean of the end-of-year distribution represented by the transition matrix.

Or, like the answer given in the practice paper, we can alternatively solve for the absolute VaR which is given by:

• current value of $110.00 - 0.050 quantile of bond value = $9.00.
  • This is an absolute (C)VaR as it represents the worst loss relative to the current value.
    
  • Absolute VaR is normally smaller than relative VaR, but it's reversed here due to the unusual situation of a negative drift (bond value of $110.00 is expected to drop to $108.16).

Two of the sentences in the Rationale confuse me:

• The Rationale says "5% credit VaR corresponds to the unexpected loss at the 95th percentile minus the expected loss." This is incorrect. UL-EL is not meaningful. The choice is CVaR = UL(α)+EL or CVaR = UL(α).
• The Rationale also says "Since cash flows for the bond are not provided, we cannot derive the precise expected and unexpected losses ..." This is confusing, at least to me, because at this level of assumption it is natural to assume the bond values (already) incorporate the expected losses. This is related to how can we infer default probabilities from bond yields, we do that because the bond market prices (hence their yields) already incorporate expected losses! Actually, I've been writing C(VaR) because this question seems to me to be an MVaR in disguise as a CVaR: the VaR is a function of value change due to credit deterioration, not portfolio defaults.

You can see why I think this question is interesting--it invokes several issues. Hopefully, you can see why I can agree with GARP's answer but I just think, to avoid ambiguity, the VaR method (relative or absolute) should be specified; also, the border-bin could be easily avoided.

I hope these thoughts are helpful!

 

[NNath]

Hi David, Thanks you for the detail explanation. Please share the detail excel sheet with your calculations as per the slide. thank you.

I found the excel sheet but I am not sure why actuarial put us using the ROA instead of Rf.

 

[David Harper CFA FRM]

Hi @NNath You make an excellent point. I don't have a better spreadsheet (than the one you found) to share yet due exactly to the issue you mention. Please see the recent discussion here at https://forum.bionicturtle.com/threads/p2-t6-305-credit-value-at-risk-cvar.6816/#post-37949. I am going to contact Malz to confirm he text has a typo and/or, more likely, that I misunderstand before I further modify as I do agree with you: the actuarial put should use a risk-free rate. Thanks,