Expected loss and credit var

[afterworkguinness]

Sorry to be that guy who keeps posting questions, this really is by far the best resource out there.

The answer to practice question 305.3 for Malz says the following statement is true:

CVaR (alpha) = Unexpected loss (alpha), where alpha is a significance or confidence level

Q: if CVaR is Expected loss - Unexpected loss, how is this true ?

(can't view/post in the thread for this question set as my BT package expired)

 

[[email protected]]

Hi,

Actually CVaR is calculated as:

CVaR= Unexpected Loss= Loss at (alpha or significance level) - Expected Loss

Hope this helps.

 

[afterworkguinness]

Hi Gyilmaz,
I went back and double checked the notes and can see that, as you said, CVaR= Unexpected Loss= Loss at (alpha or significance level) - Expected Loss. I guess it was a typo in the practice question answers.
Given that CVaR = UL - EL, I'm assuming it's incorrect / imprecise to state that CVaR is simply unexpected loss ?

Further adding to my confusion, are the other practice questions in this set. Multiple of them make reference to Credit VaR being Expected Loss - Unexpected loss. I've included the questions and answers below, I think I'm missing something.

Practice Questions:

305.1. A firm has current asset value of $1.0 billion with asset volatility of 25.0%. Its sole
debt issue is a zero-coupon bond with face value of $800 million due in one year. The risk
free rate is only 3.0% but the firm's expected return (ROA) is 12.0%. (Note: these
assumptions given, so far, complete the set of necessary inputs into the Merton.
Technically, the problem given below can be solved at this point. For a difficult challenge,
far more difficult than you would encounter on the exam, you can attempt the problem with
only the assumptions given so far). Here are the additional implications of Merton model:
- Merton probability of default = 10.6%
- The actuarial expected loss (EL; i.e., the future value of the "actuarial default put") =
$8.40 million
- The expected loss given default (LGD) = $8.40 million EL / 10.6% PD = $79.25 million
- The 0.001 quantile of the future bond value = $503.00 million

Which is nearest to the implied Credit VaR (CVaR) at the 99.9% confidence level?
a) $288.60 million
b) $394.11 million
c) $437.04 million
d) $501.25 million

305.2. Which of the following best summarizes the 99.9% Credit VaR (CVaR)?

a) 99.9% CVaR = (Current bond value - PV of expected loss) - Current 0.001 Quantile of
Bond Value
b) 99.9% CVaR = (Bond's par value - Future expected loss) - Future 0.001 Quantile of
Bond Value
c) 99.9% CVaR = (Bond's par value - Future 0.001 Quantile of Bond Value)
d) 99.9% CVaR = (Bond's expected terminal value - Future expected loss) - Future
0.001 Quantile of Bond Value

305.3. According to Malz, each of the following is true about Credit VaR EXCEPT:

a) CVaR (alpha) = Unexpected loss (alpha), where alpha is a significance or confidence
level
b) For a fixed quantile of future bond value (e.g., $503 million) and increase in expected
loss implies a decrease in CVaR; i.e., CVaR excludes EL
c) Like market risk VaR (MVaR), CVaR compares a future value with a current value
d) An increase in the firm's expected return (mu) will, ceteris paribus, increase the
CVaR

...and the given solutions

305.1. A. $288.60 million

CVaR (99.9%) = Expected future value of debt - (the 0.001 quantile of the future bond
value). In this case, The expected future value of debt = $800 face value - $8.40 EL =
$791.60 million, such that: CVaR (99.9%) = 791.60 - $503.00 = $288.60 million
i.e., we don't require the LGD assumption as we already have the EL, which is the mean of
the loss distribution

305.2. B. 99.9% CVaR = (Bond's par value - Future expected loss) - Future 0.001 Quantile
of Bond Value

- In regard to (A), this is false: "[Credit VaR] 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."

- In regard to (C), this is false because it includes the expected loss, but Malz's CVaR is
an unexpected loss: "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."

- In regard to (D), this is wrong because it double-counts EL: (Bond's expected
terminal value - Future expected loss) - Future 0.001 Quantile of Bond Value.
However, since Bond's expected terminal value = (Bond's par value - Future expected
loss).

- It is true that 99.9% CVaR = (Bond's par value - Future expected loss) - Future 0.001
Quantile of Bond Value = (Bond's expected terminal value - Future 0.001 Quantile of
Bond Value)

305.3. C. Like market risk VaR (MVaR), CVaR compares a future value with current value
- In regard to (A) and (B), they are true and similar: Malz defines CVaR(%) = EL -
UL(%).
- In regard to (D), an increase in ROA both increases the expected FV of the debt and
decreases the quantile, such that CVaR increases!

 

[David Harper CFA FRM]

Hi @afterworkguinness I disagree that any of the answers say that Credit VaR is "Expected Loss - Unexpected loss". The source Q&A is located here (https://forum.bionicturtle.com/threads/p2-t6-305-credit-value-at-risk-cvar.6816). I would say something that I have often written, including to GARP: across the FRM, strictly speaking CVaR can be either UL+EL (e.g., some Basel references) or, more typically, CVaR = UL. So just to be careful, because Basel has referred to CVaR as UL+EL, a fair question must specifically state that is assumes CVaR is net of EL (or not).

That said, in the context of the assigned Malz (and these practice questions), I think @Gyilmaz is basically correct if you are careful to note that "Loss at (alpha or significance level)" refers to, in Malz terms, "(Bond's par value - Future 0.001 Quantile of Bond Value)" i.e., the expected loss plus the unexpected loss. Put again,

• "CVaR= Unexpected Loss= Loss at (alpha or significance level) - Expected Loss" is the same as my correct 305.2.B:
  
• CVaR = (Bond's par value - Future expected loss) - Future 0.001 Quantile of Bond Value;
  
  in Malz' diagram (below) they both have CVaR as the distance between the (99.9th) percentile and the expected terminal value (not the par value). The difference between par value and expected terminal value is expected loss. In Malz, you can see visually that CVaR = UL.

Here is Malz text associated with this figure (emphasis mine):

"Credit losses can be decomposed into three components: expected loss, unexpected loss, and the loss “in the tail,” that is, beyond the unexpected.

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. [davidh: notice he even seems to be aware of alternative definitions of CVaR given he calls this the "standard definition:]

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.

To make this concept clearer, let’s continue the Merton model example. The results are illustrated in Figure 6.5, the probability density function of the bond’s future value."

 

[afterworkguinness]

@David Harper CFA FRM CIPM , thanks for clearing that up. I had (incorrectly) thought loss at (alpha or significance level) to be the same as unexpected loss.

 

[afterworkguinness]

Sorry @David Harper CFA FRM CIPM , I've got another question on the same topic of calculating CVaR. I'm having trouble wrapping my head around the second half of this chapter, thanks in advance for the help.

In practice question 311.2 we are given the following inputs and asked to calculate a 99.9% CVaR:

$100MM portfolio, 2% 1 year probability of default, 50% recovery rate (and thus 50% loss given default), and a correlation parameter of .13

I'm lost when the CVaR is then calculated as [Exposure At Default * N(-1)(m) * LGD] - EL

Questions:

1. Malz says "The probability of the loss level is equal to the probability of the market factor", but how is that since we have a PD given of 2% and the probability of observing the market factor of -1.00730 is 15.6895%? Same goes in his example 8.6 where we have a probability of default of 1% and we find the probability of observing the market factor of -0.6233 is 26.65%

2. Can you explain why CVaR is being calculated like this ? It looks like we're calculating an expected loss and then subtracting a different expected loss from it.

3. Malz says the probability of default pi "sets the unconditional expected value". How does this work?

 

[Tipo]

Hi @afterworkguinness I disagree that any of the answers say that Credit VaR is "Expected Loss - Unexpected loss". The source Q&A is located here (https://forum.bionicturtle.com/threads/p2-t6-305-credit-value-at-risk-cvar.6816). I would say something that I have often written, including to GARP: across the FRM, strictly speaking CVaR can be either UL+EL (e.g., some Basel references) or, more typically, CVaR = UL. So just to be careful, because Basel has referred to CVaR as UL+EL, a fair question must specifically state that is assumes CVaR is net of EL (or not).

That said, in the context of the assigned Malz (and these practice questions), I think @Gyilmaz is basically correct if you are careful to note that "Loss at (alpha or significance level)" refers to, in Malz terms, "(Bond's par value - Future 0.001 Quantile of Bond Value)" i.e., the expected loss plus the unexpected loss. Put again,

• "CVaR= Unexpected Loss= Loss at (alpha or significance level) - Expected Loss" is the same as my correct 305.2.B:
  
• CVaR = (Bond's par value - Future expected loss) - Future 0.001 Quantile of Bond Value;
  
  in Malz' diagram (below) they both have CVaR as the distance between the (99.9th) percentile and the expected terminal value (not the par value). The difference between par value and expected terminal value is expected loss. In Malz, you can see visually that CVaR = UL.

Here is Malz text associated with this figure (emphasis mine):

Im looking at Gregory's chapter and am wondering if the 99th percentile is what Gregory calls the PFE and the expected terminal value the EE? or are they entirely different concepts whereby one does MTM (Gregory) and Malz doesnt?

From Gregory p39 of the BT notes (ch3)

 

[ShaktiRathore]

Yes PFE is the potential future exposure its the maximum exposure that can occur in the future,its the 99th percentile quantile on the exposure distribution above.Whereas the EE is the expected exposure is the mean of the positive future exposures. The expected terminal value that Malz found is the expected mean value of the Bond its not the mean of the exposures,Malz is finding the maximum loss that can occur on the bond through Credit Var from bond value distribution,whereas Gregory is fining the maximum exposure from exposure distribution.Please don't confuse.
Thanks