CVAR calculation doubt

[Kavita.bhangdia]

Hi David,

Please can you help me with this problem.
I am struggling with how do I take the confidence interval into consideration while calculating the CVAR.

A portfolio has n credit, total portfolio value is 1,0000,000.
The probability of default is 2% for each credit. Assume zero recovery and default correlation is 1.


Find the CVAR at 95% confidence interval.


Solution
My expected loss =.02*1000000= 20000

My credit loss at 95% is zero. So my Cvar is -20000
Is it correct??
What will be my Cvar at 99% confidence.

 

[QuantMan2318]

Hi @Kavita.bhangdia

This CVAR is one hell of a topic to understand, I think you mean Portfolio CVAR; In that case, of the portfolio has a correlation of one among its constituent credits, then either the entire portfolio defaults or doesn't default at all ( the words of Malz )

So, there are two case, as above the PD being 2%, the Probability of Non Default being 98%, but the Confidence level for our VaR is 95%, since the Bond doesn't default 98% of the time for something we need to find 95% of the time, I think the Total Loss (beyond our quantile of 0.05% including the CVAR and the EL) is zero, therefore our CVAR becomes negative or a gain, viz:

0 - 0.02*10,000,000 = -200,000

However, if the CVAR is at a 99% Confidence Level, then the Probability of Non default at 98% changes the game, Our estimate for the CVAR focusses on the 1% Tail with more precision than before, but the Bond doesn't default only 98% of the time, so here, the Total Loss is the full exposure (CVAR+Expected loss) of 10,000,000, therefore our CVAR becomes 10,000,000 - 200,000 = 9,800,000

This is only for Correlation of one, if correlation is zero, then thats another story involving a Binomial distribution and number of losses

 

[QuantMan2318]

As a key, I assume Total Loss = EL+ CVAR where, for me the (Bonds Par Value - 0.001 Quantile) is the Total Loss (CVAR+EL) and therefore, subtract the EL to get CVAR;

To be frank, previously, I got around this mess by assuming for Malz alone UL as the TL and hence UL = CVAR+EL and hence UL = Par-0.001 Quantile, it didn't make sense, however I was able to differentiate this and the Basel Readings more easily

 

[David Harper CFA FRM]

Hi @Kavita.bhangdia I agree with @QuantMan2318 ! (I just have fewer zeros, is all ....). This is Malz. I just tweaked our spreadsheet for your, I needed an XLS to grok it: http://trtl.bz/0428-cvar-binomial (snapshot below)

The non-intuitive result is due (mathematically) to the fact we are retrieving a loss quantile from the binomial (i.e., discrete) distribution. As QuantMan implies, in the case of a single bond, if the default probability is less than the CVaR significance (1 - confidence), the worst expected loss (the quantile) is zero, so we get an awkward outcome where the "tail" is not where we expect it relative to the mean. In short, the 95% quantile (ie., inverse binomial CDF) of a 2% pd bond is zero!? Note correlation = 1.0 is unrealistic and is tantamount to n = 1 position. I hope that helps!

 

[Kavita.bhangdia]

Hi David,

In the above spreadsheet and your video on malz chapter 8, I have two doubts.

1. how have you calculated "number of defaults"
I mean given
PD= 2% at 95% Confidance, for granularity =10(2nd column), why is the number of default=1?

2. for column 1 , at 95% the credit loss is 0, the expected loss is -20,000, so the CVAR should be -20,000, something that myself and Quantman2318 were discussing above..

Where am I going wrong? Please advice..

Thanks
Kavita

 

[David Harper CFA FRM]

Hi @Kavita.bhangdia

I quickly added three charts to the XLS above (same location: http://trtl.bz/0428-cvar-binomial) see below:

1. The number of defaults is returned by =BINOM.INV(trials = n, probability = pd, alpha = confidence); eg, in the first column 5.0 =BINOM.INV(trials =100, prob = 0.02, alpha = 0.95). This is an inverse cumulative distribution function. Here is a (hopefully familiar) analogy, given a standard normal distribution, what is the 95% quantile? As we know, it is 1.645, which is NORM.S.INV(95%) = 1.645. You can see, in the three distributions below--which differ only in their N, everything else is identical--given the distribution, we looking for cumulatively where the 0.95 quantile falls (I charted pdfs rather than CDFs because i think they are slightly more intuitive visually). In the case of n = 100, the 0.95 quantile falls at X = 5 or where there are 5.0 defaults (barely, because cumulatively 4 defaults goes out to 94.92% while 5 defaults goes out to 98.45%; but we are not doing fancy interpolation, and the 95.0% falls barely into the 5 defaults). In English, what's this telling us? It's saying: if we have 100 bonds with i.i.d. pd = 2.0%, and if we are sufficiently unlucky such that the 95.0% worst thing happens (not the absolutely worst which is all 100 defaulting but that's virtually impossible will prob of 0.02^100), then what will happen is 5.0 defaults.
2. Yes, you and @QuantMan2318 are correct. But this spreadsheet is always assuming zero correlation. So, if there are 100 bonds (n = 100) and they are independent (i.i.d.) then the first column applies (the portfolio value is the same $1.0 million). But if they are perfectly correlated, then we really have one position, which is the third column. In the case of one bond, the 0.95 quantile is zero defaults (because the pd is only 2.0%), so the CVaR = 0.95 quantile - EL = 0 - 20,000 = -20,000. I hope that clarifies!

 

[Kavita.bhangdia]

Yes..that makes sense now..do you think the exam will ask us to find BINOMIAL INV(pd,alpha,trials)

Thanks
Kavita

 

[QuantMan2318]

No, the exam doesn't generally ask us to find the BINOMIAL INV. However, we can get around that using our TI BA II plus, use the Binomial PDF formula for 0,1,2,3 and 4 defaults and add them up, you get 94.92% (nCr p^r*q^(n-r)) which means, the number of defaults is more than 4, 5 in our case.
Note: This is just for your understanding, don't try this in the exam, you won't get any time

And as I mentioned earlier, I only explained the correlation = 1 scenario, if its so, either one Bond defaults or all default together corresponding to one huge hulk of default, hence the n=1 in David's Excel;

If the correlation is zero, then, there may be any number of Bonds that do default, though our overall probability of default is 2%, there are Hundred Bonds and for 95% confidence level, we have to find out the number of Bonds that may default at that Confidence Level for our overall PD, think of this as number of Trials where there may be a PD equal to 2%.
These are the columns corresponding to n=10 and n=100

Anyway, the above is only superfluous, there is nothing more than can be added to David's excellent explanation. To make things a little more interesting, I have provided the CDF version of the same chart as David's PDF