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# FRM 2008 Practice PI question 30 - One large vs. many small exposures

#### fullofquestions

##### New Member
QUESTION
You are hired as the credit risk manager for a large bank. You find that the bank’s credits are poorly
diversified. The bank has an extremely large exposure to one firm with a BB rating. All its other loans
have the equivalent of an AAA rating. You recommend that the bank diversify its credit exposures. After
the bank follows your advice, you are summoned to the CEO’s office and fired. The CEO says that they
followed your advice, acquired many small exposures to firms with BB ratings to replace the large
exposure, and all it did was to make the bank riskier because its credit VaR increased. The bank measures
its credit VaR as the maximum loss of principal over one year at the 1% level. You seek advice from
a consultant to make sure not to repeat the mistake you made. Which of the following explanations
provided by the consultant is correct?
a. VaR necessarily falls as diversification increases. Consequently, the bank’s software to compute VaR
must be flawed.
b. The bank did not diversify since it replaced one exposure with a BB rating with multiple exposures
with a BB rating.
c. The VaR would not have increased had the bank measured it as a shortfall relative to the expected
value of the banking book.
d. The VaR would not have increased had the bank not used the normal distribution for the portfolio
return.
Explanation: By diversifying, the bank swaps the small probability of a large loss for the certainty of a
small loss. Yet, the expected value of the banking book is unchanged and the volatility of the terminal
value of the banking book has fallen.

In broad terms, changing a large exposure with exactly rated smaller exposures that, we can only assume, have the same expected value, should not change the RWA calculation since PD, EAD, LGD and M add up to the same value. Is the explanation also confirming this by saying 'the expected value of the banking book is unchanged'?
In any event, how is the following true? "the volatility of the terminal value (I guess another way of saying at time t = T) of the banking book has fallen"
Anyway, the answer clearly states that Credit VaR increased. What I don't understand is what answer c tries to accomplish. To compute Credit VaR we are indeed computing the shortfall relative to the value of the Banking Book (Trading book does not factor into credit risk). Please advise.

#### ajsa

##### New Member
Hi David,

Could you take a look this pls? I am also confused. Does it mean diversification makes both EL and absolute VAR increase, while it makes relative VAR decrease?

But according to Ong, the portfolio's EL is the sum of each postions' ELs. so I feel EL should not change. (Here does Ong assume PDs are perfectly correlated? It seems there are 2 kinds of correlations now. One is correlation between PDs and the other is between assets..)

Thanks.

#### fullofquestions

##### New Member
Again I searched the forum and could not find this topic as being already covered. Anyone have any suggestions?

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi FoQ & asja,

This question honestly irritates/saddens me, I'm not sure which; I am going to send this to GARP Monday.
@asja: you have done the hard work, you ask good questions that reflect a strong understanding. I encourage you to trust yourself. just b/c somebody publishes a question shouldn't make you re-doubt your learnings... maybe you are better than the question writer? (I think you are)

I know what it intends: this is about VaR's lack of subadditivity.
(where my example is PD = 2% and 95% VaR)

we can imagine a similar case to the one i have in the XLS: say the single BB exposure has PD = 0.5%
now what is the 99% VaR of this one bond?
0, right? ...because it's a Bernouilli, the 99% VaR is 0 and the 99.6% VaR is loss of (1-recovery)

now replace the single BB with, say, 10 smaller bonds, each with PD = 0.5%
now the 99% VaR is > 0. Because the probability of zero losses = 1 - 99.5%^10 = ~ 95%
...i assume independence (an issue avoided by the question!) but see how the "diversification" of zero correlation now creates a higher probability that at least something will be lost?

see the illustration of VaR's lack of sub-additivity:
the one bond at 0.5% PD has a zero Var @ 99% (due to the Bernouilli)
but replace it with 10 0.5% PDs and the 99% VaR becomes non-zero (we'd need, it appears, a 94% VaR to get a zero VaR)

so the question intends to ask about VaR lack of sub-additivity (i.e., VaR counter-intuitively "punishing" the diversification)
then (c) is a horrible wreck of phrasing that intends to suggest "Expected Shortfall (ES) would not have increased the risk" ... or, if you had used ES, the ES would not have increased due to the diversification

(you can tell the question is about sub-additvity b/c both (a) and (d) are variations: (a) is the property of sub-additivity and (d) VaR would be sub-additive if the distribution is normal, but we don't assume that for bonds)

the rest of the question is a mess, IMO.
of course, you are both right about the EL: it is linear.
The phrase "bank measured it as a shortfall relative to the expected value of the banking book" is just wrong combination of terms.
(also notice the implication that the wrong metric makes the bank riskier??)

David

#### ajsa

##### New Member
Hi David,

Thank you very much for your encouragements!!! They are HUGE for me!

"i assume independence (an issue avoided by the question!) " That is actually my another confusion.. Since EL% = PD * LGD and the portfolio’s EL is the sum of each postions’ ELs, does it assume ELs are perfectly correlated? If they are independent, should portfolio’s EL be smaller than the sum of each postions’ ELs (subadditivity)?

Thank you again!

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi asja,

EL doesn't depend on correlation (that's what i really meant by linear). As an average, just like expected return, both are "first moments" and simply add:
if the one BB exposure = $1,000 with PD = 0.5%, the EL =$1,000 * 0.5% = $5; replace with 10 BB exposures, each with PD = 0.5%, the portfolio EL is still$10: 10 * \$100 * 0.5% = 10 * 0.5 REGARDLESS of 1.0 correlation or 0 correlation
...if we replace with 10 BBs, although the EL (1st moment) is unchanged, the distribution is very different depending on the correlation!

the correlation will impact the "shape" of the distribution (i.e., 2nd moment variance, 4th moment kurtosis); the UL is just the 2nd moment, it changes (is nonlinear) but EL does not change
...this is related to why the ES is sub-additive: it is an (conditional) AVERAGE as opposed to VaR which is a quantile

David