VaR and the choice of confidence intervall

tara · Post by **tara** » Wed Oct 24, 2007 9:55 am

What is the best way to compute portfolio 1Y VaR with 99.97% quantile if 5Y time frame is used for the simulations of defaults and 99.97% is a target quantile, chosen in accordance with 1Y survival probability due to the Moody's def. table?

Does it make sense to relate the quantile to the time frame, i.e. if 5Y horizon then VaR_5Y is computed with lower quantile ~ 99.32 (1-PD(5Y)) and then annualised, or this computation should be quantile-independent i .e we simple compute the VaR_1Y from VaR_5Y(99.97) by applying the square root rule?

sfca · Post by **sfca** » Wed Oct 24, 2007 5:34 pm

Model risk makes any VAR percentiles around 99 and time frames more than 10 days rather suspect. To do a 99.97% over a number of years is just a test of the particular model you used and says little to nothing about your actual market risk.

aaron · Post by **aaron** » Thu Oct 25, 2007 12:19 am

I agree with sfca, although I don't single out model risk as the culprit. 99% 1-day VaR is observable, 99.97% 1 year VaR would take about 15,000 years to observe with any confidence, assuming you were 100% sure parameters were constant over that period.

To answer your question in general, if you simulate only defaults then you need to estimate the effect of future unknown defaults on credit spreads at the one year point. The simple approach is to regress credit losses in years 2-5 on credit losses in year 1 in your simulation. If you find $1 of default losses in year 1 implies $1.50 extra default losses in years 2-5, it's reasonable to assume credit spreads will increase to realize that $1.50 loss.

No, do not use the square root rule on VaR.

nnja · Post by **nnja** » Thu Oct 25, 2007 2:47 pm

As pointed out above, you can't easily move from 5 yr VaR to 1 yr VaR in the general case.

If you are using Moody's default table instead of market spreads, wouldn't that imply constant spreads, and therefore miss out on market losses? Can you tell us more about the model you are using?

tara · Post by **tara** » Thu Oct 25, 2007 3:20 pm

Thanks for your replays,

about the model, I'm using MC simulations to determine the portfolio loss probability distribution. Of course, I'm interested in the higher quantiles and therefore I need a lot of simulations, in same cases over 1 million. For the simulation of defaults till time horizon which is in my case more than 1Y, I can use Moody's default table or any other default tables, doesn't matter, but I do not derive default probabilities from the market spreads. Again, the thing which bothers me is whether I need to adapt the quantile when I simulate more than 1Y and at the same time want to determine my 1Y VaR 99.97. Do I need to change the quantile according to the time horizon and if yes then how?

nnja · Post by **nnja** » Thu Oct 25, 2007 4:02 pm

It seems that you are just interested in VaR due to credit losses; why can't you just use CDOROM - IIRC, it has the loss distribution as an output.

To answer your question more directly, you never "need" to change the quantile for VaR - if the comfort level you want is very high, e.g. 99.97%, then that is an acceptable level at 1 yr or 5 yr. However, as scfa alludes to, that is a high quantile and its meaning as a probability is dubious.

So therefore, I would guess that 99.97% corresponds to the 1 year default probability on the Moody's table at some rating rather than some arbitrarily decided comfort level. If that is the case, then (obviously depending on your specific application), you are probably targeting that rating level rather than the number 99.97%, and therefore to be theoretically consistent, you should use the table at that rating level to determine the quantile for any tenor that you want to run at.

tara · Post by **tara** » Thu Oct 25, 2007 4:59 pm

Ok, this implies that my desired rating together with time horizon should be an indicator for a quantile. Meaning, with time horizon 5Y and rating A2 I can compute VaR with a quantile specific for that rating level. But how can I deduce from 5Y VaR my 1Y VaR if not by using square root rule...

sfca · Post by **sfca** » Thu Oct 25, 2007 7:02 pm

I really don't think you will get a defensible result from all this, but as long as you are going to try that approach you might as well use some interesting math so you could Google roots of markov transition matrices and look at [url=/Show%20Post.aspx?PostIDKey=96798]discussion[/url] noting Aaron's warning not to do that either.

aaron · Post by **aaron** » Fri Oct 26, 2007 4:31 pm

You can't determine five-year VaR from one-year VaR except in textbook examples.

Suppose for example, you held only one asset, a five year bond. As long as the bond's one-year default probability is greater than your VaR confidence point, the VaR over one year and five years is about the same. Even with a portfolio of bonds the one-year and five-year VaR are not so different, defaults are clustered in time and the bonds that survive a credit crunch are very likely to repay.

At the other extreme, suppose you held an option with a one-year forward start. It has very little one-year VaR, but have any amount of five-year VaR.

If you hold a diversified portfolio with a reasonably Normal return distribution and reasonably constant volatility, you might not be too far off with the square root rule. In my experience, however, if you graph VaR versus time you get something closer to a log curve than a square root curve.

sfca · Post by **sfca** » Fri Oct 26, 2007 5:45 pm

Well, yeah, but my assumption was that we gave up on reality already. For example, while he did not say so explicitly, I infer that he is using some kind of 5 year transition matrix. Such a matrix would be an average over one or more business cycles and not provide any useful information for a distribution of default probabilities and doesn't scratch the surface of issues such as sector clustering of defaults. However, thats just my guess and it may not be correct.