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# Problem with VaR calculation

Discussion in 'Strategy Development' started by trd, Aug 7, 2009.

1. ### trd

Problem with VaR calculation example:

Value-at-Risk (VaR) measures the worst expected loss
under normal market conditions over a specific time interval
at a given confidence level (ie. VaR answers the question:
"how much can I lose with x% probability over a pre-set horizonâ).

Suppose portfolio manager manages a portfolio which consists
of a single asset. The return of the asset is normally distributed
with annual mean return 10% and annual standard deviation
(ie. volatility) 30%. The value of the portfolio today is \$100 million.
Assume year has 250 trading days. We want to answer the following questions:

Q1) With 1% probability what is the maximum loss in percent at the end of the year?
Q2) With 1% probability what is the maximum loss in percent for 1 day, 5 days, 21 days?

The document I got this example from gives the following answers:

A1) 47.4237%
A2) 0.256831%, 1.27758%, 5.25717%

I'm not sure if these results are correct, because simple intuition
says that for example the 1 day risk should be more than 0.256831%, isn't it?

Can someone please check these numbers.

Thx

2. ### black diamond

It looks like you are missing something about their methodology. From the normal distribution your worst 1% loss should be ~2.33 standard deviations below the mean. I get a return of around -60%. Do they tell you anything about the return type or distribution to use? They might be using exponential returns or some other distribution that cuts off at 0 value/-100% return. Also it is common to use a mean of 0 for VAR calcs.

And just using a normal dist of returns and scaling to 1 day, you get a daily std dev of .0188. 2.33 stds below 0 would be a 4.4% loss.

What are you trying to do with this?

3. ### trd

Thank you, this seems to confirm my observations, ie. that the last 3 of their numbers seem to be wrong, isn't it?
I'm giving you the link to the document for details of their method:
http://finance.wharton.upenn.edu/~benninga/mma/MiER74.pdf
(referenced from the wiki page http://en.wikipedia.org/wiki/Value_at_risk )

BUT: now I'm a little bit confused about your value of 60% for annual VaR,
because IMHO their annual VaR seems to be correct I think.

I simply want to use VaR, and would like to take the "most correct method" , (ie. lognormal?)
Which method would you suggest?

4. ### black diamond

I actually like historical VAR. You simulate how your porfolio would have performed over the last X time periods and rank the results. If you have 100 returns, then the worst is you 1% VAR, the 5th is your 5% VAR, etc. It gets you out of making questionable distribution assumptions. But it is misleading if the historical period is not a represenative sample - so make sure you are not taking all your data from a low vol period or any "wierd" period.

But I don't think there is a best VAR, it depends on your situation and preferences. My only strong opion is that the extreme tails of whatever VAR you use aren't very informative.

5. ### Martinghoul

If you read what they're doing, you should be able to arrive at the same conclusions.

For A, they're taking into account the initial portfolio value and the mean return, which is why they get their number.

For B, it's actually lognormal var they're calculating, which might be why they get the numbers they're showing.

6. ### trd

Yes, lognormal and taking into account the mean return are my goals as well.
I'm pretty sure their said 3 numbers are buggy.
I get these numbers:

Annual VaR at 1%: 47.431%
21-day VaR at 1%: 17.938%
5-day VaR at 1%: 9.300%
1-day VaR at 1%: 4.298%

And here are some alternate numbers from a slightly different calc method:

Annual VaR at 1%: 45.011%
21-day VaR at 1%: 17.627%
5-day VaR at 1%: 9.219%
1-day VaR at 1%: 4.281%

(The first method uses Ito's lemma, the second does not.)

These numbers differ much from their numbers, especially the last 3 numbers.

8. ### panzerman

The failures of the normal distribution are well known, and this is what Taleb has become famous for dispelling. However a financial law of the universe is that with no risk there is no return. Someone please tell me a better way to control risk on a huge portfolio worth millions or billions of dollars, than by using a mathematical model, even with its flawed assumptions.

I know what Taleb would say, take big risk and at the same time take no risk i.e. buy treasuries and use the income to purchase options.

9. ### trd

I would say: diversification and active trading (ie. tight stop loss),
and closely watching the stats (stddev should be <= 5% or <= 7.5%)

BTW, here's a good read on risk mgmt, enjoy! :
"'Perfect Storms' â Beautiful & True Lies In Risk Management" by Satyajit Das
http://www.wilmott.com/blogs/satyajitdas/enclosures/perfectstorms(may2007)1.pdf

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