May 17th, 2002, 1:55 pm
I'm not sure at what point you are proposing to apply asymptotic methods.One approach to estimating VaR is to simulate a one-day price movement for every asset in the portfolio and add them up to a firm P&L. Repeat that, say, 10,000 times, and take the 100th biggest loss. This is your 99%, one-day VaR.Another approach says with lots of assets, and enough independence among them, only mean and standard deviation matter. So all I care about is the mean return on each asset and its Beta with the total portfolio P&L. So I can estimate the mean and standard deviation of the one-day portfolio P&L and set VaR at 2.33*SD - Mean. I think this is what you have in mind as an asymptotic approximation, although no doubt you are interested in more sophisticated methods than alpha and Beta.Neither one of these pure approaches is adequate for practical VaR computation in large portfolios (say 1,000,000 positions in 100,000 different assets of 1,000 different types). There's no practical way to simulate 1,000,000 different assets, you don't know what all the statistical relationshps are. On the other hand, there's no way to determine Beta (or other price relationships) for 1,000 different asset types on one central factor.Therefore, the portfolio is broken up into sectors such that within each sector all assets can be valued based on a small number of factors, possibly with some idiosyncratic risk (this is significant, even a very large multinational bank will have significant VaR due to exposure to a few companies or market factors). Asymptotic approximations are used if appropriate (such as for large-cap US equities), but do not work in many markets with lumpier risk.Once you know the distribution of P&L within market sector, you can simulate each sector using the first method and estimate an overall VaR. Generally you will have a sparse covariance matrix for this step. Some sectors (like FX and Interest Rates, or Equities and Equity Implied Volatility) will have stable correlations, but for most pairs of sectors you assume zero (not because it is, but because it isn't stable enough to measure). Asymptotic methods are not appropriate because the number of sectors, 10 to 100, is not large.
Last edited by
Aaron on May 16th, 2002, 10:00 pm, edited 1 time in total.