Not Kurtosis, but someone else who sat down to write this thinking it would be relatively easy, and ended up with an essay. I haven't referenced it, since I can't be bothered. Happy to take comments and amend as appropriate...Value at Risk (VaR) is one of the many measures of risk. Put simply, the VaR is the expected loss over a certain number of days for a given confidence level. Say a trading desk with a one day 99% VaR of USD 10million. This desk would expect its trading profit from t to t+1 to be worse than USD 10million on only one day out of 100. The VaR value gets bigger for a higher confidence level (say 1 day in 1000 – 99.9% confidence), and for a longer time horizon. VaR has a number of features, that make it a useful tool:- VaR is a single number that can be easily understood by senior management (see below). - VaR can be calculated for any product. - VaR can be calculated at any level, from position up to firm. - VaR can be aggregated. - When aggregated, VaR is sub-additive – the aggregate VaR for a trading desk should be less than the sum of the VaRs of each strategy on the desk. Hedges, or offsetting exposures should be effective in a VaR model. VaR has a number of weaknesses, among which are:- It does not tell you what the loss will be on the day that you lose more than the VaR value. - It does not tell you when the loss will happen. - Since VaR is a statistical measure, once you have had your one day’s big loss, there is still the same probability of losing more than your VaR figure on subsequent days. - VaR deals with “expected” market conditions, not when things go badly wrong and markets move in a more extreme fashion. - VaR does not identify weaknesses in a portfolio. - The accuracy of a VaR model calculation depends on the data you feed in to it, particularly how quickly a volatile market can be incorporated. VaR is used by bank regulators as a measure of market risk capital adequacy for trading (as opposed to banking), interest rate and foreign exchange books. Since the first Basel accord (mid 1990’s), a bank could use the standard model, which required a capital reserve of 8% of the notional of each position (long or short), or an internal VaR model (subject to approval). If VaR is used, the bank’s market risk capital requirement is based on the 99% confidence, 10 day VaR, with a multiplier of at least 3. The multiplier is set for each bank by the regulator, and is intended to act as a qualitative measure of the bank’s market risk policy, systems and controls. I once heard a story that during the Basel accord discussions, American regulators were keen on a multiplier of 1, Europeans wanted 5, and 3 was a British compromise. VaR is also published in the annual reports of financial organisations. To calculate VaR, you need to be able to build a statistical model of the P&L that would arise from your positions at the end of the trading day. There are three main approaches to calculating VaR. In order of increasing complexity they are:- Historical simulation- Covariance matrix- Monte Carlo simulationFor the simulation methods P&L calculation can be done using a full revaluation, partial revaluation (interpolating a set of market factor P&L points), or approximations based on market factor sensitivities or greeks - delta, gamma (optionally) and vega. Positions are revalued for a range of market factor moves, producing a distribution of P&L, from which percentiles can be read. The difference between historical and Monte Carlo simulation is the input data – actual market data moves for the former case, and theoretical ones, based on statistical models for the latter. Covariance matrix models use greeks and a matrix of correlations or covariances to calculates a portfolio standard deviation which can be used to calculate percentiles. Usually, VaR is calculated based on one day market factor moves. To move to longer time horizons, a square root of time approach is used. For example, 10-day VaR = 1-day VaR * sqrt(10). VaR became most popular during the 1980’s and particularly in the mid 1990’s, primarily through JP Morgan’s publication of their RiskMetrics methodology and tools. VaR like approaches to credit risk and operational risk are also used, and the Basel 2 Capital Accord is moving toward an internal model for credit risk capital. VaR models should be back tested against a firm’s P&L. Ideally the P&L should be “clean” from accrued interest, day trading, fees and so on – we only want to test against the pure “market” side of the P&L. The VaR model confidence level sets an expected level of VaR exceptions in a given number of days. Regulators look at exceptions and may use them to adjust multipliers etc. If you have fewer exceptions than the model predicts (no exceptions in 100 days), then regulators are happy, though you may find you are retaining too much regulatory capital as your model is over cautious. A few exceptions may mean either you are unlucky, or the model is a little lenient – you get put on amber watch. More than a few exceptions are you are in the red zone, where penalties may be applied. VaR has played a role, but in general not been responsible for some of the major financial catastrophes of the past twenty years or so. Some prominent voices have written that a “herd” mentality of VaR exceptions followed by position closures, causing further market disturbance causing more VaR exceptions. Many of the bigger failures have been due to operational risks (rogue traders). LTCM is one failure where VaR models played some kind of role. A key note about LTCM is that a number of their positions were exposed to a set of extreme events, particularly a sovereign defaulting on debt denominated in its own currency. Such events lay outside the VaR model’s scope and would have been very far into the tail of the model’s P&L distribution. VaR is not the only risk management tool a firm should use. VaR has to be used for regulatory and accounting disclosure purposes. VaR should be used as a measure of aggregating risk across disparate product lines, businesses and locations. VaR should be used by people who know what it does and doesn’t provide. There is no substitute for insight and experience to identify the weaknesses in a trading strategy. Stress Testing should support VaR in order to identify weaknesses, re-run extreme market scenarios, run hypothetical scenarios.