QuoteOriginally posted by: Paul...including weaknessesPPaul, You have to get kurtosis to contribute to this.

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.

excellent summary... I wouldn't say that VaR is the expected loss at a given CI - it's an estimate of the expectation. This estimate may be accurate, conservative, or conditional. Sqrt-t scaling is not always used, but it can be used... if the distribution looks more like TLF then this would be the wrong answer. In practice, VaR's inputs are mostly historical values. This is a sticking point for many traders, who think this makes no sense when spot market parameters are predicting something different in the future.In my experience, regulators may not be easily convinced that a conservative model is a good model. Strictly speaking, a conservative VaR may fail backtests because of insufficient bandbreaks. Those same regulators may demand extra risk factors added to the VaR ingredients list which make an accurate VaR impossible to achieve. Any real-life VaR will include allowances for modelling assumptions and data errors. This becomes obvious when one recognizes that taking CIs magnifies errors.VaR can also be used in a marginal sense - i.e. one might want to know what the change in VaR will be when a given trade is done. This kind of idea is a good one, but it does expose the firm to practices which tend to game the VaR. For example, if the trader's compensation is risk-adjusted with the VaR number then he/she may ultimately end up putting on aggressive trades whose risk the VaR system does not capture (example: basis risks). Another version of this is when trades are rebooked for better VaR treatment. Different books may even have different VaR methodology. This is not really an inconsistency, especially if the risk management practices are not the same.

One might add that one criticism of VaR has been the absence of subadditivity. Artzner et al. (1998) defined a set of criteria that a risk measure ro(X) should satisfy. 1 Translation invariance:ro(X+alpha*r)=ro(X)-alpha with alpha any real number and r is a strictly positive total return2 Subadditivity: ro(X+Y)<=ro(X)+ro(Y)3 Positive homogeneity: ro(lambda*X)=lambda*ro(X)4 Monotonicity:X<Y ----> ro(Y) <= ro(X)5 Relevance: X<=0, X not identically 0 -------> ro(X)>0A risk measure satifying the first 4 axioms is called "coherent".VaR fails to satisfy axiom 2 - subadditivity.This happens, e.g. for certain option positions as demonstrated by Artzner et al. However, if you use historical VaR (or Monte Carlo) you do not need non-linear instruments for subadditivity to vanish.In historical VaR we apply historical changes to our risk factors to their current values for each day in our data window. Then we order the resulting daily changes in value an pick our desired quantile. The operations of ordering and extracting the quantile do not commute. We can look at a simple example. Lets assume that the size of our data window is such that our VaR corrensponds to the fourth largest loss, i.e. after ordering the daily losses we can read of our VaR as the fourth number. Assume that we have two portfolios A and B that happen to have their four largest losses on the same days but the not "in the same order".Day Loss in A Loss in B total loss A+B 1 20 5 25 2 15 10 25 3 10 15 25 4 5 20 25Othef days result in smaller losses for both A and B.We see that VaR(A)=5 and VaR(B)=5 but VaR(A+B)=25 > 5+5.Some people argue that because VaR is not coherent we should abandon VaR as a risk measurement tool in favour of measures like expected shortfall which does satisfy all the axioms of coherence. However, VaR is by now so widely used that I believe this to be unrealistic. Regards,Niclas

nsande does indeed highlight that VaR is not definitely sub-additive - my mistake, I had blindly used the Artzner criterion without looking for a pathological counter example. I guess this raises the question whether Artzner's criteria are all correct. Depending on what exactly your X and Y represent in the formulae. I would certainly question the monotonicity argument - have you got the inequalities the correct way round. Surely it should read: X<Y --> ro(X) <= ro(Y)The bigger your "exposure", the more risk. If X and Y represent the "size" of a position, then a small position in a highly non-linear product may have a larger risk measure than a larger position in a linear position. If X and Y correspond to market factor moves, or we are considering short positions, then the inequality X<Y should probably use absolute values.

kr makes some good points too. The "historical" inputs gives rise to the arguement that using VaR is like driving a car by looking in the rear view mirror only. Historical data is used for both VaR simulation and covariance matrix calculation presumably because it is easier (and regulators may be more convinced) than building, continually developing and back testing an alternative model. Remember, in theory yuo have to model every single market to which your portfolio is exposed. kr's mention of marginal VaR also leads me to mention component VaR - you can break down your VaR number into its contributions from different positions and market factors. Not that due to the offsetting nature of VaR (can't call it sub-additivity any more), then component VaR of a position is not necessarily the same as the VaR of the position. Cooking the VaR books does happen - the easiest way to do it is to move a position from a trading to a banking book, where it becomes subject to only the regulatory credit capital charge. The idea of booking trades to reduce "visible" risks such as VaR, but transfer them to hidden ones is also well known. This is where the experience and judgement comes in. At least one firm (whose name begins with JP) has a system whereby traders have to enter into a web based application all of the "other" risks of a trade. The idea is that if a position loses money and the trader knew about it and entered it into the system, then he had done his part. If the potential loss scenario wasn't on the system, then career or bonus enhancement potential was severely reduced. More comments welcome...

Regarding the axiom of monotonicity.X and Y are in G - the set of all risks which is the set of all real valued functions on your state space.X<Y means that X is dominated by Y:Change in value of risk (portfolio) X < Change in value of risk (portfolio) Yi.e. if you loose money you loose more on X and if you gain you will gain less on X. Then the requirement of monotonicity becomes fairly obvious.Regards,Niclas

Excellent work from all of you but in my opinion you have ommited a critical point. Its the assumption that within the horizon the VaR is estimated, the positions can be totally hedged or totally closed. In my opinion this is the main deception of VaR methodologies.Any comments are wellcome

I think about 3 components of Value at Risk; credit, market and operational risks.And each component has risk that can can be hedged and risk that can't. In my view, once a position has been established, hedging is mostly an operational risk.Rare event risk is almost impossible to hedge in many cases unless you move the risk to an insurance company or become the insurer yourself.

I'm sure there are much to say about VaR. To me, VaR is just a standard measurement, which is 1. used for limiting the open position by each trader, each group of trader, and so on.2. a number that goes out with quarterly earning report.Why VaR? As soon as one of your competitors publish VaR number in its earning report, you don't have any other choice. It is the term that Wall street analysts understand (they claim!). Internally, it's the same: a measurement that traders agree to obey.More surprisingly, people use 1 point whatever standard deviation regardless of distribution.And you should do the same. Otherwise, you won't be happy, when you see your company stock isfalling down. Why do you obey traffic lights? Get it?

VaR limits are possible, but they are liable to "model risk" - take on a couple of outrageous positions that net in the VaR model, but otherwise are extremely likely to go bad. They can be useful at "high" level - desk and above. Individual traders should have a more detailed limit structure that specialises on the products and markets they trade. Traders, Risk Managers and firms vary in their opinion of VaR. A bank has regulatory VaR commitments, so tends to be far more serious than a pure broker-dealer. Don't stick to one confidence interval and holding period - you might want to know 1-day VaR if you are a day trader, but if you are an asset manager looking at a long term gain then holding periods of months or longer are more appropriate. The regulatory 10 day holding period is a compromise that covers the full service one would expect of a bank. VaR does appear in annual reports for financial firms and others with significant trading exposure. It is often buried though, and I would question how much the retail investor (despite Professor Jorion's best efforts) would know and ask about the VaR numbers rather than the P&L...

HA -I disagree... Even after this no-jaywalking furor in NYC people still disobey the traffic lights, even stopping traffic because you can't run somebody over even if the light is green. VaR is not a stoplight if you know how to game it. When you are making steady profits, your mind is not focused on how to arb the VaR, but when the future doesn't look as great as it used to, it is tempting to increase your risk by stealth, increasing the value of your trader's option without having it hit the reporting.

kr,You are right. I was a bit too sarcastic about the gap between the ideal risk measurement andwhat monkeys can understand. As I said it's a rule, and as you said, people try to arb on it.My question is where do all this new development of alternative risk measures (like coherentmeasure of risk) fit in to practioners world? At the moment, you can't replace VaR even if youknow it's not as ideal as CMR, because people do publish this number as an indication of their goodrisk management. If your number is higher than others, regardless of how you calculated it, analysts will say negative about it. As a consequence, your stock will drop, for sure. This loss to your companyis often a lot bigger than those created by some mispriced derivative transactions. Not a trivialissue, and certainly a very important problem to your business. After all, we practioners are hiredby these firms to help them make money. Given this task, what is the optimal strategy?

As one often finds in this forum, Aaron has done this all (or at least some of it) before...Look here...