VAR (Value At Risk)

VAR is defined as the maximum possible amount of loss for a given financial portfolio held by a firm (financial organization, traders, investors, speculators) with a given confidence interval. To be more specific, let's say that a risk management report states that 95% - 1 day VAR is 5.34 million US dollars. What does it means ? It means that there are approximately 5 days out of 100 trading days resulting in a loss of more than 5.34 million US dollars. (This is just a definition! You will see below that this definition is really meaningless without additional statement). So, one can get a rough idea of how much a portfolio can lose. So, that is great ! It is so simple! This is one of the reason why VAR is established as the most popular tool in describing financial risk. The strength comes from its ability to express the risk value in one simple number. Its simplicity has many drawbacks as well.

Most importantly, is it always true ? In other words, should management regard this number as an absolute guideline? The answer is NO! (For example, argument given by Mr. Nassim Taleb against Dr. Joridon- the author of "Value-AT-Risk") When you look at actual history of any given portfolio, you can quickly discover that some times you lose only 1-2 days more than 95%-1day VAR out of 100 days (VAR is "OVER-ESTIMATING" the actual market risk). On the other hand, you lose 10-15 days more than 95%-1day VAR out of 100 days (VAR is "UNDER-ESTIMATING" the actual market risk). What is wrong?

There is nothing wrong with VAR's definition itself. It is the underlying assumption that is the problem. Most VAR system uses Variance -Covariance Method (See below for the explanation) which ignores the non-linear nature of option books. Therefore, it becomes meaningless. Also, VAR (unless one uses historical simulation or Gaussian simulation with jumps) uses an assumption that the market prices/rates follow s Gaussian process. This creates a problem even for Monte Carlo which uses correlated Gaussian random deviates. In reality, the market does not quite follow the simple Gaussian distribution. Historical simulation gives the "exact price/rate" movement of the past history (whether 260 days or 760 days). However, this approach is quite useless as well in case you have extremely volatile market which only occurs once every 50 years. So what one can do?

( SOLUTION)

First, one should restate the VAR as "If the market follow the Gaussian process at least for 1 day given today's prices/rates, then using the correlation matrix obtained from the past, one will most likely lose only 5 days more than 95%-1 day VAR out of 100 trading days". Then the above confusion will be solved. Then next question, manager can ask is, "What happens if the market does not follow Gaussian process?", "What happens if the market crashed? Any jumps?", "Any unexpected Political, Socio Economical News?", "How much loss do we most likely to lose under these circumstances?" Well, it is where stress test comes in. But stress test, by nature is static (not dynamic). Therefore, one can not really implement systematically. So, what is the solution? Moreover, even if there is no jump, what method is the best method?

The answer is still under intensive research. Meanwhile, please enjoy the following technological advances in financial risk management history.

Summary of Major VAR Calculation Methods

Simple Volatility Multiplication Method(maybe 1970??-Now)

Last resort kind of method used by many financial risk managers for getting rough idea of the maximum potential loss. It is obtained by simply multiplying the cashflow by risk factors such as daily volatility.
Its weakness comes from its ignoring the cross correlation between each asset classes, therefore over estimating the actual risk exposure.

Variance-Covariance Method (1996-Now)

It is an improvement of the former method by incorporating covariance matrix (correlation effects between each asset classes) primarily developed by J.P.Morgan Risk Management Advisory Group in 1996. It is often called Risk Matrics Methodology.
Major method used by many financial organizations due to its simplicity (Uses only delta).
Many 3rd party financial software vendors supports this method using industry standard RiskMatrics (TM) Daily Data sets.
Reasonably good method for portfolio with no option type products
Thus far, the computationally fastest method known today.
Not suited for portfolio with major option type financial products.

Higher Order Variance Covariance Method (Late1996-Now)

Improvement of the previous Variance-Covariance Method by incorporating the Gamma & Theta effects.
Answers some questions of portfolio with some optionality, but not quite well suited if any of the option is extremely near expiration (The Gamma/Theta approximation not good enough to capture the extremely non-linearity near the expiration)
Considered as the add-on for the typical Variance-covariance Method
Not widely used yet due to lack of industry consensus

Simulation Method (Historical Simulation) (1996-Now)

Uses the historical market prices/rates to simulate the differences in MTM (Mark-to-Market)
Works for virtually any portfolio.
Not widely used due to the lack of industry support.

Simulation Method (Monte Carlo Method) (1996-Now)

Actually generating the entire MTM (Mark-to-Market) any thousands of times from correlated random numbers.
Considered as the industry standard for simulation method.
Works for virtually any portfolio.
Problem is its slow computational speed, and also subject to bias if not enough sample are run.

Simulation Method (Speeded Up Monte Carlo Method) (1997-Now)

Improvement of the above method by using mathematically much faster techniques such as variance reduction methods. Therefore, this method is computationally much faster, and yet still gives the identical result theoretically as the above method.
Subject of much intense research. Many researchers (including myself) are investigating this methodology.
Gaining market support due to its flexibility & ability to combine customized distribution with fat tails (rather than traditional correlated Gaussian distribution) applied to virtually all portfolio types (whether linear or non-linear instruments) .

CVAR (Credit Value At Risk)

For the past 10 years, the issue of credit risk has been getting many attentions. Many firms such as J.P.Morgan (CreditMatrix, Credit manager), CS First Boston (CreditRisk+), and KMV (EDF- Expected Default Frequency Method) have come up with their own ways of quantifying the credit risk. However, there seems to be no consensus on how one can incorporating credit risk with market risk. Credit risk is an very important risk one must quantify given the recent Asian crisis (Japan, China), and emerging markets crisis which plagued many financial institution.

TVAR (Total Value At Risk)

This name is more like my own creation than anything else. To my best knowledge, there is no word "TVAR" in financial literatures. I will explain this terminology in the coming update