Value at Risk (VaR)

Look for any security on BSEINDIA.COM or NSEINDIA.COM and u'll find this term called VaR mentioned.

I always wondered what it is. I just had a vague idea that its a measure of risk. And then, as I m crazy about finding out things that interest me, I ventured to quench my thirst to know what is VaR. Read on and am sure u'll find it interesting too. Lots of technicals though, but for analysts/CAs/Portfolio Managers/Risk managers and those in the capital markets... its imperative to know what is VaR.

Introduction to VAR
Define VAR
VAR summarizes the predicted maximum loss (or worst loss) over a target horizon within a given confidence interval.

Consider a trading portfolio. Its market value in Rupees today is known, but its market value tomorrow is not known. The investment bank holding that portfolio might report that its portfolio has a 1-day VaR of Rs 1.7 million at the 95% confidence level. This implies that under normal trading conditions the bank can be 95% confident that a change in the value of its portfolio would not result in a decrease of more than Rs 1.7 million during 1 day. This is equivalent to saying that there is a 5% confidence level that the value of its portfolio will decrease by Rs 1.7 million or more during 1 day. A 95% confidence interval does not imply a 95% chance of the event happening, the actual probability of the event cannot be determined.
The key point to note is that the target confidence level (95% in the above example) is the given parameter here; the output from the calculation ($Rs 1.7 million in the above example) is the maximum loss (the value at risk) at that confidence level.

How can I compute VAR?
Assume you hold Rs.100 million in medium-term notes. How much could you lose in a month? As much as Rs.100,000? Or Rs.1 million? Or Rs.10 million? Without an answer to this question, investors have no way to decide whether the return they receive is appropriate compensation for risk.

To answer this question, we first have to analyze the characteristics of medium-term notes. We obtain monthly returns on medium-term bonds from 1993 to 2005.
Returns ranged from a low of -6.5% to a high of +12.0%. Now construct regularly spaced ``buckets'' going from the lowest to the highest number and count how many observations fall into each bucket. For instance, there is one observation below -5%. There is another observation between -5% and -4.5%. And so on. By so doing, you will construct a ``probability distribution'' for the monthly returns, which counts how many occurrences have been observed in the past for a particular range.

For each return, you can then compute a probability of observing a lower return. Pick a confidence level, say 95%. For this confidence level, you can find on the graph a point that is such that there is a 5% probability of finding a lower return. This number is -1.7%, as all occurrences of returns less than -1.7% add up to 5% of the total number of months, or 26 out of 516 months. Note that this could also be obtained from the sample standard deviation, assuming the returns are close to normally distributed.

Therefore, you are now ready to compute the VAR of a Rs.100 million portfolio. There is only a 5% chance that the portfolio will fall by more than Rs.100 million times -1.7%, or Rs.1.7 million. The value at risk is Rs.1.7 million. In other words, the market risk of this portfolio can be communicated effectively to a non-technical audience with a statement such as:
Under normal market conditions, the most the portfolio can lose over a month is Rs.1.7 million.

What is the effect of VAR parameters?
In the previous example, VAR was reported at the 95% level over a one-month horizon. The choice of these two quantitative parameters is subjective.
(1) Horizon
For a bank trading portfolio invested in highly liquid currencies, a one-day horizon may be acceptable. For an investment manager with a monthly rebalancing and reporting focus, a 30-day period may be more appropriate. Ideally, the holding period should correspond to the longest period needed for an orderly portfolio liquidation.
(2) Confidence Level
The choice of the confidence level also depends on its use. If the resulting VARs are directly used for the choice of a capital cushion, then the choice of the confidence level is crucial, as it should reflect the degree of risk aversion of the company and the cost of a loss of exceeding VAR. Higher risk aversion, or greater costs, implies that a greater amount of capital should cover possible losses, thus leading to a higher confidence level. In contrast, if VAR numbers are just used to provide a company-wide yardstick to compare risks across different markets, then the choice of the confidence level is not too important.

How can we convert VAR parameters?
If we are willing to assume a normal distribution for the portfolio returns, then it is easy to convert one horizon or confidence level to another.
As returns across different periods are close to uncorrelated, the variance of a T-day return should be T times the variance of a 1-day return. Hence, in terms of volatility (or standard deviation), Value-at-Risk can be adjusted as:

VAR(T days) = VAR(1 day) x SQRT(T)

Conversion across confidence levels is straightforward if one assumes a normal distribution. From standard normal tables, we know that the 95% one-tailed VAR corresponds to 1.645 times the standard deviation; the 99% VAR corresponds to 2.326 times sigma; and so on. Therefore, to convert from 99% VAR (used for instance by Bankers Trust) to 95% VAR (used for instance by JP Morgan),

VAR(95%) = VAR(99%) x 1.645 / 2.326.

How can I use VAR?
This single number summarizes the portfolio's exposure to market risk as well as the probability of an adverse move. It measures risk using the same units as the bottom line---Rupeess. Investors can then decide whether they feel comfortable with this level of risk.
If the answer is no, the process that led to the computation of VAR can be used to decide where to trim risk. For instance, the riskiest securities can be sold. Or derivatives such as futures and options can be added to hedge the undesirable risk. VAR also allows users to measure incremental risk, which measures the contribution of each security to total portfolio risk. Overall, it seems that VAR, or some equivalent measure, is an indispensable tool for navigating through financial markets.