To answer this question, we first have to analyze the characteristics of medium-term notes. We obtain monthly returns on medium-term bonds from 1953 to 1995. Plot History of Returns

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. Plot Distribution

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 $100 million portfolio. There is only a 5% chance that the portfolio will fall by more than $100 million times -1.7%, or $1.7 million. The value at risk is $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 $1.7 million.

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.

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.

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Contents (c) 1996 - Philippe Jorion