4.3 Value-at-Risk
The Value-at-Risk (VaR) is a measure of downside risk commonly used by banks, insurers, and investment companies. The commonly used VaR indicators are the 5% and 1%. The 5% VaR is simply the 5th percentile of a probability distribution, that is the value of the random variable such that at least 5% of all observations lie to the left of it or are lower than it. For a normally distributed random variable, the 5% and 1% VaR can be easily calculated as follows:[1]
Example 9: VaR for Normal Distribution
We can apply the formulas above to find the 5% VaR using the BioTech-X distribution depicted in Figure 3. The 5th percentile of the distribution is -1.58% (5% – 1.645 x 4%). VaR is expressed in dollars, so to find the VaR in this example, you need to specify the amount invested in BioTech-X stock. If you have invested $100,000, then your loss when the rate of return is -1.58% will be equal to the product of the amount invested and the 5th percentile of the return distribution or $1,580 (-1.58% x $100,000). The 5% VaR in this example shows that over the next year, your loss on your investment in BioTech-X stock will be $1,580 or more with a 5% probability.
One drawback of the VaR as a measure of downside risk is that it is the best of the worst possible outcomes. We may be lulled into false sense of security if we believe that the VaR is the maximum possible loss that we can face. Another pitfall is computing VaR under the assumption that a variable is normally distributed while in really, the distribution is not normal.
A more conservative measure of tail risk is the expected shortfall (ES) or Conditional Value at Risk (CVaR), which is the average of the worst possible cases measured at a given VaR. For example, the 5% ES is the average of the 5% worst cases. In this context where negative deviations from the mean are undesirable, the ES is lower than VaR or if we compare ES and VaR in absolute values, the ES is greater than VaR.
- To find a given percentile of a normal distribution, you can use the relationship between the normal, X, and standard normal, Z, variables: X = μ Zσ where μ and σ are the mean and standard deviation of the normally distributed variable X. ↵
A measure of downside risk commonly used by banks, insurers, and investment companies.
The average of the worst possible cases measured at a given Value-at-Risk (VaR).
