Value at Risk and Expected Shortfall

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Expected Shortfall is a risk measure defined to evaluate the financial risk both Market and Credit Risk under economic stress. Expected Shortfall is also known as conditional Value at risk as expected shortfall helps to quantify the loss; when the threshold of probability of loss for Value at Risk is crossed over or the amount of loss is bigger than the amount predicted by Value at Risk.

Value at Risk: Value at Risk predicts the maximum loss which will not be exceeded with a certain level of confidence. For example, if a bank’s 10-day 99% VAR is $5 million, then there is only 1% chance that losses will exceed $5 million in 10 days.  As per the above explanation; VAR the best choice to predict the possible loss but with one constraint and that is what will happen if probability of 99% is crossed and we move into the extreme tail.

Expected Shortfall: Expected Shortfall helps to quantify the losses in the extreme tail; in the above image Expected losses are lying in the shaded region beyond the threshold of Value at Risk. Expected loss or Conditional Value at risk is calculated as the average of the extreme losses lying in the tail of the probability distribution of returns beyond the value at risk threshold.

Example: Let us suppose; we have invested in a portfolio where 10-day 99% VAR is $5 million; which suits our risk appetite or loss absorption capabilities. But, what if portfolio is constructed in such a way that certain financial assets in the bundle have the risk of huge potential losses under economic downturn; let us say there is certain probability that losses might go up to the $ 200 million. Even though the probability of $ 200 million is very less say .01% but still under extreme cases we might have to face such huge losses.

Expected Shortfall gives that potential losses number; while probability of potential losses lying in the extreme tail of probability distribution function.

Calculation of Expected Loss:

p(x)dx = the probability density of getting a return with value “x”

c = the cut-off point on the distribution where the analyst sets the VaR breakpoint

VAR = the agreed-upon VAR level

 

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