Measures of risk - Value at Risk | Vose Software

# Measures of risk - Value at Risk

Value at Risk (VaR) is defined as the amount which, over a predefined amount of time, losses won't exceed at a specified confidence level. As such, VaR represents a worst-case scenario at a particular confidence level. It does not include the time value of money, as there is no discount rate applied to each period's cashflows. VaR is very often used at banks and insurance companies to give some feel for the riskiness of an investment strategy. In that case, the time horizon should be the time required to be able to liquidate the investments in an orderly fashion.

VaR is very easy to calculate using Monte Carlo simulation on a cashflow model. You set up a cell to sum the cashflows over the period of interest, say in Cell A1. Then in another cell, to get the VaR at the 95% confidence level you run a simulation with A1 as an output, read off the 5th percentile and put a minus sign in front of it.

The main problem with VaR is that it is not subadditive (Embrechts, 2000) meaning that it is possible to design two portfolios, X and Y, in such a way that VaR (X + Y) > VaR(X) + VaR(Y). That is counter-intuitive because we would normally consider that by investing in independent portfolios we have reduced risk. For example:

Consider a defaultable corporate bond. A bond is a contract of debt: the corporation owes to the bond holder an amount called the face value of the bond and has to pay at a certain time - unless they go bust in which case they default and the bond holder gets nothing. Let's say that the probability of default is 1%, the face value is \$100 and the current price of the bond is \$98.5. If you buy this bond, the total cashflow at exercise can be modelled using a Bernoulli distribution:

Bernoulli(99%)*\$100 - \$98.5

i.e. a 1%  chance of -\$98.5 and a 99% chance of \$1.50: a mean of \$0.50.

You could buy 50 such bonds for the same company which means that you will either get 50*\$100 or nothing, i.e. a cashflow of:

(Bernoulli(99%)*\$100 - \$98.5) * 50

with a mean of \$25. Alternatively, perhaps you could buy 50 bonds with the same face value and default probability but each with different companies that have no connection between them so the default events are completely independent, in which case your cashflow is:

Binomial(50,99%)*\$100 - \$98.5*50

but has the same mean of \$25. Obviously the latter is less risky since one would expect most bonds to be honoured. This figure plots the two cumulative distributions of revenue together.

Cumulative distributions for the two bond portfolios. Probability scale is in logs in the right pane to show details at low probability.

The relationship between required confidence and VaR for these two portfolios is shown in the figure below:

Relationship between required confidence and VaR for two bond portfolios in the example

In this example the VaR is larger for the diversified portfolio (the 50 bonds) until the required confidence is greater than (1-default probability). For more traditional investments models where revenues can be consider approximately Normal or T-distributed (elliptical portfolios) this was not a problem even with correlations. However, financial instruments like derivatives and bonds are now traded very heavily and these are either discrete variables or a discrete-continuous mix, and non-subadditivity then applies. For example, if one uses optimizing software (as is often done) to find a portfolio that minimizes the ratio of VaR to mean return the analyst can inadvertently end up with a highly risky portfolio. The expected shortfall method of valuing risk discussed next avoids this problem.