Value at Risk (VaR)
Value at Risk is one of the basic measures of the risk of a loss in a particular portfolio. It is
typically stated in a form like, "there is a one percent chance that, over a particular day of
trading, the position could lose" some dollar amount. We can state it more generally as
VaR(X,N) = V, where V is the dollar amount of the possible loss, X is the percentage (managers
might commonly be interested in the 1%, 5%, or 10%), and N is the time period (one day, one
week, one month). To get from one-day VaR to N-day VaR, if the risks are independent and
identically distributed, we multiply by
.
Or, in one irreverent definition, VaR is "
a number invented by purveyors of panaceas for
pecuniary peril intended to mislead senior management and regulators into false confidence that
market risk is adequately understood and controlled
" (Schachter, from gloriamundi.org).
measure risk exposure; ensure correct capital allocation; provide information to counterparties,
regulators, auditors, and other stakeholders; evaluate and provide incentives to profit centers
within the firm; and protect against financial distress. Note that the final desired outcome
(protection against loss) is not the only desired outcome. Being too safe costs money and loses
business!
VaR is an important component of bank regulation: the Basel Accord sets capital based on the
10-day 1% VaR (so, if risks are iid, then the 10-day Var is
times larger than the
one-day). The capital is set at least 3 times as high as the 10-day 1% VaR, and can be as much as
four times higher if the bank's own VaR calculations have performed poorly in the past year. If a
bank hits their own 1% VaR more than 3 times in a year (of 250 trading days) then their capital
adequacy rating is increased for the next year. Poor modeling can carry a high cost!
If we graph the distribution of possible returns of a portfolio, we can interpret VaR as a measure
of a percentile. If we plot the cumulative distribution function (cdf) of the portfolio value, then
the VaR is simply the inverse of the probability, X: