A Note on Value at Risk
The safety first approach can be used to calculate the value-at-risk of the portfolio. Value-at-risk is an increasingly popular measure of the potential for loss over a given time horizon. It is
applied in the banking industry to calculate capital requirements, and it is applied in the
investment industry as a risk control for portfolios of securities.
Consider the problem of estimating how big a loss your portfolio could experience over the next
month. If the distribution of portfolio returns is normal, then a three standard deviation drop is
possible, but not very likely. Typically, the estimate of the maximum expected loss is defined for
a given time horizon and a given confidence interval. Consider the type of loss that occurs one in
twenty months. If you know the mean and standard deviation of the portfolio, and you specify
the confidence interval as a 5% event (1 in twenty months) or a 1% event (1 in a hundred
months) is straightforward to calculate the "Value at Risk."
Let Rp be the portfolio return and STDp be the portfolio standard deviation. Let T be the t-statistic associated with the confidence interval. T of 1.64 corresponds to a one in 20 month event. Let Rvar be the unknown negative return portfolio return that we expect to occur one in twenty times.
The equation for the line is: Rp = Rvar + T*STDp and thus, Rvar = Rp - T*STDp. Rvar
multiplied times the value of the assets in the portfolio is the Value at Risk.
A VAR Example
Suppose you are considering the VAR of a $100 million pension portfolio over the monthly horizon. It is composed of 60% stocks and 40% bonds, and you are interested in the 95% confidence interval.
Let us assume that the monthly expected stock return is 1% and the expected bond return is .7%,
and their standard deviations are 5% and 3% respectively. Assume that the correlation between
the two asset classes is .5. First we calculate the mean and standard deviation of the portfolio:
Rp = (.6)*(.01) + (.4)(.007) = .0088
STDp = sqrt[ .6^2*.05^2 + .4^2*.03^2 + 2*.5*.6*.4*.05*.03] = .038
Then Rvar = .0088 - 1.64*.038 = -.054
Thus, the monthly value-at-risk of the portfolio is ($100 million)(.054) = $5.4 million.
Note that, despite the terminology, this does not really mean that $94.6 is not at risk. The
analysis only means that you expect a loss at least as large as $5.4 million one month out of 20.
This approach to calculating value-at-risk depends on key assumptions. First, returns must be
close to normally distributed. This condition is often violated when derivatives are in the
portfolio. Second, historically estimated return distributions and correlations must be
representative of future return distributions and correlations. Estimation error can be a big
problem when you have statistics on a large number of separate asset classes to consider. Third,
returns are not assumed to be auto-correlated. When there are positive trends in the data, losses
should be expected to mount up from month to month.
In summary, value at risk is becoming pervasive in the financial industry as a summary measure
of risk. While it has certain drawbacks, its major advantage is that it is a probability-based
approach that can be viewed as a simple extension of safety-first portfolio selection models.