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Skew Densities and Ensemble Inference for Financial Economics

# 5. Value at Risk

The VaR is a popular measure of potential loss used by investors, bankers, and regulators. Let be the change in the value of a portfolio at time for horizon and let (<0.5) denote a probability. The VaR is defined by the probability statement:

where the negative VaR is designed to measure losses in positive dollars. Intuitively, VaR measures a worst-case scenario loss associated with 'long' positions (buying side). For example, consider an investor with a time horizon of one year buying \$100,000 worth of mutual fund shares, . Now assume that the fund could lose 25% or more of its value in a year with probability . Then , implying that VaR = \$25,000 is an upper bound on the loss. Note that we can compute the VaR for any density whose cdf is known, at least numerically.

For our original data in terms of returns, the low return of suggests VaR = \$9629. It is customary not to use the estimate based on the empirical cdf, since the investor is concerned with worst-case scenarios, beyond what was experienced. One way to do this has been to use the lower one percentile of , namely .

This suggests that VaR = \$8033. It is somewhat surprising that the VaR from is less than \$9629, from the quantile of the original data. We have not yet computed the VaR from . First, let us check its cdf as:

Since inverting this analytically is not feasible, we use numerical methods as follows. A numerical version of the density is:

The numerical cdf is obtained by integrating as follows:

Next, we ask Mathematica to search for the value of at which the numerical cdf is equal to 0.01.

Hence the VaR estimated from the location-scale SN density is \$9222, rounded to the nearest dollar. Vinod and Morey [9] note that financial economists usually ignore the estimation risk. Since this is also true in calculated VaR estimates, there is a need to develop inference methods for VaR. In future work, using tools described in Section 2.6 of [3] we seek to compute pseudo-random number generation from our and also from the kernel density plotted in Figure 1. The large number (, say) of realizations of VaR can then be ordered from the smallest to the largest and denoted by with . An obvious possibility is to choose the VaR(10) as the estimate of VaR which incorporates both estimation risk and investment risk.