The basic historical simulation (HS) method, when applied to 100 Profit/Loss (P/L) observations, only allows us to estimate Value at Risk (VaR) at discrete confidence levels, such as 95%. This is a significant limitation of the basic HS method because it does not allow for the estimation of VaR at non-discrete intervals. For instance, with 100 P/L observations, the VaR at the 95% confidence level is given by the sixth-largest loss. However, the VaRs at 95.1%, 95.9%, and 95.5% confidence levels cannot be obtained because there are no corresponding loss observations. This limitation restricts the flexibility and precision of risk estimation, which is crucial in financial risk management. Therefore, this example accurately demonstrates the drawback of the basic HS method in non-parametric density estimation.

Choice B is incorrect. The VaR at the 95% confidence level is not given by the seventh-largest loss when we have 100 P/L observations. In fact, it should be given by the fifth largest loss because in a list of 100 observations, the top 5% (or top five) represent losses beyond the 95% confidence level.

Choice C is incorrect. Similarly to choice B, this statement incorrectly identifies that VaR at a 95% confidence level would be represented by the fourth-largest loss in a set of 100 P/L observations. This again misrepresents how VaR calculations are made as it should be represented by the fifth largest loss.

Choice D is incorrect. In a basic historical simulation with 100 observations, the 95% confidence level VaR should correspond to the sixth-largest loss, not the ninth.

Things to Remember

• Non-parametric density estimation is a method used to estimate the probability density function of a random variable without assuming a specific distribution.