Historical simulation and historical VaR

In short

• Historical simulation applies actual past changes in risk factors to current positions; it assumes past changes are a reasonable guide to future ones and treats the past data as the full population.
• It makes no distributional assumption, so it keeps real-world features such as fat tails, skewness and volatility clustering.
• Risk factors: log returns $\ln(P_t/P_{t-1})$ for equities; yield changes for bonds, which are then revalued by discounting cash flows.
• Each scenario's P/L on a position = position value × $(e^{r} - 1)$. Summing across positions gives the portfolio P/L.
• Historical VaR: rank the P/L outcomes from worst to best; the 5% VaR sits at position 0.05 × n, interpolating between neighbours if needed.
• It often differs from parametric VaR (z × $\sigma_p$, zero mean), because real returns are not normal and history contains drift.