Historical Simulation VaR from an Ordered Loss Sample
Historical simulation estimates a loss quantile directly from observed losses. The calculation does not require fitting a parametric distribution: order the observations, identify the position corresponding to the requested confidence level, and report the loss at that position or an interpolated value. The method is simple, but the answer depends on the quantile convention and on how much relevant history is available.
All examples use hypothetical loss observations measured in thousands of dollars. Assume each observation represents a loss over the same specified horizon, on a comparable exposure basis. Historical simulation gives equal weight to each observation unless a different weighting scheme is stated.
This tutorial uses that inverse empirical distribution rule when it calls a result empirical VaR. It also shows a commonly used linear interpolation convention, with plotting position \(h=(n+1)p\). If \(h=k+\gamma\), where \(k\) is an integer and \(0<\gamma<1\), the interpolated estimate is \((1-\gamma)x_{(k)}+\gamma x_{(k+1)}\). This rule is defined only when \(1\leq h\leq n\); it does not authorize extrapolation beyond the observed losses. Different interpolation conventions can produce different estimates, so a result should identify the convention used.
Why the Empirical Rank Rule Works
The empirical distribution assigns probability \(1/n\) to each observation. For a value \(x_{(k)}\), at least \(k\) observations are no greater than it, so \(F_n(x_{(k)})\geq k/n\). The inverse empirical quantile is the smallest observed loss whose cumulative empirical probability reaches \(p\). Therefore, the required integer rank is the smallest \(k\) satisfying \(k/n\geq p\).
This argument also explains why rounding \(np\) to the nearest integer is not the rule: the rank must be large enough that the empirical cumulative probability is at least \(p\). If losses are tied, the ordered list still contains all \(n\) observations, including repeated values, and the selected rank may have the same dollar amount as a neighboring rank.
Worked Example: Selecting Empirical Ranks at Three Confidence Levels
A hypothetical insurer records ten comparable annual losses, in thousands of dollars. In ascending order, they are \(8, 11, 14, 17, 19, 23, 27, 34, 46, 70\). Find empirical VaR at confidence levels \(80\%\), \(90\%\), and \(95\%\).
Method. Use the inverse empirical distribution rule \(x_{(\lceil np\rceil)}\). Here \(n=10\), and losses are already ordered.
At \(p=0.80\), \(np=10(0.80)=8\), so the rank is \(\lceil8\rceil=8\). Thus \(VaR_{0.80}=x_{(8)}=34\) thousand dollars.
At \(p=0.90\), \(np=10(0.90)=9\), so the rank is \(\lceil9\rceil=9\). Thus \(VaR_{0.90}=x_{(9)}=46\) thousand dollars.
At \(p=0.95\), \(np=10(0.95)=9.5\), so the rank is \(\lceil9.5\rceil=10\). Thus \(VaR_{0.95}=x_{(10)}=70\) thousand dollars.
Check. The selected ranks have empirical cumulative probabilities \(8/10=0.80\), \(9/10=0.90\), and \(10/10=1.00\), respectively; each is at least its requested confidence level. The estimates are nondecreasing as confidence increases. A common mistake is to round \(9.5\) down to rank 9, which would give \(46\) and cumulative probability \(0.90\), below the requested \(0.95\).
Linear Interpolation Between Observed Losses
An interpolated estimate can avoid restricting the reported quantile to one of the observed loss amounts. It is a convention for estimating a quantile between ordered observations, not a claim that the unobserved losses between them were actually recorded. In particular, it should not be silently substituted for the inverse empirical distribution estimate.
The interpolation formula is a weighted average of the two adjacent losses. Starting at \(x_{(k)}\), the gap to \(x_{(k+1)}\) is \(x_{(k+1)}-x_{(k)}\); taking the fraction \(\gamma\) of that gap gives \(x_{(k)}+\gamma(x_{(k+1)}-x_{(k)})\). Equivalently, the weights \(1-\gamma\) and \(\gamma\) sum to one. Thus the interpolated estimate lies between the two losses whenever \(0\leq\gamma\leq1\).
Worked Example: Interpolating a Quantile Between Two Losses
A hypothetical portfolio has eight comparable quarterly losses, in thousands of dollars: \(9, 12, 15, 18, 21, 29, 40, 65\). Find the \(80\%\) VaR using both the inverse empirical distribution rule and the stated linear interpolation convention.
Empirical estimate. Here \(n=8\), so \(np=8(0.80)=6.4\), and the empirical rank is \(\lceil6.4\rceil=7\). Therefore, inverse empirical distribution VaR is \(x_{(7)}=40\) thousand dollars.
Interpolated estimate. Calculate the plotting position \(h=(n+1)p=9(0.80)=7.2\). This lies between ranks 7 and 8, so \(k=7\) and \(\gamma=0.2\). The adjacent losses are \(x_{(7)}=40\) and \(x_{(8)}=65\). Their difference is \(65-40=25\). The interpolated estimate is
Thus the interpolated estimate is \(45\) thousand dollars, while the empirical estimate is \(40\) thousand dollars. Check. The interpolated answer is between \(40\) and \(65\), as required. A common mistake is to report \(45\) as though it were the seventh observed loss; it is not. It is an estimate produced by the stated interpolation convention.
A Complete Historical Simulation Calculation
A sound calculation starts by checking that observations refer to a common loss measure and horizon. If the exposures or time periods are not comparable, merely sorting the numbers does not make them suitable for historical simulation. Once the data are comparable, the mechanical steps are straightforward.
Identify the loss definition, units, horizon, and number \(n\) of observations. Do not mix, for example, monthly and annual losses without an explicitly justified adjustment.
Write all observations from smallest to largest as \(x_{(1)},\ldots,x_{(n)}\), retaining ties and checking that none were omitted.
For inverse empirical VaR, calculate \(\lceil np\rceil\). For an interpolated estimate, calculate \(h=(n+1)p\) and verify that \(h\) lies within the available sample ranks.
State the confidence level, convention, units, and result. Check that the selected rank reaches the target empirical probability, or that an interpolated estimate lies between the adjacent observations.
Worked Example: Comparing Two Conventions at a High Confidence Level
A hypothetical risk manager has twelve ordered annual losses, in thousands of dollars: \(4, 5, 7, 9, 10, 12, 14, 16, 19, 24, 31, 55\). Calculate the \(90\%\) VaR under the inverse empirical distribution rule and the \((n+1)p\) interpolation convention.
Inverse empirical distribution. The sample size is \(n=12\). Then \(np=12(0.90)=10.8\), so the selected rank is \(\lceil10.8\rceil=11\). The empirical VaR is \(x_{(11)}=31\) thousand dollars. Its cumulative empirical probability is \(11/12\), or approximately \(0.9167\), which is at least \(0.90\).
Interpolation. The plotting position is \(h=(12+1)(0.90)=11.7\), so \(k=11\) and \(\gamma=0.7\). The adjacent losses are \(x_{(11)}=31\) and \(x_{(12)}=55\), with difference \(55-31=24\). Therefore,
The interpolated estimate is \(47.8\) thousand dollars. Check. The plotting position \(11.7\) is within the sample, and \(47.8\) lies between \(31\) and \(55\). The difference between \(31\) and \(47.8\) is not an arithmetic error: the methods use different quantile conventions. A common mistake is to call both answers simply “the historical VaR” without stating which convention was used.
Interpretation and Limits of the Sample
Historical simulation VaR is a quantile estimate based on the selected historical observations. It is not the largest possible loss, and it does not say how severe losses beyond the estimated quantile might be. Its interpretation depends on the chosen confidence level, the loss horizon, the sample, and the quantile convention. A higher confidence level generally selects a higher order statistic, but a small sample can make the estimate change sharply when the selected rank moves by one.
The finite-sample boundary matters. With the interpolation rule above, a high confidence level may produce \(h>n\). For example, if \(n=8\) and \(p=0.95\), then \(h=9(0.95)=8.55\), which is beyond the eighth observation. The rule supplies no interpolated estimate there; extending the line past the largest observed loss would be extrapolation, not interpolation. By contrast, the inverse empirical rule selects rank \(\lceil8(0.95)\rceil=8\), the largest observed loss. These are different statements about what the sample convention permits.
Historical observations also need to be relevant to the risk being estimated. Changes in exposure size, policy terms, market conditions, or the environment can make older observations less comparable with the current risk. The calculation itself does not correct for such changes. Any adjustment or weighting would need to be specified and justified before the ordered sample is used.
Check Your Understanding
For each problem, use the stated units and convention. Ordered losses are listed from smallest to largest.
- A hypothetical reinsurer has nine comparable losses, in thousands of dollars: \(6, 10, 13, 17, 22, 28, 35, 44, 60\). What is inverse empirical VaR at \(75\%\)? (Answer: \(35\) thousand dollars.)
- A sample of ten hypothetical losses, in thousands of dollars, is \(5, 8, 12, 15, 19, 24, 30, 38, 49, 75\). Calculate inverse empirical VaR at \(80\%\) and \(90\%\). (Answer: \(38\) thousand dollars at \(80\%\); \(49\) thousand dollars at \(90\%\).)
- Six ordered hypothetical losses are \(10, 14, 20, 27, 39, 60\), in thousands of dollars. Using \(h=(n+1)p\), calculate the interpolated VaR at \(70\%\). (Answer: \(37.8\) thousand dollars.)
- Eight ordered hypothetical losses are \(3, 6, 9, 13, 18, 26, 41, 70\), in thousands of dollars. At \(75\%\), find both inverse empirical VaR and the \((n+1)p\) interpolated estimate. (Answer: inverse empirical VaR \(=26\) thousand dollars; interpolated estimate \(=37.25\) thousand dollars.)
- For eight observations and a \(95\%\) confidence level, calculate the inverse empirical rank and the interpolation plotting position \(h=(n+1)p\). Is the stated interpolation rule defined? (Answer: inverse empirical rank \(=8\); \(h=8.55\), so the interpolation rule is not defined within the sample and must not be extrapolated.)