Empirical Quantile Conventions in VaR

Why a Finite-Sample VaR Needs a Convention

Historical loss data give a finite set of observations, while a VaR estimate is a quantile at a stated confidence level. To calculate it, first sort the losses and then apply a rule for selecting or interpolating between observations. At some confidence levels, different standard rules produce different answers from exactly the same sample.

This difference is not necessarily a calculation error. It can arise because a finite sample does not specify one universal interpolation rule. A report should therefore identify the quantile convention as well as the confidence level, sample size, and estimated VaR. This tutorial focuses on those finite-sample conventions; it does not change the underlying meaning of a population VaR.

Definition: Let $X_1,\ldots,X_n$ be observed losses, with larger values representing larger losses. Write their sorted values, retaining repeated observations, as $X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}$. The empirical distribution function is $\hat F_n(x)=n^{-1}\sum_{j=1}^{n}\mathbf{1}\{X_j\leq x\}$. The inverse empirical distribution, or empirical VaR, at confidence level $p$, where $0<p\leq1$, is $\hat q_p=\inf\{x:\hat F_n(x)\geq p\}$.

The notation $X_{(k)}$ means the observation in sorted position $k$, not the $k$th original observation. Each observed loss contributes one count, including when several losses have the same value. For the inverse empirical distribution, the rank is $k=\lceil np\rceil$, so $\hat q_p=X_{(\lceil np\rceil)}$. This is the empirical-distribution quantile; it is sometimes called the nearest-rank or inverse-empirical-CDF convention.

Other commonly used sample-quantile estimates interpolate between adjacent order statistics. These estimates can be useful as smooth summaries, but an interpolated value need not be one of the losses actually observed. The formulas below make the conventions precise. Names and default settings can vary across software, so the formula is more informative than a label alone.

Three Finite-Sample Conventions

The inverse empirical distribution selects the first observed loss at which the cumulative sample proportion reaches or exceeds $p$. Two common interpolation rules assign positions that may fall between order statistics. If $h=j+g$, where $j$ is an integer and $0\leq g<1$, linear interpolation between $X_{(j)}$ and $X_{(j+1)}$ is $X_{(j)}+g\bigl(X_{(j+1)}-X_{(j)}\bigr)$.

Quantile conventions:

Inverse empirical CDF: $k=\lceil np\rceil$, and $\hat q_p=X_{(k)}$.

Type 7 interpolation: Set $h=1+(n-1)p$. If $j=\lfloor h\rfloor$ and $g=h-j$, use $X_{(j)}+g\bigl(X_{(j+1)}-X_{(j)}\bigr)$ when $h$ is between 1 and $n$; at an integer position, use that order statistic. This convention places the endpoints at positions 1 and $n$.

Type 6 interpolation: Set $h=(n+1)p$, then interpolate by the same rule when $1\leq h\leq n$. For this tutorial, positions below 1 are assigned $X_{(1)}$, and positions above $n$ are assigned $X_{(n)}$. This endpoint rule must be stated because implementations can handle out-of-range positions differently.

The rank formula for the inverse empirical CDF follows directly from the definition. At the $k$th sorted observation, the empirical distribution has reached at least $k/n$; immediately below that observation it has counted at most $k-1$ sample values. Thus the first rank whose cumulative proportion reaches $p$ is the smallest integer $k$ satisfying $k/n\geq p$, which is $k=\lceil np\rceil$. This reasoning also explains why an exact equality $np=k$ selects rank $k$, not rank $k+1$.

\frac{k-1}{n}<p\leq\frac{k}{n} \quad\Longrightarrow\quad \hat q_p=X_{(k)},\qquad k=\lceil np\rceil

Interpolation follows a different principle: it assigns an estimated quantile position to each sorted observation and joins neighboring positions linearly. Type 7 uses $h=1+(n-1)p$; Type 6 uses $h=(n+1)p$. Because these positions generally differ from $\lceil np\rceil$, their interpolated estimates can differ from the inverse empirical-CDF estimate and from one another.

Worked Comparisons

Worked Example: Three Conventions at the Same Confidence Level

A hypothetical risk team has recorded eight annual losses, in thousands of dollars: $31, 9, 60, 22, 100, 4, 45, 15$. Calculate the 75% VaR using the inverse empirical CDF, Type 7, and Type 6. Report the answers in thousands of dollars to two decimal places.

Method. Sort the observations first, retaining all eight values. The ordered sample is $4,9,15,22,31,45,60,100$, so $n=8$, $p=0.75$, and the order statistics are in thousands of dollars.

Inverse empirical CDF. The rank is $\lceil np\rceil=\lceil8(0.75)\rceil=\lceil6\rceil=6$. Therefore, $\hat q_{0.75}=X_{(6)}=45$.

Type 7. The position is $h=1+(n-1)p=1+7(0.75)=6.25$. Thus $j=6$ and $g=0.25$. The estimate is $X_{(6)}+0.25(X_{(7)}-X_{(6)})=45+0.25(60-45)=45+3.75=48.75$.

Type 6. The position is $h=(n+1)p=9(0.75)=6.75$. Thus $j=6$ and $g=0.75$. The estimate is $X_{(6)}+0.75(X_{(7)}-X_{(6)})=45+0.75(60-45)=45+11.25=56.25$.

The three estimates are, respectively, $\$45.00$ thousand, $\$48.75$ thousand, and $\$56.25$ thousand. The values are ordered and lie between the sixth and seventh observations for the interpolated methods. Common mistake avoided: using a single formula without naming it. Here the data and confidence level are fixed, but the conventions still give different estimates.

Worked Example: Ties and an Exact Cumulative Proportion

A hypothetical insurer has five observed claim losses, in thousands of dollars: $25,8,70,8,14$. Find the 80% VaR under all three conventions, using the same endpoint rule for Type 6. Explain why the repeated loss must remain in the sample.

Method. Sort the five observations, including both occurrences of 8: $8,8,14,25,70$. Here $n=5$ and $p=0.80$.

Inverse empirical CDF. The rank is $\lceil np\rceil=\lceil5(0.80)\rceil=4$, so $\hat q_{0.80}=X_{(4)}=25$. The empirical distribution at 25 is $4/5=0.80$, which reaches the requested confidence level.

Type 7. The position is $h=1+(5-1)(0.80)=4.2$. Interpolate with $j=4$ and $g=0.2$: $25+0.2(70-25)=25+9=34$.

Type 6. The position is $h=(5+1)(0.80)=4.8$. Interpolate with $j=4$ and $g=0.8$: $25+0.8(70-25)=25+36=61$.

The estimates are $\$25.00$ thousand, $\$34.00$ thousand, and $\$61.00$ thousand. Both copies of the loss 8 count because they are separate observations; deleting a duplicate would change $n$, the cumulative proportions, and potentially every rank. Common mistake avoided: rounding an interpolated position to an integer. Type 7's position is 4.2, so its specified rule interpolates rather than rounding to the fourth observation.

Worked Example: A High Confidence Level and the Sample Maximum

A hypothetical portfolio has ten annual loss observations, in thousands of dollars: $41,12,160,29,95,18,68,24,52,33$. Calculate the 95% VaR using the inverse empirical CDF, Type 7, and Type 6 with the endpoint rule stated above. Give the results to two decimal places.

Method. The sorted losses are $12,18,24,29,33,41,52,68,95,160$. Thus $n=10$ and $p=0.95$.

Inverse empirical CDF. The rank is $\lceil10(0.95)\rceil=\lceil9.5\rceil=10$, so the estimate is $X_{(10)}=160$.

Type 7. The position is $h=1+9(0.95)=9.55$. Interpolate between $X_{(9)}=95$ and $X_{(10)}=160$: $95+0.55(160-95)=95+0.55(65)=95+35.75=130.75$.

Type 6. The position is $h=11(0.95)=10.45$, which is above the largest available order-statistic position. Under the endpoint rule specified here, assign $X_{(10)}=160$.

The estimates are $\$160.00$ thousand, $\$130.75$ thousand, and $\$160.00$ thousand. The Type 6 result depends on the stated endpoint rule; it is not obtained by inventing an eleventh observation. Common mistake avoided: extrapolating beyond the sample maximum without an explicit method that calls for extrapolation.

How to Choose and Communicate a Convention

There is no finite-sample interpolation rule that should be silently assumed to be universal. For a risk report, internal policy, or comparison of model outputs, apply the prescribed convention consistently. If no convention is prescribed, state the formula used and, when the difference is material, show how an alternative convention changes the estimate. This makes a reported VaR reproducible.

Convention	How the position is determined	What the estimate represents
Inverse empirical CDF	$\lceil np\rceil$	An observed loss at the first rank whose cumulative sample proportion reaches $p$
Type 7	$1+(n-1)p$	A linearly interpolated sample quantile between endpoint positions 1 and $n$
Type 6	$(n+1)p$	A linearly interpolated sample quantile using positions based on $n+1$, with endpoint handling stated

These rules are not interchangeable with a change in the population definition of VaR. They are ways to estimate a quantile from a finite sample. In particular, an interpolated estimate may be useful as a numerical summary, but it does not assert that the interpolated loss was observed or that the empirical distribution assigns probability to every value between adjacent observations.

Common Exam Traps

Using the wrong tail. For a loss VaR at confidence $p$, use the $p$th quantile of losses, not the $(1-p)$th quantile. A 95% VaR is not the 5th percentile of loss.
Rounding $np$ down. In the inverse empirical-CDF rule, the rank is $\lceil np\rceil$. For $np=6.2$, the rank is 7, not 6.
Sorting incorrectly or discarding ties. Sort losses in ascending order and retain each observation. A duplicate is still a separate sample value.
Mixing rank rules and interpolation rules. The rank $\lceil np\rceil$ belongs to the inverse empirical CDF; it is not a replacement for the Type 6 or Type 7 position formula.
Leaving endpoint treatment unstated. A plotting position can fall outside the observed ranks at extreme confidence levels. Specify how the convention handles that case.

Confirm the target.
Identify the loss variable, sample size $n$, and confidence level $p$.

Order the observations.
Sort from smallest loss to largest and preserve repeated values.

Apply the named convention.
Use its rank or position formula, and interpolate only if that convention requires it.

Check and report.
Verify that the estimate is consistent with adjacent order statistics and state the convention and units.

Key takeaway: Empirical VaR is convention-sensitive in finite samples. The inverse empirical CDF selects $X_{(\lceil np\rceil)}$; interpolation conventions instead calculate a position from their own formulas. Sorting, retaining ties, applying the specified endpoint rule, and naming the convention are all part of the calculation.

Check Your Understanding

For each problem, sort the sample before applying the stated rules. Unless specified otherwise, use the Type 6 endpoint rule defined in this tutorial. Report monetary amounts in the units given.

A hypothetical fund records losses of $44,5,27,12,90,18$, in thousands of dollars. Find the inverse empirical-CDF VaR at $p=0.50$ and the Type 7 estimate. (Answer: Inverse empirical CDF $=\$18.00$ thousand; Type 7 $=\$22.50$ thousand.)
A hypothetical claims unit observes losses $80,20,10,30$, in thousands of dollars. At $p=0.75$, calculate the inverse empirical-CDF, Type 7, and Type 6 estimates. (Answer: $\$30.00$ thousand, $\$42.50$ thousand, and $\$67.50$ thousand, respectively.)
A hypothetical operation records losses $40,2,75,15,28,6,9$, in thousands of dollars. At $p=0.60$, calculate all three estimates. (Answer: Inverse empirical CDF $=\$28.00$ thousand; Type 7 $=\$22.80$ thousand; Type 6 $=\$25.40$ thousand.)
A hypothetical insurer has losses $17,50,4,17,11$, in thousands of dollars. At $p=0.40$, calculate all three estimates and explain briefly why the two observations of 17 must both be retained. (Answer: Inverse empirical CDF $=\$11.00$ thousand; Type 7 $=\$14.60$ thousand; Type 6 $=\$13.40$ thousand; both 17s are separate observations and affect the sample size and rank positions.)
A hypothetical portfolio has six losses $3,14,21,35,48,120$, in thousands of dollars. At $p=0.95$, calculate all three estimates and identify which methods return a value not observed in the sample. (Answer: Inverse empirical CDF $=\$120.00$ thousand; Type 7 $=\$102.00$ thousand; Type 6 $=\$120.00$ thousand; Type 7 returns an interpolated, unobserved value.)

Convention	How the position is determined	What the estimate represents
Inverse empirical CDF	\(\lceil np\rceil\)	An observed loss at the first rank whose cumulative sample proportion reaches \(p\)
Type 7	\(1+(n-1)p\)	A linearly interpolated sample quantile between endpoint positions 1 and \(n\)
Type 6	\((n+1)p\)	A linearly interpolated sample quantile using positions based on \(n+1\), with endpoint handling stated

What You'll Learn