Historical Simulation VaR

Historical Simulation VaR: The Basic Idea

Historical simulation estimates portfolio VaR by asking how the portfolio held today would have performed under risk-factor changes observed in the past. It is a “what if” calculation: historical changes provide the scenarios, while current positions provide the portfolio to which those scenarios are applied. This differs from simply taking a quantile of past portfolio profits and losses, which may reflect positions that are no longer held.

The method does not require fitting a parametric probability distribution to losses. Instead, it treats the selected historical scenarios as an empirical distribution, typically giving each scenario equal weight. The estimate depends on the chosen historical window, the risk factors represented, the scenario horizon, the way positions are repriced, and the empirical quantile convention.

Definition: In historical simulation, each selected historical change in risk factors is applied to the current portfolio to produce a hypothetical portfolio profit and loss. Scenario loss is the negative of that profit and loss. Historical simulation VaR at confidence level $p$ is an empirical $p$-quantile of the resulting scenario losses.

For a position with current signed market value $V_j$ in a risk factor with simple return $r_j$, the linear approximation to profit and loss is $V_jr_j$. A long position has positive signed value; a short position has negative signed value. For a portfolio with $m$ such positions and scenario $s$,

\operatorname{P\&L}_s=\sum_{j=1}^{m} V_jr_{j,s}, \qquad L_s=-\operatorname{P\&L}_s.

A positive $L_s$ is a loss; a negative $L_s$ is a gain. The signed-value rule is important: a positive return raises the value of a long position but reduces the value of a short position. The simple-return formula is appropriate when the risk-factor move is a return and the position’s response is linear in that return. For options or other nonlinear positions, the scenario should generally be applied by repricing the position under the changed risk factors rather than relying on a linear exposure.

From Historical Changes to Scenario Losses

A practical calculation has four main stages. First, select the risk factors and a set of historical observation dates. Second, calculate or obtain the changes in those factors over the intended horizon. Third, apply each scenario to the current portfolio and calculate its loss. Finally, sort the losses and select the empirical quantile corresponding to the desired confidence level.

Define the portfolio and horizon.
Record current positions and specify, for example, whether the scenarios represent one-day or ten-day changes.

Build aligned scenarios.
Use changes in the relevant risk factors over matching historical intervals. A scenario is a vector of changes, so contemporaneous movements in different factors remain together.

Revalue and calculate losses.
Apply each scenario to the current positions using the stated valuation method. Define loss as negative profit and loss, consistently across all scenarios.

Sort and select the quantile.
Order the scenario losses from smallest to largest and use the stated empirical quantile convention to identify the VaR estimate.

For a price or index, a historical simple return is usually a relative change. For a yield or spread, the scenario is often an absolute change, such as an increase of 0.001 (10 basis points), rather than a percentage return on the yield. The position valuation must match the risk-factor scale. Mixing percentage changes with absolute changes, or combining changes from different horizons, can produce meaningless scenario results.

Selecting an Empirical Quantile

For $n$ equally weighted scenario losses, write the sorted values as $L_{(1)}\leq L_{(2)}\leq\cdots\leq L_{(n)}$. This tutorial uses the inverse empirical CDF convention: choose the smallest observed loss for which the empirical cumulative proportion is at least $p$. Equivalently, use order statistic $L_{(k)}$, where $k=\lceil np\rceil$. This is a stated convention for the calculations here; quantile implementations can use other conventions, a distinction addressed separately in material on empirical quantile conventions.

\widehat{\operatorname{VaR}}_p=L_{(\lceil np\rceil)}.

The reason is direct. At the $k$th ordered value, at least $k$ of the $n$ scenario losses are no greater than that value, so the empirical cumulative proportion is at least $k/n\geq p$. Immediately below that value, at most $k-1$ observations have been reached, and $(k-1)/n<p$ because $k=\lceil np\rceil$. Thus the $k$th order statistic is the smallest observed loss attaining cumulative proportion $p$. Ties do not invalidate this definition; they may cause the empirical cumulative proportion at the selected loss to exceed $p$.

Worked Example: Two Linear Positions With a Short Exposure

A hypothetical portfolio is long $800,000 of an equity index and short $300,000 of a commodity. Five historical one-day scenarios give the paired index and commodity returns below. Estimate the 80% one-day historical simulation VaR. Treat each scenario as equally likely in the empirical distribution. No interest-rate or mortality assumption is needed, and no parametric loss distribution is fitted.

Scenario	Index return	Commodity return
1	−2%	+1%
2	+1%	−4%
3	−3%	−2%
4	−1%	+5%
5	+2%	+3%

Method and signed exposures. The signed values are $+$800{,}000$ for the index and $-$300{,}000$ for the commodity. For every scenario, calculate portfolio profit and loss as the sum of signed value times return, then take its negative to obtain loss.

Scenario 1. Index profit and loss is $800{,}000(-0.02)=-$16{,}000$. Commodity profit and loss is $(-300{,}000)(0.01)=-$3{,}000$. Total profit and loss is $-$19{,}000$, so loss is $19,000.

Scenario 2. Index profit and loss is $800{,}000(0.01)=+$8{,}000$. Commodity profit and loss is $(-300{,}000)(-0.04)=+$12{,}000$. Total profit and loss is $+$20{,}000$, so loss is $-$20{,}000$, a gain.

Scenario 3. The two profit-and-loss contributions are $800{,}000(-0.03)=-$24{,}000$ and $(-300{,}000)(-0.02)=+$6{,}000$. Total profit and loss is $-$18{,}000$, so loss is $18,000.

Scenario 4. The contributions are $-$8{,}000$ from the index and $-$15{,}000$ from the short commodity position. Total profit and loss is $-$23{,}000$, so loss is $23,000.

Scenario 5. The contributions are $+$16{,}000$ and $-$9{,}000$. Total profit and loss is $+$7{,}000$, so loss is $-$7{,}000$.

The losses in scenario order are $19,000, $-$20,000$, $18,000, $23,000, and $-$7,000$. Ordered from smallest to largest, they are $-$20,000$, $-$7,000$, $18,000, $19,000, and $23,000. Here $n=5$, $p=0.80$, and $\lceil np\rceil=\lceil4\rceil=4$, so the 80% VaR is $19,000. As a check, four of the five losses are no greater than $19,000, giving an empirical cumulative proportion of 0.80.

Common mistake avoided: Treating the commodity position as long because its market value is stated as $300,000 would reverse its scenario profit and loss. A short position is represented by signed value $-$300,000$; for example, a positive 1% commodity return causes a $3,000 loss to this portfolio.

Yield Changes and Approximate Revaluation

Some portfolios include instruments whose risk factors are yields rather than asset returns. One common first-order approximation for a bond is $\Delta V\approx -VD\Delta y$, where $V$ is current value, $D$ is modified duration, and $\Delta y$ is the absolute yield change in decimal form. The approximation assumes a sufficiently small yield change and a locally linear price response; it ignores convexity and other valuation effects. If exact repricing is available, it can better represent larger moves or nonlinear responses.

Worked Example: Combining an Equity Return and a Bond Yield Move

A hypothetical portfolio is long $500,000 of an equity index and holds a bond portfolio worth $2,000,000 with modified duration 5. Use the first-order duration approximation for the bond and the simple-return formula for equity. Four equally weighted historical one-day scenarios are supplied. Estimate the 75% VaR. No mortality or separate interest-accumulation assumption is used; the yield changes are the scenario inputs, and no parametric distribution is fitted.

Scenario	Equity return	Bond yield change
1	−2%	+0.001
2	+1%	−0.002
3	−3%	+0.003
4	+2%	−0.001

Method. For each row, calculate equity profit and loss as $500{,}000r$, and bond profit and loss as $-2{,}000{,}000(5)\Delta y$. The yield changes are absolute decimal changes: for example, 0.001 is a rise of 10 basis points.

In scenario 1, equity profit and loss is $500{,}000(-0.02)=-$10{,}000$. Bond profit and loss is $-2{,}000{,}000(5)(0.001)=-$10{,}000$. Total profit and loss is $-$20{,}000$, so loss is $20,000.

In scenario 2, equity profit and loss is $500{,}000(0.01)=+$5{,}000$. Bond profit and loss is $-2{,}000{,}000(5)(-0.002)=+$20{,}000$. Total profit and loss is $+$25{,}000$, so loss is $-$25,000$.

In scenario 3, equity profit and loss is $500{,}000(-0.03)=-$15{,}000$. Bond profit and loss is $-2{,}000{,}000(5)(0.003)=-$30{,}000$. Total profit and loss is $-$45{,}000$, so loss is $45,000.

In scenario 4, equity profit and loss is $500{,}000(0.02)=+$10{,}000$. Bond profit and loss is $-2{,}000{,}000(5)(-0.001)=+$10{,}000$. Total profit and loss is $+$20{,}000$, so loss is $-$20,000$.

The ordered losses are $-$25,000$, $-$20,000$, $20,000, and $45,000. Since $\lceil4(0.75)\rceil=3$, the 75% VaR is $20,000. Three of the four losses are no greater than $20,000, confirming the empirical cumulative proportion is 0.75. The sign of the bond contribution also passes a basic check: a yield rise produces an approximate bond loss, while a yield fall produces an approximate gain.

Common mistake avoided: Treating a yield change of 0.001 as a 0.1% return on the bond value, or omitting the minus sign in the duration approximation, gives the wrong bond profit and loss. The specified formula uses the absolute yield change and the bond’s negative price sensitivity to yield.

What the Estimate Depends On

Historical simulation is transparent, but its simplicity does not remove modeling choices. The result can change when the historical window changes, even if the current portfolio does not. A short window may omit unusual market conditions; a long window may include conditions that are less relevant to the current environment. Equal weighting gives every selected scenario the same empirical probability, regardless of its age.

The method also relies on the chosen risk factors and valuation method. A factor omitted from the scenario set cannot contribute to the simulated portfolio loss. Applying old changes to today’s positions captures the current portfolio’s exposure to those changes, but does not make the past a complete representation of future possibilities. It may miss shocks larger than those in the sample, structural changes, and future dependence patterns not represented by the historical observations.

Several controls help keep the calculation interpretable:

Keep scenarios aligned. Apply each historical date’s joint vector of factor changes as one scenario. Independently reshuffling factors would create combinations that were not observed and would alter their joint historical behavior.
Match the horizon. A one-day change should not be applied as though it were a ten-day change without an explicit scaling or scenario construction method.
Use current exposures consistently. Apply the same current portfolio and valuation date to every historical scenario.
Check signs and units. Define loss as negative profit and loss, use signed values for short positions, and distinguish relative returns from absolute yield or spread changes.
Use suitable revaluation. A linear approximation may be adequate for small moves and linear exposures, but may be inaccurate for nonlinear instruments or large changes.

Historical simulation VaR is therefore an estimate conditional on the selected scenarios and assumptions, not a guarantee that losses will remain below the reported amount. Its practical value is that the scenario calculations can be inspected and traced from each risk-factor movement through to portfolio profit and loss.

Key takeaway: Apply each historical risk-factor scenario to the current portfolio, calculate loss as negative profit and loss, sort the resulting losses, and select the required empirical quantile. Signed positions, compatible units, aligned scenarios, and a clearly stated quantile convention are essential to a correct estimate.

Check Your Understanding

For each problem, use the inverse empirical CDF convention and treat the listed scenarios as equally weighted. Negative loss denotes a gain.

A portfolio is long $400,000 of an index and short $100,000 of a second asset. Paired one-day returns are $(-3\%,-2\%)$, $(+1\%,+4\%)$, $(-2\%,+3\%)$, and $(+2\%,-1\%)$. Calculate the four losses and the 50% VaR. (Answer: losses are $10,000, $0, $11,000, and $-$9,000$; ordered losses are $-$9,000$, $0, $10,000, $11,000; 50% VaR is $0.)
A portfolio is long $700,000 of an equity index and short $200,000 of a commodity. The paired returns are $(-1\%,+2\%)$, $(+2\%,-1\%)$, $(-3\%,-2\%)$, $(0\%,+5\%)$, and $(+1\%,+3\%)$. Find the 80% VaR. (Answer: losses are $11,000, $-$16,000$, $17,000, $10,000, and $-$1,000$; ordered losses are $-$16,000$, $-$1,000$, $10,000, $11,000, $17,000; 80% VaR is $11,000.)
A portfolio is long $500,000 of an asset with a scenario return of −4%, and short $150,000 of a second asset with a scenario return of +2%. What is the scenario loss? (Answer: profit and loss is $-$20,000-$3,000=-$23,000$, so loss is $23,000.)
Eight equally weighted scenario losses, already sorted, are $2,000, $4,000, $5,000, $7,000, $9,000, $12,000, $15,000, and $21,000. Find the 87.5% VaR under the stated convention. (Answer: $\lceil8(0.875)\rceil=7$, so VaR is $15,000.)
A historical simulation uses 250 equally weighted scenarios. In the sorted sample, the 247th, 248th, 249th, and 250th losses are $40 million, $43 million, $44 million, and $60 million. What is the 99% VaR, and what feature of the estimate should be kept in mind? (Answer: $\lceil250(0.99)\rceil=248$, so VaR is $43 million; it is determined by an extreme upper-tail order statistic and can be sensitive to the selected historical scenarios.)

What You'll Learn