Historical Simulation VaR For Insurer Portfolios

Why Historical Simulation Uses Repricing

A market-risk estimate should reflect how a portfolio would respond to plausible joint changes in market factors. Historical simulation takes observed changes in those factors, applies each change to the portfolio’s current exposures, and revalues the resulting scenario. It therefore asks what today’s portfolio might have lost had a selected historical market move occurred now.

For an insurer, the quantity of interest is often surplus: the market value of assets less the market value of liabilities. A scenario can reduce surplus even when some assets rise in value, because liabilities may also change. Historical simulation records the resulting change in the whole portfolio rather than treating each exposure as an isolated risk estimate.

Definition: Let $V_0$ be current portfolio value and $V_j$ its value after repricing under historical scenario $j$. Define scenario loss as $L_j=V_0-V_j$. For an insurer measured on surplus, $V$ means assets minus liabilities. A positive $L_j$ is a loss; a negative $L_j$ is a gain. Historical simulation VaR at confidence level $\alpha$ is the $\alpha$-quantile of the scenario-loss distribution.

The repricing starts from current exposures, not from the portfolio that existed on the historical observation date. Historical observations supply the market-factor changes; current positions and valuation methods determine the scenario values. For example, a historical yield change is added to today’s yield, while an observed equity return is applied multiplicatively to today’s equity value. The factors must be defined consistently with the exposures being repriced.

Constructing and Ranking Historical Scenarios

A historical observation is a vector of factor changes over a specified horizon. It might include yield changes, equity returns, credit-spread changes, and foreign-exchange returns recorded over the same dates. Keeping the changes from the same observation together preserves their historical co-movement. Each vector is then applied to the corresponding current factors, and the entire portfolio is revalued.

For rate factors, a historical change is usually an absolute change in the rate. If today’s rate is $r_0$ and the historical change is $\Delta r_j$, the scenario rate is $r_0+\Delta r_j$. For a price or exchange-rate factor represented by a return $R_j$, the scenario factor is its current value multiplied by $1+R_j$. A currency exposure valued in dollars may need both the local-asset return and the exchange-rate return.

This tutorial uses the nearest-rank empirical quantile. Given $n$ equally weighted scenario losses, sort them from smallest to largest, $L_{(1)}\leq\cdots\leq L_{(n)}$. The selected rank is the ceiling of $n\alpha$.

\widehat{\operatorname{VaR}}_{\alpha}=L_{(k)}, \qquad k=\lceil n\alpha\rceil

This rule follows directly from the empirical cumulative distribution function. At the $k$-th ordered loss, at least $k$ of the $n$ observations are no larger than that loss. Thus its empirical cumulative probability is at least $k/n$. Taking $k=\lceil n\alpha\rceil$ makes $k/n\geq\alpha$, so $L_{(k)}$ is the first observed loss whose cumulative probability reaches the requested level under this convention.

Other quantile conventions may interpolate between observations and produce a different result. State the convention whenever the sample is small or results are being compared. Historical simulation does not require a normal loss distribution, but its estimate still depends on the historical window, the selected horizon, and the representativeness of the observed changes.

Repricing Insurer Surplus

For an insurer portfolio, reprice assets and liabilities separately under every scenario, then calculate the scenario surplus. The loss is current surplus less scenario surplus. The following example uses a two-year zero-coupon asset, a two-year liability payment, and an equity holding. The specified discounting model assumes annual compounding and a flat yield for each scenario.

Worked Example: Historical Yield and Equity Scenarios

A hypothetical insurer holds a two-year bond that pays $441,000 at maturity and $100,000 of equity. It has a two-year liability payment of $275,625. The current annual yield is 5%, and the current equity value is $100,000. Three historical scenarios change the yield to 6%, 4.5%, and 7%, respectively; the corresponding equity values under those observations are $92,000, $103,000, and $88,000. Find the 90% VaR using nearest rank.

Method and current position. Discount each two-year payment at the applicable annual yield, using $PV=C/(1+y)^2$. At the current 5% yield, the bond is $441{,}000/(1.05)^2=\$400{,}000.00$, since $(1.05)^2=1.1025$. The liability is $275{,}625/1.1025=\$250{,}000.00$. Current surplus is therefore $\$400{,}000+\$100{,}000-\$250{,}000=\$250{,}000.00$.

Scenario at 6%. The discount denominator is $(1.06)^2=1.1236$. The bond value is $441{,}000/1.1236=\$392{,}488.43$, and the liability value is $275{,}625/1.1236=\$245{,}305.27$, each rounded to cents. Scenario surplus is $\$392{,}488.43+\$92{,}000-\$245{,}305.27=\$239{,}183.16$. The loss is $\$250{,}000.00-\$239{,}183.16=\$10{,}816.84$.

Scenario at 4.5%. The denominator is $(1.045)^2=1.092025$. The bond value is $441{,}000/1.092025=\$403{,}836.91$, and the liability value is $275{,}625/1.092025=\$252{,}398.07$. Scenario surplus is $\$403{,}836.91+\$103{,}000-\$252{,}398.07=\$254{,}438.84$. The loss is $\$250{,}000.00-\$254{,}438.84=-\$4{,}438.84$, a gain.

Scenario at 7%. The denominator is $(1.07)^2=1.1449$. The bond value is $441{,}000/1.1449=\$385{,}186.48$, and the liability value is $275{,}625/1.1449=\$240{,}741.55$. Scenario surplus is $\$385{,}186.48+\$88{,}000-\$240{,}741.55=\$232{,}444.93$. The loss is $\$250{,}000.00-\$232{,}444.93=\$17{,}555.07$.

Quantile and check. In ascending order the losses are $-\$4{,}438.84$, $\$10{,}816.84$, and $\$17{,}555.07$. The selected rank is $\lceil0.90(3)\rceil=3$, so the 90% VaR is $17,555.07. The selected scenario is the largest loss, as expected when the requested rank is the third of three observations. The main mistake this method avoids is measuring asset losses alone: the liability value also changes under each yield scenario.

Applying Returns to Foreign-Currency Exposures

A foreign investment’s dollar value can change because of its local market return, its exchange-rate return, or both. If the exchange-rate factor is quoted so that a positive return means more dollars per unit of foreign currency, and the returns over the same horizon are $R_{\mathrm{local}}$ and $R_{\mathrm{FX}}$, the revalued dollar amount is the current dollar amount multiplied by both gross-return factors.

V_j=V_0(1+R_{\mathrm{local},j})(1+R_{\mathrm{FX},j})

The product is important: it reflects the compounded effect of the two returns. Adding the returns is only an approximation and omits their cross-product.

Worked Example: Joint Equity and Foreign-Bond Repricing

A hypothetical insurer has $120,000 in domestic equities and a foreign bond currently worth $80,000 in dollars. Four equally weighted historical observations give, respectively, domestic equity returns of −5%, 3%, −8%, and 1%; foreign-bond local returns of 2%, −1%, 4%, and −3%; and exchange-rate returns of −4%, 2%, 1%, and −5%. Calculate the 75% VaR under nearest rank.

Method. The current portfolio value is $\$120{,}000+\$80{,}000=\$200{,}000$. Apply each equity return to $120,000. Revalue the foreign bond using the product of its local and exchange-rate gross returns.

Observation 1. Equity value is $120{,}000(0.95)=\$114{,}000$. Foreign-bond value is $80{,}000(1.02)(0.96)=80{,}000(0.9792)=\$78{,}336$. Total value is $\$192{,}336$, so loss is $\$200{,}000-\$192{,}336=\$7{,}664$.

Observation 2. Equity value is $120{,}000(1.03)=\$123{,}600$. Foreign-bond value is $80{,}000(0.99)(1.02)=80{,}000(1.0098)=\$80{,}784$. Total value is $\$204{,}384$, so loss is $-\$4{,}384$.

Observation 3. Equity value is $120{,}000(0.92)=\$110{,}400$. Foreign-bond value is $80{,}000(1.04)(1.01)=80{,}000(1.0504)=\$84{,}032$. Total value is $\$194{,}432$, so loss is $\$5{,}568$.

Observation 4. Equity value is $120{,}000(1.01)=\$121{,}200$. Foreign-bond value is $80{,}000(0.97)(0.95)=80{,}000(0.9215)=\$73{,}720$. Total value is $\$194{,}920$, so loss is $\$5{,}080$.

Quantile and check. The sorted losses are $-\$4{,}384$, $\$5{,}080$, $\$5{,}568$, and $\$7{,}664$. The selected rank is $\lceil0.75(4)\rceil=3$; hence the 75% VaR is $5,568. As a trap check, adding the first observation’s returns gives a foreign-bond factor of $1+0.02-0.04=0.98$, or $78,400, which is $64 too high. The correct compounded factor is $1.02(0.96)=0.9792$.

Repricing Fixed Cash Flows and Selecting the Tail

The next example combines a market exposure with an asset and liability whose values respond to the same yield. This illustrates why the scenario loss should be computed from the change in surplus. It also makes the ranking step explicit: a VaR is the selected order statistic, not an average of the most adverse observations.

Worked Example: Five Yield and Equity Observations

A hypothetical insurer has $250,000 in equity, a one-year bond paying $157,500, and a one-year liability payment of $105,000. The current annual yield is 5%, and the current equity value is $250,000. Thus the bond is currently worth $150,000, the liability is worth $100,000, and current surplus is $300,000. Five historical observations produce yield levels of 4%, 6%, 7%, 3%, and 5%, and equity returns of −4%, 2%, −7%, 1%, and −1%, respectively. Find the 80% VaR using nearest rank.

Method. Under annual compounding, reprice each one-year cash flow as $C/(1+y)$. For each observation, scenario surplus is the equity value plus bond value less liability value. The scenario loss is $300,000 less that surplus.

The scenario bond and liability values, before rounding, have difference $157{,}500/(1+y)-105{,}000/(1+y)=52{,}500/(1+y)$. Thus the scenario surplus can be calculated as the repriced equity value plus that net fixed-cash-flow value.

Observation at 4%. Equity value is $250{,}000(0.96)=\$240{,}000$. The bond is $157{,}500/1.04=\$151{,}442.31$, and the liability is $105{,}000/1.04=\$100{,}961.54$. Net fixed-cash-flow value is $52{,}500/1.04=\$50{,}480.77$. Scenario surplus is $\$290{,}480.77$, giving a loss of $\$9{,}519.23$.

Observation at 6%. Equity value is $\$250{,}000(1.02)=\$255{,}000$. The bond is $\$148{,}584.91$, the liability is $\$99{,}056.60$, and their net value is $52{,}500/1.06=\$49{,}528.30$. Scenario surplus is $\$304{,}528.30$, giving a loss of $-\$4{,}528.30$.

Observation at 7%. Equity value is $\$250{,}000(0.93)=\$232{,}500$. The bond is $\$147{,}196.26$, the liability is $\$98{,}130.84$, and their net value is $52{,}500/1.07=\$49{,}065.42$. Scenario surplus is $\$281{,}565.42$, giving a loss of $\$18{,}434.58$.

Observation at 3%. Equity value is $\$250{,}000(1.01)=\$252{,}500$. The bond is $\$152{,}912.62$, the liability is $\$101{,}941.75$, and their net value is $52{,}500/1.03=\$50{,}970.87$. Scenario surplus is $\$303{,}470.87$, giving a loss of $-\$3{,}470.87$.

Observation at 5%. Equity value is $\$250{,}000(0.99)=\$247{,}500$. The bond is $150,000, and the liability is $100,000, so net fixed-cash-flow value is $50,000. Scenario surplus is $297,500, giving a loss of $2,500.

Quantile and check. The sorted losses are $-\$4{,}528.30$, $-\$3{,}470.87$, $\$2{,}500$, $\$9{,}519.23$, and $\$18{,}434.58$. The selected rank is $\lceil0.80(5)\rceil=4$, so the 80% VaR is $9,519.23. The fourth loss is $9,519.23, not the largest loss; the largest is the fifth order statistic. The result is consistent with the stated nearest-rank rule.

Assumptions, Uses, and Common Traps

Historical simulation is attractive because it uses observed joint market moves and can incorporate nonlinear valuation when each scenario is fully repriced. It does not require choosing a normal distribution for portfolio losses. Its reliability nevertheless depends on the scenario design and the data. A historical window may omit severe events, contain too few tail observations, or reflect market conditions unlike those facing the current portfolio. The exposure mapping and valuation models must also remain appropriate when applying old changes to current positions.

A historical VaR is conditional on the chosen holding period and confidence level. A one-day factor change should not be treated as a one-year change, and the estimate should not be annualized without a separate, justified model. If observations are not equally weighted, use the stated weights and the corresponding cumulative-weight quantile rule rather than the nearest-rank formula for equal weights.

Wrong base portfolio: applying historical market levels to historical positions answers a different question. Apply the observed changes to current exposures.
Wrong sign: with loss defined as current value minus scenario value, a gain is a negative loss. Do not reverse the subtraction when ranking.
Wrong factor arithmetic: returns compound when applied sequentially to a value; rate changes are normally added to the current rate.
Wrong tail rank: sort losses from smallest to largest and use $k=\lceil n\alpha\rceil$. Do not choose the $1-\alpha$ rank or silently interpolate.
Incomplete repricing: for insurer surplus, revalue liabilities as well as assets. A change in liability value can offset or amplify the asset-side movement.

Key takeaway: Historical simulation VaR is obtained by applying each historical factor-change vector to today’s portfolio, fully repricing assets and liabilities, calculating scenario losses, and selecting the loss quantile under an explicitly stated convention. This treatment is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each problem, use the nearest-rank convention where requested and define loss as current value or surplus less scenario value or surplus.

A hypothetical portfolio is currently worth $900,000. Four equally weighted historical scenarios reprice it to $910,000, $870,000, $895,000, and $860,000. Find the 75% VaR. (Answer: $30,000)
A foreign asset is currently worth $50,000 in dollars. In one historical observation its local return is 3% and its exchange-rate return is −2%. Reprice it using both returns. (Answer: $50,470)
Six equally weighted scenario losses, in thousands of dollars, are −4, 1, 3, 8, 11, and 20. Find the 90% VaR and selected rank. (Answer: $20,000; rank 6)
A hypothetical insurer has current surplus of $1,200,000. Five historical scenarios produce surpluses of $1,180,000, $1,230,000, $1,150,000, $1,190,000, and $1,160,000. Find the 80% VaR. (Answer: $40,000)
A portfolio currently worth $400,000 has three equally weighted scenario values of $390,000, $415,000, and $380,000. Find the 67% VaR using nearest rank. (Answer: $20,000)

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