Sensitivity-Based Estimates Of Market VaR

Why Combine Sensitivity and Volatility?

A portfolio’s sensitivity tells us how much its value changes when a market factor moves. The factor’s volatility tells us how large a move may be over a specified period. Neither quantity alone estimates a loss: a highly sensitive portfolio may have modest VaR if the factor is stable, while a volatile factor may have little effect on a portfolio with low sensitivity. A sensitivity-based estimate combines both quantities and then selects a loss quantile at the required confidence level.

The method is useful for a rapid, transparent estimate when one factor dominates and the portfolio’s value is approximately linear in that factor over the horizon. It is a model-based estimate, not a guarantee that losses cannot exceed the reported amount. The estimate is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: Let $F$ be a single market factor and let $V$ be portfolio value. The portfolio’s dollar sensitivity to $F$ at the current market level is $g=\partial V/\partial F$. For a small factor change $\Delta F$, the first-order approximation to the change in value is $\Delta V\approx g\Delta F$. Define loss as $L=-\Delta V$. A confidence-level VaR, for $0<\alpha<1$, is the loss quantile $\operatorname{VaR}_{\alpha}=\inf\{l:P(L\leq l)\geq\alpha\}$.

The sensitivity’s units depend on the factor. If $F$ is an equity return expressed as a decimal, $g$ is dollars of value change per unit return. If $F$ is a yield expressed as a decimal, $g$ is dollars per unit yield change. The factor volatility must use the same factor units and refer to the same time period as the modeled change.

One-Factor Normal VaR

Suppose the portfolio is exposed to one factor and its value change is adequately approximated by the linear sensitivity $g$. Assume the factor change over the horizon is normally distributed with mean zero and standard deviation $\sigma_F$. Then the approximate loss is $L\approx-g\Delta F$. Multiplication by $-g$ changes the sign and scales the standard deviation, so the loss has mean zero and standard deviation $|g|\sigma_F$. Its $\alpha$-quantile is therefore the standard normal quantile $z_\alpha$ times that standard deviation.

\operatorname{VaR}_{\alpha}\approx z_{\alpha}|g|\sigma_F, \qquad z_{\alpha}=\Phi^{-1}(\alpha).

For a multi-period horizon, let each period’s factor change be independent and identically distributed as $N(0,\sigma^2)$. The sum over $n$ periods is then $N(0,n\sigma^2)$; its standard deviation is $\sigma\sqrt{n}$, not $n\sigma$. Thus the horizon estimate is

\operatorname{VaR}_{\alpha,n}\approx z_{\alpha}|g|\sigma\sqrt{n}.

This derivation depends on the stated assumptions: independent normal changes with constant volatility, a fixed sensitivity, and a linear value response. The square-root-of-time rule does not generally hold for dependent or changing-volatility factor changes. It also does not correct for nonlinear portfolio behavior. For confidence levels commonly used in examples, $z_{0.95}$ is approximately 1.645 and $z_{0.99}$ is approximately 2.326; these are rounded standard normal quantiles.

Worked Example: Equity Exposure Over Five Days

A hypothetical insurer holds an equity position with current dollar sensitivity $g=\$15{,}000{,}000$ per unit of index return. The index’s daily returns are independent and normally distributed as $N(0,0.014^2)$, so the daily standard deviation is 1.4%. Estimate the five-day 95% VaR using $z_{0.95}=1.645$. Report dollars to the cent.

Method. Use the linear one-factor model and scale the daily standard deviation by $\sqrt{5}$. The factor return is measured as a decimal, so 1.4% is $0.014$.

The five-day standard deviation of the index return is

\sigma_5=0.014\sqrt{5}=0.031304951\ldots

The corresponding one-standard-deviation dollar change is

|g|\sigma_5 =\$15{,}000{,}000(0.014\sqrt{5}) =\$469{,}574.275\ldots

The 95% VaR estimate is

\operatorname{VaR}_{0.95,5} =1.645(\$15{,}000{,}000)(0.014\sqrt{5}) =\$772{,}449.682\ldots

Rounded only at the final step, the estimate is $772,449.68. The calculation checks: the VaR is 1.645 times the five-day dollar standard deviation, as required for the stated normal model. The common mistake avoided here is rounding the intermediate dollar standard deviation and then carrying that rounded amount into the VaR calculation.

Applying the Method Across Factor Types

The same calculation applies whether the factor is an equity return, an exchange-rate return, or a yield change. The essential checks are that the sensitivity is expressed per unit of the chosen factor and that the volatility is expressed in those same units. For fixed-income positions, modified duration provides a first-order yield sensitivity: $g\approx-D_{\mathrm{mod}}V$, where $V$ is current value and $D_{\mathrm{mod}}$ is modified duration. The negative sign indicates that, locally, a yield increase reduces bond value.

Worked Example: Foreign-Exchange Exposure Over Ten Days

A hypothetical insurer has a portfolio whose value sensitivity to a currency’s percentage return is $g=\$4{,}320{,}000$ per unit return. The currency’s daily returns are independent and normally distributed as $N(0,0.008^2)$. Estimate the ten-day 99% VaR using $z_{0.99}=2.326$. Report dollars to the cent.

Method. Apply the one-factor formula with ten-day volatility $\sigma_{10}=0.008\sqrt{10}$. The return standard deviation of 0.8% is $0.008$ in decimal units.

The ten-day factor standard deviation is

\sigma_{10}=0.008\sqrt{10}=0.025298221\ldots

The one-standard-deviation dollar change is

|g|\sigma_{10} =\$4{,}320{,}000(0.008\sqrt{10}) =\$109{,}288.3159\ldots

Therefore,

\operatorname{VaR}_{0.99,10} =2.326(\$4{,}320{,}000)(0.008\sqrt{10}) =\$254{,}204.6228\ldots

Rounded only at the final step, the estimate is $254,204.62. The result is checked by confirming that it equals the stated 99% quantile multiplier times the ten-day dollar standard deviation. The method avoids treating a 0.8% daily volatility as 0.8 rather than $0.008$.

Worked Example: Bond Yield Risk Over Five Days

A hypothetical bond portfolio is worth $5,000,000 and has modified duration 3.6. Daily yield changes are independent and normally distributed as $N(0,0.0008^2)$, where the yield change is in decimal units. Estimate the five-day 99% VaR using $z_{0.99}=2.326$. Report dollars to the cent.

Method. Approximate the portfolio’s yield sensitivity by $g=-D_{\mathrm{mod}}V$. First calculate the sensitivity, then combine its absolute value with the five-day yield standard deviation.

g=-(3.6)(\$5{,}000{,}000)=-\$18{,}000{,}000 \quad\text{per unit yield change}.

The five-day yield-change standard deviation is

\sigma_5=0.0008\sqrt{5}=0.001788854\ldots

Thus the one-standard-deviation dollar change and VaR are

|g|\sigma_5 =\$18{,}000{,}000(0.0008\sqrt{5}) =\$32{,}199.3789\ldots

\operatorname{VaR}_{0.99,5} =2.326(\$18{,}000{,}000)(0.0008\sqrt{5}) =\$74{,}895.755\ldots

Rounded only at the final step, the estimate is $74,895.76. The sign of $g$ indicates the direction of the value response, while VaR uses $|g|$ for the loss magnitude under a symmetric, zero-mean factor model. The estimate is checked against the product of the 99% multiplier and the five-day dollar standard deviation.

Nonzero Mean and Model Boundaries

If the factor change has a nonzero mean, the loss distribution is shifted. Suppose independent period changes each have distribution $N(\mu,\sigma^2)$, while $g$ remains fixed. Over $n$ periods, the sum has mean $n\mu$ and standard deviation $\sigma\sqrt{n}$. Since $L\approx-g\sum_{t=1}^{n}\Delta F_t$, its mean is $-gn\mu$; its standard deviation remains $|g|\sigma\sqrt{n}$. The resulting normal quantile is

\operatorname{VaR}_{\alpha,n} \approx -gn\mu+z_{\alpha}|g|\sigma\sqrt{n}.

For short-horizon risk estimates, a zero-mean factor change is often used to focus on uncertainty rather than expected return. If a problem specifies a nonzero mean, it must not be silently discarded. Conversely, the mean adjustment above applies only under the stated normal, fixed-sensitivity model; it is not a general correction for all market-risk distributions.

Sensitivity-based VaR is most useful when a single factor is a reasonable representation of the dominant exposure and small changes produce approximately linear value changes. It is less reliable when the portfolio has material curvature, embedded options, large potential factor moves, or significant exposure to several factors. In those cases, the single-factor estimate may omit important effects; a more suitable valuation or risk model is needed.

Common Exam Traps

Confusing factor units. A change of 8 basis points is $0.0008$ in decimal yield units, not $0.008$. Using $0.008$ in the bond example would make the estimated VaR ten times too large.
Scaling volatility incorrectly. Under independent, identically distributed period changes, volatility scales by $\sqrt{n}$. Multiplying by $n$ incorrectly scales the standard deviation as if the same-direction movement accumulated with certainty.
Using signed sensitivity as a negative VaR. The sensitivity sign describes whether value rises or falls as the factor rises. For the zero-mean normal model, the VaR magnitude uses $|g|$.
Mixing a percentage with a decimal. A daily return volatility of 1.4% is $0.014$; 8 basis points is $0.0008$. Convert before multiplying by a sensitivity stated per unit return or yield.
Ignoring the approximation. The calculation uses first-order sensitivity. It is not an exact repricing result when value responds nonlinearly or sensitivity changes materially over the modeled factor range.

Key takeaway: For a linear portfolio exposed to one normally distributed factor with independent, zero-mean period changes, multiply the absolute dollar sensitivity by the factor’s horizon standard deviation and the appropriate normal quantile. State the factor units, time-horizon assumptions, and limitations alongside the result.

Check Your Understanding

For each problem, use the stated linear normal model and round the final dollar VaR to the nearest cent.

A hypothetical equity account has sensitivity of $7,500,000 per unit index return. Daily index returns are independent $N(0,0.012^2)$. Estimate three-day 95% VaR using $z_{0.95}=1.645$. (Answer: $256,430.12)
A bond portfolio has value $8,000,000 and modified duration 4.2. Daily yield changes are independent $N(0,0.0005^2)$. Estimate one-day 99% VaR using $z_{0.99}=2.326$. (Answer: $39,076.80)
A hypothetical currency position has sensitivity $-\$2,400,000$ per unit currency return. Daily returns are independent $N(0,0.006^2)$. Estimate four-day 95% VaR using $z_{0.95}=1.645$. (Answer: $47,376.00)
A bond portfolio is worth $12,000,000 with modified duration 2.5. Daily yield changes are independent $N(0.0001,0.0006^2)$, with changes in decimal yield units. Estimate five-day 95% VaR using $z_{0.95}=1.645$, including the mean loss. (Answer: $81,209.97)
A hypothetical short equity position has sensitivity $-\$1,800,000$ per unit market return. Daily market returns are independent $N(0,0.02^2)$. Estimate two-day 99% VaR using $z_{0.99}=2.326$. (Answer: $118,420.59)

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