Parametric VaR With Correlated Market Factors

From Factor Changes to Portfolio VaR

A portfolio exposed to several market factors can gain or lose value as rates, equity returns, exchange rates, or spreads change. A sensitivity estimate describes how the portfolio responds to a factor move. Parametric VaR combines those sensitivities with assumptions about the factors’ distributions and dependence to estimate a loss quantile. The dependence matters: offsetting exposures may reduce risk when their factors move together, while exposures that respond in opposite directions may lose that offset when their factors move in opposite directions.

The calculation below uses a first-order, linear loss model and normally distributed factor changes. It is useful when the portfolio’s sensitivities provide a reasonable approximation over the risk horizon and the factor distribution assumptions are acceptable. It is not a full repricing of nonlinear positions, and the normal model can be unsuitable when losses have strong skewness or heavy tails.

Definition: Let $\Delta\mathbf{Z}=(\Delta Z_1,\ldots,\Delta Z_k)^\mathsf{T}$ be the vector of market-factor changes over a specified horizon. Let $\mathbf{b}=(b_1,\ldots,b_k)^\mathsf{T}$ be the signed loss sensitivities, so that $b_j$ is the approximate change in loss, in dollars, for a one-unit increase in factor $j$. The linearized portfolio loss is $L=\mathbf{b}^\mathsf{T}\Delta\mathbf{Z}$. If sensitivities are instead stated as changes in portfolio value, their signs must be reversed to obtain loss sensitivities.

Assume the factor-change vector is multivariate normal with mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. A covariance matrix is symmetric; its diagonal entries are factor variances and its off-diagonal entries are covariances. The loss is then normally distributed, with mean $\mathbf{b}^\mathsf{T}\boldsymbol{\mu}$ and variance $\mathbf{b}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{b}$.

Theorem: Under the multivariate normal factor model and linear loss assumption, the confidence-level $\alpha$ VaR is the $\alpha$-quantile of the loss distribution:

\operatorname{VaR}_{\alpha}(L) = \mathbf{b}^\mathsf{T}\boldsymbol{\mu} + z_{\alpha}\sqrt{\mathbf{b}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{b}}, \qquad z_{\alpha}=\Phi^{-1}(\alpha).

Here $\Phi$ is the standard normal cumulative distribution function. For example, $z_{0.95}=1.6449$ and $z_{0.99}=2.3263$. VaR is a loss quantile, not the expected loss in the worst tail. If the loss mean is zero, the formula reduces to $z_\alpha$ times the loss standard deviation. If the mean is not zero, omitting it gives the wrong quantile.

Why Covariance Enters the Formula

The variance of a weighted sum includes both each factor’s individual variance and the covariance between every pair of factors. This is the mathematical reason that factor correlation cannot generally be ignored.

Expand the loss.
$L=\sum_{j=1}^k b_j\Delta Z_j$. Its mean is $\sum_j b_j\mu_j=\mathbf{b}^\mathsf{T}\boldsymbol{\mu}$.

Center each factor.
$L-E[L]=\sum_j b_j(\Delta Z_j-\mu_j)$.

Take the variance.
Expanding the square gives $\operatorname{Var}(L)=\sum_j\sum_m b_jb_m\operatorname{Cov}(\Delta Z_j,\Delta Z_m)=\mathbf{b}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{b}$.

For two factors with standard deviations $\sigma_1,\sigma_2$ and correlation $\rho_{12}$, the covariance is $\rho_{12}\sigma_1\sigma_2$. The loss variance becomes

b_1^2\sigma_1^2+b_2^2\sigma_2^2 +2b_1b_2\rho_{12}\sigma_1\sigma_2.

The sign of the cross-term depends on both the correlation and the signed sensitivities. Equivalently, define the sensitivity-weighted standard deviations $q_j=b_j\sigma_j$; then each pair contributes $2\rho_{jm}q_jq_m$. This form makes the sign logic visible.

Worked Calculations

Worked Example: Nonzero Mean and Two Correlated Factors

For a hypothetical one-day market-risk estimate, let the two factor changes be jointly normal with means $(0.2,-0.1)$, standard deviations $(3,2)$, and correlation $0.5$. The signed loss sensitivities are $(\$2{,}500,-\$1{,}500)$ per factor unit. Calculate 99% VaR using $z_{0.99}=2.3263$. The stated distribution parameters are for this one-day horizon.

Method. First calculate the expected loss, then calculate the variance including the covariance term, and finally apply the normal quantile formula.

The mean loss is

E[L] =2{,}500(0.2)+(-1{,}500)(-0.1) =500+150 =\$650.

The sensitivity-weighted standard deviations are $q_1=2{,}500(3)=\$7{,}500$ and $q_2=(-1{,}500)(2)=-\$3{,}000$. Therefore,

\operatorname{Var}(L) =7{,}500^2+(-3{,}000)^2 +2(0.5)(7{,}500)(-3{,}000) =56{,}250{,}000+9{,}000{,}000-22{,}500{,}000 =42{,}750{,}000.

The standard deviation is $\sqrt{42{,}750{,}000}=\$6{,}538.35$ (rounded to cents). Thus,

\operatorname{VaR}_{0.99}(L) =650+2.3263\sqrt{42{,}750{,}000} =\$15{,}860.16.

Check. The covariance is negative because the factor correlation is positive but the two loss sensitivities have opposite signs. It reduces the variance below the sum of the stand-alone variances. The calculation avoids the mistake of assuming positive factor correlation must increase portfolio loss variance.

Worked Example: Negative Correlation Can Increase Loss Risk

For a hypothetical one-day portfolio, two factor changes are jointly normal with zero means, standard deviations $1.5$ and $2$, and correlation $-0.25$. The signed loss sensitivities are $(\$4{,}000,-\$2{,}000)$ per factor unit. Calculate 95% VaR using $z_{0.95}=1.6449$, and compare it with the result if correlation were set to zero.

Method. Weight each standard deviation by its signed sensitivity. The resulting exposures are $q_1=4{,}000(1.5)=\$6{,}000$ and $q_2=(-2{,}000)(2)=-\$4{,}000$. With correlation,

\operatorname{Var}(L) =6{,}000^2+(-4{,}000)^2 +2(-0.25)(6{,}000)(-4{,}000) =36{,}000{,}000+16{,}000{,}000+12{,}000{,}000 =64{,}000{,}000.

The loss standard deviation is $\$8{,}000$, so the correlated VaR is

\operatorname{VaR}_{0.95}(L) =1.6449(8{,}000) =\$13{,}159.20.

If correlation is set to zero, the covariance term disappears. The variance is $36{,}000{,}000+16{,}000{,}000=52{,}000{,}000$, and the standard deviation is $\$7{,}211.10$. The resulting VaR is $1.6449\sqrt{52{,}000{,}000}=\$11{,}861.54$.

Common sign trap. A mistaken calculation might treat the negative correlation as making the cross-term negative and use $52{,}000{,}000-12{,}000{,}000=40{,}000{,}000$, producing a VaR of about $\$10{,}403.26$. That is wrong: $q_1q_2$ is already negative, so multiplying it by the negative correlation makes the covariance contribution positive. The correct correlated VaR is $\$13{,}159.20$, greater than the zero-correlation estimate.

Worked Example: Using a Three-Factor Covariance Matrix

A hypothetical portfolio has three normally distributed factor changes over one specified horizon, all with zero means. Their covariance matrix and the portfolio’s signed loss sensitivities are

\boldsymbol{\Sigma} = \begin{pmatrix} 4&1&0\\ 1&9&-3\\ 0&-3&4 \end{pmatrix}, \qquad \mathbf{b} = \begin{pmatrix} 500\\-200\\300 \end{pmatrix}.

Calculate 95% VaR using $z_{0.95}=1.6449$. The factor units are defined so that the stated sensitivities are dollars per factor unit and the matrix entries are factor-change covariances in corresponding units. The matrix is a valid covariance matrix: its leading principal minors are $4$, $35$, and $104$, all positive.

Method. Compute the matrix product before taking the quadratic form:

\boldsymbol{\Sigma}\mathbf{b} = \begin{pmatrix} 4(500)+1(-200)+0(300)\\ 1(500)+9(-200)-3(300)\\ 0(500)-3(-200)+4(300) \end{pmatrix} = \begin{pmatrix} 1{,}800\\-2{,}200\\1{,}800 \end{pmatrix}.

Then

\operatorname{Var}(L) =\mathbf{b}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{b} =500(1{,}800)+(-200)(-2{,}200)+300(1{,}800) =900{,}000+440{,}000+540{,}000 =1{,}880{,}000.

The loss mean is zero, and the standard deviation is $\sqrt{1{,}880{,}000}=\$1{,}371.13$. Therefore,

\operatorname{VaR}_{0.95}(L) =1.6449\sqrt{1{,}880{,}000} =\$2{,}255.37.

Check. The matrix method incorporates every variance and covariance term once through the quadratic form. It avoids omitting an off-diagonal entry or including only one of the two equal cross-terms.

Interpreting the Estimate and Its Limits

A correlation matrix $\mathbf{R}$ can be used with factor standard deviations. If $\mathbf{D}$ is the diagonal matrix of those standard deviations, then $\boldsymbol{\Sigma}=\mathbf{D}\mathbf{R}\mathbf{D}$. This construction separates the scale of each factor from the dependence between factors. It also provides a useful check: the matrix of correlations must be symmetric, have ones on its diagonal, and be a valid correlation matrix. Pairwise correlations that look plausible individually can still combine into an invalid matrix.

The examples show why correlation has no fixed effect on portfolio VaR. Its contribution depends on the product of correlation and the two signed, volatility-weighted sensitivities. With same-direction loss exposures, positive correlation tends to increase variance; with opposite-direction exposures, positive correlation tends to reduce it. A negative correlation reverses these effects. The full calculation, rather than a rule of thumb about “diversification,” determines the result.

Parametric VaR is conditional on its model. It relies on the selected horizon, estimated factor means and covariance matrix, normally distributed factor changes, and a sufficiently accurate linear sensitivity approximation. Large market moves can make local sensitivities less representative, and a normal distribution may not capture all tail behavior. The estimate should therefore be interpreted as a model-based quantile under stated assumptions, not as a guaranteed maximum loss. These calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Key takeaway: For a linear portfolio loss under normally distributed correlated factors, calculate the mean loss as $\mathbf{b}^\mathsf{T}\boldsymbol{\mu}$, the variance as $\mathbf{b}^\mathsf{T}\boldsymbol{\Sigma}\mathbf{b}$, and add the confidence-level normal quantile times the loss standard deviation. Preserve the signs of sensitivities and covariances throughout.

Check Your Understanding

For each problem, use the stated horizon and normal model. Round dollar VaR to the nearest cent.

A two-factor portfolio has zero-mean normal factor changes with standard deviations $0.02$ and $0.03$, correlation $0.4$, and signed loss sensitivities $\$1{,}000$ and $\$600$ per factor unit. Find 95% VaR using $z_{0.95}=1.6449$. (Answer: $\$52.33$)
Two normally distributed factor changes have means $0.1$ and $0.05$, standard deviations $1$ and $2$, and correlation $0.3$. A portfolio has signed loss sensitivities $\$3{,}000$ and $-\$2{,}500$. Find 99% VaR using $z_{0.99}=2.3263$. (Answer: $\$11{,}806.50$)
A zero-mean bivariate normal factor vector has standard deviations $2$ and $2$, with correlation $0.75$. Signed loss sensitivities are $\$1{,}000$ and $-\$1{,}000$. Find 95% VaR using $z_{0.95}=1.6449$, and state whether correlation increases or decreases variance relative to zero correlation. (Answer: $\$2{,}326.24$; correlation decreases variance.)
A hypothetical three-factor portfolio has zero-mean normal changes with covariance matrix $\begin{pmatrix}4&1&0\\1&9&-3\\0&-3&4\end{pmatrix}$ and signed loss sensitivities $(500,-200,300)^\mathsf{T}$. Calculate 95% VaR using $z_{0.95}=1.6449$, showing the variance before applying the quantile. (Answer: variance $1{,}880{,}000$; VaR $\$2{,}255.37$.)

What You'll Learn