Delta-Normal Portfolio VaR

From Market-Factor Changes to Portfolio VaR

A portfolio’s market value changes when interest rates, exchange rates, equity prices, or other market factors move. A useful first approximation is to multiply each factor change by the portfolio’s sensitivity to that factor, then add the resulting value changes. If the factor changes are modeled as jointly normal, this linear approximation produces a normal model for portfolio loss, from which a VaR quantile can be calculated.

This method is called delta-normal VaR: “delta” refers to the portfolio’s sensitivities, and “normal” refers to the assumed distribution of the resulting loss. It is most appropriate when the portfolio’s value is reasonably well approximated by a linear function of the factors over the horizon being studied.

Definition: Let $Z_j$ be the change in market factor $j$ over a specified horizon, and let $g_j$ be the portfolio’s sensitivity to that factor, measured as change in portfolio value per unit change in the factor. The first-order approximation to the portfolio’s market-value change is $\Delta V \approx \sum_{j=1}^{n} g_j Z_j$. Define the corresponding loss as $L=-\Delta V$, so a fall in portfolio value is a positive loss.

The sensitivity $g_j$ can be positive or negative. For example, a portfolio may gain value when a factor rises, giving a positive sensitivity, or lose value, giving a negative sensitivity. Keep the sign of every sensitivity in the calculation. Factor changes, sensitivities, and the resulting dollar amounts must also use compatible units.

The Delta-Normal VaR Formula

Assume the vector of factor changes is jointly normal, with mean $m_j=E[Z_j]$, standard deviation $\sigma_j$, and pairwise correlation $\rho_{jk}$. These are assumptions about changes over the same stated horizon. Since $L$ is a linear combination of jointly normal variables, it is normally distributed. Its mean and variance are

E[L]=-\sum_{j=1}^{n} g_j m_j, \qquad \operatorname{Var}(L) = \sum_{j=1}^{n} g_j^2\sigma_j^2 + 2\sum_{j<k}g_jg_k\sigma_j\sigma_k\rho_{jk}.

For confidence level $\alpha$, let $z_\alpha$ be the $\alpha$-quantile of the standard normal distribution. The delta-normal estimate of VaR is

\operatorname{VaR}_{\alpha}(L) = -\sum_{j=1}^{n}g_jm_j + z_\alpha \sqrt{ \sum_{j=1}^{n}g_j^2\sigma_j^2 + 2\sum_{j<k}g_jg_k\sigma_j\sigma_k\rho_{jk} }.

For example, $z_{0.95}\approx1.6449$ and $z_{0.99}\approx2.3263$. VaR is the loss quantile at the stated confidence level; it is not automatically a nonnegative number. If the estimated loss quantile is negative, the model’s quantile at that level is a gain rather than a loss.

Why the variance formula holds: Write the linearized loss as $L=-\sum_j g_jZ_j$. The variance of a sum includes each individual variance and twice each distinct pairwise covariance. Since $\operatorname{Var}(Z_j)=\sigma_j^2$ and $\operatorname{Cov}(Z_j,Z_k)=\rho_{jk}\sigma_j\sigma_k$, expanding the variance gives the displayed expression. The two negative signs in a covariance term cancel, so the loss has the same variance as the market-value change.

A convenient way to keep the signs straight is to calculate the signed volatility exposure $g_j\sigma_j$ for each factor. The covariance contribution for factors $j$ and $k$ is then twice the product of their signed volatility exposures times $\rho_{jk}$. A negative correlation does not always reduce portfolio risk: whether it increases or decreases variance depends also on the signs of the sensitivities.

Worked Calculations

Worked Example: One-Factor Equity Sensitivity

A hypothetical portfolio has sensitivity $g=-\$250{,}000$ per unit change in an equity factor. Over the one-month risk horizon, the factor change has a normal distribution with mean $0.004$ and standard deviation $0.02$. Estimate the portfolio’s 99% delta-normal VaR. Use $z_{0.99}=2.32635$, and report dollars to the nearest cent.

Method. Convert the factor’s mean and standard deviation into the mean and standard deviation of loss. There is one factor, so no correlation term is needed.

The mean portfolio-value change is $gE[Z]=-\$250{,}000(0.004)=-\$1{,}000$. Therefore the mean loss is $E[L]=-\bigl(-\$1{,}000\bigr)=\$1{,}000$.

The loss standard deviation is the absolute value of the sensitivity multiplied by the factor standard deviation:

\sigma_L=|g|\sigma_Z =\$250{,}000(0.02)=\$5{,}000.

Thus the 99% VaR is

\operatorname{VaR}_{0.99} = \$1{,}000+2.32635(\$5{,}000) = \$1{,}000+\$11{,}631.75 = \$12{,}631.75.

Check. The positive mean loss reflects the factor’s positive expected change combined with the portfolio’s negative sensitivity. The result is reported to the nearest cent. A common mistake is to use the factor’s mean as the mean loss directly; the loss mean must include the negative sign in $L=-\Delta V$.

Worked Example: Two Correlated Factors and Signed Sensitivities

A hypothetical portfolio has sensitivities of $+\$100{,}000$ and $-\$60{,}000$ per unit to two factor changes. Over the same one-month horizon, the factor means are $0.02$ and $-0.05$, their standard deviations are $0.10$ and $0.20$, and their correlation is $+0.25$. Estimate the 95% VaR in thousands of dollars using $z_{0.95}=1.6449$. Round the final result to two decimal places.

Method. First calculate the mean loss. Then calculate each signed volatility exposure in thousands of dollars and include the correlation term.

The expected market-value change is

E[\Delta V] = \$100{,}000(0.02) + (-\$60{,}000)(-0.05) = \$2{,}000+\$3{,}000 = \$5{,}000.

Consequently, the mean loss is $-\$5{,}000$, or $-5$ thousand dollars. For the variance calculation, the signed value-change volatility exposures are $100(0.10)=10$ and $(-60)(0.20)=-12$, in thousands of dollars. The variance, in squared thousands of dollars, is

\sigma_L^2 = 10^2+(-12)^2+2(10)(-12)(0.25) = 100+144-60 = 184.

Thus $\sigma_L=\sqrt{184}\approx13.5647$ thousand dollars. The VaR estimate is

\operatorname{VaR}_{0.95} = -5+1.6449(13.5647) \approx -5+22.3125 = 17.31 \text{ thousand dollars}.

Check and trap. The estimated 95% VaR is $\$17.31$ thousand. A common wrong calculation drops the negative sign on the second sensitivity and uses $2(10)(12)(0.25)$, giving variance $304$, standard deviation $17.4356$, and VaR about $\$23.68$ thousand. That calculation is wrong because the second portfolio sensitivity is negative: its signed volatility exposure is $-12$, not $+12$.

Worked Example: Three Factors With Different Correlation Effects

A hypothetical portfolio has sensitivities, in thousands of dollars per unit, of $80$, $-50$, and $30$ to three market factors. The factor changes have mean zero and standard deviations $0.05$, $0.10$, and $0.20$. Their pairwise correlations are $\rho_{12}=-0.20$, $\rho_{13}=0.10$, and $\rho_{23}=0.25$. Estimate the 99% VaR in thousands of dollars using $z_{0.99}=2.3263$. Round to two decimal places.

Method. Since the factor means are zero, the mean loss is zero. Calculate the three signed volatility exposures: $80(0.05)=4$, $(-50)(0.10)=-5$, and $30(0.20)=6$, all in thousands of dollars.

The variance in squared thousands of dollars is the sum of the squared exposures plus the three pairwise covariance contributions:

\begin{aligned} \sigma_L^2 &=4^2+(-5)^2+6^2 +2\bigl[(4)(-5)(-0.20)+(4)(6)(0.10)+(-5)(6)(0.25)\bigr]\\ &=16+25+36+2(4+2.4-7.5)\\ &=77-2.2\\ &=74.8. \end{aligned}

Therefore $\sigma_L=\sqrt{74.8}\approx8.6487$ thousand dollars, and

\operatorname{VaR}_{0.99} = 0+2.3263(8.6487) \approx 20.12 \text{ thousand dollars}.

Check. The 99% VaR is approximately $\$20.12$ thousand. The pair with correlation $-0.20$ contributes positively to variance because its signed exposures have opposite signs. The other two pairwise terms do not both have the same effect: their signs depend on both correlation and exposure signs.

Assumptions, Interpretation, and Common Traps

Delta-normal VaR is an approximation, not a statement that every portfolio loss is actually normal. The calculation relies on a first-order linearization of market-value changes and on a joint normal model for the factor changes. It also treats the sensitivities and factor distribution as appropriate for the chosen horizon. State that horizon explicitly: a daily sensitivity should not be combined with monthly factor volatility unless the sensitivity has been defined for that same horizon and unit.

The method can be useful when the portfolio value responds approximately linearly to modest factor movements. It can be less reliable when value changes are strongly nonlinear, as can occur for options or for large market moves. In those cases the linear approximation may miss material effects. The result should therefore be interpreted as a model-based estimate under the stated assumptions, not as a guarantee that losses cannot exceed VaR.

Before finalizing a calculation, check the following:

Loss sign: If loss is defined as $L=-\Delta V$, its mean is the negative of the mean market-value change.
Sensitivity sign: Keep positive and negative sensitivities in covariance terms. Correlation alone does not determine whether two factors diversify the portfolio.
Units: If sensitivities are dollars per unit and factor changes are in units, the product is dollars. If sensitivities are stated in thousands, keep all portfolio amounts in thousands until the final conversion.
Confidence level: Use the standard normal quantile for the requested tail probability; do not use the confidence level itself as a multiplier.
Horizon and assumptions: Use factor means, volatilities, and correlations for a consistent horizon, and identify the normal and linearization assumptions.

Key takeaway: Delta-normal VaR combines signed portfolio sensitivities with factor means, volatilities, and correlations. The factor correlation must be applied to the signed exposures, and the mean loss must follow from the stated loss convention.

Check Your Understanding

For each problem, use the stated horizon and normal quantile, and keep sensitivities and factor changes in compatible units.

A portfolio has sensitivity $-\$40{,}000$ per unit to one factor. The factor change has mean $0.01$ and standard deviation $0.03$. Estimate 95% VaR using $z_{0.95}=1.6449$, in dollars to the nearest cent. (Answer: $\$2{,}373.88$)
A portfolio has sensitivities of $50$ and $20$ thousand dollars per unit to two factors. Both factor means are zero; their standard deviations are $0.04$ and $0.10$, and their correlation is $-0.50$. Estimate 95% VaR using $z_{0.95}=1.6449$, in thousands of dollars. (Answer: $3.29$ thousand dollars)
A portfolio has sensitivity $+\$120{,}000$ per unit to a factor whose change has mean $-0.01$ and standard deviation $0.05$. Estimate 99% VaR using $z_{0.99}=2.3263$, in dollars to the nearest cent. (Answer: $\$15{,}157.80$)
A portfolio has sensitivities of $30$ and $-40$ thousand dollars per unit to two factors. Both means are zero; standard deviations are $0.10$ and $0.05$, and correlation is $0.40$. Estimate 95% VaR using $z_{0.95}=1.6449$, in thousands of dollars. (Answer: $4.71$ thousand dollars)
A portfolio has sensitivities of $80$ and $-25$ thousand dollars per unit to two factors. Their means are $0.01$ and $0.02$, standard deviations are $0.05$ and $0.08$, and correlation is $0.20$. Estimate 95% VaR using $z_{0.95}=1.6449$, in thousands of dollars. (Answer: $6.44$ thousand dollars)

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