Variance-Covariance VaR

A risk manager needs a Value at Risk number for a 200-position equity book before the market opens tomorrow. Reprising every historical scenario or running ten thousand simulations both take real computing time. Variance-covariance VaR skips all of that: assume portfolio returns are normally distributed, and the entire loss distribution collapses into two numbers, a mean and a standard deviation, that plug straight into a formula.

The analogy before any symbols

Predicting how far a dropped ball will roll usually means watching many actual drops and measuring where they land. But if you already know the ball is a perfect sphere on a flat floor with known friction, physics gives you the stopping distance directly from one equation, no experiment needed. Variance-covariance VaR makes the equivalent bet: assume the "shape" of possible portfolio outcomes is a known, well-behaved curve (the normal distribution), and skip straight to the formula. The bet pays off in speed and costs you accuracy exactly when the ball turns out not to be a perfect sphere, when real returns have fatter tails than a bell curve allows.

The mechanics

If daily portfolio profit and loss $\Delta V$ is normally distributed with mean $\mu$ (often set to zero for short horizons) and standard deviation $\sigma$, then Value at Risk at confidence $\alpha$ is:

$\text{VaR}_\alpha = -\big(\mu + z_\alpha \, \sigma\big),$

where $z_\alpha$ is the standard normal quantile, $-1.645$ for 95% confidence, $-2.326$ for 99%. In words: start from the average expected change, add a multiple of the standard deviation, and the multiple is however many standard deviations of cushion the chosen confidence level demands; flip the sign because VaR is quoted as a positive loss number.

For a portfolio, $\sigma$ itself comes from the asset weights and their covariance matrix: $\sigma_p = \sqrt{w^\top \Sigma w}$, where $w$ is the dollar (or weight) exposure to each risk factor and $\Sigma$ is the factor covariance matrix. In words: portfolio variance is not just a weighted average of individual variances, it also depends on every pairwise covariance, which is exactly how diversification and hedges show up in the number.

Slide the mean and standard deviation here and watch the tails: the normal curve always thins out at the same predictable rate on both sides, which is precisely the assumption variance-covariance VaR needs and real return data routinely breaks.

Worked example: a single-asset book

A $10m position in an index with daily volatility $\sigma = 1.2\%$ and $\mu \approx 0$. The 99% one-day VaR is $2.326 \times 0.012 \times 10{,}000{,}000 = 279{,}120$, i.e. $279,120. At 95% confidence, $1.645 \times 0.012 \times 10{,}000{,}000 = 197{,}400$, i.e. $197,400. The 99% number is larger purely because $z_{0.99} > z_{0.95}$; nothing about the position changed between the two calculations.

Worked example: two positions, and where diversification enters

Add a second $10m position in a different index with $\sigma = 1.5\%$ and correlation $\rho = 0.3$ to the first. Portfolio variance is $\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2$ in dollar terms: $(10{,}000{,}000 \times 0.012)^2 + (10{,}000{,}000\times0.015)^2 + 2(10{,}000{,}000\times0.012)(10{,}000{,}000\times0.015)(0.3)$, which works out to $\sigma_p \approx 210{,}000$, i.e. $210,000. The 99% VaR on the combined $20m book is $2.326 \times 210{,}000 = 488{,}460$, i.e. $488,460. The second position's standalone 99% VaR is $2.326\times0.015\times10{,}000{,}000 = 348{,}900$, i.e. $348,900, so the sum of the two standalone VaRs is $279,120 + $348,900 = $628,020, well above the $488,460 diversified figure. The correlation term is the entire reason the combined number is smaller than the sum of the parts.

What this means in practice

Variance-covariance VaR is the default for large multi-asset books where speed matters more than tail precision, particularly intraday risk checks, because it is a handful of matrix multiplications rather than a simulation. It is at its weakest for option books, where payoffs are not linear in the underlying and a single $\sigma$ cannot capture curvature, see Delta-Gamma VaR for Option Books, and during market stress, where correlations move toward one just when diversification is needed most.