Parametric VaR

Historical VaR makes you do the dishes: gather a pile of past returns, sort them, and count to the percentile you want. It’s honest but laborious, and it can only ever tell you about losses that already happened. Parametric VaR — the variance–covariance method — takes a shortcut so brazen it almost feels like cheating: assume the return distribution is a perfect bell curve, and then the entire VaR calculation collapses into a single line of algebra. No sorting. No simulation. Just a mean, a standard deviation, and a number off a z-table.

That assumption is both the method’s superpower and its Achilles’ heel — but let’s earn the payoff before we audit the bill.

Assume a bell, then it’s just algebra

Analogy. Historical VaR is a hand-drawn portrait — you trace every wrinkle of the real data. Parametric VaR is a passport photo: you summarize the whole face with two numbers (location and spread) and trust that the rest follows a known template. If the face really is symmetric and well-behaved, the photo is plenty. If it has a scar in the tail, the photo airbrushes it out.

The idea. Suppose your portfolio’s P&L (or return) over the chosen horizon is normally distributed with mean $\mu$ and standard deviation $\sigma$. The defining magic of the normal distribution is that every quantile is the same fixed number of standard deviations from the mean. Formally, the value at cumulative probability $c$ is

where $z_c$ is the standard-normal quantile (the number of $\sigma$‘s that leaves probability $c$ below it). Once you accept normality, you never touch the data again — you just plug in $\mu$, $\sigma$, and the right $z$. The shape is pre-baked.

This is why the method is also called variance–covariance: the only statistics it needs are the variance of each position and the covariances between them. Everything else is the bell curve doing the work.

Single-asset parametric VaR

Analogy. Think of $\sigma$ as the width of the dartboard and $z_c$ as how many board-widths out you have to walk before only $1-c$ of the darts land beyond you. VaR is just that distance, flipped into a positive loss number.

Definition. For a portfolio of value $V$ whose return has mean $\mu$ and standard deviation $\sigma$, the VaR at confidence level $c$ is

where $z_c$ is the lower-tail standard-normal quantile — a negative number, because we’re looking at the left (loss) side of the bell. The leading minus sign flips the loss into a positive figure (VaR is quoted as a positive dollar amount). The key quantiles:

The $\mu \approx 0$ convention. Over a single trading day, the expected return $\mu$ is minuscule next to the daily $\sigma$ — a stock might drift up 0.04% a day while wobbling 2%. So desks routinely drop $\mu$ for short horizons, leaving the clean workhorse formula $\mathrm{VaR}_c = -z_c\,\sigma\,V = |z_c|\,\sigma\,V$.

Worked example. A $10M book with daily $\sigma = 2\%$ and $\mu \approx 0$:

• 1-day 95% VaR $= 1.645 \times 0.02 = 0.0329$, times $10M $\Rightarrow$ $329,000.
• 1-day 99% VaR $= 2.326 \times 0.02 = 0.04652$, times $10M $\Rightarrow$ $465,200.

Read that out loud: “On a normal day, we’re 95% confident we won’t lose more than $329k, and 99% confident we won’t lose more than $465k.” Same book, same $\sigma$ — only the $z$ changed. Crank the confidence and you walk farther out the same tail.

Scaling across time: the √t rule

Analogy. Volatility compounds like a drunkard’s walk, not a march. Take ten random steps left-or-right and you don’t end up ten paces from home — you end up about $\sqrt{10} \approx 3.2$ paces out, because the steps partly cancel. Risk over time behaves the same way: ten days of randomness doesn’t pile up to ten times the one-day risk.

The rule. For independent, identically distributed (i.i.d.) returns, variance adds across time. Over $h$ days the variance is $h\sigma^2$, so the standard deviation is $\sqrt{h}\,\sigma$. Since VaR is linear in $\sigma$,

Worked example. Our $10M book had a 1-day 95% VaR of $329k. The 10-day VaR is $\sqrt{10} \approx 3.162$ times that: $3.162 \times 329\text{k} \approx 1{,}040\text{k}$, i.e. about $1.04M.

Not $10 \times 329\text{k} = 3{,}290\text{k}$, i.e. $3.29M. Multiplying by the horizon instead of its square root would triple-count the risk — a wildly conservative, capital-wasting error.

The fine print. The $\sqrt{t}$ rule rides on two assumptions: returns are i.i.d. and have zero autocorrelation (today’s move tells you nothing about tomorrow’s). Reality bends both ways:

• Under mean reversion (moves tend to reverse), true multi-day risk is lower than $\sqrt{t}$ predicts — so $\sqrt{t}$ overstates it.
• Under momentum / trending (moves tend to persist), true risk is higher — so $\sqrt{t}$ understates it.

Portfolio VaR with correlation

Analogy. Two rowdy toddlers in one room aren’t twice as loud as one — sometimes they shout together, sometimes one naps while the other yells. Correlation is the “do they tantrum in sync?” dial. Crank it to $+1$ and the noise simply adds; turn it below $1$ and their chaos partly cancels. That cancellation is diversification, and parametric VaR captures it exactly through the covariance term.

Definition. For two assets with weights $w_1, w_2$ (dollar exposures), volatilities $\sigma_1, \sigma_2$, and correlation $\rho$, the portfolio standard deviation is

Then portfolio VaR is just the single-asset formula wearing $\sigma_p$:

The whole game lives in that $2w_1 w_2\rho\sigma_1\sigma_2$ cross-term. When $\rho < 1$, it’s smaller than it would be at perfect correlation, so $\sigma_p$ — and therefore portfolio VaR — comes in below the sum of the standalone VaRs.

Worked example. Two $5M positions (so total $V$ is $10M, with each weight $5M), with daily $\sigma_1 = 2\%$, $\sigma_2 = 3\%$, correlation $\rho = 0.3$. Work in thousands of dollars, $z_{95} = 1.645$.

First the standalone dollar volatilities:

• Asset 1: $5\text{M} \times 0.02 = 100\text{k}$, i.e. $100k.
• Asset 2: $5\text{M} \times 0.03 = 150\text{k}$, i.e. $150k.

Portfolio variance (in units of $\text{k}^2$):

So $\sigma_p = \sqrt{41{,}500} \approx 203.7$, i.e. about $203.7k. The 95% portfolio VaR is $1.645 \times 203.7 \approx 335.1$, i.e. about $335.1k.

Now compare against the sum of standalone VaRs:

The portfolio VaR ($335k) is well under the naive sum ($411k) — a $76k diversification benefit, entirely because $\rho = 0.3 < 1$. Push $\rho$ to $1$ and the cross-term swells until $\sigma_p$ equals $250k and VaR climbs to exactly the $411k sum: at perfect correlation, diversification evaporates.

Strengths and weaknesses

Why desks love it. Parametric VaR is fast — it’s a closed-form formula, no sorting a million scenarios or simulating paths. It’s analytic: you can see exactly how VaR responds to a wiggle in $\sigma$ or $\rho$, which makes it perfect for marginal-risk and “what-if” attribution. And it’s frugal with data — feed it a covariance matrix and you’re done, where historical VaR demands a long, clean return history.

Why it can lie to you. Every one of those virtues rests on the normality assumption, and real markets are emphatically not normal:

• Fat tails. Real return distributions have far more mass in the extremes than a bell curve allows. The normal model treats a $-5\sigma$ day as a once-in-millennia freak; markets serve them up every few years. So parametric VaR systematically underestimates the size and frequency of the worst losses — the exact losses you built a risk model to catch.
• Nonlinear payoffs. The formula treats portfolio P&L as a linear function of the risk factors. For a book stuffed with options, that’s only a local approximation — the so-called delta-normal method captures the first-order delta exposure but ignores gamma, the curvature. A big move makes that linearization wrong, and for a heavily optioned portfolio it can be wrong by a lot.

Putting it together

Parametric VaR is the algebraic shortcut of the VaR world: assume a normal P&L, and every loss quantile is $\mu + z_c\sigma$. Drop the drift for a single day, scale across horizons with $\sqrt{t}$, and stitch positions together through the covariance cross-term so correlation below $1$ pays you a diversification discount. It’s fast, transparent, and data-light — and it quietly lies in exactly the tail and the convexity you most need it to tell the truth about.

Next up — Monte Carlo VaR — we stop trusting a single tidy bell curve and instead manufacture the distribution ourselves: simulate thousands of random future scenarios from whatever dynamics we like (fat tails, jumps, nonlinear option payoffs and all), revalue the portfolio in each, and read VaR off the resulting pile. It’s the heavy artillery for exactly the tails and convexity that parametric VaR airbrushes away.