Value at Risk (VaR) Explained: Parametric, 95/99%, and CVaR
7 min read·
Value at Risk estimates how much a portfolio could lose on a bad day at a given confidence level. Here is how parametric VaR, 95/99%, and CVaR actually work — with a worked example and their real limits.
What Value at Risk actually measures
Value at Risk (VaR) answers one question: over a chosen horizon, how much could your portfolio lose at a given confidence level? A 1-day 95% VaR of $4,000 means that on 95% of days you would not expect to lose more than $4,000 — and on roughly 1 day in 20 (the worst 5%), losses could exceed it. VaR is a threshold, not a worst case. It deliberately ignores what happens in that tail beyond the cutoff, which is both its appeal (one clean number) and its biggest weakness. Always state all three parts: horizon (e.g. 1-day), confidence (95% or 99%), and currency or percentage of the book.
Parametric (variance-covariance) VaR
The parametric method assumes returns are roughly normally distributed, so VaR is just a multiple of volatility. The formula is: VaR = Portfolio Value × daily volatility × z. The z is the standard-normal critical value for your confidence level: 1.645 for 95% and 2.326 for 99%. You estimate daily volatility as the standard deviation of recent daily returns (StoqPulse uses one year of daily closes, ~252 trading days). It is fast, transparent, and needs only a volatility estimate. The trade-off: real market returns have fatter tails than a normal curve, so parametric VaR tends to understate extreme losses.
95% vs 99%: choosing a confidence level
The confidence level sets how deep into the loss tail you cut. 95% VaR uses z = 1.645 and is breached about 1 day in 20; 99% VaR uses z = 2.326, is breached about 1 day in 100, and is therefore a larger number. Neither is 'correct' — they answer different questions. 95% is a practical day-to-day gauge of typical bad days; 99% is a more conservative, tail-focused view favoured for stress and capital planning. Reporting both (as the Risk page does) frames a range: the routine-bad-day loss and the rarer, deeper loss, so you are never anchored to a single figure.
Worked example
Suppose a $100,000 portfolio with a daily return standard deviation of 1.5% (0.015). Parametric 95% VaR = 100,000 × 0.015 × 1.645 = $2,468. The 99% VaR = 100,000 × 0.015 × 2.326 = $3,489. So on a typical bad day you would not expect to lose more than ~$2,500, and the rarer 1-in-100 day threshold sits near ~$3,500. Note both scale linearly with volatility: if daily vol doubles to 3%, both VaR figures double. That linearity is exactly why VaR is a useful relative gauge — it moves directly with how turbulent your holdings have become.
CVaR / expected shortfall: looking past the cliff
VaR tells you the threshold; it says nothing about how bad the worst days get beyond it. Conditional VaR (CVaR), also called expected shortfall, fixes that by averaging the losses in the tail past the VaR cutoff. For a normal distribution at 95%, the expected-shortfall multiple is φ(z) ÷ (1 − 0.95), where φ(z) is the normal density at z = 1.645 — about 2.063, versus 1.645 for VaR. So CVaR is always larger than the matching VaR. In the example above, 95% expected shortfall ≈ 100,000 × 0.015 × 2.063 = $3,095. CVaR is the better number when tail severity, not just frequency, is what you care about.
The limits you should respect
Parametric VaR assumes normal, stable, and independent returns — none of which hold perfectly. Real returns are fat-tailed, so true 95%/99% losses are breached more often than the model implies; volatility clusters, so a calm trailing year understates risk just before a shock; and correlations between holdings spike toward 1 in a crash, concentrating losses VaR computed in calmer times. VaR is also horizon- and window-dependent: a 1-day estimate is not a crisis forecast. Treat VaR and CVaR as a disciplined gauge of current exposure, cross-checked against beta, drawdown, and concentration — not a guarantee or a prediction.
How StoqPulse computes your portfolio VaR
On the StoqPulse Risk page, VaR is computed deterministically — never by an AI. We pull one year of daily closes for each holding and SPY, build a portfolio value series over the dates common to all positions, and derive its daily returns. From that we report 1-day VaR three ways — parametric (z = 1.645 at 95%, z = 2.326 at 99%), historical from the empirical return distribution, and a seeded Monte-Carlo bootstrap — plus 95% CVaR, each in dollars and as a percentage of your book. Around it sit non-normal tail stats (Sortino, skewness, excess kurtosis), benchmark-relative analytics vs SPY (tracking error, active return, information ratio, up/down capture), per-holding risk contribution (marginal and component VaR, share of portfolio volatility) and a correlation heatmap, alongside beta, annualised volatility, max drawdown, and Sharpe. The numbers are pure math; an optional AI narrative only explains them in plain English. US coverage is free for 14 days (no card).
FAQ
What does a 95% 1-day VaR of $2,500 mean?
On about 95% of days you would not expect to lose more than $2,500 over one day. On roughly 1 day in 20 — the worst 5% — losses could exceed it, and VaR does not say by how much. CVaR/expected shortfall answers that part.
Why is 99% VaR larger than 95% VaR?
Because a higher confidence level cuts deeper into the loss tail. Parametric VaR multiplies volatility by a z-score: 1.645 at 95% and 2.326 at 99%. The larger z makes the 99% figure bigger and more conservative, capturing rarer, deeper losses.
What is the difference between VaR and CVaR?
VaR is the loss threshold at a confidence level — the edge of the cliff. CVaR (conditional VaR / expected shortfall) is the average loss in the tail beyond that threshold. CVaR is always larger and better reflects how severe the worst days could be.
What are the main weaknesses of parametric VaR?
It assumes normally distributed, stable returns. Real markets have fat tails (so extremes happen more than modelled), volatility clusters (a calm window understates risk), and correlations spike in crashes. So parametric VaR tends to understate true tail losses and should be cross-checked with drawdown and concentration.
Does StoqPulse calculate VaR automatically?
Yes. The Risk page computes 1-day VaR three ways — parametric, historical and a seeded Monte-Carlo bootstrap — at 95% and 99% plus 95% CVaR, directly from one year of your holdings' daily prices, alongside tail stats, benchmark capture vs SPY, per-holding risk contribution, a correlation heatmap, beta, volatility, drawdown, and Sharpe. The math is deterministic; any AI commentary only explains it. US coverage is free for 14 days (no card).
Put this to work in StoqPulse
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