Using the parametric method, compute Value at Risk: VaR = |expected return − z-score × √days × standard deviation| × portfolio value.
Input Data
Results
At a glance:Parametric VaR scales the horizon volatility (daily sigma times √days) by the z-score, subtracts the expected return, and applies it to the portfolio value.
Formula
varResult = |expectedReturn − zScore × sqrt(days) × standardDeviation| × portfolioValue
$$$\\mu$ $\\sigma$ $t$ $V$$$$$$z_{95\\%}=1.645,\\; z_{99\\%}=2.326$$$How to Use
- Enter the portfolio value and expected return.
- Enter the standard deviation, horizon days, and z-score.
- Read the Value at Risk.
FAQ
What does VaR actually mean?
At a 95% confidence level with a VaR of HK$30,000, it means there is a 95% chance the loss will not exceed HK$30,000 over that horizon; in other words, only about 5% of the time will the loss surpass that amount. It gives an intuitive estimate of the maximum likely loss.
What are the z-scores for 95% and 99% confidence?
Under the normal-distribution assumption, the one-tailed z-score is 1.645 at 95% confidence and 2.326 at 99%. A higher confidence level gives a larger z-score and a larger (more conservative) VaR.
What are the limits of VaR?
VaR does not measure the extreme tail loss beyond the threshold, and it relies on the assumption that returns are normally distributed; market shocks or fat tails can make it understate risk. It should be used together with stress testing and Expected Shortfall (ES/CVaR).
What is the difference between parametric VaR, historical simulation, and Monte Carlo—and which does this calculator use?
There are three main VaR methods, each with pros and cons; this calculator uses the 'parametric method'. First, 'parametric (variance-covariance, also called variance-covariance method)': the method here, from RiskMetrics. It 'assumes returns are normally distributed'; knowing just the mean (expected return) and standard deviation (volatility), multiplied by the z-score for the confidence level, yields the VaR. Advantages: simple, fast, little data needed (just mean and standard deviation). Disadvantage: it leans heavily on the 'normal distribution' assumption, while real financial returns often have 'fat tails'—extreme events occur more often than the normal distribution predicts—so parametric VaR tends to 'understate' tail risk in turbulent markets. Second, 'historical simulation': assumes no distribution; it takes the actual returns over a past period, sorts them worst to best, and takes the quantile at the confidence level as the VaR (e.g. 95% VaR takes the worst 5% quantile in history). Advantage: no distribution assumption, naturally reflects real historical fat tails. Disadvantage: fully dependent on historical data—if a certain extreme never happened in the window, it cannot be estimated; also sensitive to the chosen window. Third, 'Monte Carlo simulation': sets a statistical model for returns (normal, or a more complex fat-tail-capturing one), uses a computer to randomly simulate many (e.g. tens of thousands) possible future scenarios, then takes the quantile from the simulated distribution. Advantage: most flexible, handles complex portfolios and non-linear products (e.g. options), can embed realistic distribution assumptions. Disadvantage: heavy computation, needs modelling skill, and quality depends on the chosen model. The choice depends on need: quick and simple with a plain portfolio uses parametric; to reflect the real historical distribution uses historical simulation; a complex or derivative-containing portfolio seeking rigour uses Monte Carlo. This calculator uses the parametric method—good for a quick estimate and understanding VaR conceptually—but always remember its normal-distribution assumption.
What are the limits of VaR, and why is it said not to reflect the 'worst case'? How to make up for it?
VaR is intuitive and widely used in regulation and risk management, but it has important limits users must know, or its 'sense of security' can mislead. First, and most critical: 'VaR does not tell you how much you lose beyond the threshold'. VaR answers 'in 95% (or 99%) of cases, the loss will not exceed X', but for the remaining 5% (or 1%)—the 'tail', the true 'worst case'—VaR says nothing. Example: two portfolios may both have a 95% VaR of HK$1m, but one loses at most HK$1.2m in the worst 5% while the other could lose HK$5m—VaR shows them as identical, masking the huge tail difference. This is VaR's most criticised flaw; in the 2008 crisis, over-reliance on VaR while ignoring tail risk was a key lesson. Second, 'depends on the distribution assumption': parametric VaR assumes normality, but real markets have fat tails, so VaR often understates extreme risk. Third, 'depends on history/inputs': VaR uses historical volatility or set standard deviations, but 'a calm past does not mean a calm future'—structural regime changes make historical parameters inaccurate. Fourth, 'non-subadditivity': in some cases, the combined VaR of two portfolios can exceed the sum of their individual VaRs, violating the 'diversification should lower risk' intuition, so VaR is not a 'coherent risk measure' mathematically. How to make up for these? First, 'pair with Expected Shortfall (ES, also CVaR)': ES computes exactly 'when the loss exceeds the VaR threshold, on average how much is lost', directly filling the tail VaR cannot see—this is why Basel has gradually replaced VaR with ES for market-risk regulation. Second, 'conduct stress testing and scenario analysis': actively assume extreme but plausible scenarios (rate spikes, market crashes) and see the loss under those 'worst cases',compensate for VaR's blindness to the tail. Third, 'cross-validate with multiple methods and confidence levels': view parametric, historical and different-confidence VaRs together to avoid single-assumption bias. Fourth, 'combine with real downside metrics like maximum drawdown'. In short, VaR is a useful 'daily risk dashboard' but not the whole of risk—especially not for judging 'how much can I lose at worst', which must be answered by ES and stress testing.
References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.