Value at Risk Calculator

Understanding Value at Risk (VaR)

Value at Risk (VaR) is a widely used risk management metric in finance that estimates the maximum potential loss an investment portfolio could experience over a given time frame for a set confidence level.

For instance, if a portfolio has a 1-day 95% VaR of $10,000, it means there is a 95% probability that the portfolio's losses will not exceed $10,000 in a single day. Conversely, there is a 5% chance that the losses will be greater than $10,000.

VaR Calculation Methods

This calculator supports two primary methods for calculating Value at Risk:

1. Parametric (Variance-Covariance) Method

The parametric method assumes that the returns of the portfolio follow a normal distribution. It utilizes the mean (expected return) and standard deviation (volatility) of returns.

First, the expected return and volatility are adjusted for the time horizon $t$ (in days):

\[ \mu_{adj} = \mu \times t \] \[ \sigma_{adj} = \sigma \times \sqrt{t} \]

Then, the Value at Risk percentage is calculated as:

\[ \text{VaR (\%)} = -(\mu_{adj} - Z \times \sigma_{adj}) \]

Where $Z$ is the critical value (Z-score) corresponding to the desired confidence level:

For 90% confidence level: $Z = 1.282$
For 95% confidence level: $Z = 1.645$
For 99% confidence level: $Z = 2.326$

2. Historical Simulation Method

The historical simulation method is non-parametric and does not assume a normal distribution. It uses actual historical daily returns to predict potential future losses.

Collect a series of historical returns.
Sort them from worst (largest loss) to best (largest gain).
Identify the percentile rank corresponding to the significance level $100 - \text{Confidence Level}$.
Interpolate the value if the percentile falls between two data points.
Scale the daily VaR percentage by the square root of time horizon ($\sqrt{t}$).

Frequently Asked Questions

What are the limitations of Value at Risk (VaR)?

VaR only tells you the threshold of losses at a specific confidence level; it does not indicate the magnitude of losses beyond that threshold. For example, if your 95% VaR is $10,000, you don't know if the worst 5% of outcomes average a loss of $11,000 or $1,000,000.

Why is time horizon important in VaR?

Risk accumulates over time. A 10-day VaR is larger than a 1-day VaR because there is more time for market prices to swing. By default, VaR scales with the square root of time: $\text{VaR}_{t\text{-day}} = \text{VaR}_{1\text{-day}} \times \sqrt{t}$.

What is the difference between Parametric and Historical VaR?

Parametric VaR assumes returns are normally distributed and is easy to compute but can underestimate the risk of extreme events (fat tails). Historical VaR does not assume a distribution but relies entirely on past data, which may not repeat in the future.

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