Value at Risk

What Is VaR?

Value at Risk (VaR) answers a simple question: "What is the most I can expect to lose over a given period at a given confidence level?" For example, a 1-day 95% VaR of $1M means there is a 5% chance of losing more than $1M in a single day.

Formally, VaR at confidence level $\alpha$ is the $\alpha$-quantile of the loss distribution:

$P(L > \text{VaR}_\alpha) = 1 - \alpha.$

Three Approaches

Parametric (Variance-Covariance)

Assume returns are normally distributed with mean $\mu$ and standard deviation $\sigma$:

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

where $z_\alpha$ is the standard normal quantile (e.g., $z_{0.95} = 1.645$, $z_{0.99} = 2.326$).

For a portfolio with value $V$:

$\text{VaR}_\alpha = V \cdot (z_\alpha \, \sigma \sqrt{\Delta t} - \mu \, \Delta t).$

Historical Simulation

Sort the last $n$ daily P&L observations. The VaR at 95% is the 5th percentile of that empirical distribution. No distributional assumptions needed, but it relies on the past being representative of the future.

Monte Carlo

Simulate thousands of scenarios from a fitted model, compute the portfolio P&L for each, and take the appropriate quantile. Most flexible: handles non-linearities, fat tails, and complex portfolios.

Scaling VaR

Under the i.i.d. assumption, scale from 1-day to $T$-day VaR:

$\text{VaR}_T = \text{VaR}_1 \cdot \sqrt{T}.$

This $\sqrt{T}$ rule comes from the CLT and is widely used despite its limitations (autocorrelation, volatility clustering).

Worked Example

A portfolio has daily returns with $\mu = 0$ and $\sigma = 1.5\%$. Portfolio value is $10M. Compute the 1-day 99% VaR.

$\text{VaR}_{99\%} = 10{,}000{,}000 \times 2.326 \times 0.015 = 348{,}900.$

There is a 1% chance of losing more than $349K in a single day.

One-day P&L of the $10M portfolio: the 1% tail beyond the 99% VaR

Limitations

• VaR says nothing about how bad losses can be beyond the threshold.
• It is not subadditive: combining two portfolios can increase VaR, violating the diversification principle.
• These limitations motivate Expected Shortfall (next lesson).

Interview tip: Always mention VaR's limitations unprompted. Interviewers want to see you understand it's a flawed but practical tool, not a complete risk measure.