Value at risk from the normal distribution

VaR is a one-sided confidence interval

Value at risk (VaR) is the lower bound of a one-sided confidence interval for the change in value: a loss that will be exceeded only with a small probability over a stated horizon.

There are two ways to state it. As a percentage: a one-day 95% VaR of 2% means a 5% chance of losing more than 2% of the value in one day. As an amount: a one-month 99% VaR of EUR 2.8 million means a 1% chance of losing more than EUR 2.8 million in a month. Every VaR has three parts: an amount or percentage, a horizon and a confidence level.

It is one-sided because a risk manager cares about the loss tail only, so all of α sits in the left tail.

The parametric formula

Memorize

The parametric variance–covariance method assumes returns are normal and describes them with the standard deviation. Under the zero-mean assumption , which is common practice, especially over short horizons, the formula is:

$$\text{VaR} = Z \times \sigma \times P$$

Z is the one-tailed critical value for the confidence level, σ is the standard deviation of returns over the VaR horizon and P is the current value of the position. Match σ to the horizon: daily σ for one-day VaR, monthly σ for one-month VaR.

Confidence	One-tailed Z	Chance of a bigger loss
95%	1.645	5%
99%	2.33	1%

These are one-tailed values: all of α is in the loss tail. 1.96 and 2.576 are the two-sided values used for confidence intervals.

Switch between 95% and 99%, then turn on the two-sided Z to see how far the common slip moves the threshold.

Three ways to estimate VaR

Ways to estimate VaRthree methods; only the first is computed in this lesson

• Parametric variance–covariance Use it assumes normal returns; this lesson
• Historical simulation Know the idea replays past returns; Learning Module 9
• Monte Carlo simulation Know the idea simulates many random paths; Learning Module 9

This lesson computes only the parametric version. Historical simulation and Monte Carlo simulation VaR are built in Learning Module 9.

What does the formula leave out?

Use it

The limitations of parametric VaR all come from its assumptions.

Why parametric VaR can mislead

• Normality Use it real returns have skew and fat tails, so extreme losses are understated
• Same distribution at every horizon Use it return behavior can change with the horizon
• History Use it parameters from the past may not fit an unprecedented market
• Linearity Use it options and embedded options respond non-linearly

• Normality. Fat tails make large losses more frequent than the normal curve says, so VaR is understated.
• One distribution at every horizon. The shape of returns can differ between a day and a year.
• Reliance on history. A calm history understates the risk of a crisis nobody has seen.
• Linearity. An option’s value does not move in proportion to its underlying, so Z × σ × P misstates its risk.

Exam tip VaR is a threshold in the loss tail, never the worst case. Read “99% VaR of X” as “a 1% chance of losing more than X”.

Go deeper VaR in risk management

Parametric VaR is one tool in a risk manager’s kit. The risk-management lesson on VaR, conditional VaR and sensitivities places it beside measures that describe what happens beyond the threshold, which VaR itself does not. No new formulas are needed here.