Calculating and Applying VaR

FRM Part I · FRM · 5 min read · Updated: 8 Nov 2026

Welcome to the World of VaR!

Hello there! Today, we are diving into one of the most famous (and essential) concepts in finance: Value at Risk (VaR). If you’ve ever wondered how a bank or hedge fund answers the question, "What is the most we could lose over a given horizon under normal conditions?", you are in the right place. VaR provides a standardized framework to quantify downside risk in a single metric. Let's break it down step-by-step using clear logic and practical examples.

1. What Exactly is VaR?

At its heart, Value at Risk (VaR) measures the minimum loss expected over a specified time horizon at a given significance level, or equivalently, the maximum loss not exceeded with a given confidence level.

To define VaR, you always need two parameters:

1. The Time Horizon ( $T$): The period examined (e.g., 1 day, 10 days, or 1 year).
2. The Confidence Level ( $1 - \alpha$): How sure we want to be (e.g., 95% or 99%, where $\alpha$ is the significance level).

Example: If a portfolio has a 1-day 95% VaR of \$1 million, it means that under normal market conditions, there is only a 5% probability that the portfolio will lose more than \$1 million over a single day.

Three Equivalent Ways to Interpret VaR

• We are 95% confident that the portfolio will not lose more than \$1M over the day.
• There is a 5% probability that the loss will equal or exceed \$1M over the day.
• Across 100 trading days, we expect losses to exceed \$1M on about 5 days.

Key Takeaway: VaR specifies a threshold cutoff along the loss distribution, but it does not specify the magnitude of losses when that threshold is breached.

2. Calculating VaR: The Parametric (Delta-Normal) Method

The Parametric Method (often termed Delta-Normal) assumes that risk-factor returns follow a specific distribution, typically the Normal Distribution.

The Formula

When measuring dollar loss at a significance level $\alpha$ with critical value $z_{\alpha}$, standard deviation $\sigma_{p}$, expected return $\mu_{p}$, and portfolio value $V$:

$ VaR = (z_{\alpha} \times \sigma_{p} - \mu_{p}) \times V $

Note: For short horizons (such as 1 day), the expected return $\mu_{p}$ is often assumed to be zero, simplifying the formula to:

$ VaR = z_{\alpha} \times \sigma_{p} \times V $

Standard Normal Critical Values ( $z$-scores) to Memorize

• For 95% Confidence (one-tailed 5% tail): $ z_{0.05} = 1.645 $
• For 99% Confidence (one-tailed 1% tail): $ z_{0.01} = 2.326 $ (or 2.33)

For the same loss distribution and horizon, 99% VaR is at least as large as 95% VaR. The two can be equal when both confidence levels select the same loss quantile, as can occur with discrete losses.

3. Scaling VaR: The Square Root of Time Rule

When returns are assumed to be independent and identically distributed (i.i.d.) with constant volatility, standard deviation scales with the square root of time:

$ \sigma_{T} = \sigma_{1} \times \sqrt{T} $

Assuming a zero mean return across horizons:

$ VaR_{T} = VaR_{1} \times \sqrt{T} $

Warning: Scaling by $\sqrt{T}$ is only valid under the i.i.d. assumption. If returns exhibit autocorrelation (mean reversion or trending) or volatility clustering, this rule understates or overstates risk.

4. Linear vs. Non-Linear Portfolios and Derivatives

Applying VaR depends heavily on the linear or non-linear profile of the portfolio assets.

Linear Derivatives (Forwards, Futures, Swaps)

Linear contracts have payoffs that scale directly with the underlying asset price. The portfolio value change is approximately:

$ \Delta P \approx \Delta \times \Delta S $

Because the relationship is linear, normally distributed underlying returns yield normally distributed portfolio changes, making the Delta-Normal approach exact and straightforward.

Non-Linear Derivatives (Options)

Options feature asymmetric payoffs driven by curvature, measured by Gamma ($\Gamma$):

$ \Delta P \approx \Delta \times \Delta S + \frac{1}{2} \Gamma (\Delta S)^2 $

• Delta-Normal VaR: Disregards the $\Gamma$ term and treats the option as a linear position equal to its delta equivalent ($\Delta \times S$). While computationally fast, it introduces substantial error for large moves. It understates the risk of short option positions (negative gamma) and overstates the risk of long option positions (positive gamma).
• Full Revaluation: Re-prices options completely using pricing models (e.g., Black-Scholes-Merton) across each simulated scenario or historical shock. This accurately captures gamma and higher-order sensitivities.

5. Historical Simulation and Monte Carlo Simulation

Historical Simulation

This non-parametric approach applies actual historical percentage changes of market risk factors over a look-back window (e.g., past 500 days) to the current portfolio, constructing an empirical distribution of hypothetical profits and losses.

• Strengths: Requires no distributional assumption; naturally incorporates fat tails, skewness, and non-linear option payoffs.
• Weaknesses: Completely backward-looking; cannot anticipate unprecedented shocks; assumes historical observations are equally weighted and independent.

Monte Carlo Simulation

Monte Carlo constructs parametric distributions for risk factors, draws thousands of pseudo-random correlated paths, and applies full revaluation to value the portfolio across each path.

• Strengths: Maximum flexibility; models path-dependent and complex non-linear derivatives; can handle multi-asset interactions.
• Weaknesses: High computational expense; subject to severe model risk and parameter estimation error.

6. Correlation Breakdown and Worst-Case Scenario Analysis

Correlation Breakdown

In standard VaR models, diversification benefits rely on historical correlation estimates between assets. However, in periods of acute market distress, empirical correlations frequently break down:

• Assets that are normally uncorrelated or negatively correlated often exhibit sharp, synchronized declines, driving correlations toward +1.
• This phenomenon, known as correlation breakdown, eliminates expected portfolio diversification precisely when protection is most needed, causing realized losses to far exceed model-projected VaR.

Worst-Case Scenario Analysis vs. VaR

While VaR answers what loss can occur with a specified statistical confidence under normal market conditions, worst-case scenario (WCS) analysis examines the maximum potential loss under severe, plausible stress scenarios regardless of historical probability distributions.

• VaR: Distribution-dependent, percentile-based, and silent on tail severity beyond the cutoff.
• Worst-Case Scenario Analysis: Focuses on extreme tail outcomes, stress scenarios, and structural regime shifts where statistical relationships and correlations fail.

7. Summary of VaR Limitations

1. Tail Blindness: VaR indicates only the threshold loss, revealing nothing about the magnitude of losses beyond that point. Expected Shortfall (ES) is used alongside VaR to measure the average loss in the tail.
2. Non-Subadditivity: VaR is not generally a coherent risk measure because it can violate subadditivity: the VaR of a combined portfolio can exceed the sum of the individual VaRs.
3. Model Risk: Inaccurate assumptions regarding normality, stable correlations, or volatility dynamics lead to distorted risk estimates.

References

2026 FRM Exam Part I Learning Objectives — GARP
2026 FRM Candidate Guide — GARP

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the core definition and interpretation of Value at Risk (VaR)?

VaR is the minimum expected loss over a specified horizon at a given significance level (or the maximum loss not exceeded at a given confidence level).

How is dollar VaR calculated under the Parametric (Delta-Normal) method?

Dollar VaR is given by $ VaR = (z_{\alpha} \times \sigma_{p} - \mu_{p}) \times V $, or $ VaR = z_{\alpha} \times \sigma_{p} \times V $ assuming zero mean return over short horizons.

What are the standard normal critical values ( $z$-scores) for 95% and 99% one-tailed confidence levels?

For 95% confidence, $ z_{0.05} = 1.645 $; for 99% confidence, $ z_{0.01} = 2.326 $ (or 2.33).

What is the square root of time rule and under what condition is it valid?

It scales risk across horizons via $ VaR_{T} = VaR_{1} \times \sqrt{T} $, and it is valid only when returns are independent and identically distributed (i.i.d.).

Why is Delta-Normal VaR well-suited for linear derivatives but problematic for options?

Linear derivatives have payoffs that scale directly with underlying returns, preserving normality; options introduce non-linear curvature (gamma) that distorts the distribution and leads to estimation errors.

What is Historical Simulation and what are its key strengths and weaknesses?

Historical Simulation applies past percentage changes of risk factors to the current portfolio; it requires no distributional assumptions but is entirely backward-looking and assumes equally weighted past observations.

How does Monte Carlo Simulation compute VaR, and what is its primary drawback?

It generates thousands of simulated random price paths based on assumed distributions and fully revalues the portfolio; its main drawback is heavy computational expense and high model/parameter risk.

What is correlation breakdown and how does it impact portfolio VaR during market distress?

Correlation breakdown occurs when assets that are normally uncorrelated move together toward +1 during crises, eliminating expected diversification benefits and causing actual losses to exceed model VaR.

How does Worst-Case Scenario (WCS) analysis differ from Value at Risk (VaR)?

VaR is a probabilistic percentile based on historical distributions under normal conditions, whereas WCS assesses the maximum loss under extreme stress scenarios regardless of statistical probabilities.

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

VaR suffers from tail blindness (ignoring losses beyond the cutoff), non-subadditivity (violating coherent risk criteria), and model risk from incorrect assumptions.

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