Value-at-Risk (VaR): Concept and Computations

Key Takeaways
• Value-at-Risk (VaR) is a widely used risk metric that estimates the maximum ex-
pected loss over a specified horizon at a given confidence level.

• It answers the question: “With X% confidence, I will not lose more than $Y over the
next Z days.”

• VaR is simple to communicate and can be computed via parametric (variance–covariance),

historical simulation, or Monte Carlo approaches.

• Although VaR provides a single, intuitive number for potential losses, it relies on as-
sumptions (e.g., normality of returns) and does not capture extreme tail events beyond
the chosen confidence level.

Terminology
• Confidence Level (1 − α): The probability threshold (e.g., 95%, 99%) up to which
losses are not expected to exceed the VaR estimate. For example, a 99% VaR means
there is a 1% chance the loss will exceed the VaR.

• Holding Period (Horizon): The time frame over which VaR is calculated (e.g., 1
day, 5 days, 10 days). Losses are projected over this period.

• VaR Horizon Scaling: When returns are assumed to be independent

√ and identically
distributed with zero drift, multi-day VaR typically scales by H (square-root of time
rule) under the normality assumption.

• Parametric (Variance–Covariance) VaR: Assumes that portfolio returns are nor-

mally (or elliptically) distributed, using mean and covariance of asset returns.

• Historical Simulation VaR: Builds an empirical distribution of portfolio returns

using historical data, then picks the loss at the chosen percentile.

• Monte Carlo VaR: Simulates many future price paths based on assumed stochastic
processes and computes the loss distribution.

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 • VaR Backtesting: Compares realized losses with VaR estimates to check how often
losses exceed the predicted VaR, ensuring the model’s reliability over time.

• Tail Risk: Probability of very large losses beyond the VaR cutoff (e.g., the worst 1%
of outcomes if VaR uses 99% confidence).

Scenario (Illustrative Example)

Portfolio Setup
• Total value: $200,000

• Two assets (A and B), each receiving 50% allocation ($100,000 in A, $100,000 in B)

• Daily volatility of each asset: σA = σB = 0.02

• Correlation between A and B: ρ = 0.30

• Desired horizon: 5 days

• Confidence level: 99%

1. Portfolio Weights
wA = wB = 0.5

2. Daily Portfolio Variance (Parametric approach)

2 2 2 2
Vardaily = wA σA + wB σB + 2 wA wB ρ σA σB
= (0.5)2 (0.02)2 + (0.5)2 (0.02)2 + 2 × 0.5 × 0.5 × 0.30 × 0.02 × 0.02
= 0.25 × 0.0004 + 0.25 × 0.0004 + (0.5 × 0.30 × 0.0004)
= 0.00010 + 0.00010 + 0.00006 = 0.00026
Thus, the daily portfolio volatility is
√
σdaily = 0.00026 ≈ 0.0161245 (≈ 1.6125%).

3. Five-Day Scaling (time rule)

Assuming zero drift and independent returns,
√ √
σ5d = σdaily × 5 = 0.0161245 × 5 ≈ 0.0161245 × 2.23607 ≈ 0.0360555 (≈ 3.6056%).

4. Z-Score for 99% Confidence

For a one-tailed 99% VaR (i.e., 1% in the left tail),

z0.99 ≈ 2.33 (since Φ(−2.33) = 1%).

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5. Dollar VaR Calculation
VaR5d, 99% = z0.99 × σ5d × Portfolio Value
≈ 2.33 × 0.0360555 × 200,000
≈ 2.33 × 7,211.10
≈ $16,800 (rounded).
Result: Over a 5-day holding period, with 99% confidence, the portfolio’s maximum
expected loss is approximately $16,800.

Advantages When Applied

• Simplicity & Communicability

– VaR reduces a complex distribution of potential portfolio outcomes to a single,

intuitive number.
– Nontechnical stakeholders and senior management often find “$X at 99% over 5
days” straightforward to interpret.

• Standardization & Comparability

– Widely adopted by banks, asset managers, and regulators (e.g., under Basel II/III
rules).
– Allows for benchmarking: different desks or portfolios can be compared on a
common risk metric.

• Regulatory Compliance

– Many regulators require VaR-based capital charges (e.g., market risk capital under
Basel III).
– Backtesting frameworks (comparing actual losses vs. predicted VaR) hold firms
accountable.

• Flexibility in Methodology

– You can choose between parametric, historical, or Monte Carlo approaches based
on data availability, computational resources, and desired level of model complex-
ity.
– Parametric (variance–covariance) VaR can be calculated quickly with estimates
of means, variances, and covariances.
– Historical simulation requires no distributional assumptions—directly reuses past
returns.
– Monte Carlo allows for sophisticated, non-normal return distributions and non-
linear payoffs (e.g., options).

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Risks & Limitations When Applied
• Dependence on Distributional Assumptions

– Parametric VaR typically assumes normal (or at least elliptical) returns, which
underestimates tail risk if actual returns exhibit fat tails or skewness.
– During market crises, correlations tend to spike, making the historical covariance
matrix a poor predictor of joint moves.

• Lack of Tail-Risk Information

– VaR only tells you the threshold loss at the chosen percentile. It does not indicate
how large losses can be if the threshold is breached.
– A 99% VaR of $16,800 does not say anything about the magnitude of that 1%
worst loss—losses could be far larger.

• Horizon Scaling Pitfalls

√
– The “ t” scaling rule holds only if returns are independent, identically distributed
with zero autocorrelation.
– Volatility clustering and serial correlation violate this assumption, especially in
stressed markets, leading to under- or overestimation of multi-day VaR.

• Static Correlation Estimates

– Correlations are often unstable, especially under stress when they approach 1
(“correlation breakdown”).
– Using a static correlation (e.g., ρ = 0.30) may underestimate how assets co-move
during extreme market moves, distorting VaR.

• Over-reliance & False Sense of Security

– Solely relying on VaR can mask concentration risks or nonlinear exposures (e.g.,
complex derivative payoffs).
– Firms might optimize portfolios to minimize VaR, inadvertently increasing other
risks (e.g., tail risk or liquidity risk).

Bottom Line
Value-at-Risk is a cornerstone of modern risk management:

• Provides a clear, numeric benchmark for potential portfolio losses at a given

confidence level and horizon.

• Enables comparability across portfolios, desks, and firms, and is integral to regula-
tory capital frameworks.

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 • Easy to communicate but must be complemented by additional metrics (e.g., Ex-
pected Shortfall, stress testing) to capture extreme tail events.

• Model risk is real: reliance on normality, static covariances, and horizon scaling can
lead to underestimation of losses during periods of market stress.

In the illustrative example, a $200,000 portfolio equally split between two 2%-volatility
assets with ρ = 0.30 yields a 5-day, 99% VaR of roughly $16,800. While this single num-
ber provides a convenient “loss ceiling” estimate, it should always be paired with scenario
analyses and stress tests to ensure robustness, especially in out-of-sample or crisis conditions.

Summary
• What is VaR? A quantitative metric estimating the maximum expected loss for a
portfolio over a specified horizon at a certain confidence level.

• Key Components:

1. Horizon: Number of days (e.g., 5 days).

2. Confidence Level: Typically 95% or 99%.
3. Methodology:
– Parametric (variance–covariance): uses mean, volatility, and covariance; easy
and fast but assumes normality.
– Historical Simulation: nonparametric; re-samples past returns but requires a
sufficiently long history.
– Monte Carlo: simulates draws from assumed distributions; flexible but com-
putationally heavier.

• Illustrative Example Recap:

– Portfolio of $200,000, split 50/50 between two assets (daily σ = 2%, ρ = 0.30).
– Daily portfolio σ ≈ 1.6125%; 5-day σ ≈ 3.6056%.
– 99% VaR (5 days) = 2.33 × 3.6056% × $200,000 ≈ $16,800.

• Advantages: Simplicity, standardization, regulatory alignment, and multiple compu-

tational approaches.

• Risks:

– Underestimation of tail risk (fat tails, skewness).

– Static correlations that break down in crises.
– Independence assumptions for horizon scaling.
– Potential misuse if portfolios are optimized solely to minimize VaR.

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• Bottom Line: VaR is a powerful, standardized tool for quantifying market risk but
must be used alongside stress testing, backtesting, and complementary risk measures
(e.g., Expected Shortfall, scenario analysis) to capture the full spectrum of potential
losses.