VaR and CVaR: Calculating Maximum Losses

VaR and CVaR: calculating maximum losses. Value at Risk (VaR) is the main quantitative indicator of market risk, used by institutional investors, banks, and regulators to assess the potential losses of an investment portfolio. VaR answers the question: “What is the maximum amount that a portfolio can lose over a given period of time with a specified probability under normal market conditions?” For example, VaR(95%, 1 day) = $2.5M means: with a probability of 95%, portfolio losses over one trading day will not exceed $2.5 million. Or equivalently: in 5% of trading days (roughly 12–13 days a year), losses may exceed $2.5 million. VaR was formalized by JP Morgan’s quantitative analysis group (RiskMetrics, 1994) and became a risk management standard after the adoption of the Basel Accord (Basel II, 2004), which required banks to calculate market risk using VaR to determine capital adequacy.

Historical VaR Calculation Method (Historical VaR)

Historical VaR (Historical Simulation VaR) is the most intuitive method, not requiring assumptions about the distribution of returns. Algorithm: collect historical daily portfolio returns over a certain period (usually 1–3 years, 250–750 trading days); sort the returns from worst to best; VaR(95%) = the value at the 5th percentile (for 250 observations — the 12th or 13th worst return).

Advantages:

• Does not require assumptions about normality of the distribution;
• Automatically accounts for fat tails, skewness, and non-linearities (relevant for portfolios containing options);
• Easy to explain to non-technical management.

Disadvantages:

• Completely dependent on the historical period — VaR calculated on 2017–2019 data (low volatility) will be radically different from VaR based on 2020–2022 data (high volatility);
• Cannot predict events with no historical precedent (for example, COVID-19 pandemic for data before 2020);
• Ghost Effect — an extreme event “drops out” of the observation window after 1–3 years, which leads to a sharp and often unjustified decrease in VaR.

Parametric VaR Calculation Method (Parametric VaR)

Parametric VaR (Parametric VaR, Variance-Covariance VaR) assumes a normal distribution of portfolio returns and calculates VaR using statistical parameters — mean return ($\mu$) and standard deviation ($\sigma$). The formula:

$ VaR(\alpha) = \mu - z(\alpha) \times \sigma \times \sqrt{T}, $

where $z(\alpha)$ is the quantile of the standard normal distribution ($z(95%) = 1.645$, $z(99%) = 2.326$), $\sigma$ is the portfolio volatility, $T$ is the time horizon in days.

For a portfolio with daily volatility of 1.5% and value of $100M: $ VaR(95%, 1\ \text{day}) = $100\text{M} \times 1.645 \times 1.5% = $2.47\text{M}. $

For 10-day VaR: $ $2.47\text{M} \times \sqrt{10} = $7.8\text{M}. $

Advantages:

• Fast calculation, suitable for large portfolios with thousands of positions;
• Easily decomposed into Component VaR (the contribution of each position to total VaR) and Marginal VaR (change in VaR from adding one unit of a position).

Critical disadvantage: the assumption of normality. Real financial returns exhibit leptokurtosis — a sharper peak and fatter tails than the normal distribution. This means systemic underestimation of tail risks: a “–5$\sigma$” event, which by normal distribution should happen once in 14,000 years, in practice occurs once every 50–70 years.

Monte Carlo Method (Monte Carlo VaR)

The Monte Carlo Method (Monte Carlo Simulation VaR) generates thousands or tens of thousands of random scenarios of future returns based on a given model (distribution, correlation, volatility) and estimates VaR as the corresponding percentile of the resulting distribution of outcomes.

Advantages:

• Can use any distribution ($t$-distribution with fat tails, mixture distributions, regime-switching models);
• Correctly evaluates non-linear positions (options, structured products);
• Allows modeling of stochastic volatility (Stochastic Volatility — Heston, SABR models) and jumps (Jump-Diffusion — Merton models).

For portfolios with options and structured products, Monte Carlo is the only correct VaR calculation method, as the non-linearity of the payoff function makes the parametric approach inapplicable.

Disadvantages:

• Computational complexity (for a portfolio of 500 positions with 10,000 scenarios — 5 million revaluations);
• Model risk — the quality of VaR is entirely determined by the model’s quality (“garbage in, garbage out”).

CVaR: Overcoming VaR Limitations

Conditional Value at Risk (CVaR, also Expected Shortfall, ES) is a tail risk metric that answers the question: “What is the expected average loss if the loss exceeds VaR?” If VaR(95%) = $2.5M, then CVaR(95%) = $4.0M means that in those 5% worst days, the average loss is $4.0M. CVaR is always greater than VaR and is especially informative for portfolios with fat-tailed distributions.

Mathematically: $ CVaR(\alpha) = E[\text{Loss} \mid \text{Loss} > VaR(\alpha)] $ — the conditional expectation of losses greater than VaR.

Advantages of CVaR over VaR:

• CVaR is a coherent risk measure, satisfying the property of subadditivity: $CVaR(A + B) \leq CVaR(A) + CVaR(B),$ that is, diversification always reduces CVaR. VaR does not possess this property: there are situations when the VaR of a portfolio exceeds the sum of the VaRs of its individual positions — diversification “increases” risk according to VaR, which is illogical.
• The Basel Committee (Basel III.1, also known as FRTB — Fundamental Review of the Trading Book) replaced VaR with Expected Shortfall (CVaR) as the standard market risk metric for banks starting in 2025.

VaR Limitations and Tail Risk Hedging

VaR has a number of fundamental limitations, which are critical to understand. VaR tells nothing about the magnitude of losses beyond the threshold — VaR(95%) = $2.5M does not reveal whether the losses in the worst 5% of cases will be $3M or $30M. VaR assumes “normal market conditions” and is not applicable to extreme events (Black Swan events). VaR is procyclical: in calm periods, VaR decreases, investors take on more risk, enhancing the future shock; during crises, VaR spikes, investors cut positions, worsening the decline.

Tail risk hedging through OTM (Out-of-The-Money) options is a practical tool for protection against events lying beyond VaR. Buying deep OTM put options on the S&P 500 (strike 80–90% of spot) provides nonlinear protection: negligible cost in normal times and huge payoff in a crisis. The Tail Risk Parity (TRP) strategy, developed by Universa Investments’ Mark Spitznagel, allocates 1–3% of the portfolio to systematic OTM put purchases, allowing a 10–15% increase in the main portfolio’s risk allocation. The Put Spread Collar (buying a put + selling a call + selling a deep OTM put) reduces hedge cost to 0.5–1.0% of notional by restricting protection range and giving up potential gains above a certain level. Variance Swap — a derivative on realized volatility — allows one to monetize volatility increases without needing to predict market direction: a long variance swap with notional $100K/vega point yields ~$100K per point rise in realized volatility above the strike.

Practical Recommendation for a Large Portfolio Manager

Use VaR and CVaR as complementary metrics. Daily monitoring: VaR(95%, 1 day) and CVaR(95%, 1 day) for each asset class and for the overall portfolio. Limits: set maximum VaR(99%, 10 days) no more than 10–15% of NAV. Backtesting: quarterly VaR model review — if actual losses exceed VaR more often than intended by the confidence level (over 5–6 breaches out of 250 days for 95% VaR), the model needs recalibration. Stress-testing: supplement VaR/CVaR with scenario analysis — applying historical crisis scenarios (2008, 2020, 2022) and hypothetical shocks to the current portfolio.