VaR / CVaR Methodology

Multi-asset risk engine: 3-day 99% Conditional Value at Risk with GFC stress augmentation.

Overview

PrimeRisk computes Conditional Value at Risk (CVaR), also known as Expected Shortfall (ES), for portfolios spanning four asset classes:

• Equities (stocks, ETFs)
• US Treasuries (on-the-run and off-the-run)
• Corporate Bonds (investment grade and high yield)
• Convertible Bonds (hybrid equity + credit)

CVaR answers the question: "Given that we are in the worst 1% of outcomes, what is the average loss?" This is more conservative than VaR alone, which only identifies the threshold.

Default Parameters

Methodology by Asset Class

Equities

Risk factor: Individual stock return volatility.

dailyVol = annualVol / sqrt(252)
3-day VaR = |MV| × dailyVol × sqrt(3) × z_0.99

Where z_0.99 = 2.326 (99th percentile of standard normal).

US Treasuries

Risk factor: Yield volatility by maturity bucket, translated to dollar P&L via DV01.

dailyYieldVol = bucketYieldVol / sqrt(252)    [in basis points]
dailyDollarVol = DV01 × dailyYieldVol
dailyReturnVol = dailyDollarVol / |MV|
3-day VaR = |MV| × dailyReturnVol × sqrt(3) × z_0.99

Yield Volatility by Maturity Bucket (annualized, basis points):

DV01 (Dollar Value of a Basis Point):

DV01 = MV × ModifiedDuration × 0.0001

If modified duration is not supplied, it is approximated from years to maturity and coupon using a semi-annual compounding model with a 4.25% yield proxy.

Corporate Bonds

Risk factor: Combined rate risk + credit spread risk.

totalYieldVol = sqrt(rateVol² + spreadVol²)

Rate risk uses the same Treasury yield volatility table. Spread risk uses a rating-dependent spread volatility.

Credit Spread Volatility by Rating (annualized, basis points):

The combined daily volatility is:

dailyTotalVol = sqrt(rateVol² + spreadVol²) / sqrt(252)
dailyDollarVol = DV01 × dailyTotalVol

This assumes low correlation between rate moves and spread moves — they can move independently (rates fall while spreads widen in a flight-to-quality scenario).

Convertible Bonds

Risk factor: Hybrid equity delta + credit spread risk with negative correlation.

Convertible bonds have both equity sensitivity (through the embedded call option) and credit risk (as a corporate bond). These are combined using a negative correlation of -0.30, reflecting the empirical observation that equity rallies often coincide with spread tightening and vice versa.

equityComponent = delta × equityDailyVol
creditComponent = spreadDailyVol × modDur × 0.0001
combinedVol = sqrt(equity² + credit² + 2 × (-0.30) × equity × credit)

VaR Computation

Two methods are computed for each position; the conservative (higher) value is used.

Parametric VaR

Assumes normally distributed returns.

VaR_99 = |MV| × dailyVol × sqrt(holdingPeriod) × z_0.99
CVaR_99 = |MV| × dailyVol × sqrt(holdingPeriod) × phi(z_0.99) / (1 - 0.99)

Where:

• z_0.99 = 2.326 (inverse normal CDF at 99%)
• phi(z) = e^(-z²/2) / sqrt(2π) (standard normal PDF)
• phi(2.326) / 0.01 ≈ 2.665

For a $10M position with 25% annual equity vol:

• Daily vol = 25% / sqrt(252) = 1.575%
• 3-day vol = 1.575% × sqrt(3) = 2.728%
• VaR_99 = $10M × 2.728% × 2.326 = $634,734
• CVaR_99 = $10M × 2.728% × 2.665 = $727,024

Monte Carlo VaR

Generates synthetic P&L distributions with a seeded PRNG for reproducibility.

1. Generate ~2,520 normal-period daily returns using position-specific daily vol
2. Generate ~378 GFC-stub returns at 2x the daily vol (stress period)
3. Scale daily returns to 3-day returns via sqrt(3)
4. Sort the combined sample (2,898 observations)
5. VaR_99 = loss at the 1st percentile
6. CVaR_99 = average of all losses beyond VaR_99 (~29 observations)

The stub ensures the GFC is always represented in the tail, even though it may be outside the 10-year lookback window.

Portfolio VaR with Diversification

Position-level VaRs are aggregated using a correlation-adjusted approach:

Portfolio_VaR² = Σ_i Σ_j sign_i × sign_j × VaR_i × VaR_j × ρ_ij
Portfolio_VaR = sqrt(Portfolio_VaR²)

Where sign accounts for long (+1) vs. short (-1) positions, and ρ_ij is the estimated correlation between positions.

Cross-Asset Correlation Matrix

Key relationships:

• Equity ↔ Treasury (-0.20): Flight-to-quality effect — when stocks fall, Treasuries rally
• Equity ↔ Corporate (0.40): Credit and equity risk are correlated
• Equity ↔ Convertible (0.60): Converts have high equity delta sensitivity
• Treasury ↔ Corporate (0.30): Corporates have rate duration but also spread risk

Diversification Benefit

Diversification = 1 - (Portfolio_VaR / Undiversified_VaR)

Where Undiversified_VaR is the simple sum of all position VaRs. A portfolio with long equities and long Treasuries will show meaningful diversification due to the negative equity-Treasury correlation.

GFC Stub Period

The 18-month stress window (January 2008 – June 2009) captures the most severe fixed income and credit market dislocations in modern history:

The stub is applied at 2x normal volatility, which is conservative relative to actual GFC vol (2-3x in equities, 3-5x in credit spreads, 1.5-2x in Treasuries).

By always including this period, the VaR estimate never "forgets" the GFC even as the 10-year rolling window moves forward.

Comparison: CVaR vs. Haircut Margin

The FICC margin calculator shows both:

If CVaR exceeds the haircut margin, it suggests the published haircut schedule may understate risk for that portfolio (e.g., concentrated duration exposure or high-vol maturity buckets).

Implementation

import { computeVaR } from '@/lib/var'

const report = computeVaR([
  {
    id: 'pos-1',
    ticker: 'AAPL',
    assetClass: 'equity',
    side: 'long',
    marketValue: 10_000_000,
    vol: 0.28,
  },
  {
    id: 'pos-2',
    ticker: '912810TV0',
    assetClass: 'treasury',
    side: 'long',
    marketValue: 50_000_000,
    modifiedDuration: 18.5,
    yearsToMaturity: 28,
  },
  {
    id: 'pos-3',
    ticker: 'XYZ-CONVERT',
    assetClass: 'convertible',
    side: 'long',
    marketValue: 5_000_000,
    equityDelta: 0.65,
    rating: 'BB',
    underlyingTicker: 'XYZ',
  },
])

console.log(report.portfolioCvar99)    // 3-day 99% CVaR
console.log(report.diversificationPct) // diversification benefit %

Module Structure

Limitations

1. Parametric vol proxies: Volatility estimates use built-in lookup tables, not live market data. For production use, feed actual implied vols and historical yield series.

2. Normal distribution assumption: Parametric VaR assumes normality. The Monte Carlo component with GFC stub partially addresses fat tails, but extreme events can still exceed CVaR.

3. Static correlations: Cross-asset correlations are fixed constants. In practice, correlations spike during crises (correlation breakdown). The GFC stub partially compensates.

4. No term structure modeling: Treasury VaR uses parallel yield shifts per bucket, not a full term structure model (e.g., Nelson-Siegel or PCA-based).

5. No jump-to-default: Corporate bond VaR captures spread widening but not sudden default events. For distressed credits (CCC and below), a separate jump-to-default model would be more appropriate.

6. Convertible simplification: The convert model uses a fixed delta and linear equity sensitivity. A full convertible model would incorporate gamma, vega, and credit-equity coupling dynamics.

Sources

• Basel Committee on Banking Supervision — "Minimum capital requirements for market risk" (FRTB), Jan 2019
• DTCC/FICC — "GSD Clearing Fund Methodology" v2.1
• RiskMetrics — J.P. Morgan Technical Document, 4th Edition (parametric VaR)
• Carol Alexander — "Market Risk Analysis, Volume IV: Value-at-Risk Models" (Wiley, 2009)
• Acerbi & Tasche — "On the coherence of Expected Shortfall" (Journal of Banking & Finance, 2002)
• Federal Reserve — DFAST Severely Adverse Scenario specifications