Sujitn Convex Bond Risk Metrics

Bond Risk Metrics Implementation

High-performance fixed income risk analytics targeting Bloomberg YAS parity. All calculations use dirty price basis and 1bp symmetric shifts unless otherwise noted.

Core Calculation Workflow

1. Determine bond type → Select appropriate duration method
2. Build/obtain yield curve → Required for spread and KRD calculations
3. Calculate base metrics → Duration, DV01, convexity
4. Calculate spread metrics → If credit bond: OAS, Z-spread, CS01
5. Calculate KRD → If curve risk decomposition needed
6. Validate → Sum of KRDs ≈ Total duration (within convexity tolerance)

Duration Metrics

DV01 and Modified Duration

Use central finite difference with 1bp shift:

DV01 = (P₋ - P₊) / 2
Modified_Duration = DV01 × 10000 / P₀

Where P₋ = price at yield - 1bp, P₊ = price at yield + 1bp.

Macaulay Duration (for immunization):

Mac_D = Σ(t × PV_t) / Σ(PV_t)
Modified_D = Mac_D / (1 + y/n)  // n = compounding frequency

Effective Duration (Bonds with Embedded Options)

Critical: Hold OAS constant, shift benchmark curve:

Effective_D = (P₋ - P₊) / (2 × P₀ × Δy)

Use 25-50bp shifts for optioned bonds. Recalculate interest rate tree at each shifted curve.

Bloomberg field: OAD

Convexity

Standard Convexity

Convexity = (P₊ + P₋ - 2×P₀) / (P₀ × Δy²)

Bloomberg scaling note: Verify if Bloomberg reports convexity/100. Adjust for Taylor expansion:

ΔP/P ≈ -Duration × Δy + 0.5 × Convexity × Δy²

Negative Convexity (Callables)

Occurs when rates fall below coupon. Price ceiling at call price causes concave price-yield curve.

Detection: Compare up/down price sensitivities. If |ΔP_down| < |ΔP_up|, bond exhibits negative convexity.

Key Rate Duration (KRD)

Standard Tenors

[3M, 6M, 1Y, 2Y, 3Y, 5Y, 7Y, 10Y, 15Y, 20Y, 25Y, 30Y]

For portfolio reporting: [2Y, 5Y, 10Y, 30Y]

Triangular (Tent) Bump Method

Δr(t) = h × max(0, 1 - |t - T_key| / span)

Where span = distance to adjacent key rate tenor.

Critical relationship: Σ KRD_i ≈ Effective_Duration (verify as sanity check)

Curve Interpolation Impact

Prefer piecewise constant forwards for clean bucket attribution.

Spread Metrics

Spread Hierarchy (Most to Least Theoretically Precise)

CS01 (Credit Spread 01)

CS01 = Spread_Duration × Price × 0.0001

Key distinction:

• Fixed-rate corporates: CS01 ≈ DV01
• Floating-rate notes: CS01 >> DV01 (rate duration ≈ 0)

OAS to Z-spread Relationship

Z-spread = OAS + Option_Cost

• Option-free: Z = OAS
• Callable: Z > OAS
• Putable: Z < OAS

Bond-Type Specific Calculations

See references/bond-types.md for detailed formulas.

Quick Reference

Bloomberg Field Mapping

See references/bloomberg-fields.md for complete mapping.

Essential Fields

Day Count Conventions

Rust Implementation Patterns

Core Traits

pub trait RiskMetrics {
    fn dv01(&self, curve: &YieldCurve) -> f64;
    fn modified_duration(&self, curve: &YieldCurve) -> f64;
    fn convexity(&self, curve: &YieldCurve) -> f64;
}

pub trait SpreadRisk {
    fn z_spread(&self, curve: &YieldCurve, price: f64) -> f64;
    fn cs01(&self, curve: &YieldCurve) -> f64;
}

pub trait KeyRateRisk {
    fn key_rate_durations(&self, curve: &YieldCurve, tenors: &[f64]) -> Vec<f64>;
}

Performance Optimizations

1. Parallel KRD: Use rayon for independent tenor calculations
2. Cache discount factors: Reuse across bump scenarios
3. SIMD: Vectorize discount factor array operations
4. Jacobian reuse: Transform between rate types without full reprice

Finite Difference Defaults

const DV01_SHIFT_BP: f64 = 1.0;
const EFFECTIVE_DUR_SHIFT_BP: f64 = 25.0;  // For optioned bonds
const KRD_SHIFT_BP: f64 = 1.0;

Validation Checklist

• DV01 × 10000 / Price ≈ Modified Duration
• Σ KRD ≈ Effective Duration (within 0.1%)
• OAS ≤ Z-spread for callable bonds
• Convexity positive for vanilla bonds
• FRN interest rate duration ≈ time to next reset

Regulatory Context (2025)

ISDA SIMM v2.8

• Delta tenors: 2w, 1m, 3m, 6m, 1y, 2y, 3y, 5y, 10y, 15y, 20y, 30y
• Sub-curves: OIS, SOFR, inflation, cross-currency basis
• Semiannual calibration cycle

FRTB SA

• Sensitivities-Based Method: Delta + Vega + Curvature
• Three correlation scenarios (high/medium/low)

Reference Files

• references/formulas.md - Complete mathematical formulas with derivations
• references/bloomberg-fields.md - Full Bloomberg field mapping and conventions
• references/bond-types.md - Type-specific calculation details (FRN, TIPS, callables)

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