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QuantYield - Quantitative Models Reference

This document describes the mathematical models and conventions used in the pricing engine, curve builder, and risk analytics.

Day Count Conventions

Convention Key Description Formula
Actual/Actual actual/actual Actual calendar days over actual year length days / (366 if leap else 365)
Actual/360 actual/360 Actual days over fixed 360-day year days / 360
Actual/365 actual/365 Actual days over fixed 365-day year days / 365
30/360 30/360 Each month counts as 30 days (360*(y2-y1) + 30*(m2-m1) + (d2-d1)) / 360

Coupon Frequencies

Label Periods Per Year Months Between Payments
annual 1 12
semiannual 2 6
quarterly 4 3
monthly 12 1
zero 0 N/A (discount bond)

Bond Pricing

Dirty Price

The dirty (full) price is the present value of all future cash flows discounted at YTM:

P_dirty = sum( C/m / (1 + y/m)^(t_i * m) ) + F / (1 + y/m)^(T * m)

Where:

  • C = annual coupon
  • m = coupon frequency (periods per year)
  • y = yield to maturity (annual, compounded at frequency m)
  • t_i = time in years to the i-th coupon from settlement
  • F = face value
  • T = time in years to maturity from settlement

Clean Price

P_clean = P_dirty - AI

Where AI is the accrued interest.

Accrued Interest

AI = (C/m) * (t_elapsed / t_period)

Where t_elapsed is the day-count fraction since the last coupon date and t_period is the full coupon period fraction.

Zero Coupon Bond

P_dirty = F / (1 + y)^T

No accrued interest applies to zero coupon bonds.

YTM Solver

YTM is solved numerically using Brent's method on the objective:

f(y) = P_dirty(y) - P_dirty_target = 0

Parameters:

  • Bracket: y in [-0.5, 5.0]
  • Convergence tolerance: 1e-10
  • Maximum iterations: 500

Brent's method guarantees convergence when a root exists within the bracket and does not require derivatives.

Duration Measures

Macaulay Duration

D_mac = (1 / P_dirty) * sum( t_i * PV_i )

Where PV_i is the present value of the cash flow at time t_i.

Modified Duration

D_mod = D_mac / (1 + y/m)

This is the approximate percentage price change per unit change in yield:

dP/P ~ -D_mod * dy

Convexity

C = (1 / P_dirty) * sum( t_i * (t_i + 1/m) * PV_i / (1 + y/m)^2 )

Second-Order Price Approximation

dP ~ P * (-D_mod * dy + 0.5 * C * dy^2)

DV01

Dollar value of a 1 basis point change in yield:

DV01 = (P(y - 0.0001) - P(y + 0.0001)) / 2

DV01 is always reported as a positive number.

Key Rate Duration

KRD measures sensitivity to a shift at a single point on the yield curve, holding all other rates constant. QuantYield uses triangular bumps:

bump_weight(t_i) = max(0, 1 - |t_i - kt|)

Where kt is the key tenor. This means the bump fades linearly to zero at tenors more than 1 year away from the key tenor.

Standard key tenors: 0.25, 0.5, 1, 2, 3, 5, 7, 10, 20, 30 (years)

Z-Spread

The Z-spread is the constant spread added to the entire spot rate curve such that the resulting discount rates reprice the bond exactly:

P_dirty = sum( C/m / (1 + r(t_i) + Z)^t_i ) + F / (1 + r(T) + Z)^T

Solved using Brent's method on Z in [-0.5, 5.0].

OAS (Option-Adjusted Spread)

For callable bonds, OAS is computed via Monte Carlo simulation:

  1. Generate N short-rate paths using a lognormal model: r_{t+1} = r_t * exp(-kappa * r_t * dt + sigma * sqrt(dt) * epsilon)

  2. For each path, exercise the call option at each call date if: call_price <= continuation_value(r_path, spread)

  3. Price all paths at the given spread. OAS is the spread that makes the average simulated price equal to the market dirty price.

For non-callable bonds, OAS equals Z-spread.

Parameter Default Description
n_paths 500 Monte Carlo simulation paths
rate_vol 1% Short rate lognormal volatility
mean reversion kappa 0.10 Speed of mean reversion

Total Return Analysis

Horizon total return accounts for:

  1. Reinvested coupons compounded at the reinvestment rate
  2. Price appreciation or depreciation to the horizon date
  3. Accrued interest at the horizon
TV = P_horizon + sum( C_i * (1 + r_reinv)^((H - t_i)/365) )

TR_annualised = (TV / P_purchase)^(1/H) - 1

Where:

  • TV = terminal value
  • H = horizon in years
  • P_purchase = purchase dirty price
  • r_reinv = reinvestment rate

Yield Curve Models

Nelson-Siegel

y(t) = beta0 + beta1 * ((1 - e^(-t/lambda)) / (t/lambda))
             + beta2 * ((1 - e^(-t/lambda)) / (t/lambda) - e^(-t/lambda))
Parameter Economic Interpretation
beta0 Long-run yield level (t -> infinity)
beta1 Short-run slope (negative if upward sloping)
beta2 Curvature / hump magnitude
lambda1 Decay factor (location of hump)

Svensson

Extends Nelson-Siegel with a second curvature term to capture multiple humps:

y(t) = beta0 + beta1 * L(t, lambda1) + beta2 * C(t, lambda1) + beta3 * C(t, lambda2)

Where L and C are the standard NS loading functions applied at two decay rates.

Bootstrap

Extracts zero-coupon spot rates from par yields using no-arbitrage:

DF(T) = (1 - coupon * sum(DF(t_k), k=1..T-1)) / (1 + coupon)
s(T) = -ln(DF(T)) / T

Assumes semi-annual coupon frequency (FRED convention). Intermediate discount factors are linearly interpolated.

Cubic Spline

Fits a natural cubic spline through all market data points with not-a-knot boundary conditions. Exact at knot points, smooth (C2 continuous) everywhere.

Regime Detection

Regime Condition Description
flat 2s10s in [-15, +15] bps Near-zero slope
inverted 2s10s < -10 bps Short rates above long rates
steep 2s10s > 120 bps Very steep upward slope
humped 2s5s10s butterfly > 20 bps Peak in medium tenors
normal All other cases Moderate upward slope

The ML regime classifier uses these thresholds as labels for training and augments them with momentum, volatility, and level features.

VaR Methodologies

Historical VaR

  1. Collect DV01 for the portfolio.
  2. Compute daily yield changes over the lookback window.
  3. Scale: P&Li = -DV01 * yieldchange_i * 10,000.
  4. For multi-day VaR, use overlapping n-day changes (not sqrt-n scaling).
  5. VaR = abs(percentile(P&L, 1 - confidence_level)).
  6. CVaR (Expected Shortfall) = mean of observations below VaR.

Parametric VaR

Assumes normally distributed P&L:

VaR = DV01 * z * sigma * sqrt(H) * 10000

Where:

  • z = normal quantile at the confidence level
  • sigma = daily yield volatility (annualised / sqrt(252))
  • H = holding period in days

Scenario Analysis - Standard Set

Scenario Parallel Twist Short Twist Long Credit
+100bps parallel +100 0 0 0
-100bps parallel -100 0 0 0
+200bps parallel +200 0 0 0
-200bps parallel -200 0 0 0
Bear flattener 0 +50 -50 0
Bull steepener 0 -50 +50 0
Bear twist +50 +50 +25 0 0
Bull twist -50 -50 -25 0 0
+300bps shock +300 0 0 0
Credit widening 0 0 0 +100

All shifts are in basis points.