Thematic Pillar Hub · 20 Published Briefings
Bond Duration Convexity Math research topics and source records from QuantCalc Tools. Linear Modified Duration understates bond gains and overstates losses by 82 bps on a 10-year note during a 150 bps yield shock; this quantitative guide unifies DV01, convexity, KRD, and repo carry math.
A named author is recorded on some briefings in this hub, including Elena Rostova, FRM, CQF . Reviewer records and automated-check status appear on each briefing when available.
Linear Modified Duration understates bond gains and overstates losses by 82 bps on a 10-year note during a 150 bps yield shock; this quantitative guide unifies DV01, convexity, KRD, and repo carry math.
Quoting a 4.50% coupon bond on annual ICMA 251 European conventions versus semiannual US Street conventions shifts Modified Duration by 17 bps even when Macaulay cashflow timing matches.
Multiplying Modified Duration by quoted clean price instead of full dirty invoice price understates DV01 by up to 2.5% on high-coupon bonds just before coupon payment dates.
Bond convexity scales with squared cashflow maturity t*(t + 1/k), generating strictly positive quadratic price protection that rescues linear duration estimates from 150+ bps errors.
Because borrowers hold an embedded call option to refinance when mortgage rates drop, agency MBS pass-throughs suffer negative convexity that slashes rally upside and triggers extension risk.
Matching market value and Modified Duration between a 2Y/30Y Treasury barbell and a 10Y bullet boosts portfolio convexity by +114% while introducing calculable yield-give-up and 2s30s twist exposure.
Thomas Ho’s Key-Rate Duration (KRD) partitions parallel curve risk across localized triangular zero-rate tent shocks (2Y, 5Y, 10Y, 30Y) that sum identically to total Effective Duration.
Standard 50/50 DV01-weighted 2s5s10s butterfly trades suffer hidden front-end volatility bias; empirical regression and PCA weighting immunize 5-year belly curvature against 2s10s slope steepeners.
CME 10-Year Note futures do not track a fictitious 6% coupon bond; exact futures DV01 equals the Cheapest-to-Deliver cash Treasury dirty DV01 divided by its 6%-yield Conversion Factor.
Forward fixed-income total return equals net repo carry plus yield-curve rolldown; dividing their sum by forward DV01 defines the exact basis-point yield spike a leveraged position survives.
Matching asset and liability PV and Modified Duration satisfies only two of Redington’s three immunization rules; asset convexity must strictly exceed liability convexity while minimizing M-squared dispersion.
Stripping all intermediate coupons locks Macaulay Duration exactly to maturity M and scales convexity quadratically as M*(M + 0.5)/(1 + y/2)^2—making 30Y STRIPS more than twice as convex as 30Y par bonds.
Risk engines compute sensitivities to bootstrapped zero-coupon spot rates, whereas trading desks execute hedges in par SOFR OIS swaps; inverting the lower-triangular Jacobian matrix bridges the two.
Day-count mismatches between ACT/ACT ICMA sovereigns and 30/360 corporate bonds distort accrued interest across 31-day months and February, shifting fractional-period dirty DV01 and stub yields.
Hedging $100M of 10-year TIPS 1-for-1 with nominal 10-year Treasuries over-hedges interest-rate risk by ~38% because real yields move only ~0.62 bps per 1.00 bp move in nominal yields.
Credit spreads widen proportionally to their starting spread level rather than in flat parallel basis points; Duration-Times-Spread (DTS = D_spread * OAS) captures cross-rating credit risk 3x more accurately than raw CS01.
Confusing near-zero interest-rate reset duration with credit spread duration on Floating-Rate Notes is a classic fixed-income blunder; a 5-year SOFR FRN still loses ~4.55% per +100 bps widening in Discount Margin.
Issued at 0.85% coupon in 2020, Austria’s 2120 century bond carried 56 years of Modified Duration and 3,980 years-squared convexity—losing 72% when yields normalized yet surging asymmetrically on rate cuts.
Litterman and Scheinkman’s Principal Component Analysis proves that three orthogonal eigenmodes—Level (86%), Slope (10%), and Curvature (2.8%)—explain ~98.8% of yield-curve variance across all tenors.
Realized compound bond return equals initial Yield-to-Maturity ONLY when holding horizon H matches Macaulay Duration; the Duration Gap (H - D_Mac) determines whether rising rates boost or hurt terminal wealth.
CAGR calculator →
Compound annual growth rate between two values, with a year-by-year projection.
FX forward rate calculator →
Forward exchange rate and forward points from spot and two interest rates (covered interest parity).
Value at Risk (VaR) calculator →
Parametric VaR and Expected Shortfall for a position, at 95% or 99% confidence.
SOFR overnight compounding & carry calculator →
Daily compounded SOFR in arrears (ACT/360), effective annual yield and net levered carry.
Kelly criterion optimal fraction calculator →
Full, Half and Quarter Kelly position sizing, expected edge and geometric log growth rate.
Sharpe & Sortino ratio benchmark calculator →
Risk-adjusted performance across Sharpe ratio, Sortino downside ratio and Information ratio.
Macaulay Duration measures the present-value-weighted average time (in years) to receive a bond’s cashflows. Dividing Macaulay Duration by (1 + y/k)—where y is annual yield-to-maturity and k is coupon frequency—converts time-to-receipt into Modified Duration (-1/P * dP/dy).
Actual cash settlement and present-value discounting operate on full dirty price (Clean Price + Accrued Interest). Because accrued interest is invariant to instantaneous yield changes (dP_clean/dy = dP_dirty/dy), differentiating P_dirty produces -D_mod * P_dirty.
Modified Duration assumes cashflows remain fixed regardless of yield shifts. For callable corporate bonds, putable notes, or agency mortgage-backed securities (MBS), falling rates trigger borrower refinancing, requiring finite-difference OAS Effective Duration.
Because Ho’s triangular tent shocks across key maturities (such as 2Y, 5Y, 10Y, and 30Y) form a partition of unity that sums to a flat +1 bp parallel shift everywhere on the zero curve, the sum of all partial Key-Rate Durations equals total Effective Duration.
European sovereign bonds follow ICMA Rule 251 annual compounding (k = 1), dividing Macaulay Duration by (1 + y_1), whereas US Treasuries use semiannual Street convention (k = 2), dividing by (1 + y_2 / 2). One basis point of annual yield represents a slightly smaller shock.
No for positive-yield option-free fixed-rate bonds: Macaulay Duration is bounded at most by final maturity M (achieved only by a zero-coupon bond), and dividing by (1 + y/k) > 1 makes Modified Duration strictly smaller than M unless yields are deeply negative!