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IntermediateOptions, Derivatives & FinancePython

Run this module

cd "Finance - Duration Convexity"
python "duration_convexity.py"

View source on GitHub

Bond Duration, Convexity, and DV01

Fixed income sensitivity measures that quantify how bond prices respond to changes in interest rates.

Functions

Function Description
bond_price(cashflows, times, ytm) PV of cash flows at given yield
macaulay_duration(cashflows, times, ytm) Weighted average time to cash flows (years)
modified_duration(cashflows, times, ytm) % price change per 1% yield change
convexity(cashflows, times, ytm) Second-order yield sensitivity
dv01(cashflows, times, ytm) Dollar value of 1 basis point
price_change_approx(mod_dur, conv, price, dy) Taylor expansion price estimate
build_cashflows(face, coupon_rate, maturity, freq) Generate coupon bond cash flows

Key Concepts

  • Macaulay Duration: Measures the weighted average maturity of cash flows. Zero-coupon bond has duration = maturity.
  • Modified Duration: D_mod = D_mac / (1 + y). A bond with mod duration 7 loses ~7% in price per +100bp yield move.
  • Convexity: The curve in the price-yield relationship. Positive convexity benefits investors — price rises more than duration predicts when yields fall.
  • DV01: Practical measure for hedging. "My portfolio has DV01 of $5,000" means a +1bp move costs $5,000.

Example

from duration_convexity import build_cashflows, bond_price, modified_duration, dv01

cfs, ts = build_cashflows(face=1000, coupon_rate=0.05, maturity=10)
price = bond_price(cfs, ts, ytm=0.04)     # ~1081.11
mod_dur = modified_duration(cfs, ts, 0.04)  # ~7.99
dv = dv01(cfs, ts, 0.04)                    # ~0.086 per $1000

Duration Approximation

ΔP ≈ -D_mod × P × Δy + 0.5 × Convexity × P × Δy²

For a +100bp shock: duration term dominates. For large moves, convexity correction matters.

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