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Duration, convexity, and curve hedges

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Start with the idea

Duration and convexity approximate the slope and curvature of a bond’s price-yield relationship. They make a large cash-flow schedule easier to reason about locally, but a single parallel-yield sensitivity cannot capture every curve move.

Symbols, units & horizon

P: current currency price or book value as specified
y: decimal annual effective yield
∂: derivative with other inputs fixed
D_mod: modified duration in years
C: convexity, not coupon here, in years squared under this convention
Δy: absolute decimal yield change
ΔP/P: fractional price change
DV01: positive magnitude of price decline for a 1 bp yield rise for an ordinary long bond
10⁻⁴: one basis point as a decimal
N_hedge: signed hedge contract count
CFₜ: currency cash flow at year t

When and why to use this

Use DV01 to compare and hedge rate risk across positions. Add key-rate, spread and optionality scenarios when those are material.

D_{\mod} = - \frac{1}{P} \frac{\partial P}{\partial y}, C = \frac{1}{P} \frac{\partial^{2} P}{\partial y^{2}}

Ordinary differentiation

Differentiate discounted cash flows

For $P (y) = \sum C F_{t} (1 + y)^{- t}$ , differentiate to get $P' = - \sum t C F_{t} (1 + y)^{- t - 1}$ .
Differentiate again: $P'' = \sum t (t + 1) C F_{t} (1 + y)^{- t - 2}$ . Divide −P′ and P″ by price for modified duration and convexity.

Work it by hand

For a one-year zero-coupon bond, Dmod=1/(1+y) and convexity=2/(1+y)². At 5% they are .952381 and 1.814059.

\frac{Δ P}{P} \approx - D_{\mod} Δ y + \frac{1}{2} C (Δ y)^{2}, D V 01 \approx P D_{\mod} 10^{- 4}

Taylor approximation + unit conversion

Use Taylor expansion and convert a basis point

$Δ P \approx P' Δ y + \frac{P'' (Δ y)^{2}}{2}$ . Divide by P and substitute duration and convexity.
For a one-basis-point rise, Δy=.0001. The magnitude of the first-order dollar loss is $P D_{m o d} 10^{- 4}$ . The sign of a long conventional bond’s price change is negative.

Work it by hand

A $2m position, D=6,C=45,Δy=.005 has approximate return −.03+.0005625=−2.94375%; DV01=$1,200.

Duration is the first-order sensitivity to a yield move and convexity is the second-order correction. DV01 is the positive magnitude of a small loss from a one-basis-point yield increase for a conventional long bond. Use position value for dollar risk; use yield changes as decimals.

A single duration assumes a parallel shift. Key-rate durations separate sensitivities to curve tenors, exposing steepener and butterfly risks. Equal total DV01 can hide a long two-year/short thirty-year mismatch. Optionality and changing spreads can make a fixed-duration approximation unreliable.

N_{hedge} \approx - \frac{D V 01_{book}}{D V 01_{o n e h e d g e c o n t r a c t}}

Algebra and arithmetic

Solve a first-order hedge equation

Require signed book sensitivity plus N times one contract’s sensitivity to equal zero: $D V 01_{b o o k} + N D V 01_{c o n t r a c t} = 0$ .
Subtract the book sensitivity and divide by contract sensitivity. Round to executable contract counts, then recalculate the remaining exposure.

Work it by hand

Book DV01=$1,200, hedge contract=$80 per bp → N=−15. The short hedge offsets only the chosen first-order risk.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def bond_sensitivities(cashflows, yield_rate):
    """Annual periods at t=1,...; final cash flow includes principal."""
    if yield_rate <= -1:
        raise ValueError("Yield must exceed -1")
    p = sum(cf/(1+yield_rate)**t for t,cf in enumerate(cashflows,1))
    first = -sum(t*cf/(1+yield_rate)**(t+1) for t,cf in enumerate(cashflows,1))
    second = sum(t*(t+1)*cf/(1+yield_rate)**(t+2) for t,cf in enumerate(cashflows,1))
    duration, convexity = -first/p, second/p
    return p, duration, convexity, p*duration*1e-4

def duration_pnl(price, duration, convexity, yield_change):
    return price*(-duration*yield_change+.5*convexity*yield_change**2)

def hedge_contracts(book_dv01, one_contract_dv01):
    return -book_dv01/one_contract_dv01  # round subject to residual risk

print(bond_sensitivities([5,105], .05))

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