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Modified duration and price sensitivity

The elasticity of price with respect to the interest rate: the first-order estimate of how much a portfolio loses when rates rise.

The formulas

Definition
DMod=−P′(i)P(i)=DMac1+iD_{Mod} = -\dfrac{P'(i)}{P(i)} = \dfrac{D_{Mac}}{1+i}
First-order estimate
P(i+Δi)≈P(i)(1−DMod Δi)P(i+\Delta i) \approx P(i)\left(1 - D_{Mod}\,\Delta i\right)
Percentage change
ΔPP≈−DMod Δi\dfrac{\Delta P}{P} \approx -D_{Mod}\,\Delta i

Where it comes from

  1. Differentiating P(i)=∑Ct(1+i)−tP(i) = \sum C_t (1+i)^{-t} gives P′(i)=−∑tCt(1+i)−t−1=−11+i∑tCtvtP'(i) = -\sum t C_t (1+i)^{-t-1} = -\frac{1}{1+i}\sum t C_t v^t.
  2. Dividing by −P-P produces DMac1+i\frac{D_{Mac}}{1+i}, which is the modified duration.
  3. The estimate is the first-order Taylor expansion of PP about the current rate, so it is exact only in the limit and always OVERSTATES the loss for a rise (price is convex).

Worked example

The 5-year 6% par bond has Macaulay duration 4.4651 at a 6% yield. Estimate its price if the yield rises to 6.5%.

  1. DMod=4.4651/1.06=4.212357D_{Mod} = 4.4651/1.06 = 4.212357.
  2. Δi=0.005\Delta i = 0.005, so ΔP/P≈−4.212357×0.005=−0.021062\Delta P/P \approx -4.212357 \times 0.005 = -0.021062.
  3. Estimated price =1,000(1−0.021062)=978.94= 1{,}000(1 - 0.021062) = 978.94.
  4. The true price at 6.5% is 979.15 - the first-order estimate is 0.21 low, exactly the convexity correction.

Answer: 978.94

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Modified = Macaulay / (1 + i). One division, and the units change from years to 'percent per percent'.
  • The first-order estimate always UNDERSTATES the price, because the true price curve is convex.

Traps

  • Using Macaulay duration directly in ΔP/P ≈ −D·Δi.
  • Applying a duration computed at one yield to a large rate move - the estimate degrades quadratically.

Related

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