Exam FMInterest rate riskFree to read

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

$$D_{Mod}=-\frac{P^{\prime }(i)}{P(i)}=\frac{D_{Mac}}{1+i}$$

First-order estimate

$$P(i+\Delta i)\approx P(i)\left(1-D_{Mod}\Delta i\right)$$

Percentage change

$$\frac{\Delta P}{P}\approx -D_{Mod}\Delta i$$

Where it comes from

1. Differentiating $P(i)=\sum C_t(1+i{)}^{-t}$ gives $P^{\prime }(i)=-\sum t{C}_t(1+i{)}^{-t-1}=-\frac{1}{1+i}\sum t{C}_t{v}^t$.
2. Dividing by $-P$ produces $\frac{D_{Mac}}{1+i}$, which is the modified duration.
3. The estimate is the first-order Taylor expansion of $P$ 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

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

• Macaulay duration
• Convexity
• Immunisation

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