Key rate duration

What if the yield curve does not move in parallel?

Effective duration answers one question: what if every rate on the benchmark curve moves by the same amount? Real curves rarely do that. A non-parallel shift moves some maturities more than others. Two common cases are steepening , where long rates rise relative to short rates (the gap between them widens), and flattening , where short rates rise relative to long rates (the gap narrows). In a twist, different parts of the curve move in different directions, for example the middle falling while both ends rise.

Benchmark curve changeswhat key rate durations are built to tell apart

• Parallel shift Use it every maturity moves by the same amount
  • Effective duration is enough one number covers it
• Non-parallel shift Use it maturities move by different amounts
  • Steepening Memorize the gap between long and short rates widens
  • Flattening Memorize the gap between long and short rates narrows
  • Twist maturities move in different directions, bending the curve

A bond’s sensitivity to these changes in shape is its shaping risk . Effective duration cannot see it, because it assumes one parallel move. Key rate duration (also called partial duration ) can: it measures a bond’s sensitivity to a change in the benchmark yield at one maturity, with all other rates held constant. It asks “what if only the 5-year rate moves?”.

How is key rate duration measured?

Use it

The procedure is the effective duration calculation, repeated one maturity at a time. Pick the key rates, for example the 2-, 5-, 10- and 30-year points. Shift one key rate up and down by 1 bp, leave the others alone, reprice the bond, and apply the duration formula. Then do the next one.

$$KRD_k = -\frac{1}{PV} \times \frac{\Delta PV}{\Delta r_k}$$

The key rate durations, taken across all the key rates, add back up to the effective duration. That is the check on every set of numbers.

$$\sum_{k=1}^{n} KRD_k = \text{EffDur}$$

How do you estimate a price change from key rate durations?

Rearrange the definition. For each key rate, multiply its duration by its own rate change, add the products, and flip the sign.

$$\%\Delta PV^{Full} \approx -\sum_{k} KRD_k \times \Delta r_k$$

A manager who tracks an index can use key rate durations to take a view on the shape of the curve. Expecting the long end to rise, for example, the manager can shift duration from the 10-year point to the 2-year point, so the portfolio keeps the same total effective duration but is less exposed to the part of the curve expected to move most.

Exam tip Match each key rate duration with the shift at its own maturity, keep the sign of every shift, and add. Never multiply the total duration by an average.

Go deeper Which bonds load on which key rate

A 10-year zero-coupon bond has almost all its exposure at the 10-year point, because it pays one cash flow there. A coupon bond spreads its exposure across earlier points too, since its coupons arrive sooner. For a portfolio, each key rate duration is the weighted sum of the bonds’ exposures at that point. Say a portfolio holds 2-, 5- and 10-year zero-coupon bonds in weights of 20%, 35% and 45%, with durations of 1.9, 4.8 and 9.6. The key rate durations are 0.38, 1.68 and 4.32, and they sum to 6.38, the portfolio’s duration.

Go deeper Can a key rate duration be negative?

For some structured securities, yes: the exposure at one maturity can have the opposite sign to the rest, so that a rise in that one rate raises the value. You do not calculate this at Level I. Just know that the individual slices need not all be positive, even though they add to the effective duration.

Go deeper Why this matters for the term structure

Key rate durations make the curve shapes in the term structure module concrete. A flattening, a steepening and a twist are just different patterns of Δr across maturities, and the same key rate durations price each of them.