8.6 Managing Yield-Curve Risk: Key Rate Durations, One-Sided Durations & Yield Volatility Structure

8.6 Managing Yield-Curve Risk: Key Rate Durations, One-Sided Durations & Yield Volatility Structure

How this fits: section 8.1 built the term structure and section 8.3 built effective duration and convexity for bonds with embedded options. This section covers the remaining risk-measurement learning outcomes: how a bond's exposure to each factor driving the yield curve is measured, how one-sided durations work, and how the maturity structure of yield volatilities converts duration into actual price risk.

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1. Why Effective Duration Is Not Enough

Effective duration measures sensitivity to a parallel shift in the benchmark yield curve:

$ED = \frac{PV_- - PV_+}{2 \times \Delta \text{Curve} \times PV_0}$

The problem is that yield curves rarely move in parallel. Empirically, principal component analysis of curve movements identifies three factors that together explain more than 95% of the variation:

Effective duration captures the level factor only. Two portfolios with identical effective duration can have opposite reactions to a steepening or a butterfly move, which is exactly the situation vignettes construct.

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2. Key Rate Duration

Key rate duration (partial duration) measures the percentage price change from a 1 basis-point change in the spot rate at one specific maturity, holding all other spot rates constant:

$KRD_k = -\frac{1}{PV_0} \times \frac{\Delta PV}{\Delta r_k}$

Two properties define its use:

1. The key rate durations sum to the effective duration. $\sum_k KRD_k = ED$. This is the check the exam expects you to apply to a table of partial durations.
2. A zero-coupon bond has essentially all of its key rate duration at its own maturity. A coupon bond spreads exposure across every date on which it pays, with the bulk at the maturity date where principal is repaid.

Bullet versus barbell

Illustration. Two portfolios each have an effective duration of 6.0.

Now apply a steepening twist: the 2-year rate falls 25 bp and the 30-year rate rises 25 bp, with the middle unchanged.

• Bullet: $-[0.10 \times (-0.25%) + 0.10 \times (0.25%)] \approx 0.00%$ — nearly immune
• Barbell: $-[2.80 \times (-0.25%) + 2.90 \times (0.25%)] = -[-0.70% + 0.725%] = \textbf{-0.025%}$

Under a flattening twist the barbell gains where the bullet is flat. And under a butterfly — the middle of the curve rising relative to both ends — the bullet loses substantially while the barbell gains, because the bullet's exposure sits precisely where the rate rises.

The general rules:

• A barbell outperforms a bullet when the curve flattens, and underperforms when it steepens.
• A bullet outperforms a barbell when the curve is hit by a butterfly move that lowers the middle, and underperforms when the middle rises.
• Matching effective duration alone does not immunize a portfolio against non-parallel shifts; matching key rate durations across the curve does.

Using key rate durations to manage risk

An asset-liability manager whose liabilities have concentrated key rate exposure at 12 and 20 years cannot hedge with a duration-matched portfolio built from 2-year and 30-year instruments. The hedge must match the profile, not merely the total. This is the operational meaning of the learning outcome about measuring a bond's exposure to each factor driving the yield curve and using those exposures to manage yield curve risk.

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3. One-Sided Durations

Effective duration averages the price responses to an up-move and a down-move. For a bond whose cash flows are not symmetric in rates — anything with an embedded option — that average conceals the asymmetry.

One-sided durations compute the two halves separately:

$D_{\text{up}} = \frac{PV_0 - PV_+}{\Delta \text{Curve} \times PV_0} \qquad D_{\text{down}} = \frac{PV_- - PV_0}{\Delta \text{Curve} \times PV_0}$

For a callable bond trading near its call price, effective duration materially overstates the price gain from a rate decline. A manager expecting rates to fall and relying on effective duration will be disappointed; the one-sided down-duration is the relevant number. This asymmetry is the same phenomenon as negative convexity measured on a different axis.

Worked example. A callable bond priced at 101.20 with a 25 bp curve shock:

• $PV_- = 101.55$, $PV_+ = 100.55$
• Effective duration: $(101.55 - 100.55)/(2 \times 0.0025 \times 101.20) = 1.00/0.5060 = \textbf{1.98}$
• One-sided down: $(101.55 - 101.20)/(0.0025 \times 101.20) = 0.35/0.2530 = \textbf{1.38}$
• One-sided up: $(101.20 - 100.55)/(0.0025 \times 101.20) = 0.65/0.2530 = \textbf{2.57}$

The bond loses at a rate consistent with duration 2.57 when rates rise and gains at a rate consistent with only 1.38 when they fall. The symmetric 1.98 describes neither direction.

4. The Maturity Structure of Yield Volatilities

Duration answers "how much does the price move per unit of yield change". It says nothing about how large a yield change is likely at that point on the curve. The term structure of yield volatilities supplies the second half:

$\sigma(t) = \text{annualized standard deviation of the change in the yield at maturity } t$

Empirical shape and its drivers:

• Short-maturity volatility is typically highest, driven by uncertainty about monetary policy: the market's view of the next several central bank meetings changes frequently and sharply.
• Intermediate-maturity volatility is lower, sitting between the two regimes.
• Long-maturity volatility is driven by expectations about inflation and the real economy, which evolve slowly, so it is usually lower than short-end volatility — though it rises sharply in inflation-regime shifts.

The practical implication is that a 1 basis-point shift is not equally likely at every maturity. Combining the two measures:

$\text{Price volatility}_k \approx |KRD_k| \times \sigma(k)$

Illustration. Two positions:

Duration still dominates here, but note that the volatility term compresses the differences: the 30-year has 10 times the duration of the 2-year but only 5 times the price volatility. In periods when short-rate volatility spikes — a policy inflection — the ranking can compress much further, and a short-duration position can become the larger contributor to portfolio risk than its duration suggests.

For bonds with embedded options this interacts with option value. Higher assumed interest rate volatility raises the value of both the call and the put option, which:

• lowers the value of a callable bond (the investor is short the call), and
• raises the value of a putable bond (the investor is long the put).

Consequently a rise in the term structure of yield volatilities widens the option-adjusted spread on a callable bond held at a constant price, because more of the yield spread is being attributed to the option. This connection between the volatility term structure and OAS is the single most tested cross-link between sections 8.3 and this one.

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5. Establishing a View on Rates, Spreads, and Curve Shape

The final learning outcome asks how key economic factors are used to form a view on benchmark rates, spreads, and yield curve changes.

Composing a trade from a view. If an analyst expects the central bank to cut faster than the market has priced while long-run inflation expectations remain anchored, the correct expression is a steepener — long the short maturity, short the long maturity, duration-neutral — rather than an outright long-duration position. A vignette that gives you a macro view and four portfolio actions is testing whether you can translate the view into the correct key rate exposure rather than into a directional duration bet.

Level II traps in this area

1. Assuming that matching effective duration immunizes against all curve movements. It immunizes only against parallel shifts.
2. Forgetting that key rate durations sum to effective duration; a table that does not sum has an error or an omitted maturity.
3. Applying symmetric effective duration to a callable bond near its call price, and overstating the upside from falling rates.
4. Concluding that a barbell is always safer than a bullet; it is safer under a flattening and worse under a steepening.
5. Ranking risk by duration alone when the yield volatilities at the relevant maturities differ materially.