CFA® Level III: how do convexity and key rates guide a curve view?

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A Level III curve position has two separate dimensions: where along the curve it responds to yield changes, and how its price response bends as yields move. Key rate duration locates the first dimension; convexity helps describe the second. A manager can match a benchmark's overall duration yet still make a substantial bet on a twist or a change in rate volatility.

Start with the risk question, not the instrument

The CFA Institute yield curve reading distinguishes a change in the curve's level from a change in its slope or shape. If all relevant yields move together, effective duration gives a useful first estimate of price response. If short and long yields move by different amounts, a single duration figure conceals the location of the exposure. The Level III task is therefore to turn a stated view about particular maturities into a profile of sensitivities and then check whether the proposed trade actually has that profile.

Imagine two portfolios with the same starting value and the same effective duration. One holds mostly intermediate maturity exposure. The other combines short and long maturity exposure. A parallel change can make their first order estimated returns look similar. A twist that moves the intermediate yield differently from both ends can make their returns diverge. Equal total duration does not mean equal curve risk, just as equal total exposure to an equity market does not identify the sectors that supplied it.

What key rate duration measures

To find sensitivity at one selected maturity, revalue the position after shifting that point on the benchmark curve while holding the other selected points fixed. The resulting key rate duration is an approximate percentage price sensitivity per unit change in that key yield. The set of key rates forms a profile. The sum of a consistently defined profile approximates effective duration for a parallel move, but the separate entries are what matter for a non-parallel scenario. Interpolation between selected points, the valuation model and the size of the shock all affect the measurement.

\frac{\Delta P}{P} \approx -\sum_k KRD_k\,\Delta y_k

The minus sign is essential: positive duration at a maturity loses value when the yield there rises, all else equal. When comparing a portfolio with its benchmark, subtract benchmark key rate duration from portfolio key rate duration at each maturity. The result is active exposure. Positive active exposure at one key point helps relative performance if that key yield falls relative to what was priced into the benchmark; negative active exposure helps if it rises. This is an exposure statement, not proof that a manager's forecast will be correct.

An illustrative curve decision

Suppose a manager studies three key yields: short, intermediate and long. This is an invented teaching example, not a market quote. The portfolio has key rate durations of 1, 2 and 4; its benchmark has 2, 3 and 2. Both have total duration of 7, yet active duration is minus 1 at the short point, minus 1 at the intermediate point and plus 2 at the long point. The manager has kept parallel-shift duration neutral while concentrating a relative preference at the long end.

Now suppose the short yield rises by 0.10 percentage points, the intermediate yield rises by 0.20 points and the long yield falls by 0.10 points. In decimal form those changes are plus 0.001, plus 0.002 and minus 0.001. The portfolio's first order estimated price change is minus 0.1%, minus 0.4% and plus 0.4%, for minus 0.1% overall. The benchmark's corresponding estimate is minus 0.2%, minus 0.6% and plus 0.2%, for minus 0.6% overall. The portfolio's estimated relative gain is therefore 0.5 percentage points before carry, spread changes, transaction costs and convexity.

You can check the same result using active durations alone: a short-point active duration of minus 1 times a plus 0.001 move contributes plus 0.1%; the intermediate contribution is plus 0.2%; and long-point active duration of plus 2 times a minus 0.001 move contributes plus 0.2%. The contributions total plus 0.5%. The arithmetic shows why a duration-neutral trade can still express a substantial curve view. It also makes the risk concrete: if the long yield rises instead, the largest active position works against the manager.

Where convexity changes the answer

Duration draws a tangent to the price-yield relationship. Convexity describes how that slope changes. The CFA Institute convexity reading explains why an option-free fixed-rate bond's positive convexity improves the linear estimate for a yield change. A larger move makes the second order term more consequential. It does not transform a weak rate forecast into a profitable trade; price paid for the exposure, carry and other market moves still matter.

\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2}C(\Delta y)^2

For another wholly illustrative calculation, take two option-free portfolios with duration 7 but convexities 50 and 30. If the same parallel yield rises by 0.01, duration predicts a 7% decline for each. The convexity adjustments are plus 0.25% and plus 0.15%, so the approximations become declines of 6.75% and 6.85%. If yields instead fall by 0.01, the approximations become gains of 7.25% and 7.15%. The more convex portfolio has a more favourable curvature in either direction for that symmetrical shock, but this comparison excludes any difference in price, income and financing.

Do not paste the parallel convexity calculation mechanically onto a twist. Different curve points and cash flows are moving by different amounts. A full revaluation under the proposed shape scenario is preferable, especially when the move is large or when there are options. Key rate durations offer an interpretable first order decomposition; they are not a complete non-linear model of every possible curve path.

Embedded options change the measurement

A callable bond can lose some of the price upside from falling yields because the issuer becomes more likely to call it. This can create negative effective convexity in relevant rate ranges. A putable bond gives its holder a floor in appropriate circumstances. The CFA Institute reading on embedded options treats effective duration, one-sided duration, key rate duration and effective convexity as tools for these securities. The cash-flow possibilities must be reconsidered under each shocked scenario; assuming fixed promised cash flows would miss the option's effect.

For a callable position, a symmetric up-and-down shock may hide asymmetry. A candidate should ask whether the call is likely to matter after rates fall, and whether the upward shock changes expected exercise differently. One-sided sensitivities can reveal that the response to a fall is weaker than the response to an equal rise. Negative convexity is a local property of a valuation and rate range, rather than a guarantee that every observed rate move will produce the same shape of return.

A manager can also change the profile with stand-alone options or swaptions. The Level III reading identifies these as tools for altering duration and convexity exposure. Buying an option introduces premium cost and an expiry; selling one creates a different risk profile. A derivative also brings implementation details such as collateral and liquidity. In an exam explanation, name the intended sensitivity first, then explain why the chosen instrument delivers it and what new risks it adds.

How to answer a curve-positioning question

A reliable sequence is to identify the benchmark and the manager's view, separate level from shape, compare key rate durations at each stated maturity, and use the sign of the forecast yield move to judge each active exposure. Add the active contributions before reaching for a label such as barbell or butterfly. Then check convexity and options where the scenario requires more than a small linear shock. This sequence prevents a visually appealing trade name from replacing the actual price logic.

Keep the unit convention visible. A 0.10 percentage point yield move is 0.001 in decimal form, while 1.0 percentage point is 0.01. Duration multiplies the decimal change. If an answer supplies basis points, convert them before calculating. If it gives percentages of portfolio value, distinguish the price change of a bond from the contribution of the bond's portfolio weight. The most common mistakes are mixing these units and assuming that matching total duration removes curve risk.

For study context, the CFA program page identifies the programme, while the methodology page explains Omni's study approach. For this concept, the useful self-check is to invent two duration-matched profiles and test a twist. If the relative result can be explained from the signs and magnitudes of the active key rates, the curve view has been translated into risk rather than merely named.