Yield Curve Risk

Yield curve risk is the risk that the value or income of a position changes because interest rates at different maturities move by different amounts, altering the shape of the yield curve. A portfolio can be protected against a parallel move in all rates and still lose money if short-term rates fall while long-term rates rise, or the reverse. Bond investors, swap dealers, banks and corporate treasurers all face it whenever their assets, liabilities and hedges are spread across different maturities.

The concept fills a gap left by single-number measures such as portfolio duration and basis point value, which assume parallel shifts. In banking, it is one of the components of interest rate risk in the banking book, alongside repricing, basis and option risk.

Imagine a rope held at both ends by two people. If both lift their ends by the same amount, the whole rope rises evenly. If one lifts and the other lowers, the rope tilts, and different parts move by different amounts even though the average height may not change.

Interest rates for different lengths of time behave like points along that rope. Sometimes they all rise or fall together. Often, though, short-term rates move one way and long-term rates another. Anyone whose money is spread unevenly along the rope can win or lose from the tilt, even if they protected themselves against the whole rope going up or down.

• Steepening. Long-term rates rise relative to short-term rates, widening the gap, as described under curve steepening.
• Flattening. The gap narrows, which can end in a flat yield curve or an inverted yield curve.
• Twist. Short and long rates move in opposite directions around a pivot maturity.
• Butterfly or curvature change. Medium-term rates move differently from both short and long rates.

Statistical studies of government bond markets have found that three broad factors, often called level, slope and curvature, explain most changes in the curve, with level the largest. Slope and curvature together make up yield curve risk.

The standard tool is to split rate sensitivity by maturity. Key rate durations or key rate basis point values measure the change in value when a single point on the curve, such as two, five, ten or 30 years, moves by one basis point while the rest stay fixed. The estimated change in value for any curve move is then:

• $\text{BPV}_k$ is the basis point value at key maturity $k$, in currency per basis point.
• $\Delta y_k$ is the change in yield at that maturity, in basis points.

Suppose a fund holds ten-year bonds with a BPV of $30,000 and hedges by shorting two-year notes with a BPV of −$30,000. Its total BPV is zero, so a parallel move does nothing. In a steepening where two-year yields fall 20 basis points and ten-year yields rise 30 basis points, the change is −[(−30,000) × (−20) + 30,000 × 30] = −(600,000 + 900,000) = −$1.5 million. The position was duration-neutral but exposed to slope.

Banks also run prescribed scenarios. The Basel standards for IRRBB include steepener, flattener, short-rate-up and short-rate-down shocks in addition to parallel moves, and the results feed measures of economic value of equity.

Yield curve risk arises naturally from maturity transformation, when a bank funds long fixed-rate loans with short deposits, and from hedges whose maturities do not match the exposures they cover. Monetary policy often drives it: rate cuts tend to steepen the curve by lowering short rates, while expectations about inflation and government borrowing move long rates. Managers control the risk by setting limits on key rate exposures, matching hedges more closely to the maturity profile, using swaps at several points on the interest rate swap curve, and taking deliberate curve positions, such as steepener or flattener trades, when they want exposure.

Key rate measures depend on the choice of key points and on how moves are interpolated between them, so two systems may report different figures for the same portfolio. Historical scenarios may not capture future shocks, and relationships between maturities can change when central banks intervene, as with yield curve control. Measures of curve risk also need to be combined with credit spread, basis and option risk to give a full picture of interest rate risk.