Term Structure Models

A Treasury desk hedges a 7-year bond with the 5-year and 10-year on-the-run. DV01-neutral against the 5-year leaves residual risk; DV01-neutral against both legs produces a smaller residual but still can move 10bp on a curve twist. Regression hedging quantifies the residual; Vasicek and Gauss+ price the curve dynamics that produced it. The exam tests both the hedge math and the model that justifies it.

A trader buys a 7-year bond and wants to hedge interest rate risk. The standard DV01 hedge sells an offsetting position (say, a 5-year on-the-run) with matched dollar duration. If both bonds' yields move by exactly the same amount (a parallel shift), the hedge is perfect. Curves don't move that way in practice. The 7-year and 5-year tenors move related but not identically; their moves have a documented systematic relationship.

Run a regression of historical 7-year yield changes on 5-year yield changes:

$$\Delta y_7=\alpha +\beta \Delta y_5+\epsilon$$

The slope $\beta$ is the hedge adjustment factor. A typical $\beta$ is between 0.85 and 1.05 depending on the curve segment.

Common mistakes

• Treating reverse regression as the inverse of standard regression. Beta of Y on X is NOT 1 / Beta of X on Y under measurement error. The regressions answer different questions.
• Confusing Vasicek's stationary variance with the conditional variance. The conditional variance of $r_T$ given $r_0$ is time-dependent: ${\sigma }^2/(2k)\cdot (1-e^{-2k(T-t)})$. The stationary variance ${\sigma }^2/(2k)$ is the limit at infinite horizon. Trap: a question asking "what is the variance of the 5-year-ahead rate" with the right answer being the conditional variance, not the stationary one.
• Using risk-neutral drift for VaR. Risk-neutral drift is for derivative pricing; true (P-measure) drift is for forecasting and VaR. These are systematically different by the risk premium. Trap: a question gives a risk-neutral parameterization and asks for 1-year forecast: the true-measure drift adds back the risk premium.

Bottom line

• Regression hedge scales the DV01 hedge by a beta coefficient capturing the systematic ratio of position-tenor to hedge-tenor moves; two-variable regression additionally handles curve twists.
• Principal Component Analysis (PCA) decomposes yield-curve covariance into orthogonal factors: level (~80%), slope (~12%), curvature (~5%); PCA-neutral hedging kills the largest factor exposures.
• Risk-neutral (Q) probabilities price derivatives via no-arbitrage; true (P) probabilities forecast and drive VaR, differing from Q by the risk premium.
• Vasicek model: $dr=k(\theta -r)dt+\sigma dW$. Mean-reverts to $\theta$ at speed k, admits negative rates, gives closed-form bond and option prices; stationary variance ${\sigma }^2/(2k)$.

Exam shortcut

When a question contrasts a regression hedge with a DV01-neutral hedge, the regression hedge is correct unless the question states $\beta =1$. When PCA is mentioned, expect three components (level, slope, curvature) with the variance shares 80/12/5. When a model's negative-rate behavior is questioned, Vasicek allows it, CIR doesn't, lognormal doesn't. Time-varying drift (Ho-Lee, Hull-White) makes a model arbitrage-free against today's curve. Memory aid: "Level Slopes Curve" (the three PCA factors).

The full lesson (about 3,423 words, 23 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 1
• 2
• 3
• 4
• 5
• 6
• 7
• 8
• 9
• 10
• 11
• 12
• 13
• 14
• 15
• 16
• 17
• 18

Browse all free FRM Part II lessons or jump into free FRM Part II practice questions.