Set level regressions and change regressions side by side, noting their similarities and differences.
The idea
Level regressions regress the yield level of one security on the yield level of another. Change regressions regress the change in one yield on the change in the other over a fixed interval such as a day or a week. Both fit an intercept, a slope and an error term, and both can drift between sample periods. Yield levels trend and are persistent, so level regressions show very high R² and serially correlated errors, which overstates how well one yield explains the other. Hedging is about profit and loss from moves in rates. The change regression, with its lower but more honest R², is therefore the one to size a hedge. Level regressions suit judging whether a security is rich or cheap against a fitted relationship. Exam point: size hedges from changes, not levels.
A practice question
What is the basic difference between a level regression and a change regression of two yields?
- A level regression uses bond prices; a change regression uses yields.Why not: Both use yields; the distinction is levels versus changes, not prices versus yields.
- A level regression relates the yield levels of the two securities; a change regression relates the changes in the two yields over a fixed interval. - correct
- A level regression is estimated on one date only; a change regression uses a time series.Why not: Both are estimated on a time series; the level regression uses the series of levels.
- A level regression has no intercept; a change regression always includes one.Why not: Either regression can include an intercept; this is not what defines them.
Level regression: yield_A(t) on yield_B(t). Change regression: change in yield_A over the interval on change in yield_B over the same interval.
A harder one
A risk manager finds that a regression of 10-year yield levels on 2-year yield levels over five years has R² of 0.95, while a regression of daily changes has R² of 0.60. The best interpretation is:
- the level regression is better and a hedge based on it would remove 95% of P&L variance.Why not: Treats R² in levels as P&L variance removed; the P&L depends on changes, for which R² is 0.60.
- the two R² values should be equal, so one regression is mis-specified.Why not: There is no reason for them to match; level data are persistent and change data are noisy.
- an R² of 0.60 means the change regression's slope is wrong and should be replaced by the level slope.Why not: R² measures fit, not whether the slope is correct; the slope from changes is the one relevant to hedging.
- the level R² is inflated by trending, persistent yields, and the change regression's 0.60 is the better guide to hedge effectiveness. - correct
Yield levels trend and persist, which produces very high fits and autocorrelated residuals. Hedge effectiveness is judged on changes, here R² 0.60, meaning about 60% of variance is removed.
The hardest tier
A position's DV01-neutral hedge calls for USD 100 million of the hedging bond. Over three years, the regression of target yield levels on hedge yield levels has slope 0.90, while the regression of daily target yield changes on daily hedge yield changes has slope 0.70. The face amount of the regression hedge that best protects against daily rate moves is:
- USD 90 million.Why not: Uses the level slope; the hedge is meant for changes in rates.
- USD 100 million.Why not: Ignores both regressions and keeps the DV01-neutral amount.
- USD 70 million. - correct
- USD 63 million.Why not: Multiplies the two slopes (0.9 x 0.7) together.
Beta for a hedge comes from the change regression: 0.70 x 100 = USD 70 million.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Regression Hedging and Principal Component Analysis
- Account for the shortcomings of hedging a bond position with a DV01-neutral hedge.
- Set out what a regression hedge is and show how it improves on the plain DV01-neutral hedge.
- Work out beta, the adjustment factor in a regression hedge.
- Determine the face amount of the offsetting position that a regression hedge calls for.
- Compute how much face value each of the multiple swaps must carry when a regression hedge uses two explanatory variables.
- Explain the reasons for, and the mechanics of, the difference between a regression hedge and one derived from regressing in the opposite direction.
- Describe principal component analysis (PCA) and how it helps in putting together a portfolio of hedges.