*Learning objectives: Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge. Compare and contrast level and change regressions. Describe principal component analysis and explain how it is applied to constructing a hedging portfolio.*

**Questions:**

22.15.1. Consider the following regression of daily yield changes of the 20-year Treasury bond against daily yield changes in 10-year and 30-year bonds (similar to Tuckman's two-variable regression hedge):

A relative value trader perceives that the 20-year yield is too high (aka, price is too low) relative to the 10- and 30-year bonds. She executives a butterfly trade to exploit her view: she buys $100.0 million face amount of the 20-year Treasury bond and sells both the 10- and 30-year bonds.

How does the dollar value of '01 (i.e., DV01) of her hedging position compare to the DV01 of the 20-year bond (aka position being hedged)?

a. 41.% less

b. 13% less

c. +12% more

d. +27% more

22.15.2. Sam wants to regress bond Y against bond X, but he cannot decide between a level or change regression

- Regress yield level of Y(t) on yield level of X(t); aka, level-on-level regression
- Regress yield change, ΔY(t), on yield change of ΔY(t); aka, change-on-change regression

Which regression is

**most appropriate**in this circumstance?

a. Any change-on-change regression

b. Change-on-change regression but employing least-squares estimates that are BLUE

c. Level-on-level regression with autoregressive, AR(1), error dynamics, ε(t) = ρ*ε(t-1) + ν(t)

d. Level-on-level regression with coefficients reversed (aka, reverse regression) to smooth decay and enforce lower bound

22.15.3. The table below replicates (but the seven middle rows are truncated: 6 to 10, 15, and 20-year terms) Tuckman's principal component analysis (PCA) of USD Libor swap rates from June 1, 2020, to July 16, 2021. Source: Tuckman's Fixed Income Securities, 5th edition.

The dataset consists of 13 interest rate factors. About this principal component analysis (PCA), each of the following statements is TRUE

**except**which is false?

a. While the interest rate factor covariance matrix contains 78 covariances (plus 13 variances), the principal component covariance matrix is a diagonal matrix

b. A fixed income PCA is convenient because it enables the user to map term structure volatility to the desired fundamental factors that contribute to an easily explainable narrative; in particular, level (aka, parallel shift), slope, and twist

c. The PCA constructs 13 principal components (PC), but conveniently only a limited number of PCs are needed, in this case, three, to sufficiently characterize term structure fluctuations

d. The PC volatility ("PC vol") of the one-year rate is sqrt[0.23^2 + (-0.16)^2 + 0.29^2] because the PCs are orthogonal by construction

**Answers here:**