P2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition

Transcription

1 P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM

2 Tuckman, Chapter 6: Empirical Approaches to Risk Metrics and Hedging EXPLAIN THE DRAWBACKS TO USING A DV-NEUTRAL HEDGE FOR A BOND POSITION DESCRIBE A REGRESSION HEDGE AND EXPLAIN HOW IT IMPROVES ON A STANDARD DV- NEUTRAL HEDGE CALCULATE THE REGRESSION HEDGE ADJUSTMENT FACTOR, BETA CALCULATE THE FACE VALUE OF AN OFFSETTING POSITION NEEDED TO CARRY OUT A REGRESSION HEDGE

3 Tuckman, Chapter 6: Empirical Approaches to Risk Metrics and Hedging Explain the drawbacks to using a DV-neutral hedge for a bond position. Describe a regression hedge and explain how it can improve a standard DV-neutral hedge. Calculate the regression hedge adjustment factor, beta. Calculate the face value of an offsetting position needed to carry out a regression hedge. Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge. Compare and contrast between level and change regressions. Describe principal component analysis and explain how it is applied in constructing a hedging portfolio. Explain the drawbacks to using a DV-neutral hedge for a bond position. The major drawback to using a DV-neutral hedge is related to the drawback of duration: this is a single-factor model (where the single factor is yield to maturity because we are typically referring implicitly to a yield-based DV), the assumption is that movements of the entire structure can be described by a one interest rate factor. Implicitly, this is an assumption that the different rates along the curve move in parallel. Such a hedge cannot account for twists in the curve: when one rate moves more or less than another. This is referred to as curve risk, and this risk is very real. By neglecting curve risk and simplistically assuming parallel shifts in the rate curve, the DV-hedge is not necessarily a realistic hedge. If we want to hedge a given position, we seek to make the position DV-neutral. For example, a trader has a view that the spread between nominal and real interest rates will increase due to inflation. To capitalize on this, she may sell short US Treasury bonds and go long Treasury Inflation Protected Securities (TIPS). The payment from TIPS are inflation indexed, thus they are not sensitive to inflation. If a trader goes short $100m of Treasury bonds, we can solve for how big a long position she needs to enter into to make the position DV neutral. For example, if the DV of the Treasury bond is and the DV of the TIPS is 0.075, then the trader executes the following long trade in TIPS: = $100. = $73.. Neutralizing DV ensures that the trade neither makes nor loses money only if the yield on the TIPS and the nominal bond both increase or decrease by the same number of basis points. 3

4 Describe a regression hedge and explain how it improves on a standard DV-neutral hedge. We can regress changes in the nominal yield,, against changes in the real yield,, as follows (the regression below uses actual 2105 data from y = α + β y + ε Least-Squares Regression: No. of Observations 248 R-Squared 81.99% Standard Error Regression Coeff Value Std Error t-stat Constant (α) Change in Real Yield (β) A regression hedge can improve upon the standard DV hedge One problem with the DV-neutral approach is that it implicitly assumes that the T- bond and the TIPS are perfectly co-dependent, meaning they move 1:1. In reality, empirical data show this is not the case. Indeed, for our regression (with 25 data) of nominal yield changes on changes in TIPS yields produces an R 2 of 81.2%. An advantage of the regression hedge is that the trader can estimate how much the nominal yield changes, on average, given a change in the TIPS yield. Even though the beta of the regression can change over time, it g a more realistic picture than does the DV based hedging. Indeed, from section T1 on regression analysis, we recall that a least squares, regression based framework for hedging will minimize the variance of the P&L. Another advantage of this approach is that it automatically provides an estimate of the volatility of the hedged portfolio. We go on to explore the adjustment factor, beta. 4

5 Calculate the regression hedge adjustment factor, beta. We now go on to show how we can use the adjustment factor, beta to improve upon the DV-neutral hedge. The daily P&L of the portfolio is given by, & = = With the following implied values: 100. Face value of hedge, F R is given by therefore, the beta, β, of the hedge is: =, =. These formulas worth knowing. Please note that, Tuckman elaborates on the derivation of these equations in his Appendix. Calculate the face value of an offsetting position needed to carry out a regression hedge. We calculated DV-neutral hedge above, now, let us use what we have learned about the benefits of regression hedging and apply it to the same position. Data Input DV TIPS DV Nominal F TIPS $73.333M F Nominal $100m Beta = $ = $ While our DV-neutral hedge resulted in long position equivalent to $73.33 million, the regression-adjusted hedge indicates that a better hedge would be to go long million, a difference of $4.44 million, or about 6%; i.e., the difference is β-1. 5

6 Switching to Tuckman s own example (Table 6-2) of the regression hedge US T-Bonds Yield DV Nominal 30-yr 3.275% TIPS 30-yr 1.237% Least-Squares Regression: No. of Observations 229 R-Squared 56.30% Standard Error Regression Coeff Value Std Error t-stat Constant (α) Change in Real Yield (β) For this dataset, the fitted regression line is given by: y = α + β y + ε y = y Let s assume the trader plans to short $100.0 million of the nominal 3 5/8 bonds. Let s first analyze the trade that neutralizes DV; i.e., the DV-nuetral trade which assumes that both bonds (nominal and TIPS) experience the same basis point shift: = $100.. = $ Next, consider the regression hedge that incorporates the regression slope (beta): = $ = $ Because the standard error of the regression is (see table above), the standard deviation of the P&L is given by: σ = $ = $255,940. 6