P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes Reading 40 By David Harper, CFA FRM CIPM www.bionicturtle.com
TUCKMAN, CHAPTER 6: EMPIRICAL APPROACHES TO RISK METRICS AND HEDGING... 3 EXPLAIN THE DRAWBACKS TO USING A DV-NEUTRAL HEDGE FOR A BOND POSITION.... 3 DESCRIBE A REGRESSION HEDGE AND EXPLAIN HOW IT IMPROVES ON A STANDARD DV- NEUTRAL HEDGE.... 3 CALCULATE THE REGRESSION HEDGE ADJUSTMENT FACTOR, BETA.... 5 CALCULATE THE FACE VALUE OF AN OFFSETTING POSITION NEEDED TO CARRY OUT A REGRESSION HEDGE.... 5 CALCULATE THE FACE VALUE OF MULTIPLE OFFSETTING SWAP POSITIONS NEEDED TO CARRY OUT A TWO-VARIABLE REGRESSION HEDGE.... 6 COMPARE AND CONTRAST BETWEEN LEVEL AND CHANGE REGRESSIONS.... 7 DESCRIBE PRINCIPAL COMPONENT ANALYSIS AND EXPLAIN HOW IT IS APPLIED IN CONSTRUCTING A HEDGING PORTFOLIO.... 8 QUESTIONS AND ANSWERS... 9 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 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 with a DV-neutral hedge (this generally refers to a yield-based DV) is related to the drawback of duration: as a single-factor model (where the factor is yield to maturity), 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. Describe a regression hedge and explain how it improves on a standard DV-neutral hedge. If you want to hedge a given position, we have seen that this can be done with the goal of making 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. 3
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. As seen in the last chapter, if the DV of the Treasury bond is 0.055 and the DV of the TIPS is 0.075, then the trader executes the following long trade in TIPS: = $100. = $73. 1. Nominal vs Real yield changes (22 data, souce: US Dept of Treasury) Change in Nomial Yield (bps) 15 10 5 0-14 -9-4 1 6 11 16-5 -10-15 Change in Real Yield (bps) A regression hedge can improve upon the standard DV hedge One problem with the DV-neutral approach is that she 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 year-to-date (23), a regression of nominal yield changes on changes in TIPS yields produces an of 61.1%, and for the year 22 as a whole that number was 58.3%. An advantage of the regression hedge is that the trader may 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 does paint 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. 1 Note that this example differs from that in the assigned Tuckman: we are using more recent data. 4
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, & = 100 100 = 100. and the Face value of the hedge is given as so, =, and the formula for the beta of the hedge is given by, =. These are key formulas that you should know. In the appendix, the derivation of these equations are elaborated on. 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 0.075 DV Nominal 0.055 F TIPS $73.333M F Nominal $100m Beta 0.918 = $100 0.918.. = $67.32. While our DV-neutral hedge resulted in long position equivalent to $73.33million, the regression-adjusted hedge indicates that a better hedge would be to go long 67.32million, a difference of $6million, or 9%. 5
Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge. A two variable regression is similar to the single-variable case, with a few exceptions. The way we go about calculating offsetting swap positions needed to carry out a twovariable regression hedge will be a simple continuation of the single-variable case. We follow here the example in Tuckman, where a market maker has a long position in (illiquid) 20-year interest rate swaps, and wants to hedge his exposure with a combination of the more liquid 10-year and 30-year interest rate swaps in order to preserve the spread gained from buying the illiquid 20s. The trader runs a multivariable regression on yield changes as so, which yields the following output: = + + + Regression analysis result Observations 1281 R-squared 98.80% Standard error 0.14 Regression coefficients Value Std. Error Constant (alpha) -0.04 0.004 delta 10y rate (beta 10y) 0.2221 0.0034 delta 30y rate (beta 30y) 0.7765 0.0037 We can clearly see that the multivariate approach yields a much better result than did the single-variable case. Here, we are able to explain 98.8% of the change in the 20-year swap rate, with a combination of the change in the 10y and 30y swap rates. Notice also how alpha, the intercept is not significantly different from 0, thus our assumption of 0 holds up. Using our results from the single-variable case, we have that the P&L of the hedged position as: & = 100 100 100 Substituting the expected change in from the first equation into the second equation (and remembering that when taking expectations = 0), retaining only the terms that depend on and, we get: 100 100 + 100 100 6
We now choose and such that the terms in the brackets equal zero to eliminate dependence on and. It is OK if you don t follow this derivation completely, however, it is important that you understand the concept, and that you understand our results below. We get the following two equations, where we have solved for the Face value and then simply re-arranged the solution to express the beta,, as a function of the solvedfor Face value, for the two maturities, respectively: IMPORTANT FORMULA: = = = = Compare and contrast between level and change regressions. Level and price change regressions are two different animals. Since, price levels are rarely stationary, the result of a regression is spurious. A time series is stationary if it has a constant mean and variance. Looking at, e.g. stock prices; we know from past history that they tend to trend upwards with time. Thus any two series following this pattern will exhibit some degree of correlation on a price-level basis. The picture might be different entirely when we look at the returns though. For fixed income instruments, the picture is less clear: regressions of levels do not exhibit the same long-term growth pattern that stocks do. Technically, if the error terms of our regression are i.i.d. and are uncorrelated with the independent variable, we may still be able to use rate-levels. However, in practice, it is well documented that the invariants is not the levels themselves, but rather the change in yields are close to being invariant (i.i.d.), thus one would rarely see, or get auditor approval for level-on-level regressions. Generally, we do observe autocorrelation, also known as serial correlation. That is, today s estimate is not independent of yesterday s. As we have seen, models that incorporate autoregressive features are well suited to deal with the cases where there is persistent autocorrelation. 7
Describe principal component analysis and explain how it is applied in constructing a hedging portfolio. Given a large number of rates n, it turns out that we can describe the risk factors f, with a much smaller sub-set of rates. Given some 30 interest-rate factors, we can typically describe most of the change using only three principal components. Principal component analysis is a powerful statistical tool that can help solve the curse of dimensionality. Imagine for a movement that you have a 1000x1000 matric of forward curves. In this scenario, you would typically be able to reduce the number of risk factors down to some 7 factors that drive the forward curves. That is, we have reduced the dimensionality of the problem by a factor of ~ 140! Tuckman lists the standard properties of PCA as follows: 1. The sum of the variances of the principal components equals the sum of the variances of the individual rates. 2. The principal components are uncorrelated with one another. Changes in the individual rates are correlated, however, the principal components, by construction are not. 3. The key property of the principal component analysis is that s.t. the two properties above, the principal components are chosen to have the maximum possible variance, given the other principal components. The first principal component explains the largest fraction of the sum of the variance; the second explains the second largest fraction and so forth. By statistically maximizing the variance of the principal components, we pick the minimum number of rates necessary to adequately describe movements in rates. Hedging with principal components Going back to the example of the 1000x1000 matrix of forward curves, we saw that only roughly 7 of the first principal components are needed to explain, say 95% of the change in the rates. The exact same principle applies to interest rates of course. The point here, is that we were able to reduce the number of factors needed to describe 95% of the variance of our data with only 7 principal components, which we saw was a reduction by a factor of 140. The implication for hedging should be clear: by conducting a PCA on our rates, we can obtain information regarding how many rates, and which factors are needed to hedge x% of the variance in the rates, where x can be arbitrarily chosen to be any number, but is typically chosen to describe 90-98% of the variance of the dataset. Accordingly, we can focus on only the key-risk drivers, rather than a large set of rates to effectively hedge our exposure. In practice PCA analysis is used extensively, from selecting the appropriate hedging instruments, to reducing the dimensionality of a large-scale simulation. Although you are not asked to calculate any Principal components for the purpose of the exam, learning the mathematical techniques behind it is important for real-world applications. 8
Questions and Answers 321.1. A trader shorts $100.0 million of nominal US Treasury 3 5/8s bonds and, in a relative trade, wants to hedge with a long in US Treasury Inflation Protected Securities (TIPS); the relative trade will express a view on inflation as reflected in the spread between the rate of the nominal bond and a TIPS. The yields and DVs of each bond are shown below, in addition to the output from a regression of nominal yield changes against real (TIPS) yield changes: Which is nearest to the face amount of the long position in the 1 7/8s TIPS needed to carry out a regression hedge? a) $6.8 million face amount of TIPS b) $68.83 million face amount of TIPS c) $75.30 million face amount of TIPS d) $105.68 million face amount of TIPS 9
321.2. A trader shorts $100.0 million of nominal US Treasury 3 5/8s bonds and, employing a single-variable regression hedge, buys a face amount of 1 7/8s US TIPS. The bonds are show below in addition to the results of a regression of the nominal against real yield: Which is nearest to an estimate of the daily volatility of the P&L of the hedged portfolio? a) $2,380 b) $43,760 c) $67,323 d) $212,000 10
321.3. A trader shorts $100.0 million of nominal US Treasury 3 5/8s bonds and hedges with a purchase of a face amount of 1 7/8s US TIPS based on the regression results of a singlevariable hedge, which are shown here: Each of the following are true about this regression hedge EXCEPT for which is false? a) The regression model estimates a correlation between changes in the nominal and real yields of about 82.5% b) We can be 95% confident that the true value of beta is unity (1.0) c) The daily volatility of the P&L of the hedged portfolio is about $88,000 d) The difference between a DV-hedge and a regression hedge is about $17.14 million face amount of the TIPS 11
322.2. When estimating regression-based hedges, some practitioners regress the level of yields-on-yields, as given by Tuckman's 6.25 below, while others prefer to regress CHANGES in yields on CHANGES in yields, as given by Tuckman's 6.26 below: In regard to these two regression approaches, according to Tuckman, each of the following is true EXCEPT: a) If the error terms are independently and identically distributed (i.i.d.) with mean zero and uncorrelated with the independent variable, then the regression coefficient estimators are unbiased, consistent, and efficient b) Empirically, however, the error terms are likely to be correlated over time (i.e., serially correlated) c) If the error terms are serially correlated, the regression coefficients will not be efficient; although they likely remain unbiased and consistent d) Unfortunately, we cannot overcome the problem created by serial correlation and we must select one of the two model approaches as some "methodological imperfection is inevitable." 322.3. According to Tuckman, each of the following is true about principal component analysis (PCA) EXCEPT which is false? a) PCA is useful as an empirical description of the behavior of the term structure that is multi-factor but can be applied across all bonds b) The first three components, for example "level," "slope" and "short rate," will tend to explain the majority (i.e., high explanatory power of 90% or more) of interest rate volatility c) The number of components (factors) in a PCA analysis required will scale roughly with the portfolio's size; at a certain point, a regression-based hedge becomes more practical d) The qualitative shape of the USD level Principal Component (PC; i.e., where "level" is the first component) did change after the financial crisis; e.g., as observed in the October 2008 to 20 time frame 12
Answers 321.1. C. $75.30 million face amount of TIPS = $100.0 million * 0.0530 DV/0.0770 DV * 1.0940 beta Discuss here in forum: https://www.bionicturtle.com/forum/threads/p1-t4-321-fixedincome-single-variable-regression-hedge.6996/#post-24313 321.2. D. $212,000. Per Tuckman 6.13, Standard deviation of P&L = $100.0 million * 0.040/100 * 5.30 standard error = $212,000 Discuss here in forum: https://www.bionicturtle.com/forum/threads/p1-t4-321-fixedincome-single-variable-regression-hedge.6996/#post-24313 321.3. B. False, the beta of 1.30 is (1.30-1.0)/0.0510 = 5.88 standard deviations from 1.0; similarly, the 95% CI is [~ 1.2, 1.4] In regard to (A), (C), and (D), each is TRUE. Discuss here in forum: https://www.bionicturtle.com/forum/threads/p1-t4-321-fixedincome-single-variable-regression-hedge.6996/#post-24313 322.2. D. Tuckman proposes an auto-regressive error term. Tuckman: "The first lesson to be drawn from this discussion is that because the error terms in both (6.26) and (6.25) are likely to be correlated over time, i.e., serially correlated, their estimated coefficients are not efficient. But, with nothing to gainsay the validity of the other assumptions concerning the error terms, the estimated coefficients of both the level and change specifications are still unbiased and consistent. The second lesson to be drawn from the discussion of this section is that there is a more sensible way to model the relationship between two bond yields than either (6.26) or (6.25). In particular, model the behavior that the y-bond s yield will, on average, move somewhat closer from 1% to 5%. Mathematically, assume (6.25) with the error dynamics: (6.27) error(t) = rho*error(t-1) * v(t), for some constant rho < 1." In regard to (A), (B), and (C), each is TRUE. Discuss in forum here: https://www.bionicturtle.com/forum/threads/p1-t4-322- empirical-approaches-to-fixed-income-hedging.7009/#post-244 322.3. C. The number of components (factors) is likely to remain limited, per the TRUE statement (B), and PCA is better suited to large portfolios than regression In regard to (A), (B) and (D), each is TRUE. Discuss in forum here: https://www.bionicturtle.com/forum/threads/p1-t4-322- empirical-approaches-to-fixed-income-hedging.7009/#post-244 13