Additional Praise for Fixed Income Securities Tools for Today’s Markets, 2nd Edition phần 4 docx

turity is the same as computing DV01. In other words, the key rate expo- sures of the hedging securities equal their DV01s. To illustrate the conve- nience of par yield key rates with par hedging bonds, suppose in the example of the previous section that the 10-year bond sold at par like the other three hedging bonds. In that case, the hedging equations (7.3) through (7.6) reduce to the following simpler form: (7.8) (7.9) (7.10) (7.11) To compute the hedge amount in this case, simply divide each key rate ex- posure by the DV01 (per unit face value) of the hedging bond of corre- sponding maturity. The disadvantage of using par yields, particularly when combined with the assumption that intermediate yields are found by the lines drawn in Figure 7.1, is that the changes in the forward rate curve implied by these yields changes have a bizarre shape. The problem is similar to that of the forward curve emerging from linear yield interpolation in Figure 4.3: Kinks in the yield curve translate into sizable jumps in the forward curve. Changing key rates to spot rates has the same disadvantage. Setting key rates to forward rates naturally solves this problem: The shifted forward curve is only as odd as the shapes in Figure 7.1. The prob- lem with shifting forward rates is that spot rate changes are no longer local. Changing the forward rates from two to five years while keeping all other forward rates constant, for example, changes all the spot rates from two years and beyond. True, the effect on a 30-year rate is much less than the ef- fect on a five-year rate, but the key rate interpretation of shocking one part of the term structure at a time is lost. Forward shifts will, however, be the more natural set of shifts in bucket analysis, described in the next section. . . 15444 100 67 25637 30 F = . . 08308 100 42 36832 10 F = . . 04375 100 3 77314 5 F = . . 01881 100 98129 2 F = Choosing Key Rates 141 With respect to the terms of the key rates, it is clearly desirable to spread them out over the maturity range of interest. More subtly, well-cho- sen terms make it possible to hedge the resulting exposures with securities that are traded and, even better, with liquid securities. As an example, con- sider a swap market in an emerging market currency. A dealer might make markets in long-term swaps of any maturity but might observe prices and hedge easily only in, for example, 10- and 30-year swaps. In that case there would not be much point in using 10-, 20-, and 30-year par swap yields as key rates. If all maturities between 10 and 30 years were of extremely lim- ited liquidity, it would be virtually impossible to hedge against changes in those three ill-chosen key rates. If a 20-year security did trade with limited liquidity the decision would be more difficult. Including a 20-year key rate would allow for better hedging of privately transacted, intermediate-matu- rity swaps but would substantially raise the cost of hedging. BUCKET SHIFTS AND EXPOSURES A bucket is jargon for a region of some curve, like a term structure of interest rates. Bucket shifts are similar to key rate shifts but differ in two respects. First, bucket analysis usually uses very many buckets while key rate analysis tends to use a relatively small number of key rates. Second, each bucket shift is a parallel shift of forward rates as opposed to the shapes of the key rate shifts described previously. The reasons for these differences can be ex- plained in the context for which bucket analysis is particularly well suited, namely, managing the interest rate risk of a large swap portfolio. Swaps are treated in detail in Chapter 18, but a few notes are neces- sary for the discussion here. Since this section focuses on the risk of the fixed side of swaps, the reader may, for now, think of swap cash flows as if they come from coupon bonds. Given the characteristics of the swap mar- ket, agreements to receive or pay those fixed cash flows for a particular set of terms (e.g., 2, 5, and 10 years) may be executed at very low bid-ask spreads. Unwinding those agreements after some time, however, or enter- ing into new agreements for different terms to maturity, can be costly. As a result, market making desks and other types of trading accounts tend to re- main in swap agreements until maturity. A common problem in the indus- try, therefore, is how to hedge extremely large books of swaps. The practice of accumulating swaps leads to large portfolios that change in composition only slowly. As mentioned, this characteristic 142 KEY RATE AND BUCKET EXPOSURES makes it reasonable to hedge against possible changes in many small seg- ments of the term structure. While hedging against these many possible shifts requires many initial trades, the stability of the underlying portfolio composition assures that these hedges need not be adjusted very frequently. Therefore, in this context, risk can be reduced at little expense relative to the holding period of the underlying portfolio. As discussed in previous sections, liquid coupon bonds are the most convenient securities with which to hedge portfolios of U.S. Treasury bonds. While, analogously, liquid swaps are convenient for hedging portfo- lios of less liquid swaps, it turns out that Eurodollar futures contracts play an important role as well. These futures will be treated in detail in Chapter 17, but the important point for this section is that Eurodollar futures may be used to hedge directly the risk of changes in forward rates. Furthermore, they are relatively liquid, particularly in the shorter terms. The relative ease of hedging forward rates makes it worthwhile to compute exposures of portfolios to changes in forward rates. Figure 7.2 graphs the bucket exposures of receiving the fixed side of $100 million of a 6% par swap assuming, for simplicity, that swap rates are flat at 6%. (It should be emphasized, and it should become clear shortly, that the assumption of a flat term structure is not at all necessary for the computation of bucket exposures.) The graph shows, for example, that the exposure to the six-month rate 2.5 years forward is about $4,200. Bucket Shifts and Exposures 143 FIGURE 7.2 Bucket Exposures of a Six-Year Par Swap –$3,000 –$2,000 –$1,000 $0 $1,000 $2,000 $3,000 $4,000 $5,000 Years Forward Bucket Exposure 0 .5 1 2 2.5 3 3.5 4.5 5 5.5 1.5 4 In other words, a one-basis point increase in the six-month rate 2.5 years forward lowers the value of a $100 million swap by $4,200. The sum of the bucket exposures, in this case $49,768, is the exposure of the swap to a simultaneous one-basis point change to all the forwards. If the swap rate curve is flat, as in this simple example, this sum exactly equals the DV01 of the fixed side of the swap. In more general cases, when the swap rate curve is not flat, the sum of the forward exposures is usually close to the DV01. In any case, Figure 7.2 and this discussion reveal that this swap can be hedged by paying fixed cash flows in a swap agreement of similar coupon and maturity or by hedging exposures to the forward rates directly with Eurodollar futures. Table 7.3 shows the computation of the particular bucket exposure mentioned in the previous paragraph. The original forward rate curve is flat at 6%, and the par swap, by definition, is priced at 100% of face amount. For the perturbed forward curve, the six-month rate 2.5 years for- ward is raised to 6.01%, and all other forwards are kept the same. The new spot rate curve and discount factors are then computed using the rela- 144 KEY RATE AND BUCKET EXPOSURES TABLE 7.3 Exposure of a $100 million 6% Par Swap to the Six-Month Rate 2.5 Years Forward Initial forward curve flat at 6% Bucket exposure: $4,187 Years Cash Flow Perturbed Perturbed Discount Forward ($millions) Forward Spot Factor 0 3 6.00% 6.0000% 0.970874 0.5 3 6.00% 6.0000% 0.942596 1 3 6.00% 6.0000% 0.915142 1.5 3 6.00% 6.0000% 0.888487 2 3 6.00% 6.0000% 0.862609 2.5 3 6.01% 6.0017% 0.837444 3 3 6.00% 6.0014% 0.813052 3.5 3 6.00% 6.0012% 0.789371 4 3 6.00% 6.0011% 0.766380 4.5 3 6.00% 6.0010% 0.744058 5 3 6.00% 6.0009% 0.722386 5.5 103 6.00% 6.0008% 0.701346 Perturbed value ($millions): 99.995813 tionships of Part One. Next, the fixed side of the swap is valued at 99.995813% of face value by discounting its cash flows. Finally, the bucket exposure for $100 million of the swap is (7.12) Say that a market maker receives fixed cash flows from a customer in a $100 million, six-year par swap and pays fixed cash flows to another cus- tomer in a $141.8 million, four-year par swap. The cash flows of the result- ing portfolio and the bucket exposures are given in Table 7.4. A negative exposure means that an increase in that particular forward rate raises the value of the portfolio. The bucket exposures sum to zero so that the portfo- lio is neutral with respect to parallel shifts of the forward rate curve. This discussion, therefore, is a very simple example of a growing swap book that is managed so as to have, in some sense, no outright interest rate exposure. −×− () =$,, . % %$,100 000 000 99 995813 100 4 187 Bucket Shifts and Exposures 145 TABLE 7.4 Bucket Exposures for a Position Hedged for Parallel Shifts Initial forward curve flat at 6% Coupon 6.00% 6.00% Maturity 6.0 4.0 Face ($mm) 100.000 –141.801 Portfolio Bucket Years Cash Flow Flows Exposure Forward ($millions) ($millions) ($) 0.00 3.000 –4.254 –1.254 –2,029 0.50 3.000 –4.254 –1.254 –1,970 1.00 3.000 –4.254 –1.254 –1,913 1.50 3.000 –4.254 –1.254 –1,857 2.00 3.000 –4.254 –1.254 –1,803 2.50 3.000 –4.254 –1.254 –1,750 3.00 3.000 –4.254 -–1.254 –1,699 3.50 3.000 –146.055 –143.055 –1,650 4.00 3.000 0.000 3.000 3,832 4.50 3.000 0.000 3.000 3,720 5.00 3.000 0.000 3.000 3,612 5.50 103.000 0.000 103.000 3,507 Total: 0 Figure 7.3 graphs the bucket exposures of this simple portfolio. Since a six-year swap has been hedged by a four-year swap, interest rate risk re- mains from six-month rates 4 to 5.5 years forward. The total of this risk is exactly offset by negative exposures to six-month rates from 0 to 3.5 years forward. So while the portfolio has no risk with respect to parallel shifts of the forward curve, it can hardly be said that the portfolio has no interest rate risk. The portfolio will make money in a flattening of the forward curve, that is, when rates 0 to 3.5 years forward rise relative to rates 4 to 5.5 years forward. Conversely, the portfolio will lose money in a steepen- ing of the forward curve, that is, when rates 0 to 3.5 years forward fall rel- ative to rates 4 to 5.5 years forward. A market maker with a portfolio characterized by Figure 7.3 may very well decide to eliminate this curve exposure by trading the relevant for- ward rates through Eurodollar futures. The market maker could certainly reduce this curve exposure by trading par swaps and could neutralize this exposure completely by entering into a sufficient number of swap agree- ments. But hedging directly with Eurodollar futures has the advantages of simplicity and, often, of liquidity. Also, should the forward exposure pro- file change with the level of rates and the shape of the curve, adjustments to a portfolio of Eurodollar futures are preferable to adding even more swaps to the market maker’s growing book. 146 KEY RATE AND BUCKET EXPOSURES FIGURE 7.3 Bucket Exposures of a Six-Year Swap Hedged with a Four-Year Swap –$3,000 –$2,000 –$1,000 $0 $1,000 $2,000 $3,000 $4,000 $5,000 Years Forward Bucket Exposure 0 .5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 As each Eurodollar futures contract is related to a particular three- month forward rate and as 10 years of these futures trade at all times, 3 it is common to divide the first 10 years of exposure into three-month buckets. In this way any bucket exposure may, if desired, be hedged directly with Eurodollar futures. Beyond 10 years the exposures are divided according to the same considerations as when choosing the terms of key rates. IMMUNIZATION The principles underlying hedging with key rate or bucket exposures can be extrapolated to a process known as immunization. No matter how many sources of interest rate risk are hedged, some interest rate risk re- mains unless the exposure to each and every cash flow has been perfectly hedged. For example, an insurance company may, by using actuarial ta- bles, be able to predict its future liabilities relatively accurately. It can then immunize itself to interest rate risk by holding a portfolio of assets with cash flows that exactly offset the company’s future expected liabilities. The feasibility of immunization depends on the circumstances, but it is worth pointing out the spectrum of tolerances for interest rate risk as re- vealed by hedging techniques. On the one extreme are hedges that protect against parallel shifts and other single-factor specifications described in Part Three. Away from that extreme are models with relatively few factors like the two- and multi-factor models of Chapter 13, like the empirical ap- proach discussed in Chapter 8, and like most practical applications of key rates. Toward the other extreme are bucket exposures and, at that other extreme, immunization. MULTI-FACTOR EXPOSURES AND RISK MANAGEMENT While this chapter focuses on how to quantify the risk of particular changes in the term structure and on how to hedge that risk, key rate and bucket ex- posures may also be applied to problems in the realm of risk management. The introduction to Chapter 5 mentioned that a risk manager could combine an assumption that the annual volatility of interest rates is 100 basis points with a computed DV01 of $10,000 per basis point to conclude Multi-Factor Exposures and Risk Management 147 3 The longer-maturity Eurodollar futures are not nearly so liquid as the earlier ones. that the annual volatility of a portfolio is $1 million. But this measure of portfolio volatility has the same drawback as one-factor measures of price sensitivity: The volatility of the entire term structure cannot be adequately summarized with just one number. As to be discussed in Part Three, just as there is a term structure of interest rates, there is a term structure of volatil- ity. The 10-year par rate, for example, is usually more volatile than the 30- year par rate. Key rate and bucket analysis may be used to generalize a one-factor es- timation of portfolio volatility. In the case of key rates, the steps are as fol- lows: (1) Estimate a volatility for each of the key rates and estimate a correlation for each pair of key rates. (2) Compute the key rate 01s of the portfolio. (3) Compute the variance and volatility of the portfolio. The computation of variance is quite straightforward given the required inputs. For example, if there are only two key rates, R 1 and R 2 , if the key rate 01s of the portfolio are KR01 1 and KR01 2 , and if the portfolio value is P, then the change in the value of the portfolio is (7.13) where ∆ denotes a change. Furthermore, letting σ 2 with the appropriate subscript denote a particular variance and letting ρ denote the correla- tion between changes in the two key rates, the variance of the portfolio is simply (7.14) The standard deviation or volatility of the portfolio is simply, of course, the square root of this variance. Bucket analysis may be used in the same way, but a volatility must be assigned to each forward rate and many more correlation pairs must be estimated. σσσ ρσσ P 2 1 2 1 2 2 2 2 2 1212 2=++×××KR01 KR01 KR01 KR01 ∆∆ ∆PR R=+KR01 KR01 11 2 2 148 KEY RATE AND BUCKET EXPOSURES 149 CHAPTER 8 Regression-Based Hedging D V01 and duration measure price sensitivity under any given assump- tions about term structure movement. Yield-based DV01 and modified duration assume parallel yield shifts, key rates assume the particular local perturbations described in Chapter 7, and the models of Part Three make more complex assumptions. The goal of all these choices is to approxi- mate the empirical reality of how interest rates behave. When practition- ers hedge with DV01, for example, they express the view that a large part of the variation of nearby yields may be explained by parallel shifts. The general approach may be summarized as empirically analyzing term structure behavior, capturing the important features of that behavior in a relatively simple model, and then calculating price sensitivities based on that model. An alternative approach is to use empirical analysis directly as the model of interest rate behavior. This chapter shows how regression analy- sis is used for hedging. The first section, on volatility-weighted hedges, maintains the assumption of a single driving interest rate factor and, there- fore, of perfect correlation across bond yields, but allows changes to be other than parallel. The second section, on single-variable regression hedg- ing, continues to assume that only one bond is used to hedge any other bond, but allows bond yields to be less than perfectly correlated. The third section, on two-variable regression hedging, assumes that two bonds are used to hedge any other bond, implicitly recognizing that even two bonds cannot perfectly hedge a given bond. To conclude the chapter, the fourth section presents a trading case study about how 20-year Treasury bonds might be hedged and, at the same time, asks if those bonds were fairly priced in the third quarter of 2001. VOLATILITY-WEIGHTED HEDGING Consider the following fairly typical market maker problem. A client sells the market maker a 20-year bond. In the best of circumstances the market maker would immediately sell that same bond to another client and pocket the bid-ask spread. More likely, the market maker will immediately sell the most correlated liquid security, in this case a 30-year bond, 1 to hedge inter- est rate risk. When another client does appear to buy the 20-year bond, the market maker will sell that bond and lift the hedge—that is, buy back the 30-year bond sold as the hedge. A market maker who believes that the 20- and 30-year yields move in parallel would hedge with DV01, as described in Chapter 6. But what if a 1.1-basis point increase in the 20-year yield is expected to accompany a one-basis point increase in the 30-year yield? In that case the market maker would trade F 30 face amount of the 30-year bond to hedge F 20 face amount of the 20-year bond such that the P&L of the resulting position is zero. Mathematically, (8.1) where DV01 20 and DV01 30 are, as usual, per 100 face value. Note the role of the negative signs in the P&L on the left-hand side of equation (8.1). If the 20-year yield increases by 1.1 basis points then a position of F 20 face amount of 20-year bonds experiences a P&L of –F 20 ×1.1×DV01 20 /100. This number is negative for a long position in 20-year bonds (i.e., F 20 >0) and positive for a short position in 20-year bonds (i.e., F 20 <0). The hedge described in equation (8.1) is called a volatility-weighted hedge because, unlike simple DV01 hedging, it recognizes that the 20-year yield tends to fluctuate more than the 30-year yield. The effectiveness of this hedge is, of course, completely dependent on the predictive power of the volatility ratio. To illustrate, say that both yields are 5.70% and that the 20- and 30-year bonds in question sell for par. In that case, using the −× × −× =FF 20 20 30 30 11 100 100 0. DV01 DV01 150 REGRESSION-BASED HEDGING 1 The text purposely ignores bond futures contracts, discussed in Chapter 20. Since the cheapest-to-deliver security of the bond futures contract may very well have a maturity of approximately 20 years, a market maker might very well choose to sell the bond futures contract to hedge the purchase of a 20-year bond. [...]... Volatility-Weighted Hedging equations of Chapter 6, the DV01s are 11 842 8 and 142 940 , respectively With a volatility ratio equal to 1.1, solving equation (8.1) for F30 shows that the purchase of $10 million face amount 20-year bonds should be hedged with a position of F30 = − F20 × 1.1 × DV0120 DV0130 = −$10, 000, 000 × 1.1 × 11 842 8 142 940 (8.2) = −$9,113, 670 or a short of about $9.1 million face amount... 30-year bonds with equal risk weights will exhibit a P&L 0.05 120 0.00 100 –0.05 80 –0.10 Index –0.20 40 –0.25 20 –0.30 0 –0.35 –20 –0 .40 40 –0 .45 –0.50 Jan-95 Dec-95 Dec-96 Nov-97 Index Nov-98 Oct-99 Oct-00 –60 Sep-01 Curve FIGURE 8 .4 Evenly Weighted 20-Year Index and 10s–30s Curve Curve 60 –0.15 1 64 REGRESSION-BASED HEDGING profile similar to that of a simple curve trade that has nothing to do with... increases by 1 basis point, the position will change in value by −$10, 000, 000 × 1.3 11 842 8 142 940 + $9,113, 670 = −$2, 369 100 100 (8 .4) Similarly, if the ratio turns out to be 1.3 and the 30-year rate increases by 5 basis points, the supposedly hedged position will lose five times the amount indicated by equation (8 .4) or about $12,000 The simplest way to estimate the volatility ratio is to compute the... relative to other securities with fixed cash flows While many such securities do trade in fixed income markets, there are also many securities whose cash flows are not fixed Consider a European call option on $1,000 face value of a particular bond with a strike at par This security gives the holder the right to buy $1,000 face value of the bond for $1,000 on the maturity date of the option If the price of... and 30-Year Yields Number of observations R-squared Standard error 1,680 98.63% 0.6170 Regression Coefficients Value Constant Change in 10-year yield Change in 30-year yield 0.0067 0.1613 0.87 74 t-Statistic 0 .44 41 21.5978 99.0826 the 10-year So long as changes in the 10-year yield are positively correlated with changes in the 20-year yield, it is to be expected that the hedge will allocate some risk... the recommendation that the purchase of 20year bonds should be hedged with a 10-year risk weight of over 40 % and a 30-year risk below 70%.6 These weights differ qualitatively from the regression-based weights derived in the previous section: approximately 16% for the 10-year and 88% for the 30-year For any given risk weights β10 and β30, whether they derive from a regression model or not, an index of... the index fluctuated between approximately –.1 and –. 24 from the beginning of the sample until August 6 Several of the more sophisticated approaches included an additional risk weight on bonds of shorter maturity (e.g., two- or five-year bonds) 162 REGRESSION-BASED HEDGING 0.05 0.00 –0.05 Buybacks –0.10 Index –0.15 –0.20 LTCM –0.25 –0.30 –0.35 –0 .40 –0 .45 –0.50 Jan-95 Dec-95 Dec-96 Nov-97 Nov-98 Oct-99... the trading case study of Chapter 4 152 REGRESSION-BASED HEDGING 30-year yields, compute changes of these yields from one day to the next or from one week to the next, and then calculate the standard deviation of these changes This procedure requires a bit of work and a few important decisions First, in bond markets, data usually exist for particular issues and not for particular maturities So, to... function of the change in the 10- and 30-year yields: ∆y 20 ≈ 1613∆y10 + 87 74 y 30 t t t (8.21) But this relationship can also be written as a function of the 10-year yield and of the curve: ( ∆y 20 ≈ 1.0387 ∆y10 + 87 74 ∆y 30 − ∆y10 t t t t ) (8.22) Equation (8.22) says that the 20-year yield will change by about 1. 04 basis points for every basis point change in the 10-year with a fixed curve Also, the... 20-year sector relative to the 10- and 30-year sectors Unfortunately for the trade’s prospects, however, the 20-year sector does not appear particularly rich or cheap by recent historical experience as ˜ measured by the index I 166 REGRESSION-BASED HEDGING 0.30 120 0.25 100 80 0.20 60 40 0.10 20 Curve Index 0.15 0.05 0 0.00 –20 –0.05 –0.10 Jan-95 40 Dec-95 Dec-96 Nov-97 Nov-98 Index Oct-99 Oct-00 –60 . 3.000 4. 2 54 –1.2 54 –1,857 2.00 3.000 4. 2 54 –1.2 54 –1,803 2.50 3.000 4. 2 54 –1.2 54 –1,750 3.00 3.000 4. 2 54 -–1.2 54 –1,699 3.50 3.000 – 146 .055 – 143 .055 –1,650 4. 00 3.000 0.000 3.000 3,832 4. 50. 100.000 – 141 .801 Portfolio Bucket Years Cash Flow Flows Exposure Forward ($millions) ($millions) ($) 0.00 3.000 4. 2 54 –1.2 54 –2,029 0.50 3.000 4. 2 54 –1.2 54 –1,970 1.00 3.000 4. 2 54 –1.2 54 –1,913 1.50. 6.0017% 0.83 744 4 3 3 6.00% 6.00 14% 0.813052 3.5 3 6.00% 6.0012% 0.789371 4 3 6.00% 6.0011% 0.766380 4. 5 3 6.00% 6.0010% 0. 744 058 5 3 6.00% 6.0009% 0.722386 5.5 103 6.00% 6.0008% 0.701 346 Perturbed