89 CHAPTER 5 One-Factor Measures of Price Sensitivity T he interest rate risk of a security may be measured by how much its price changes as interest rates change. Measures of price sensitivity are used in many ways, four of which will be listed here. First, traders hedging a position in one bond with another bond or with a portfolio of other bonds must be able to compute how each of the bond prices responds to changes in rates. Second, investors with a view about future changes in in- terest rates work to determine which securities will perform best if their view does, in fact, obtain. Third, investors and risk managers need to know the volatility of fixed income portfolios. If, for example, a risk man- ager concludes that the volatility of interest rates is 100 basis points per year and computes that the value of a portfolio changes by $10,000 dollars per basis point, then the annual volatility of the portfolio is $1 million. Fourth, asset-liability managers compare the interest rate risk of their as- sets with the interest rate risk of their liabilities. Banks, for example, raise money through deposits and other short-term borrowings to lend to corpo- rations. Insurance companies incur liabilities in exchange for premiums that they then invest in a broad range of fixed income securities. And, as a final example, defined benefit plans invest funds in financial markets to meet obligations to retirees. Computing the price change of a security given a change in interest rates is straightforward. Given an initial and a shifted spot rate curve, for example, the tools of Part One can be used to calculate the price change of any security with fixed cash flows. Similarly, given two spot rate curves the models in Part Three can be used to calculate the price change of any de- rivative security whose cash flows depend on the level of rates. Therefore, the challenge of measuring price sensitivity comes not so much from the computation of price changes given changes in interest rates but in defining what is meant by changes in interest rates. One commonly used measure of price sensitivity assumes that all bond yields shift in parallel; that is, they move up or down by the same number of basis points. Other assumptions are a parallel shift in spot rates or a parallel shift in forward rates. Yet another reasonable assumption is that each spot rate moves in some proportion to its maturity. This last assump- tion is supported by the observation that short-term rates are more volatile than long-term rates. 1 In any case, there are very many possible definitions of changes in interest rates. An interest rate factor is a random variable that impacts interest rates in some way. The simplest formulations assume that there is only one fac- tor driving all interest rates and that the factor is itself an interest rate. For example, in some applications it might be convenient to assume that the 10-year par rate is that single factor. If parallel shifts are assumed as well, then the change in every other par rate is assumed to equal the change in the factor, that is, in the 10-year par rate. In more complex formulations there are two or more factors driving changes in interest rates. It might be assumed, for example, that the change in any spot rate is the linearly interpolated change in the two-year and 10-year spot rates. In that case, knowing the change in the two-year spot rate alone, or knowing the change in the 10-year spot rate alone, would not allow for the determination of changes in other spot rates. But if, for example, the two-year spot rate were known to increase by three basis points and the 10-year spot rate by one basis point, then the six-year rate, just between the two- and 10-year rates, would be assumed to in- crease by two basis points. There are yet other complex formulations in which the factors are not themselves interest rates. These models, however, are deferred to Part Three. This chapter describes one-factor measures of price sensitivity in full generality, in particular, without reference to any definition of a change in rates. Chapter 6 presents the commonly invoked special case of parallel 90 ONE-FACTOR MEASURES OF PRICE SENSITIVITY 1 In countries with a central bank that targets the overnight interest rate, like the United States, this observation does not apply to the very short end of the curve. yield shifts. Chapter 7 discusses multi-factor formulations. Chapter 8 shows how to model interest rate changes empirically. The assumptions about interest rate changes and the resulting mea- sures of price sensitivity appearing in Part Two have the advantage of sim- plicity but the disadvantage of not being connected to any particular pricing model. This means, for example, that the hedging rules developed here are independent of the pricing or valuation rules used to determine the quality of the investment or trade that necessitated hedging in the first place. At the cost of some complexity, the assumptions invoked in Part Three consistently price securities and measure their price sensitivities. DV01 Denote the price-rate function of a fixed income security by P(y), where y is an interest rate factor. Despite the usual use of y to denote a yield, this factor might be a yield, a spot rate, a forward rate, or a factor in one of the models of Part Three. In any case, since this chapter describes one-factor measures of price sensitivity, the single number y completely describes the term structure of interest rates and, holding everything but interest rates constant, allows for the unique determination of the price of any fixed in- come security. As mentioned above, the concepts and derivations in this chapter ap- ply to any term structure shape and to any one-factor description of term structure movements. But, to simplify the presentation, the numerical ex- amples assume that the term structure of interest rates is flat at 5% and that rates move up and down in parallel. Under these assumptions, all yields, spot rates, and forward rates are equal to 5%. Therefore, with re- spect to the numerical examples, the precise definition of y does not matter. This chapter uses two securities to illustrate price sensitivity. The first is the U.S. Treasury 5s of February 15, 2011. As of February 15, 2001, Fig- ure 5.1 graphs the price-rate function of this bond. The shape of the graph is typical of coupon bonds: Price falls as rate increases, and the curve is very slightly convex. 2 The other security used as an example in this chapter is a one-year European call option struck at par on the 5s of February 15, 2011. This DV01 91 2 The discussion of Figure 4.4 defines a convex curve. option gives its owner the right to purchase some face amount of the bond after exactly one year at par. (Options and option pricing will be discussed further in Part Three and in Chapter 19.) If the call gives the right to pur- chase $10 million face amount of the bond then the option is said to have a face amount of $10 million as well. Figure 5.2 graphs the price-rate function. As in the case of bonds, option price is expressed as a percent of face value. In Figure 5.2, if rates rise 100 basis points from 3.50% to 4.50%, the price of the option falls from 11.61 to 5.26. Expressed differently, the change in the value of the option is (5.26–11.61)/100 or –.0635 per basis point. At higher rate levels, option price does not fall as much for the same increase in rate. Changing rates from 5.50% to 6.50%, for example, low- ers the option price from 1.56 to .26 or by only .013 per basis point. More generally, letting ∆P and ∆y denote the changes in price and rate and noting that the change measured in basis points is 10,000×∆y, define the following measure of price sensitivity: (5.1) DV01 is an acronym for dollar value of an ’01 (i.e., .01%) and gives the change in the value of a fixed income security for a one-basis point decline DV01 ≡− × ∆ ∆ P y10 000, 92 ONE-FACTOR MEASURES OF PRICE SENSITIVITY FIGURE 5.1 The Price-Rate Function of the 5s of February 15, 2011 70 80 90 100 110 120 130 2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% Yield Price in rates. The negative sign defines DV01 to be positive if price increases when rates decline and negative if price decreases when rates decline. This convention has been adopted so that DV01 is positive most of the time: All fixed coupon bonds and most other fixed income securities do rise in price when rates decline. The quantity ∆P / ∆y is simply the slope of the line connecting the two points used to measure that change. 3 Continuing with the option example, ∆P / ∆y for the call at 4% might be graphically illustrated by the slope of a line connecting the points (3.50%, 11.61) and (4.50%, 5.26) in Figure 5.2. It follows from equation (5.1) that DV01 at 4% is proportional to that slope. Since the price sensitivity of the option can change dramatically with the level of rates, DV01 should be measured using points relatively close to the rate level in question. Rather than using prices at 3.50% and 4.50% to measure DV01 at 4%, for example, one might use prices at 3.90% and 4.10% or even prices at 3.99% and 4.01%. In the limit, one would use the slope of the line tangent to the price-rate curve at the desired rate level. Fig- ure 5.3 graphs the tangent lines at 4% and 6%. That the line AA in this fig- DV01 93 FIGURE 5.2 The Price-Rate Function of a One-Year European Call Option Struck at Par on the 5s of February 15, 2011 –5 0 5 10 15 20 25 3.00% 4.00% 5.00% 6.00% 7.00% Yield Price 3 The slope of a line equals the change in the vertical coordinate divided by the change in the horizontal coordinate. In the price-rate context, the slope of the line is the change in price divided by the change in rate. ure is steeper than the line BB indicates that the option is more sensitive to rates at 4% than it is at 6%. The slope of a tangent line at a particular rate level is equal to the derivative of the price-rate function at that rate level. The derivative is written dP(y) / dy or simply dP / dy . (The first notation of the derivative empha- sizes its dependence on the level of rates, while the second assumes awareness of this dependence.) For readers not familiar with the calcu- lus, “d” may be taken as indicating a small change and the derivative may be thought of as the change in price divided by the change in rate. More precisely, the derivative is the limit of this ratio as the change in rate approaches zero. In some special cases to be discussed later, dP / dy can be calculated ex- plicitly. In these cases, DV01 is defined using this derivative and (5.2) In other cases DV01 must be estimated by choosing two rate levels, com- puting prices at each of the rates, and applying equation (5.1). As mentioned, since DV01 can change dramatically with the level of DV01 =− () 1 10 000, dP y dy 94 ONE-FACTOR MEASURES OF PRICE SENSITIVITY FIGURE 5.3 A Graphical Representation of DV01 for the One-Year Call on the 5s of February 15, 2011 –5 0 5 10 15 20 25 Yield Price A A B B 4.00% 6.00% rates it should be measured over relatively narrow ranges of rate. 4 The first three columns of Table 5.1 list selected rate levels, option prices, and DV01 estimates from Figure 5.2. Given the values of the option at rates of 4.01% and 3.99%, for example, DV01 equals (5.3) In words, with rates at 4% the price of the option falls by about 6.41 cents for a one-basis point rise in rate. Notice that the DV01 estimate at 4% does not make use of the option price at 4%: The most stable numerical es- timate chooses rates that are equally spaced above and below 4%. Before closing this section, a note on terminology is in order. Most market participants use DV01 to mean yield-based DV01, discussed in Chapter 6. Yield-based DV01 assumes that the yield-to-maturity changes by one basis point while the general definition of DV01 in this chapter al- lows for any measure of rates to change by one basis point. To avoid con- fusion, some market participants have different names for DV01 measures according to the assumed measure of changes in rates. For example, the change in price after a parallel shift in forward rates might be called DVDF or DPDF while the change in price after a parallel shift in spot or zero rates might be called DVDZ or DPDZ. A HEDGING EXAMPLE, PART I: HEDGING A CALL OPTION Since it is usual to regard a call option as depending on the price of a bond, rather than the reverse, the call is referred to as the derivative security and the bond as the underlying security. The rightmost columns of Table 5.1 − × =− − ×− () = ∆ ∆ P y10 000 8 0866 8 2148 10 000 4 01 3 99 0641 , ,.%.% . A Hedging Example, Part I: Hedging a Call Option 95 4 Were prices available without error, it would be desirable to choose a very small difference between the two rates and estimate DV01 at a particular rate as accu- rately as possible. Unfortunately, however, prices are usually not available without error. The models developed in Part Three, for example, perform so many calcula- tions that the resulting prices are subject to numerical error. In these situations it is not a good idea to magnify these price errors by dividing by too small a rate differ- ence. In short, the greater the pricing accuracy, the smaller the optimal rate differ- ence for computing DV01. list the prices and DV01 values of the underlying bond, namely the 5s of February 15, 2011, at various rates. If, in the course of business, a market maker sells $100 million face value of the call option and rates are at 5%, how might the market maker hedge interest rate exposure by trading in the underlying bond? Since the market maker has sold the option and stands to lose money if rates fall, bonds must be purchased as a hedge. The DV01 of the two securities may be used to figure out exactly how many bonds should be bought against the short option position. According to Table 5.1, the DV01 of the option with rates at 5% is .0369, while the DV01 of the bond is .0779. Letting F be the face amount of bonds the market maker purchases as a hedge, F should be set such that the price change of the hedge position as a result of a one-basis point change in rates equals the price change of the option position as a result of the same one-basis point change. Mathematically, (5.4) (Note that the DV01 values, quoted per 100 face value, must be divided by 100 before being multiplied by the face amount of the option or of the bond.) Solving for F, the market maker should purchase approximately $47.37 million face amount of the underlying bonds. To summarize this hedging strategy, the sale of $100 million face value of options risks F F ×= × =× . ,, . ,, . . 0779 100 100 000 000 0369 100 100 000 000 0369 0779 96 ONE-FACTOR MEASURES OF PRICE SENSITIVITY TABLE 5.1 Selected Option Prices, Underlying Bond Prices, and DV01s at Various Rate Levels Rate Option Option Bond Bond Level Price DV01 Price DV01 3.99% 8.2148 108.2615 4.00% 8.1506 0.0641 108.1757 0.0857 4.01% 8.0866 108.0901 4.99% 3.0871 100.0780 5.00% 3.0501 0.0369 100.0000 0.0779 5.01% 3.0134 99.9221 5.99% 0.7003 92.6322 6.00% 0.6879 0.0124 92.5613 0.0709 6.01% 0.6756 92.4903 (5.5) for each basis point decline in rates, while the purchase of $47.37 million bonds gains (5.6) per basis point decline in rates. Generally, if DV01 is expressed in terms of a fixed face amount, hedg- ing a position of F A face amount of security A requires a position of F B face amount of security B where (5.7) To avoid careless trading mistakes, it is worth emphasizing the simple implications of equation (5.7), assuming that, as usually is the case, each DV01 is positive. First, hedging a long position in security A requires a short position in security B and hedging a short position in security A re- quires a long position in security B. In the example, the market maker sells options and buys bonds. Mathematically, if F A >0 then F B <0 and vice versa. Second, the security with the higher DV01 is traded in smaller quantity than the security with the lower DV01. In the example, the market maker buys only $47.37 million bonds against the sale of $100 million options. Mathematically, if DV01 A >DV01 B then F B >–F A , while if DV01 A <DV01 B then –F A >F B . (There are occasions in which one DV01 is negative. 5 In these cases equation (5.7) shows that a hedged position consists of simultaneous longs or shorts in both securities. Also, the security with the higher DV01 in ab- solute value is traded in smaller quantity.) Assume that the market maker does sell $100 million options and does buy $47.37 million bonds when rates are 5%. Using the prices in Table F F B AA B = −×DV01 DV01 $, , . $,47 370 000 0779 100 36 901×= $,, . $,100 000 000 0369 100 36 900×= A Hedging Example, Part I: Hedging a Call Option 97 5 For an example in the mortgage context see Chapter 21. 5.1, the cost of establishing this position and, equivalently, the value of the position after the trades is (5.8) Now say that rates fall by one basis point to 4.99%. Using the prices in Table 5.1 for the new rate level, the value of the position becomes (5.9) The hedge has succeeded in that the value of the position has hardly changed even though rates have changed. To avoid misconceptions about market making, note that the market maker in this example makes no money. In reality, the market maker would purchase the option at its midmarket price minus some spread. Tak- ing half a tick, for example, the market maker would pay half of 1 / 32 or .015625 less than the market price of 3.0501 on the $100 million for a to- tal of $15,625. This spread compensates the market maker for effort ex- pended in the original trade and for hedging the option over its life. Some of the work involved in hedging after the initial trade will become clear in the sections continuing this hedging example. DURATION DV01 measures the dollar change in the value of a security for a basis point change in interest rates. Another measure of interest rate sensitivity, duration, measures the percentage change in the value of a security for a unit change in rates. 6 Mathematically, letting D denote duration, (5.10) D P P y ≡− 1 ∆ ∆ −×+×=$,, . $, , . $, ,100 000 000 3 0871 100 47 370 000 100 0780 100 44 319 849 −×+×=$,, . $, , $, ,100 000 000 3 0501 100 47 370 000 100 100 44 319 900 98 ONE-FACTOR MEASURES OF PRICE SENSITIVITY 6 A unit change means a change of one. In the rate context, a change of one is a change of 10,000 basis points. [...]... –8.15% 5.00% 96.9499 100.0000 1 03. 15% 3. 0501 3. 15% 6.00% 91.8 734 92.56 13 100.75% 0.6879 –0.75% 0.0216 2.1629 83 –146.618 0.0411 4. 238 815 –2 23. 039 0.0586 6 .37 6924 –119.5 63 1 13 A Hedging Example, Part III: The Negative Convexity of Callable Bonds $200,000 $150,000 5s of 2/15/2011 Hedge $100,000 P&L $50,000 $0 –$50,000 Callable Bond –$100,000 –$150,000 –$200,000 2.00% 3. 00% 4.00% 5.00% 6.00% 7.00% 8.00%... of the call option on the 5s of February 15, 2011, from Tables 5.2 and 5 .3 Equation (5.20) then says that for a 25-basis point increase in rates ( ) %∆P = −120.82 × 0025 + 1 2 95 03. 330 2 × 00252 = − .30 205 + 02970 = −27. 235 % (5.21) At a starting price of 3. 0501, the approximation to the new price is 3. 0501 minus 27 235 3. 0501 or 830 70, leaving 2.2194 Since the option price when rates are 5.25% is 2.2185,... Level 3. 99% 4.00% 4.01% 4.99% 5.00% 5.01% 5.99% 6.00% 6.01% Bond Price 108.2615 108.1757 108.0901 100.0780 100.0000 99.9221 92. 632 2 92.56 13 92.49 03 First Derivative Convexity –857.4290 –856.6126 75.4725 –779.8264 –779.0901 73. 6287 –709.8187 –709.1542 71.7854 Option First Price Derivative Convexity 8.2148 8.1506 8.0866 3. 0871 3. 0501 3. 0 134 0.70 03 0.6879 0.6756 –641.8096 2,800.9970 – 639 .5266 36 9.9550... 2.8125 2.8125 2.8125 2.8125 2.8125 2.8125 2.8125 2.8125 102.8125 2.7 433 2.6758 2.6100 2.5458 2.4 832 2.4221 2 .36 25 2 .30 44 2.2477 80.1444 1 .37 17 2.6758 3. 9150 5.0916 6.2079 7.2662 8.2687 9.2175 10.1146 400.7219 102. 539 1 454.8511 8/15/01 2/15/02 8/15/02 2/15/ 03 8/15/ 03 2/15/04 8/15/04 2/15/05 8/15/05 2/15/06 Sums: DV01: 0.04 436 6 An equation for DV01 more compact than (6.5) may be derived by differentiating... SENSITIVITY 130 120 5s of 2/15/2011 Price 110 100 Callable Bond 90 80 70 2.00% 3. 00% 4.00% 5.00% Yield 6.00% 7.00% 8.00% FIGURE 5.8 Price of Callable Bond and of 5s of February 15, 2011 0410 The convexity of the callable bond is the weighted sum of the individual convexities listed in Table 5 .3: 1 03. 15% × 73. 63 − 3. 15% × 9, 5 03. 33 = −2 23 (5 .31 ) A market maker wanting to hedge the sale of $100 million callable... 8.1506 8.0866 3. 0871 3. 0501 3. 0 134 0.70 03 0.6879 0.6756 Option Duration 78.60 120.82 179.70 Bond Price 108.2615 108.1757 108.0901 100.0780 100.0000 99.9221 92. 632 2 92.56 13 92.49 03 Bond Duration 7.92 7.79 7.67 100 ONE-FACTOR MEASURES OF PRICE SENSITIVITY In the case of the underlying bond, equation (5. 13) says that the percentage change in price equals minus 7.92 times the change in rate Therefore, a one-basis... 0.70 03 0.6879 0.6756 –641.8096 2,800.9970 – 639 .5266 36 9.9550 9,5 03. 330 2 36 7.0564 –124.4984 25,627. 633 5 –122. 735 5 A Hedging Example, Part II: A Short Convexity Position 1 03 in rate to get –779.0901 Then estimate the second derivative at 5% by dividing the change in the first derivative by the change in rate: ∆2 P −779.0901 + 779.8264 = = 7, 36 3 ∆y 2 5.005% − 4.995% (5.16) Finally, to estimate convexity,... relationship introduced in Chapter 3 as equation (3. 2) or equation (3. 4).1 For convenience, these equations are reproduced here2 with the face value set at 100: 1 For expositional ease, the derivations in this chapter assume that coupon flows are in six-month intervals from the settlement date The derivations in the more general case are very similar 2 In Chapter 3 the pricing function was written as... that DV01 falls as rates increase (see Figure 5.4) Fixed income securities need not be positively convex at all rate levels Some important examples of negative convexity are callable bonds (see the last section of this chapter and Chapter 19) and mortgage-backed securities (see Chapter 21) Understanding the convexity properties of securities is useful for both hedging and investing This is the topic... maturity allows for a convenient interpretation of the Macaulay duration of any other bond The previous section calculates that the Macaulay duration of the 5.625s due February 15, 2006, is 4. 435 9 But the DMac of a zero coupon bond maturing in 4. 435 9 years is also 4. 435 9 Therefore, the first order price sensitivity of the 5.625s of February 15, 2006, equals that of a zero maturing in 4. 435 9 years In other . –856.6126 8.0866 – 639 .5266 4.99% 100.0780 3. 0871 5.00% 100.0000 –779.8264 73. 6287 3. 0501 36 9.9550 9,5 03. 330 2 5.01% 99.9221 –779.0901 3. 0 134 36 7.0564 5.99% 92. 632 2 0.70 03 6.00% 92.56 13 –709.8187 71.7854. Price DV01 3. 99% 8.2148 108.2615 4.00% 8.1506 0.0641 108.1757 0.0857 4.01% 8.0866 108.0901 4.99% 3. 0871 100.0780 5.00% 3. 0501 0. 036 9 100.0000 0.0779 5.01% 3. 0 134 99.9221 5.99% 0.70 03 92. 632 2 6.00%. Price Duration 3. 99% 8.2148 108.2615 4.00% 8.1506 78.60 108.1757 7.92 4.01% 8.0866 108.0901 4.99% 3. 0871 100.0780 5.00% 3. 0501 120.82 100.0000 7.79 5.01% 3. 0 134 99.9221 5.99% 0.70 03 92. 632 2 6.00%