Bond mathematics: DV01, duration, and convexity CHAPTER SUMMARY 7.1 DV01/PVBP OR PRICE RISK CHAPTER

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1 CHAPTER Bond mathematics: DV01, 7 duration, and convexity CHAPTER SUMMARY We develop the concept of dollar value of an 01 (DV01) or price value of a basis point (PVBP) to measure risk of fixed-income securities. The concept of price elasticity of debt securities with respect to interest rates is developed, and various duration measures such as Macaulay duration and modified duration are described. The chapter develops the concept of convexity and describes its measurement. Through several examples, this chapter develops these concepts and applies them to trading and hedging applications such as yield curve trades (steepening or flattening) and butterfly strategies. Alternative measures of duration such as effective duration are described to compute the risk of securities for which cash flows are sensitive to interest rates. Fixed income securities display varying price sensitivities to changing interest rates. The purpose of this chapter is to develop certain quantitative measures of interest rate risk. These measures will enable us to compare the interest rate risks of various securities and implement risk management strategies. 7.1 DV01/PVBP OR PRICE RISK The risk of a bond is the change in its price due to changes in the interest rates in the market. DV01 or PVBP measures the price change in debt securities for a basis point (or 0.01%) change in interest rates. If P is the price of the bond and y is its yield, a measure of the bond s risk is the change in its price for a change in its yield. This is denoted by the first derivative of the bond price with respect to its yield, or dp/dy. Let s try to get some intuition behind this concept by looking at an example. Fixed Income Markets and Their Derivatives Copyright 009 by Academic Press. Inc. All rights of reproduction in any form reserved. 105

2 106 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Consider a bond with one year to maturity. It pays a 4% coupon semiannually on a par value of $100 and has a YTM of 6%. The price of the bond is P We want to know what will happen to its price if the yields change by a small amount, say, one basis point, to 6.01%. The new price will be P We can approximate the risk dp/dy by measuring the price difference P P Often debt securities are traded in units of $1 million par amounts. Hence we can express the DV01 of a $1 million par amount of this security as follows: P P ( ) 10, Note that we compute the price change taking into account the fact that prices are quoted in percentages namely, P P ( ) 1,000,000/ Roughly we are estimating the slope of the tangent to the price-yield relationship at 6% yield. This is represented in Figure 7.1. We can get a slightly better estimate of the tangent by moving the yield down by 0.5 basis point ( y 5.995%) and up by 0.5 basis point ( y 6.005%) around 6% as follows P P The estimated price risk per $1 million par may be computed as before and is given by dp/ dy The difference between the first estimate and the second estimate tends to be small but can be relatively more important for debt securities with long maturity.

3 7.1 DV01/PVBP or price risk 107 Price Price-yield curve PVBP 6% Yield to maturity FIGURE 7.1 Price-Yield Curve and PVBP Using the PRICE function of Excel it is fairly easy to compute PVBP or DV01 of debt securities as shown in the following example. Example 7.1 For all the benchmark Treasury securities shown in Table 7.1, compute the DV01 (PVBP) for $1 million par value. Explain the differences that you found. Show all relevant calculations. Assume a settlement date of September 1, 007. Table 7.1 Benchmark Treasury Quotes Maturity Date Coupon Yield to Maturity Benchmark Maturity 8/31/ % 3.933% years 5/15/ % 3.945% 3 years 8/31/ % 4.056% 5 years 8/15/ % 4.364% 10 years 5/15/ % 4.646% 30 years Source: Bloomberg.

4 108 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Table 7. Calculating DV01 or PVBP First we can compute the clean price of each security using the PRICE function. This is shown in column D of the worksheet in Table 7.. Next we recomputed the clean price at a yield, which is one basis point more than the prevailing yield for each security. The resulting (hypothetical) prices are shown in column F. Finally, we take the difference between these prices and multiply by 10,000 to get the PVBP, which is shown in column G. Note that the -year T-note changes by $ per million-dollar par when its yield changes by one basis point. On the other hand, a 30-year bond changes by $1,665.6 for a basis point change in its yield. This implies the following: If the -year yield and 30-year bond yield were to move down by exactly one basis point, the 30-year bond will appreciate by $1,666.6, whereas the -year bond will appreciate by only We can therefore say that the 30-year bond is 1,666.6/ times more risky than a -year T-note under these hypothesized assumptions. We can compute DV01 by first calculating the price at half a basis point below the prevailing yield and then computing the price at half a basis point above the prevailing yield. We can then compute the difference between these two resulting prices and evaluate the DV01. The results of such an approach are shown in Table 7.3. Note that the results of the procedure outlined in Table 7.3 differ only marginally from the results in Table 7. for a -year T-note. But the results are relatively more significant for a 30-year T-bond. Also, the estimates in Table 7.3 are systematically

5 7. Duration 109 Table 7.3 DV01 or PVBP with Variations Around Current Yield higher than the estimates in Table 7.. This is due to the convexity of the price-yield relation, which we address later in this chapter. 7. DURATION Another concept widely used to measure risk is duration. Two related measures are used in the industry: Macaulay duration and modified duration. Macaulay duration has several interpretations: Macaulay duration of a debt security is its discounted-cash-flow-weighted time to receipt of all its promised cash flows divided by the price of security. In this sense, the duration measures the average time taken by the security, on a discounted basis, to pay back the original investment: The longer the duration, the greater the risk. In this sense, the Macaulay duration can be measured in units of time. We can think of Bond X as having duration of six years and Bond Y as having duration of three years. We can then interpret Bond X being more risky than Bond Y. This measure of risk was introduced in 1938 by Macaulay to measure risk in units of time in a way that it reflects the time pattern of cash flows. Macaulay duration can also be interpreted as the price elasticity, which is the percentage change in price for a percentage change in yield; in this sense, the greater the duration of a security, the greater the risk of the security.

6 110 CHAPTER 7 Bond mathematics: DV01, duration, and convexity We consider all these interpretations in detail in the context of the following example. Interpretation 7.1 Macaulay duration is the discounted-cash-flow-weighted time to receive all its promised cash flows divided by the price of the security. Consider also a three-year bond paying a coupon of 5% per annum, also yielding 5% yield to maturity. For simplicity we assume annual coupons. 3 The price of the three-year zero is 100 / The discount factors for the first three years are: , , and As shown in Table 7.4, we find that the sum of discounted-cash-flow-weighted promised cash flows scaled by the price gives us Macaulay duration. Mathematically, we compute the duration as follows: 1 D If we set the coupon equal to zero in the previous example (and noting that the price of a three-year zero paying 100 after three years is the discounted value of 100), we find that the equation reduces to the following: 3 D This gives us the result that the Macaulay duration of a zero coupon bond is simply its time left to maturity. Table 7.4 Duration as Cash-Flow-Weighted Time to Cash Flows Divided by Price Year Discount Factors Cash Flows Cash-Flow- Weighted Time Discounted-Cash- Flow-Weighted Time Total Price 100 Sum of discounted-cash-fl ow-weighted times divided by price.859

7 7. Duration 111 Returning to coupon bonds, we can define Macaulay duration in more general terms as follows: D ic 1 ( 1 y). P i N i i (7.1) In the general definition of Macaulay duration, in Equation 7.1, N is the number of years until maturity, y is the yield to maturity, and the cash flow at period i is denoted by C i. One of the applications of this concept is in bond portfolio immunization: If we can fund liabilities with assets in such a way that their Macaulay durations are the same, such a portfolio is immune from interest rate fluctuations. This is because the price elasticity of assets is the same as the price elasticity of liabilities. Hence their fluctuations cancel each other out. Interpretation 7. Duration is the price elasticity of interest rates; duration is also the price elasticity, which is the percentage change in price for a percentage change in yield. Formally, the elasticity measure of duration is referred to as the Macaulay duration and is represented as follows: the percentage change in price of a bond D the percentage change in the yield of the bond dp P d( 1 y) ( 1 y) dp P dy ( 1 y). This leads to the expression for Macaulay duration, as shown here: dp P D ( 1 y). (7.) dy The negative sign is just a reminder that prices and yields move in opposite directions. We take the percentage change in price of a bond, denoted by dp P, and then divide that quantity by the percentage change in the yield of the bond, denoted dy by 1 y. We can rewrite the Equation 7. as follows: dp 1 D ( 1 y ). dy P With semiannual compounding, sometimes the following convention is also used: dp 1 D ( 1 y / ). dy P

8 11 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Table 7.5 Duration as Price Elasticity of Interest Rates We note that dp/ dy is just the DV01 of the bond. Using this information in Equation 7., we conclude the following: DV 01 D ( 1 y ). (7.3) P In Table 7.5 we carry out these calculations for the three-year bond and find that the price elasticity definition of duration also leads us to the same answer: D.86. In this way of thinking about risk, we say that duration measures the elasticity of the bond price to interest rates: the percentage change in bond price for a percentage change in interest rates. Note that we are not measuring the change in interest rates but the percentage change in interest rates. Another related measure is modified duration (MD). Modified duration is the percentage change in price for a change in yield. Modified duration is denoted as MD dp P. (7.4) dy Using the definition of DV01 and rearranging Equation 7.4, we get the modified duration as follows: MD DV 01. (7.5) P Note that the price, P, in Equation 7.5 is the dirty price. From Table 7.5 we can compute the DV01 as follows: Price at a yield of 4.995% is (rounded

9 7. Duration 113 to seven decimals in Table 7.5 ). Price at a yield of 5.005% is (rounded to seven decimals in Table 7.5 ). DV 01 ( ) 10, Using this equation, we compute the Macaulay duration as: D Modified duration is simply DV01 divided by the price, which leads to MD.7. What is the economic intuition behind modified duration? Let s rearrange Equation 7.4 to get: dp P MD dy. (7.6) This says that the percentage change in the bond price is the modified duration multiplied by the change in its yield: In other words, the higher the modified duration of a bond, the higher is its percentage change to a change in its yield. We can slightly rewrite Equation 7.6 to obtain an expression for the change in the bond price as follows: dp MD P dy. (7.7) Consider a bond selling at par, with MD 7. Equation 7.7 says that a 1% increase in a bond s yield will produce a decrease in price of 7% Excel applications Excel has functions for calculating duration measures. They are shown in the Excel spreadsheet in Table 7.6. Modified duration is always smaller than Macaulay duration and is more extensively used in practice. Table 7.7 provides the duration of benchmark debt securities issued by the Treasury. Note that duration is increasing with maturity but not in direct proportion: A 30-year bond has a duration of only years, whereas the 10-year note has a duration of 8.03 years. In fact, we can show that even bonds with infinite life (perpetuity) will have only a finite duration. To see this, let s consider the price of a perpetuity given in Equation.10 in Chapter : P c = 100. y

10 114 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Table 7.6 Excel Functions for Duration Table 7.7 Duration Estimates for Treasury Benchmarks Differentiating the price, P, of a perpetuity with respect to its yield, y, we get the first derivative as follows: dp dy c 100 y. and computing its modified duration, we find the following result: dp 1 1 MD. dy P y

11 7. Duration 115 This implies that the perpetuity will have a duration equal to the reciprocal of its yield to maturity. If the yield to maturity is 5%, the duration of perpetuity is just 0 years. When yields are low, say, 1%, duration reaches a high of 100 years. Duration risk measure scales the dollar size of a security, but DV01 keeps the dollar value in tact. This difference is best illustrated through an example. Example 7. Compare the modifi ed duration of a 30-year T-bond in Table 7.7 with a strip maturing on May 15, 037, that was trading at a yield of 4.677% for settlement on September 1, 007. You have $1 million par value of each security. Which is riskier? Why? We know already that the MD of 30-year T-bond is from Table 7.7. From fi rst principles, we also know that the duration of a strip, which is a zero coupon bond, is simply equal to its maturity, which in our case is 9.7 years. The modifi ed duration of strip is % Since the modified duration of strip is much greater than the modified duration of a 30-year T-bond, we might be tempted to conclude that a strip is riskier. Such a comparison based on par values might be misleading, since the 30-year strip will sell at a considerable discount to par in the market. In fact, the price of this strip can be computed as follows: % Therefore, $1 million par value of this strip will sell at $53,300 (approximately), and hence the price risk for the same par value is much lower for the strip compared to the 30-year T-bond, which has a market value in excess of $1 million, as noted in Table 7.3. (If we equalize the market values strip and the Treasury bond, the strip is clearly more risky.) To see this, let s compute the PVBP of the strip. The price of this strip when the yield goes up by one basis point is: % The PVBP is approximately $734 for $1 million par. The T-bond has a PVBP of 1, Hence on a par value basis strip is much less risky than the 30-year T-bond. This is because with same par values, the dollar exposure of strip is far less than that of the 30-year T-bond. The main message is the following: When comparing investments

12 116 CHAPTER 7 Bond mathematics: DV01, duration, and convexity with unequal dollar amounts, PVBP gives more transparent answer. In using duration, one must exercise care to reflect the difference is dollar amounts. 7.. Properties of duration and PVBP The finding that the duration of a zero coupon bond is equal to its maturity means that the zero coupon bond is the most interest-rate-elastic security for a given maturity class. It is easy to verify that duration is generally increasing in maturity and decreasing in coupons and yield to maturity, as shown earlier. The duration of coupon bonds will be less than their maturity. Clearly, as time passes, duration will change. This requires some attention in portfolios of assets and liabilities for which the durations are held the same PVBP and duration of portfolios The PVBP of a portfolio is simply the par value-weighted PVBP of individual securities in the portfolio. For a portfolio with two securities, the PVBP can be computed as shown here: PVBP n PVBP n PVBP p 1 1. It should be noted that the par value of security 1 is n 1 and the par value of security is n. This generalizes to a portfolio with N securities easily: PVBP i N n PVBP. (7.8) p i i i 1 Example 7.3 Let s refer to Table 7.7. Suppose that we construct a portfolio with $10 million par value of a -year note and $0 million par value of a 5-year note, what is the PVBP of the resulting portfolio? We compute the portfolio PVBP as follows: The par value of a -year note is denoted by n 10. The par value of a 5-year note is n 5 0. Using the information about PVBP from Table 7.3, we can compute the portfolio s dollar exposure as follows: PVBP p , In other words, this portfolio is expected to make about $10,809 if the -year yields and 5-year yields move down by one basis point. The duration of the portfolio is the market-value-weighted sum of durations of each security in the portfolio. Each weight represents the market value-based proportion of that security as a fraction of the total market value of the portfolio. We illustrate these ideas with the same example we used before.

13 7. Duration 117 Example 7.4 What is the duration of a portfolio with $10 million par value of a -year note and $0 million par value of a 5-year note? Refer to Table 7.7. We can compute the modifi ed duration of a portfolio as a weighted average of the modifi ed duration of securities in the portfolio. The weights are market value proportions: MDp xmd x5md5. More generally, when there are N securities in a portfolio, the portfolio duration can be computed as follows: MD i N x MD. (7.9) p i i i 1 The market value weights add up to 1. To compute market values, we need to work out accrued interest and dirty prices. These calculations are shown in the Excel spreadsheet of Table 7.8. The market value proportions are based on dirty prices and are, respectively, x 0. 38, and x The weighted average modifi ed duration is Table 7.8 Duration of a Bond Portfolio

14 118 CHAPTER 7 Bond mathematics: DV01, duration, and convexity 7.3 TRADING AND HEDGING Spread trades: Curve steepening or curve fl attening trades This section illustrates the way the theoretical concepts that have been developed may be applied to set up trading strategies in practice. A trader is evaluating the shape of the yield curve for settlement on September 1, 007. (Refer to Table 7.7 for information.) The yield spread between the 10- year T-note and the -year T-note stood at basis points ([4.364% 3.933%] 10, ) on September 1, 007. The trader is expecting this spread to significantly increase in a few days; in other words, the trader is expecting the yield curve to get steeper. This expectation may be motivated by many considerations, some of which we get into later in this section. The trader wants to set up a trade that will break even if the spread stays at basis points and will make money if the spread widens. Of course, the trader must be willing to accept the risk that there will be a loss if the yield curve were to flatten; that is, if the spread actually decreases and moves against his or her beliefs. How can the trader implement the trade reflecting his or her view about the yield curve? The overall yields may either go down or go up, but it is the spread that the trader is betting on. First, the trader recognizes that for the spreads to increase in a bullish market (when all rates are expected to fall), the -year yields must drop by much more than the 10-year yields. Similarly, in a bearish market (when all rates are expected to go up), the -year yields must increase by much less than the 10-year yields. This calls for a long position in the -year T-note and a short position in the 10-year T-note. Second, the trader must determine the amount of the -year T-note to buy and the amount of the 10-year T-note to short. This is where the concepts that we have developed come in handy. The trader will want to set up the trade such that the total PVBP is zero. Or: PVBP n n (7.10) p 10 In Equation 7.10, n is the number of -year T-notes and n 10 is the number of 10- year T-notes. The fact that PVBP p 0 ensures that the price risk of -year notes is offset by the price risk of 10-year T-notes for small interest rate changes. If we set n 10 to be $100 million par amount, we can compute the par value of the -year T-note from Equation 7.10 as follows: n DV DV So, the trader will go long in $433 million par amount of the -year T-note and go short in $100 million par amount of the 10-year T-note. We know from Chapter 4 that these transactions can be arranged in repo and reverse repo markets. The trader will post $433 million par amount of the -year T-note as collateral and borrow the cash. Ignoring the haircut (margin), the trader

15 7.4 Convexity 119 will borrow the entire market value at the prevailing repo rate. This way he or she is long in the -year T-note and is entitled to its coupon. In addition, the trader will borrow and sell a $100 million par amount of the 10-year T-note and post the cash proceeds as collateral. Ignoring the haircut (margin), the trader will earn, on the entire cash proceeds, an interest income at the prevailing reverse repo rate. This way he or she is short in the 10-year T-note and is obliged to make restitution for any coupon payments. The profitability of the spread trade depends on a number of factors, including the following: Bid-offer spreads. The trader buys at the offer price and sells at the bid price. The wider the bid-offer spread, the less profitable the trade. Repo rates. If the repo rates are low, the trader pays less to borrow but also receives less on the cash collateral. Special rates. If the security that is long goes special, the trader makes more money, because it is possible to borrow cheap by using that collateral. Conversely, if the security that is short goes special, the trader will lose money. Haircut (margin). The trader will have to post some margin, and this will reduce the profitability as well. The exposure is high: The trader is long $433 million of the -year T-note and short in $100 million of the 10-year T-note. Being wrong about the spread expectations could lose the trader money. The credit risk also has to be factored in. Margins (haircuts), mark-to-market provisions, and other policies should be considered in this context. What might have motivated this type of trade? One factor might be the actions of the Fed that are expected and priced in the securities and the way market expectations relate to traders own assessments. FOMC planned to meet on September 18, 007, and the market anticipated a rate cut. The actual rate cut was 50 basis points, bringing the target rate from 5.5% to 4.75%. This caused the curve to become steeper. On September 18, the -year T-note yield fell to 3.978%. On the other hand, the 10-year T-note yield fell to 4.478%. The resulting spread on September 18 was (4.478% 3.978%) 10, basis points. This was consistent, ex-post, with the premise of the trade. The concepts that we have developed thus far ignored the fact that DV01 and duration changes with yield. We take up this issue next. 7.4 CONVEXITY As we saw earlier, the slope of price-yield relationship changes with yield levels. Furthermore, the slope of the tangent becomes steeper as the interest rates (yields) fall. This leads to what is known as convexity of the price-yield curve. Convexity measures the rate at which DV01 changes as yields change. We illustrate this concept with an example.

16 10 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Example 7.5 Consider a -year zero coupon bond yielding 10%. What is its convexity? Assume annual compounding. The convexity of a bond is the change in the slope of the price-yield curve for a small change in the yield. The second derivative of the price-yield curve provides the basis for the convexity calculations. The price of a -year zero and its interest rate risk can be presented as follows: 100 P ( 1 y ). (7.11) dp dy ( 1 y ). (7.1) For a -year zero, the second derivative of price with respect to its yield is: dp dy (7.13) ( 1 y ) 4 We plot the price, first and second derivative of this -year zero for various values of y in Table 7.9. Note that the slope of the price-yield function (given by Equation 7.1) and plotted in column E of the spreadsheet in the table decreases (ignoring the negative sign) as yields increase. This change in slope is measured by the second derivative (given by Equation 7.13), which is plotted in column F. Note that the second derivative is high at low yields and small at high yields. We can estimate the first derivative by DV01 or PVBP. The second derivative can be approximated by the change in DV01 for a change in basis point in the yield. Note that in the table we have changed yields at 0.5% each time. Hence the approximation for the second derivative is as follows: d P dy dp dy y dp ( 05. %) ( dy y 0%) 00 ( [ 00]) % (7.14) In general, for any debt security that we have presented in this chapter, we can estimate the second derivative using the following formula: d P dy ( DV 01( y) DV 01( y 1bp )) 10, 000. Applying this equation to Table 7.9, we tabulate the second derivative of all benchmark debt securities in Table 7.9.

17 7.4 Convexity 11 Table 7.9 Estimating Convexity By Taylor series approximation (only using the linear and quadratic terms), we can express the percentage price change as follows: dp P dp dy dy d P 05. ( dy). (7.15) dy Using the definition of modified duration and moving the price from the denominator on the left side to the right side, we get: d P dp P MD dy 05. P ( dy). (7.16) dy

18 1 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Table 7.10 Gain from Convexity The price change of a debt security, according to Equation 7.16, consists of two terms. The first term is the duration effect, and it is negative. As the yields increase, prices decline. Note that the convexity effect on price change is positive as seen from the sign of the second term. This is referred to as the gain from convexity. We can explicitly compute gain from convexity using the PVBP estimates as shown in Table We find that the convexity contributes favorably to the price change. Holding maturity and yield to maturity fixed, the convexity decreases as the coupon increases. Convexity increases with duration Bullet versus barbell securities (butterfl y trade) Let s consider Table 7.10 and examine the following trading strategy. Is it possible to replace a 5-year T-note with a portfolio of a -year T-note and a 10-year T-note such that (a) there is no cash outlay and (b) the PVBP remains the same? If so, what is the difference between these two positions? A long position in 5-year T-note is a bullet position. A long position in a portfolio of a -year T-note and a 10-year T-note is a barbell position, reflecting the two balloon payments. Let n be the par value of the -year T-note and let n 10 be the par value of the 10- year T-note needed to replace $100 million par value of a 5-year T-note. We require that the cash proceeds from the sale of a 5-year T-note to be sufficient to buy the requisite numbers of -year and 10-year T-notes. This is the self-financing condition. np n10p (7.17) We further require that the DV01 of the 5-year T-note that is sold is equal to the PVBP of the portfolio that is purchased. n PVBP n PVBP 100PVBP. (7.18)

19 7.4 Convexity 13 FIGURE 7. Butterfl y Trade with DV01 Weights From Figure 7., using Solver, we can determine the values for n and n 10. The portfolio we have created is very similar but not identical to the 5-year T-note that we sold. To see why this is the case, we need to analyze the effects of changes in yields on the 5-year T-note in the portfolio we have created. By construction, at the prevailing market yields (underlined in Table 7.11 ), the market value of a 5-year T-note and its PVBP are exactly matched by those of the barbell portfolio. When there is a parallel shift in the yields, the value of the barbell portfolio dominates the value of the bullet security. Consider what happens to the portfolio when the yields drop. The PVBP of the barbell portfolio, given in the last column, exceeds the PVBP of strip. This indicates that the barbell portfolio will benefit more from the reduction in yields. On the other hand, as the yields go up, the PVBP of the barbell portfolio is always lower

20 14 CHAPTER 7 Bond mathematics: DV01, duration, and convexity Table 7.11 Effect of Convexity, Barbell versus Bullet than that of the 5-year T-note. As a consequence, the barbell portfolio will lose less value compared to the bullet position. Trades of this sort, in which an intermediate maturity security is sold (bought) and two securities whose maturities straddle the intermediate maturity are bought

21 7.5 Effective duration and effective convexity Barbell Value over Bullet Value of convexity % 3.760% 3.960% 4.160% 4.360% 4.560% 4.760% 4.960% Yield scenarios FIGURE 7.3 Barbell versus Bullet (sold), are known as butterfly trades. To get a better perspective, we have plotted in Figure 7.3 the amount by which the value of the barbell portfolio exceeds the value of the 5-year T-note at different levels of yield. Note that the convexity effect really kicks in only at very high or very low yield levels. In fact, for a 100 basis point change in yields, the effect of convexity is hardly evident. A critical assumption we have maintained throughout this discussion is that the shift in yields is parallel. This assumption is especially suspect when there is a large change in the levels of the yields. Hence, the analysis presented previously should not be construed to mean that convexity is necessarily a desirable attribute. 7.5 EFFECTIVE DURATION AND EFFECTIVE CONVEXITY In our analysis of interest rate risk, we have maintained an assumption that cash flows of debt securities are unaffected by changes in market interest rates. Thus in computing DV01, duration, and modified duration, we have assumed that cash flows do not change when interest rates change. In a number of circumstances, the cash flows of debt securities may depend on interest rate. Callable bonds and MBS are two obvious examples. In such situations, we need to use a concept that reflects the fact that cash flows might change when interest rates change. One such measure is known as effective duration. Another important consequence of the sensitivity of cash flows of some debt securities to interest rates is that the concept of yield to

22 16 CHAPTER 7 Bond mathematics: DV01, duration, and convexity maturity (YTM) is no longer well defined. This is due to the fact that in computing YTM, we assume a single stream of cash flows irrespective of interest rates. We illustrate the idea of effective duration with a simple example of a callable bond with a stated maturity of three years but callable at any time at 100. Let s assume that the annual coupon is 6%. Clearly, the bond will be called if the issuer can issue a similar bond at a lower coupon rate. Thus, the cash flows of this callable bond are sensitive to interest rates. To satisfactorily deal with the sensitivity of cash flows to future changes in interest rates, it is necessary to implement the following conceptual steps: 1. We project possible interest rate scenarios into the future, covering the life of the debt security, the effective duration of which we want to estimate. For example, to compute the effective duration of a 30-year MBS, we will project the interest rates out to a horizon of 30 years. For example, in Figure 7.4 we have projected one-year interest rates over the next four years. We could project such a scenario all the way out to 30 years. Interest rates can go up or down with equal probability. Arbitrage-free interest rate models are used to project not only one-year interest rates but also interest rates with different maturities at each node of the lattice. These models also ensure that the rates are chosen in a way such that there are no arbitrage opportunities. A simple motivation for us to determine interest rates of different maturities can be given using an annual coupon-paying callable bond as an example. In valuing such a callable bond, we need to know at each node the one-year interest rate, to perform discounting of cash flows. In addition, we need to know at each node the interest rates of noncallable bonds with the same stated maturity 7% 7.5% 6.5% 6.5% 6% 6% 5.5% 5.5% 5% 4.5% t 0 t 1 t t 3 FIGURE 7.4 Future Distribution of Interest Rates

23 7.5 Effective duration and effective convexity 17 as the callable bond. This latter information will be used in determining whether the bond should be called or not. Likewise, in valuing a mortgage, we need to project monthly interest rates at each node for discounting. In addition, we also need to project refinancing rates at each node to evaluate the value of prepaying the mortgage. Finally, we need to make sure that the refinancing rates and onemonth interest rates are chosen so as to preclude arbitrage. We assume that the yield curve is flat at each node. Then, it is clear that the bond will be called at all nodes where the interest rates are lower than 6%.. Next we select a random path of interest rates. In Figure 7.5 we show a possible path of interest rates, which are highlighted. The highlighted interest rates over time are the result of a randomly chosen interest rate path over the next three years. One way to choose a random path is to flip a (fair) coin at date t 0. If the result is heads, we go up; otherwise, we move down. We repeat this process a sufficient number of times to generate a path. 3. Next we estimate the cash flows along that path. For a callable bond, when interest rates go down, the bond may be called. For a mortgage, when refinancing rates go down, mortgages may be prepaid. So, at each node, cash flows will reflect the optimal behavior of bond issuers (in the case of call) or investors (in the case of mortgages). The result will be a set of cash flows at each node, as shown in Figure 7.6. We assume that the bond starts to pay cash flows from t 1 and matures at t 3. Note that the bond will be called at t 1, and the investors will receive the par value of 100 and the coupon of 6. The bond s cash flows along the interest rate path are 106 at date 1 and zero at other dates on the simulated path. 7% 7.5% 6.5% 6.5% 6% 6% 5.5% 5.5% 5% 4.5% FIGURE 7.5 Simulation of an Interest Rate Path t 0 t 1 t t 3

24 18 CHAPTER 7 Bond mathematics: DV01, duration, and convexity t 0 t 1 t t 3 FIGURE 7.6 Cash Flows of a Callable Bond Along the Simulated Path 4. Next we estimate the present value of cash flows along each simulated path. Since we know the one-year interest rates and annual cash flows, we can discount the cash flows and compute the present value of cash flows. In the example, the present value is simply 100/ In this manner we can compute the present values of many simulated paths. Note that when simulated paths have nodes with interest rates higher than 6%, we need to compute the present values, recognizing that the bond will not be called at those nodes and will just pay the promised coupons. 5. Next we compute option-adjusted spreads (OASs). Once we have the present values of all simulated paths, we average all the present values. If the average present value across all simulated paths is exactly equal to the market price, we define the OAS as zero. If the average of present values is higher than the market price, we add a constant spread z to the discount rate at each node until the average is equal to the market price. This spread z is defined as the OAS. 6. Finally, we compute effective duration. Let s denote the market price as P. We increase interest rates at all nodes by a certain amount (say, 10 basis points) and recompute the price, holding the OAS fixed. Let s denoted this price as P. Then we recalculate the price by decreasing the interest rates at all nodes by 10 basis points. Let s denote this price as P. Then the effective duration for a 1% change in interest rates is calculated as follows: ( P P ) 5.

25 Suggested readings and references 19 P Effective duration (P P ) 5 Effective convexity [(P P) (P P )] 5 Prices P P Interest rates FIGURE 7.7 Illustration of Effective Duration and Convexity The reason that we multiply the price change by 5 is simple: The price difference is over a 0-basis-point overall change. So, multiplying by 5 gives an estimate of the price change for a 100-basis-point change. The effective duration takes into account the effect of changes in interest rates on cash flows. The calculation of effective convexity is also direct and similar to the way we computed convexity earlier. We first compute the price change for a 10-basis-point decrease. We then compute the price change for a 10-basis-point increase. We take the difference between these two price changes to get a measure of effective convexity. Figure 7.7 illustrates the prices at different yields and the way effective duration and convexity measures work. One point worth remembering is that effective duration and effective convexity measures are functions of the models used to compute OAS. Different models (with differing assumptions) can produce differing estimates of effective duration and convexity. We review some of the models of interest rates in Chapter 9 so that the reader is aware of the underlying assumptions behind such models. Sometimes the OAS is changed by 10 basis points to recalculate the price. Then this price and the original market price are used to estimate spread duration. Such calculations are useful for securities such as corporate debt securities, which trade at a spread over Treasuries. SUGGESTED READINGS AND REFERENCES Kopprasch, B. ( 004 ). A look at a variety of duration measures. United States Fixed Income Research, Solomon Smith Barney, Citigroup.