Suppose baa rated bonds currently yield

166 PA

R T IV

IN THIS CHAPTER we turn to various strate- gies that bond portfolio managers can pur- sue, making a distinction between passive and active strategies. A passive investment strategy takes market prices of securities as set fairly. Rather than attempting to beat the market by exploiting superior infor- mation or insight, passive managers act to maintain an appropriate risk–return balance given market opportunities. One special case of passive management is an immuni- zation strategy that attempts to insulate or immunize the portfolio from interest rate risk. In contrast, an active investment strat- egy attempts to achieve returns greater than those commensurate with the risk borne. In the context of bond management this style of management can take two forms. Active managers use either interest rate forecasts to predict movements in the entire bond market or some form of intramarket analy- sis to identify particular sectors of the mar- ket or particular bonds that are relatively mispriced.

Because interest rate risk is crucial to for- mulating both active and passive strategies, we begin our discussion with an analysis of the sensitivity of bond prices to interest rate fluctuations. This sensitivity is measured by the duration of the bond, and we devote con- siderable attention to what determines bond duration. We discuss several passive invest- ment strategies, and show how duration- matching techniques can be used to immunize the holding-period return of a portfolio from interest rate risk. After examining the broad range of applications of the duration mea- sure, we consider refinements in the way that interest rate sensitivity is measured, focusing on the concept of bond convexity. Duration is important in formulating active investment strategies as well, and we conclude the chap- ter with a discussion of active fixed-income strategies. These include policies based on interest rate forecasting as well as intramar- ket analysis that seeks to identify relatively attractive sectors or securities within the fixed-income market.

Managing Bond Portfolios CHAPTER SIXTEEN

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516 P A R T I V Fixed-Income Securities

We have seen already that bond prices and yields are inversely related, and we know that interest rates can fluctuate substantially. As interest rates rise and fall, bondholders expe- rience capital losses and gains. These gains or losses make fixed-income investments risky, even if the coupon and principal payments are guaranteed, as in the case of Treasury obligations.

Why do bond prices respond to interest rate fluctuations? Remember that in a competitive market all securities must offer investors fair expected rates of return. If a bond is issued with an 8% coupon when competitive yields are 8%, then it will sell at par value. If the market rate rises to 9%, however, who would purchase an 8% coupon bond at par value? The bond price must fall until its expected return increases to the competi- tive level of 9%. Conversely, if the market rate falls to 7%, the 8% coupon on the bond is attractive compared to yields on alternative investments. In response, investors eager for that return would bid up the bond price until the total rate of return for someone purchasing at that higher price is no better than the market rate.

Interest Rate Sensitivity The sensitivity of bond prices to changes in market interest rates is obviously of great concern to investors. To gain some insight into the determinants of interest rate risk, turn to Figure 16.1 , which presents the percentage change in price corresponding to changes in yield to maturity for four bonds that differ according to coupon rate, initial yield to maturity, and time to maturity. All four bonds illustrate that bond prices decrease when

16.1 Interest Rate Risk

Pe rc

en ta

g e

C h

an g

e in

B o

n d

P ri

ce

Change in Yield to Maturity (%)

D C B A

A

B

C

D

5−5 −4 −3 −2 −1 0 1 2 3 4

200

150

100

50

0

−50

Initial Bond Coupon Maturity YTM

A 12% 5 years 10% B 12% 30 years 10% C 3% 30 years 10% D 3% 30 years 6%

Figure 16.1 Change in bond price as a function of change in yield to maturity

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C H A P T E R 1 6 Managing Bond Portfolios 517

yields rise, and that the price curve is convex, meaning that decreases in yields have bigger impacts on price than increases in yields of equal magnitude. We summarize these obser- vations in the following two propositions:

1. Bond prices and yields are inversely related: as yields increase, bond prices fall; as yields fall, bond prices rise.

2. An increase in a bond’s yield to maturity results in a smaller price change than a decrease in yield of equal magnitude.

Now compare the interest rate sensitivity of bonds A and B, which are identical except for maturity. Figure 16.1 shows that bond B, which has a longer maturity than bond A, exhibits greater sensitivity to interest rate changes. This illustrates another general property:

3. Prices of long-term bonds tend to be more sensitive to interest rate changes than prices of short-term bonds.

This is not surprising. If rates increase, for example, the bond is less valuable as its cash flows are discounted at a now-higher rate. The impact of the higher discount rate will be greater as that rate is applied to more-distant cash flows.

Notice that while bond B has six times the maturity of bond A, it has less than six times the interest rate sensitivity. Although interest rate sensitivity seems to increase with matu- rity, it does so less than proportionally as bond maturity increases. Therefore, our fourth property is that:

4. The sensitivity of bond prices to changes in yields increases at a decreasing rate as maturity increases. In other words, interest rate risk is less than proportional to bond maturity.

Bonds B and C, which are alike in all respects except for coupon rate, illustrate another point. The lower-coupon bond exhibits greater sensitivity to changes in interest rates. This turns out to be a general property of bond prices:

5. Interest rate risk is inversely related to the bond’s coupon rate. Prices of low-coupon bonds are more sensitive to changes in interest rates than prices of high-coupon bonds.

Finally, bonds C and D are identical except for the yield to maturity at which the bonds currently sell. Yet bond C, with a higher yield to maturity, is less sensitive to changes in yields. This illustrates our final property:

6. The sensitivity of a bond’s price to a change in its yield is inversely related to the yield to maturity at which the bond currently is selling.

The first five of these general properties were described by Malkiel 1 and are sometimes known as Malkiel’s bond-pricing relationships. The last property was demonstrated by Homer and Liebowitz. 2

Maturity is a major determinant of interest rate risk. However, maturity alone is not suffi- cient to measure interest rate sensitivity. For example, bonds B and C in Figure 16.1 have the same maturity, but the higher-coupon bond has less price sensitivity to interest rate changes. Obviously, we need to know more than a bond’s maturity to quantify its interest rate risk.

1 Burton G. Malkiel, “Expectations, Bond Prices, and the Term Structure of Interest Rates,” Quarterly Journal of Economics 76 (May 1962), pp. 197–218.

2 Sidney Homer and Martin L. Liebowitz, Inside the Yield Book: New Tools for Bond Market Strategy (Englewood Cliffs, NJ: Prentice Hall, 1972).

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518 P A R T I V Fixed-Income Securities

To see why bond characteristics such as coupon rate or yield to maturity affect interest rate sensitivity, let’s start with a simple numerical example. Table 16.1 gives bond prices for 8% semiannual coupon bonds at different yields to maturity and times to maturity, T. [The interest rates are expressed as annual percentage rates (APRs), meaning that the true 6-month yield is doubled to obtain the stated annual yield.] The shortest-term bond falls in value by less than 1% when the interest rate increases from 8% to 9%. The 10-year bond falls by 6.5%, and the 20-year bond by over 9%.

Now look at a similar computation using a zero-coupon bond rather than the 8% coupon bond. The results are shown in Table 16.2 . Notice that for each maturity, the price of the zero-coupon bond falls by a greater proportional amount than the price of the 8% coupon bond. Because we know that long-term bonds are more sensitive to interest rate movements than are short-term bonds, this observation suggests that in some sense a zero-coupon bond must represent a longer-term bond than an equal-time-to-maturity coupon bond.

In fact, this insight about the effective maturity of a bond is a useful one that we can make mathematically precise. To start, note that the times to maturity of the two bonds in this example are not perfect measures of the long- or short-term nature of the bonds. The 20-year 8% bond makes many coupon payments, most of which come years before the bond’s maturity date. Each of these payments may be considered to have its own “maturity.” In the previous chapter, we pointed out that it can be useful to view a coupon bond as a “portfolio” of coupon payments. The effective maturity of the bond is therefore some sort of average of the maturities of all the cash flows. The zero-coupon bond, by contrast, makes only one payment at maturity. Its time to maturity is, therefore, a well-defined concept.

Higher-coupon-rate bonds have a higher fraction of value tied to coupons rather than final payment of par value, and so the “portfolio of coupons” is more heavily weighted toward the earlier, short-maturity payments, which gives it lower “effective maturity.” This explains Malkiel’s fifth rule, that price sensitivity falls with coupon rate.

Similar logic explains our sixth rule, that price sensitivity falls with yield to maturity. A higher yield reduces the present value of all of the bond’s payments, but more so for more- distant payments. Therefore, at a higher yield, a higher fraction of the bond’s value is due to its earlier payments, which have lower effective maturity and interest rate sensitivity. The overall sensitivity of the bond price to changes in yields is thus lower.

Table 16.1

Prices of 8% coupon bond (coupons paid semiannually)

Yield to Maturity (APR) T 5 1 Year T 5 10 Years T 5 20 Years

8% 1,000.00 1,000.00 1,000.00 9% 990.64 934.96 907.99 Fall in price (%)* 0.94% 6.50% 9.20%

*Equals value of bond at a 9% yield to maturity divided by value of bond at (the original) 8% yield, minus 1.

Yield to Maturity (APR) T 5 1 Year T 5 10 Years T 5 20 Years

8% 924.56 456.39 208.29 9% 915.73 414.64 171.93 Fall in price (%)* 0.96% 9.15% 17.46%

*Equals value of bond at a 9% yield to maturity divided by value of bond at (the original) 8% yield, minus 1.

Table 16.2

Prices of zero-coupon bond (semiannual compounding)

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C H A P T E R 1 6 Managing Bond Portfolios 519

Duration To deal with the ambiguity of the “maturity” of a bond making many payments, we need a measure of the average maturity of the bond’s promised cash flows. We would like also to use such an effective maturity measure as a guide to the sensitivity of a bond to interest rate changes, because we have noted that price sensitivity tends to increase with time to maturity.

Frederick Macaulay 3 termed the effective maturity concept the duration of the bond. Macaulay’s duration equals the weighted average of the times to each coupon or princi- pal payment. The weight associated with each payment time clearly should be related to the “importance” of that payment to the value of the bond. In fact, the weight applied to each payment time is the proportion of the total value of the bond accounted for by that payment, that is, the present value of the payment divided by the bond price.

We define the weight, w t , associated with the cash flow made at time t (denoted CF t ) as:

wt 5 CFt /(1 1 y)

t

Bond price

where y is the bond’s yield to maturity. The numerator on the right-hand side of this equa- tion is the present value of the cash flow occurring at time t while the denominator is the value of all the bond’s payments. These weights sum to 1.0 because the sum of the cash flows discounted at the yield to maturity equals the bond price.

Using these values to calculate the weighted average of the times until the receipt of each of the bond’s payments, we obtain Macaulay’s duration formula:

D 5 a T

t51 t 3 wt (16.1)

3 Frederick Macaulay, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since 1856 (New York: National Bureau of Economic Research, 1938).

Spreadsheet 16.1 Calculating the duration of two bonds

Column sums subject to rounding error.

A C D E F G Time until PV of CF Column (C) Payment (Discount rate times

Period (Years) Cash Flow 5% per period) Weight* Column (F) A. 8% coupon bond 1 0.5 38.095 0.0395

2 1.0 36.281 0.0376 0.0376 3 1.5 34.554 0.0358 0.0537 4 2.0 855.611 0.8871 1.7741

Sum: 964.540 1.0000 1.8852

B. Zero-coupon 1 0.5 0.000 0.0000 0.0000 2 1.0 0.000 0.0000 0.0000 3 1.5 0.000 0.0000 0.0000 4 2.0 822.702 1.0000 2.0000

Sum: 822.702 1.0000 2.0000

Semiannual int rate:

0.0197

*Weight = Present value of each payment (column E) divided by the bond price.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

B

40 40 40

1040

0 0 0

1000

0.05

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520 P A R T I V Fixed-Income Securities

As an example of the application of Equation 16.1, we derive in Spreadsheet 16.1 the durations of an 8% coupon and zero-coupon bond, each with 2 years to maturity. We assume that the yield to maturity on each bond is 10%, or 5% per half-year. The present value of each payment is discounted at 5% per period for the number of (semiannual) peri- ods shown in column B. The weight associated with each payment time (column F) is the present value of the payment for that period (column E) divided by the bond price (the sum of the present values in column E).

The numbers in column G are the products of time to payment and payment weight. Each of these products corresponds to one of the terms in Equation 16.1. According to that equation, we can calculate the duration of each bond by adding the numbers in column G.

The duration of the zero-coupon bond is exactly equal to its time to maturity, 2 years. This makes sense, because with only one payment, the average time until payment must be the bond’s maturity. In contrast, the 2-year coupon bond has a shorter duration of 1.8852 years.

Spreadsheet 16.2 shows the spreadsheet formulas used to produce the entries in Spreadsheet 16.1 . The inputs in the spreadsheet—specifying the cash flows the bond will pay—are given in columns B–D. In column E we calculate the present value of each cash flow using the assumed yield to maturity, in column F we calculate the weights for Equation 16.1, and in column G we compute the product of time to payment and payment weight. Each of these terms corresponds to one of the values that is summed in Equation 16.1. The sums com- puted in cells G8 and G14 are therefore the durations of each bond. Using the spreadsheet, you can easily answer several “what if ” questions such as the one in Concept Check 1.

Suppose the interest rate decreases to 9% as an annual percentage rate. What will happen to the prices and durations of the two bonds in Spreadsheet 16.1 ?

CONCEPT CHECK 16.1

Spreadsheet 16.2

Spreadsheet formulas for calculating duration

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Time until PV of CF Column (C) Payment (Discount rate times

Period (Years) Cash Flow 5% per period) Weight Column (F) A. 8% coupon bond 1

2 3 4

Sum:

B. Zero-coupon 1 2 3 4

0.5 1

1.5 2

0.5 1

1.5 2

Sum:

=D4/(1+$B$16)^B4 =D5/(1+$B$16)^B5 =D6/(1+$B$16)^B6 =D7/(1+$B$16)^B7 =SUM(E4:E7)

=D10/(1+$B$16)^B10 =D11/(1+$B$16)^B11 =D12/(1+$B$16)^B12 =D13/(1+$B$16)^B13 =SUM(E10:E13)

=E4/E$8 =E5/E$8 =E6/E$8 =E7/E$8 =SUM(F4:F7)

=E10/E$14 =E11/E$14 =E12/E$14 =E13/E$14 =SUM(F10:F13)

Semiannual int rate:

=F5*C5 =F6*C6 =F7*C7 =SUM(G4:G7)

=F10*C10 =F11*C11 =F12*C12 =F13*C13 =SUM(G10:G13)

=F4*C4

A C D E F GB

40 40 40

1040

0 0 0

1000

0.05

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C H A P T E R 1 6 Managing Bond Portfolios 521

Duration is a key concept in fixed-income portfolio management for at least three rea- sons. First, as we have noted, it is a simple summary statistic of the effective average matu- rity of the portfolio. Second, it turns out to be an essential tool in immunizing portfolios from interest rate risk. We explore this application in Section 16.3. Third, duration is a measure of the interest rate sensitivity of a portfolio, which we explore here.

We have seen that a bond’s price sensitivity to interest rate changes generally increases with maturity. Duration enables us to quantify this relationship. Specifically, it can be shown that when interest rates change, the proportional change in a bond’s price can be related to the change in its yield to maturity, y, according to the rule

DP

P 5 2D 3 BD(1 1 y)

1 1 y R (16.2)

The proportional price change equals the proportional change in 1 plus the bond’s yield times the bond’s duration.

Practitioners commonly use Equation 16.2 in a slightly different form. They define modified duration as D * 5 D /(1 1 y ), note that D(1 1 y ) 5 D y, and rewrite Equation 16.2 as

DP

P 5 2D*Dy (16.3)

The percentage change in bond price is just the product of modified duration and the change in the bond’s yield to maturity. Because the percentage change in the bond price is proportional to modified duration, modified duration is a natural measure of the bond’s exposure to changes in interest rates. Actually, as we will see below, Equation 16.2, or equivalently 16.3, is only approximately valid for large changes in the bond’s yield. The approximation becomes exact as one considers smaller, or localized, changes in yields. 4

Consider the 2-year maturity, 8% coupon bond in Spreadsheet 16.1 making semiannual coupon payments and selling at a price of $964.540, for a yield to maturity of 10%. The duration of this bond is 1.8852 years. For comparison, we will also consider a zero-coupon bond with maturity and duration of 1.8852 years. As we found in Spreadsheet 16.1 , because the coupon bond makes payments semiannually, it is best to treat one “period” as a half-year. So the duration of each bond is 1.8852 3 2 5 3.7704 (semiannual) periods, with a per period interest rate of 5%. The modified duration of each bond is therefore 3.7704/1.05 5 3.591 periods.

Suppose the semiannual interest rate increases from 5% to 5.01%. According to Equation 16.3, the bond prices should fall by

DP/P 5 2D*Dy 5 23.591 3 .01% 5 2.03591%

Example 16.1 Duration and Interest Rate Risk

4 Students of calculus will recognize that modified duration is proportional to the derivative of the bond’s price with respect to changes in the bond’s yield. For small changes in yield, Equation 16.3 can be restated as

D* 5 2 1

P dP

dy

As such, it gives a measure of the slope of the bond price curve only in the neighborhood of the current price. In fact, Equation 16.3 can be derived by differentiating the following bond pricing equation with respect to y:

P 5 a T

t51

CFt (1 1 y)t

where CF t is the cash flow paid to the bondholder at date t; CF t represents either a coupon payment before maturity or final coupon plus par value at the maturity date.

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522 P A R T I V Fixed-Income Securities

What Determines Duration? Malkiel’s bond price relations, which we laid out in the previous section, characterize the determinants of interest rate sensitivity. Duration allows us to quantify that sensitivity. For example, if we wish to speculate on interest rates, duration tells us how strong a bet we are making. Conversely, if we wish to remain “neutral” on rates, and simply match the interest rate sensitivity of a chosen bond-market index, duration allows us to measure that sensitiv- ity and mimic it in our own portfolio. For these reasons, it is crucial to understand the deter- minants of duration. Therefore, in this section, we present several “rules” that summarize most of its important properties. These rules are also illustrated in Figure 16.2 , where dura- tions of bonds of various coupon rates, yields to maturity, and times to maturity are plotted.

We have already established:

Rule 1 for Duration The duration of a zero-coupon bond equals its time to maturity.

We have also seen that a coupon bond has a lower duration than a zero with equal matu- rity because coupons early in the bond’s life lower the bond’s weighted average time until payments. This illustrates another general property:

Rule 2 for Duration Holding maturity constant, a bond’s duration is lower when the coupon rate is higher.

This property corresponds to Malkiel’s fifth relationship and is attributable to the impact of early coupon payments on the weighted-average maturity of a bond’s payments. The higher these coupons, the higher the weights on the early payments and the lower is the weighted average maturity of the payments. In other words, a higher fraction of the total value of the bond is tied up in the (earlier) coupon payments whose values are relatively insensitive to yields rather than the (later and more yield-sensitive) repayment of par value. Compare the plots in Figure 16.2 of the durations of the 3% coupon and 15% coupon

Now compute the price change of each bond directly. The coupon bond, which initially sells at $964.540, falls to $964.1942 when its yield increases to 5.01%, which is a per- centage decline of .0359%. The zero-coupon bond initially sells for $1,000/1.05 3.7704 5 831.9704. At the higher yield, it sells for $1,000/1.0501 3.7704 5 831.6717. This price also falls by .0359%.

We conclude that bonds with equal durations do in fact have equal interest rate sensitivity and that (at least for small changes in yields) the percentage price change is the modified duration times the change in yield. 5

5 Notice another implication of Example 16.1: We see from the example that when the bond makes payments semiannually, it is convenient to treat each payment period as a half-year. This implies that when we calculate modified duration, we divided Macaulay’s duration by (1 1 Semiannual yield to maturity). It is more common to write this divisor as (1 1 Bond equivalent yield/2). In general, if a bond were to make n payments a year, modified duration would be related to Macaulay’s duration by D * 5 D /(1 1 y BEY / n ).

a. In Concept Check 1, you calculated the price and duration of a 2-year maturity, 8% coupon bond mak- ing semiannual coupon payments when the market interest rate is 9%. Now suppose the interest rate increases to 9.05%. Calculate the new value of the bond and the percentage change in the bond’s price.

b. Calculate the percentage change in the bond’s price predicted by the duration formula in Equation 16.2 or 16.3. Compare this value to your answer for (a).

CONCEPT CHECK 16.2

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C H A P T E R 1 6 Managing Bond Portfolios 523

bonds, each with identical yields of 15%. The plot of the duration of the 15% coupon bond lies below the corresponding plot for the 3% coupon bond.

Rule 3 for Duration Holding the coupon rate constant, a bond’s duration generally increases with its time to maturity. Duration always increases with maturity for bonds selling at par or at a premium to par.

This property of duration corresponds to Malkiel’s third relationship, and it is fairly intuitive. What is surprising is that duration need not always increase with time to maturity. It turns out that for some deep-discount bonds (such as the 3% coupon bond in Figure 16.2 ), duration may eventually fall with increases in maturity. However, for virtu- ally all traded bonds it is safe to assume that duration increases with maturity.

Notice in Figure 16.2 that for the zero-coupon bond, maturity and duration are equal. However, for coupon bonds, duration increases by less than a year with a year’s increase in maturity. The slope of the duration graph is less than 1.0.

Although long-maturity bonds generally will be high-duration bonds, duration is a better measure of the long-term nature of the bond because it also accounts for coupon payments. Time to maturity is an adequate statistic only when the bond pays no coupons; then, maturity and duration are equal.

Notice also in Figure 16.2 that the two 15% coupon bonds have different durations when they sell at different yields to maturity. The lower-yield bond has longer duration. This makes sense, because at lower yields the more distant payments made by the bond have relatively greater present values and account for a greater share of the bond’s total value. Thus in the weighted-average calculation of duration the distant payments receive greater weights, which results in a higher duration measure. This establishes rule 4:

Rule 4 for Duration Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower.

Figure 16.2 Bond duration versus bond maturity

D u

ra ti

o n

( ye

ar s)

Zero-Coupon Bond

15% Coupon YTM = 15%

3% Coupon YTM = 15%

15% Coupon YTM = 6%

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Maturity

30

25

20

15

10

5

0

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524 P A R T I V Fixed-Income Securities

As we noted above, the intuition for this property is that while a higher yield reduces the present value of all of the bond’s payments, it reduces the value of more-distant payments by a greater proportional amount. Therefore, at higher yields a higher fraction of the total value of the bond lies in its earlier payments, thereby reducing effective maturity. Rule 4, which is the sixth bond-pricing relationship above, applies to coupon bonds. For zeros, of course, duration equals time to maturity, regardless of the yield to maturity.

Finally, we present a formula for the duration of a perpetuity. This rule is derived from and consistent with the formula for duration given in Equation 16.1 but may be easier to use for infinitely lived bonds.

Rule 5 for Duration The duration of a level perpetuity is

Duration of perpetuity 5 1 1 y

y (16.4)

For example, at a 10% yield, the duration of a perpetuity that pays $100 once a year forever is 1.10/.10 5 11 years, but at an 8% yield it is 1.08/.08 5 13.5 years.

Show that the duration of the perpetuity increases as the interest rate decreases in accordance with rule 4.

CONCEPT CHECK 16.3

Equation 16.4 makes it obvious that maturity and duration can differ substantially. The maturity of the perpetuity is infinite, whereas the duration of the instrument at a 10% yield is only 11 years. The present-value-weighted cash flows early on in the life of the perpetu- ity dominate the computation of duration.

Notice from Figure 16.2 that as their maturities become ever longer, the durations of the two coupon bonds with yields of 15% both converge to the duration of the perpetuity with the same yield, 7.67 years.

The equations for the durations of coupon bonds are somewhat tedious and spreadsheets like Spreadsheet 16.1 are cumbersome to modify for different maturities and coupon rates. Moreover, they assume that the bond is at the beginning of a coupon payment period. Fortu- nately, spreadsheet programs such as Excel come with generalizations of these equations that can accommodate bonds between coupon payment dates. Spreadsheet 16.3 illustrates how to

Spreadsheet 16.3

Using Excel functions to compute duration

1

2

3

4

5

6

7

8

9

10

A B C

Inputs

Settlement date

Formula in column B

Modified duration

Macaulay duration

Outputs

Coupons per year Yield to maturity

Coupon rate Maturity date

=MDURATION(B2,B3,B4,B5,B6) =DURATION(B2,B3,B4,B5,B6)

2

0.10

0.08

=DATE(2002,1,1) =DATE(2000,1,1)1/1/2000

1.7955

1.8852

2

0.10

1/1/2002

0.08

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C H A P T E R 1 6 Managing Bond Portfolios 525

use Excel to compute duration. The spreadsheet uses many of the same conventions as the bond-pricing spreadsheets described in Chapter 14.

The settlement date (i.e., today’s date) and maturity date are entered in cells B2 and B3 using Excel’s date function, DATE(year, month, day). The coupon and maturity rates are entered as decimals in cells B4 and B5, and the payment periods per year are entered in cell B6. Macaulay and modified duration appear in cells B9 and B10. The spreadsheet confirms that the duration of the bond we looked at in Spreadsheet 16.1 is indeed 1.8852 years. For this 2-year maturity bond, we don’t have a specific settlement date. We arbitrarily set the settlement date to January 1, 2000, and use a maturity date precisely 2 years later.

Coupon Rates (per Year)

Years to Maturity 6% 8% 10% 12%

1 0.985 0.981 0.976 0.972 5 4.361 4.218 4.095 3.990 10 7.454 7.067 6.772 6.541 20 10.922 10.292 9.870 9.568

Infinite (perpetuity) 13.000 13.000 13.000 13.000

Table 16.3

Bond durations (yield to maturity 5 8% APR; semiannual coupons)

6 Notice that because the bonds pay their coupons semiannually, we calculate modified duration using the semian- nual yield to maturity, 4%, in the denominator.

Use Spreadsheet 16.3 to test some of the rules for duration presented a few pages ago. What happens to duration when you change the coupon rate of the bond? The yield to maturity? The maturity? What happens to duration if the bond pays its coupons annually rather than semiannually? Why intuitively is duration shorter with semiannual coupons?