800

Cash flow

700

600

500

400

300

200 Bond duration = 5.97 years

100

0

1 2 3 4 5 6 7 8

Year

F I G U R E 10.2

Cash flows paid by 9% coupon, annual payment bond with 8-year maturity. The height of each bar is the total of interest

and principal. The shaded portion of each bar is the present value of that cash flow. The fulcrum point is Macaulay™s

duration, the weighted average of the time until each payment.

first seven years, cash flow is simply the $90 coupon payment; in the last year, cash flow is the

sum of the coupon plus par value, or $1,090. The height of each bar is the size of the cash

flow; the shaded part of each bar is the present value of that cash flow using a discount rate of

10%. If you view the cash flow diagram as a balancing scale, like a child™s seesaw, the dura-

tion of the bond is the fulcrum point where the scale would be balanced using the present

values of each cash flow as weights. The balancing point in Figure 10.2 is at 5.97 years, which

is the weighted average of the times until each payment, with weights proportional to the pres-

ent value of each cash flow. The coupon payments made prior to maturity make the effective

(i.e., weighted average) maturity of the bond less than its actual time to maturity.

To calculate the weighted average directly, we define the weight, wt, associated with the

cash flow made at time t (denoted CFt ) as:

CFt /(1 y)t

wt

Bond price

where y is the bond™s yield to maturity. The numerator on the right-hand side of this equation

is the present value of the cash flow occurring at time t, while the denominator is the value of

all the payments forthcoming from the bond. These weights sum to 1.0 because the sum of the

cash flows discounted at the yield to maturity equals the bond price.

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Fifth Edition

344 Part THREE Debt Securities

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 formula for duration, denoted D.

T

D t wt (10.1)

t 1

If we write out each term in the summation sign, we can express duration in the following

equivalent equation

D w1 2w2 3w3 4w4 TwT

Qc Qc

time until weight time until weight of

2nd cash of 2nd 4th CF 4th CF

flow CF

An example of how to apply Equation 10.1 appears in Table 10.3, where we derive the du-

rations of an 8% coupon and zero-coupon bond each with three years to maturity. We assume

that the yield to maturity on each bond is 10%. The present value of each payment is dis-

counted at 10% for the number of years shown in column B. The weight associated with each

payment time (column E) equals the present value of the payment (column D) divided by the

bond price (the sum of the present values in column D).

The numbers in column F are the products of time to payment and payment weight. Each

of these products corresponds to one of the terms in Equation 10.1. According to that equation,

we can calculate the duration of each bond by adding the numbers in column F.

The duration of the zero-coupon bond is exactly equal to its time to maturity, three

years. This makes sense for, with only one payment, the average time until payment must

be the bond™s maturity. The three-year coupon bond, in contrast, has a shorter duration of

2.7774 years.

While the top panel of the spreadsheet in Spreadsheet 10.1 presents numbers for our par-

ticular example, the bottom panel presents the formulas we actually entered in each cell. The

inputs in the spreadsheet”specifying the cash flows the bond will pay”are given in columns

B and C. In column D we calculate the present value of each cash flow using a discount rate

of 10%, in column E we calculate the weights for Equation 10.1, and in column F we compute

the product of time until payment and payment weight. Each of these terms corresponds to one

of the terms in Equation 10.1. The sum of these terms, reported in cells F9 and F14, is there-

fore the duration of each bond. Using the spreadsheet, you can easily answer several “what

if ” questions such as the one in Concept Check 1.

>

1. Suppose the interest rate decreases to 9%. What will happen to the price and du-

Concept

ration of each bond in Spreadsheet 10.1?

CHECK

Duration is a key concept in bond portfolio management for at least three reasons. First,

it is a simple summary measure of the effective average maturity of the portfolio. Second, it

turns out to be an essential tool in immunizing portfolios from interest rate risk. We will ex-

plore this application in the next section. Third, duration is a measure of the interest rate sen-

sitivity of a bond portfolio, which we explore here.

We have already noted that long-term bonds are more sensitive to interest rate movements

than short-term bonds. The duration measure enables us to quantify this relationship. It turns

out that, when interest rates change, the percentage change in a bond™s price is proportional to

its duration. Specifically, the proportional change in a bond™s price can be related to the

change in its yield to maturity, y, according to the rule

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Fifth Edition

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10 Managing Bond Portfolios

S P R E A D S H E E T 10.1

Calculation of the duration of two bonds using Excel spreadsheet

A B C D E F

1 Interest rate: 0.10

2

3 Time until Payment Column B

4 Payment Discounted

5 (Years) Payment at 10% Weight* Column E

6 A. 8% coupon bond 1 80 72.727 0.0765 0.0765

7 2 80 66.116 0.0696 0.1392

8 3 1080 811.420 0.8539 2.5617

9 Sum: 950.263 1.0000 2.7774

10

1 1 B. Zero-coupon bond 1 0 0.000 0.0000 0.0000

12 2 0 0.000 0.0000 0.0000

13 3 1000 751.315 1.0000 3.0000

14 Sum: 751.315 1.0000 3.0000

15

1 6 *W eight Present value of each payment (column D) divided by bond price

A B C D E F

1 Interest rate: 0.10

2

3 Time until Payment Column B

4 Payment Discounted

5 (Years) Payment at 10% Weight Column E

6 A. 8% coupon bond 1 80 C6/(1 $B$1)^B6 D6/D$9 E6*B6

7 2 80 C7/(1 $B$1)^B7 D7/D$9 E7*B7

8 3 1080 C8/(1 $B$1)^B8 D8/D$9 E8*B8

9 Sum: SUM(D6:D8) D9/D$9 SUM(F6:F8)

10

1 1 B. Zero-coupon 1 0 C11/(1 $B$1)^B11 D11/D$14 E11*B11

12 2 0 C12/(1 $B$1)^B12 D12/D$14 E12*B12

13 3 1000 C13/(1 $B$1)^B13 D13/D$14 E13*B13

14 Sum: SUM(D11:D13) D14/D$14 SUM(F11:F13)

(1 y)

P

s t

D (10.2)

1y

P

The proportional price change equals the proportional change in (1 plus the bond™s yield)

times the bond™s duration. Therefore, bond price volatility is proportional to the bond™s dura-

tion, and duration becomes a natural measure of interest rate exposure.1 This relationship is

key to interest rate risk management.

Practitioners commonly use Equation 10.2 in a slightly different form. They define

modified duration as D* D/(1 y) and rewrite Equation 10.2 as modified duration

Macaulay™s

P

D* y (10.3) duration divided by

P 1 yield to maturity.

Measures interest rate

The percentage change in bond price is just the product of modified duration and the change in

sensitivity of bond.

the bond™s yield to maturity. Because the percentage change in the bond price is proportional

1

Actually, as we will see later, Equation 10.3 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.

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

346 Part THREE Debt Securities

to modified duration, modified duration is a natural measure of the bond™s exposure to inter-

est rate volatility.

A bond with maturity of 30 years has a coupon rate of 8% (paid annually) and a yield to ma-

turity of 9%. Its price is $897.26, and its duration is 11.37 years. What will happen to the

10.1 EXAMPLE bond price if the bond™s yield to maturity increases to 9.1%?

Equation 10.3 tells us that an increase of 0.1% in the bond™s yield to maturity ( y .001

Duration and

in decimal terms) will result in a price change of

Interest Rate Risk

P (D* y) P

11.37

0.001 $897.26 $9.36

1.09

To confirm the relationship between duration and the sensitivity of bond price to interest

rate changes, let™s compare the price sensitivity of the three-year coupon bond in Spreadsheet

10.1, which has a duration of 2.7774 years, to the sensitivity of a zero-coupon bond with ma-

turity and duration of 2.7774 years. Both should have equal interest rate exposure if duration

is a useful measure of price sensitivity.

The three-year bond sells for $950.263 at the initial interest rate of 10%. If the bond™s yield

increases by 1 basis point (1/100 of a percent) to 10.01%, its price will fall to $950.0231, a

percentage decline of 0.0252%. The zero-coupon bond has a maturity of 2.7774 years. At the

initial interest rate of 10%, it sells at a price of $1,000/1.102.7774 $767.425. When the inter-

est rate increases, its price falls to $1,000/1.10012.7774 $767.2313, for an identical 0.0252%

capital loss. We conclude, therefore, that equal-duration assets are equally sensitive to interest

rate movements.

Incidentally, this example confirms the validity of Equation 10.2. The equation predicts

that the proportional price change of the two bonds should have been 2.7774 0.0001/1.10

0.000252, or 0.0252%, just as we found from direct computation.

>

2. a. In Concept Check 1, you calculated the price and duration of a three-year ma-

Concept

turity, 8% coupon bond for an interest rate of 9%. Now suppose the interest rate

CHECK increases to 9.05%. What is 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 10.2 or 10.3. Compare this value to your answer for (a).

What Determines Duration?

Malkiel™s bond price relations, which we laid out in the previous section, characterize the de-

terminants of interest rate sensitivity. Duration allows us to quantify that sensitivity, which

greatly enhances our ability to formulate investment strategies. 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 sensitivity and mimic it in our own

portfolio. For these reasons, it is crucial to understand the determinants of duration and con-

venient to have formulas to calculate the duration of some commonly encountered securities.

Therefore, in this section, we present several “rules” that summarize most of the important

properties of duration. These rules are also illustrated in Figure 10.3, which contains plots of

durations of bonds of various coupon rates, yields to maturity, and times to maturity.

Bodie’Kane’Marcus: III. Debt Securities 10. Managing Bond © The McGraw’Hill

Essentials of Investments, Portfolios Companies, 2003

Fifth Edition

347

10 Managing Bond Portfolios