Duration
To measure the effective maturity of a bond that makes many payments, we average over the
maturity of each of the bond’s cash flows. Frederick Macaulay (1938) called this average the
duration of the bond. Macaulay’s duration equals the weighted average of the times to each
coupon or principal payment. The weights are related to the “importance” of each payment to
the value of the bond. Specifically, the weight for 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, wt
, associated with the cash flow made at time t (denoted CFt
) as
wt = CFt/(1 + y)t
__________
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 present
value of all the bond’s payments. These weights sum to 1 because the sum of the cash flows
discounted at the yield to maturity equals the bond price.
Macaulay’s duration
A measure of the effective
maturity of a bond, defined as
the weighted average of the
times until each payment, with
weights proportional to the
present value of the payment.
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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.
D = ∑
t=1
T
t × wt (11.1)
If we write out each term in the summation sign, we can express duration as:
time until
2nd cash flow
weight of
2nd CF
time until
4th CF
weight of
4th CF
D = w1 + 2w2 + 3w3 + 4w4 + ⋯ + TwT
Spreadsheet 11.1 uses Equation 11.1 to find the durations of an 8% coupon and zero￾coupon bond, each with three years to maturity, and yield to maturity of 10%. The present
value of each payment is discounted 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 11.1. 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 11.1 presents numbers for our
particular 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 dis￾count rate of 10%, in column E we calculate the weights for Equation 11.1, 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 11.1. The sum of these terms, reported in cells
F9 and F14, is therefore the duration of each bond. Using the spreadsheet, you can easily
answer several “what if” questions such as the one in Concept Check 11.1.
CONCEPT
check 11.1 Suppose the interest rate decreases to 9%. What will happen to the price and duration of each
bond in Spreadsheet 11.1?
Duration is a key concept in bond portfolio management for at least three reasons.
First, it is a simple summary measure of the average maturity of the portfolio. Second, it turns
out to be an essential tool in immunizing portfolios from interest rate risk. We will explore
this application in the next section. Third, duration is a measure of the interest rate sensitivity
of a bond portfolio, which we explore here.
We have already seen that price sensitivity to interest rate movements generally increases
with maturity. Duration 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
___
ΔP
P = −D ×
Δ(1 + y) _______
1 + y
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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
duration, and duration becomes a natural measure of interest rate exposure.1
This relationship
is key to interest rate risk management.
Practitioners commonly use Equation 11.2 in a slightly different form. They define
modified duration as follows:
D= ____ D
1 + y (11.3)
We can therefore rewrite Equation 11.2 as
___
ΔP
P = −DΔy (11.4)
The percentage change in bond price is just the product of modified duration and the change
in the bond’s yield to maturity.
By the way, the yield used in Equation 11.3 should be consistent with the payment
period of the bond. For example, if the bond pays coupons twice a year, y should be the bond’s
semiannual yield. When the payment period is shorter, the per-period interest rate is smaller,
and the difference between modified and Macaulay duration narrows.
modified duration
Macaulay’s duration divided
by 1 + yield to maturity.
Measures interest rate
sensitivity of bond.
v
Spreadsheets are
available in Connect
SPREADSHEET 11.1
Calculation of the duration of two bonds using Excel spreadsheet
A. 8% coupon bond
Payment
Column (B)
Column (E)
×
Payment
Discounted
at 10% Weight*
Interest rate:
A
Sum:
1
2
3
1
2
3
80
80
1080
0
0
1000
Weight = Present value of each payment (column D) divided by bond price
0.000
0.000
751.315
751.315
72.727
66.116
811.420
950.263
0.0765
0.0696
0.8539
1.0000
0.0000
0.0000
1.0000
1.0000
0.0765
0.1392
2.5617
2.7774
0.0000
0.0000
3.0000
Sum: 3.0000
B
10%
C D E F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B. Zero-coupon bond
Time until
Payment
(Years)
A. 8% coupon bond
Time until
Payment
(Years) Payment
Payment
Discounted
at 10% Weight
Column (B)
Column (E)
B. Zero-coupon
1 Interest rate:
2
3
4
5
6
7
8
9
10
11
12
13
14
A
Sum:
0.1
1
2
3
80
80
1080
=C6/(1+$B$1)∧B6
=C7/(1+$B$1)∧B7
=C8/(1+$B$1)∧B8
=SUM(D6:D8)
=D6/D$9
=D7/D$9
=D8/D$9
=D9/D$9
=E6B6
=E7B7
=E8B8
=SUM(F6:F8)
=C11/(1+$B$1)∧B11
=C12/(1+$B$1)∧B12
=C13/(1+$B$1)∧B13
=SUM(D11:D13)
=D11/D$14
=D12/D$14
=D13/D$14
=D14/D$14
=E11B11
=E12B12
=E13*B13
=SUM(F11:F13)
0
0
1000
1
2
3
Sum:
B C D E F
×
1
Actually, as we will see later, Equation 11.2 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.
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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
11.1, which has a duration of 2.7774 years, to the sensitivity of a zero-coupon bond with
maturity and duration of 2.7774 years. Both should have equal interest rate exposure if
duration is a useful measure of price sensitivity.
The coupon 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 .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 interest rate increases, its price falls to $1,000/1.10012.7774 = $767.2313, for an
identical .0252% capital loss. We conclude that equal-duration assets are equally sensitive to
interest rate movements.
Incidentally, this example also confirms the validity of Equation 11.2. The equation pre￾dicts that the proportional price change of the two bonds should have been −2.7774 × .0001/
1.10 = .000252, or .0252%, just as we found from direct computation.
EXAMPLE 11.1
Duration and Interest
Rate Risk
A bond with maturity of 30 years has a coupon rate of 8% (paid annually) and a yield to maturity of
9%. Its price is $897.26, and its duration is 11.37 years. What will happen to the bond price if its yield
to maturity increases to 9.1%?
Equation 11.4 tells us that an increase of 0.1% in the bond’s yield to maturity (Δy = .001 in decimal
terms) will result in a price change of
ΔP = −(D*Δy) × P
= − _____
11.37
1.09 × .001 × $897.26 = −$9.36
CONCEPT
check 11.2 a. In Concept Check 11.1, you calculated the price and duration of a three-year maturity, 8%
coupon bond at an interest rate of 9%. Now suppose the interest rate increases to 9.05%.
What is the new value of the bond, and what is 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 11.2 or 11.4. Compare this value to your answer for (a).
The equations for the durations of coupon bonds are tedious, and spreadsheets like Spread￾sheet 11.1 are cumbersome to modify for different maturities and coupon rates. Fortunately,
spreadsheet programs such as Excel come with built-in functions for duration. Moreover,
these functions easily accommodate bonds that are between coupon payment dates. Spread￾sheet 11.2 illustrates how to use Excel to compute duration. The spreadsheets use many of
the same conventions as the bond pricing spreadsheets described in Chapter 10.
We can use the spreadsheet to reconfirm the duration of the 8% coupon bond examined
in Panel A of Spreadsheet 11.1. The settlement date (i.e., today’s date) and maturity date
are entered in cells B2 and B3 of Spreadsheet 11.2 using Excel’s date function, DATE(year,
month, day). For this three-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
three years later. The coupon rate and yield to maturity 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. Cell B9 shows that the duration of the bond is indeed
2.7774 years. The modified duration of the bond is 2.5249, which equals 2.⁷⁷⁷⁴⁄₁.10.
CONCEPT
check 11.3 Consider a 9% coupon, 8-year-maturity bond with annual payments selling at a yield to maturity
of 10%. Use Spreadsheet 11.2 to confirm that the bond’s duration is 5.97 years. What would its
duration be if the bond paid its coupon semiannually? Why intuitively does duration fall?
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A
Inputs
Outputs
Formula In column B
B C
1
Settlement date 1/1/2000
1/1/2003
2.7774
2.5249
0.08
0.10
1
=DATE(2000,1,1)
=DATE(2003,1,1)
=DURATION(B2,B3,B4,B5,B6)
=MDURATION(B2,B3,B4,B5,B6)
0.08
0.10
1
Maturity date
Coupon rate
Yield to maturity
Coupons per year
Macaulay duration
10 Modified duration
9
8
7
6
5
4
3
2
v
Spreadsheets are
available in Connect
SPREADSHEET 11.2
Using Excel functions to compute duration

概括

债券的有效期限通过计算各现金流到期时间的加权平均值来衡量，称为久期。久期反映了债券价格对利率变动的敏感度，权重由各现金流的现值占比决定。零息债券的久期等于其到期时间，而付息债券的久期通常较短。修正久期进一步简化了利率风险度量，直接关联债券价格变动与收益率变化。久期是债券组合管理的关键工具，用于衡量平均期限、免疫策略及利率风险敞口。