Principles of Corporate Finance_ 12th Edition

Chapter 3 Valuing Bonds 53

bre44380_ch03_046-075.indd 53 09/30/15 12:47 PM

measure of the average maturity, one that could be used to predict the exposure of each bond’s
price to fluctuations in interest rates. There is such a measure, and it is called duration or
Macaulay duration after its founder.

❱ TABLE 3.2 A comparison of the cash flows and prices of two Treasuries. Price is

calculated assuming annual coupon payments and a yield to maturity of 4%.

 Yield =  4%  Yield =  3%

Coupon Price ($) Price ($) Change in Price (%)

3% $ 939.98 $1,000.00 +6.4%

9% 1,300.10 1,373.82 +5.7

❱ TABLE 3.3 The effect of a 1% fall in yield on the prices of two seven-year

Treasuries.

Note: Both securities mature at the end of year 7.

● ● ● ● ●

Duration is the weighted average of the times to each of the cash payments. The times are
the future years 1, 2, 3, etc., extending to the final maturity date, which we call T. The weight
for each year is the present value of the cash flow received at that time divided by the total
present value of the bond.

Duration =

1 × PV(C 1 )

__________

PV

+

2 × PV(C 2 )

__________

PV

+

3 × PV(C 3 )

__________

PV

+ ⋅ ⋅ ⋅ +

T × PV(CT)

___

PV

Table  3.4 shows how to compute duration for the 9% seven-year Treasuries, assuming
annual payments. First, we value each of the coupon payments of $90 and the final payment
of coupon plus face value of $1,090. Of course the present values of these payments add up to
the bond price of $1,300.10. Then we calculate the fraction of the price accounted for by each

❱ TABLE 3.4 Calculating the duration of the 9% seven-year bonds. The yield to maturity is 4% a year.

Cash Payments ($)

Coupon Price ($) Year 1 Year 2  . . . Year 6 Year 7

3% $ 939.98 $30 $30 $30 $1,030

9% 1,300.10 90 90 90 1,090

Year (t )

1 2 3 4 5 6 7

Payment $90 $90 $90 $90 $90 $90 $ 1,090

PV(Ct ) at 4% $86.54 $83.21 $80.01 $76.93 $73.97 $71.13 $ 828.31 PV = $1,300.10

Fraction of total value

[PV(Ct ) / PV]

0.0666 0.0640 0.0615 0.0592 0.0569 0.0547 0.6371

Year × fraction of total

value [t × PV(Ct ) / PV]

0.0666 0.1280 0.1846 0.2367 0.2845 0.3283 4.4598 Total = duration = 5.69