Chapter 16

Chapter 16
Managing Bond
Portfolios
Change in Bond Price as a Function of Change in
Yield to Maturity
Interest Rate Sensitivity
Inverse relationship between price and
yield.
An increase in a bond’s yield to maturity
results in a smaller price decline than the
gain associated with a decrease in yield.
Long-term bonds tend to be more price
sensitive than short-term bonds.
Interest Rate Sensitivity
Price sensitivity is inversely related to a
bond’s coupon rate.
Price sensitivity is inversely related to the
yield to maturity at which the bond is
selling.
Interest Rate Sensitivity
Prices of 8% Coupon Bond (Coupons Paid Semiannually):
 As YTM rises, price falls.
 Fall in price is higher when maturity is higher
 As maturity increases, price sensitivity increases at a decreasing
rate.
Interest Rate Sensitivity
Prices of Zero-Coupon Bond (Semiannually Compounding)
 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
represents a longer-term bond than equal-time-to-maturity coupon
bond
Ambiguity with Effective Maturity
 Note that the times to maturity of the two bonds in this
example are not perfect measures of the long-term or shortterm 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
date,’ and the effective maturity of the bond is therefore some
sort of average of the maturities of all the cash flows paid out
by the bond.
 The zero-coupon bond, by contrast, makes only one payment
at maturity. Its time to maturity is, therefore, a well-defined
concept.
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 to serve as a useful summary statistic of the
effective maturity of the bond.
Duration is a measure of the effective maturity of a
bond.
Frederick Macaulay termed the effective maturity
concept the duration of the bond.
Duration
Macaulay’s duration is computed as the weighted
average of the times until each payment is received,
with the weights proportional to the present value of
the payment.
Duration is shorter than maturity for all bonds
except zero coupon bonds.
Duration is equal to maturity for zero coupon bonds.
Duration: Calculation
wt 
1  y 
CFt
t
Price
T
D   t wt
t 1
CFt  Cash Flow for period t
Duration Calculation: Example
8% coupon paid semiannually; maturity: 2 years; YTM=10%
𝑪𝟏 × 𝑪𝟒
Time
(years)
Payment
PV of CF
Weight
.5
40
38.095
.0395
.0197
1
40
36.281
.0376
.0376
1.5
40
34.553
.0358
.0537
2.0
1040
855.611
.8871
1.7742
sum
964.540
1.000
1.8852
Duration/Price Relationship
 We have seen that long-term bonds are more sensitive to
interest rate movements than are short-term bonds. The
duration measure 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 its YTM according to the rule:
∆𝑃/𝑃 = −𝐷 ×
∆ 1+𝑦
1+𝑦
𝐷∗ = modified duration
𝐷
∗
𝐷 =
1+𝑦
= 1.8852 / 1.10 = 1.71 years
∆𝑃/𝑃 = − 𝐷∗ × ∆𝑦
Rules for Duration
Rule 1 The duration of a zero-coupon bond equals its time to
maturity.
Rule 2 Holding maturity constant, a bond’s duration is higher
when the coupon rate is lower.
Rule 3 Holding the coupon rate constant, a bond’s duration
generally increases with its time to maturity.
Rule 4 Holding other factors constant, the duration of a
coupon bond is higher when the bond’s yield to maturity is
lower.
Rules for Duration (cont’d)
Rules 5 The duration of a level perpetuity is equal to:
(1  y)
y
 For example, at a 10% yield, the duration of a perpetuity
that pays $100 once a year forever is 1.10 / 0.10 = 11
years, but at an 8% yield, it is 1.08 / 0.08 = 13.5 years
 This rule makes it obvious that maturity and duration can
differ substantially. The maturity of a perpetuity is infinite,
whereas the duration at a 10% yield is only 11 years
Rules for Duration (cont’d)
Rule 6 The duration of a level annuity is equal to:
1 y
T

y
(1  y ) T  1
For example, a 10-year annual annuity with a yield of 8% will
have duration
1.08
10

 4.87 years
10
0.08 1.08  1
Rules for Duration (cont’d)
Rule 7 The duration for a coupon bond is equal to:
1  y (1  y )  T (c  y )

y
c[(1  y ) T  1]  y
 The duration of a 8% coupon (paid semiannually) bond with maturity of 2
years and YTM of 10% would be
1.05 1.05  4(0.04  0.05)
1.01

 21 
 3.7645halfyears  1.88 years
4
0.05 0.04[1.05  1]  0.05
0.0586
 Same result that we got in slide 9
Duration and Convexity
∆𝑃/𝑃 = −𝐷 ∗ × ∆𝑦
 This rule asserts that the percentage price change is
directly proportional to the change in the bond’s yield.
 If this were exactly so, a graph of the percentage change in
bond price as a function of the change in its yield would
plot as a straight line, with slope equal to – 𝐷 ∗ . Yet, we
know from the Figure presented in slide 2 that the
relationship between bond prices and yields is not linear.
 The duration rule is a good approximation for small
changes in bond yield, but it is less accurate for larger
changes.
Duration and Convexity
30-Year Maturity, 8% Coupon; Initial YTM = 8%
Correction for Convexity
1
Convexity 
P  (1  y )2
 CFt

2

t (t  t ) 
(1

y
)
t 1 

n

Correction for Convexity:
P
  D*y  12[Convexity  (y)2 ]
P
Correction for Convexity
Suppose, a bond has 30-year maturity, 8% coupon and 8% YTM. If a bond’s yield
increases from 8% to 10%, and 𝐷 ∗ =11.26 years, without adjusting for convexity,
percentage change in price
P
  D*y  11.26  0.02  22.52%
P
Convexity of the bond can be calculated to be 212.4. Adjusting for convexity,
percentage change in price
P
 11.26  0.02  1 [212.4  (0.02) 2 ]  18.27%
2
P
Why Do Investors Like Convexity?
Practice Problems
 Chapter 16:
2, 3, 4, 5, 7, CFA Problem: 3