Duration- Effective Maturity of A Bond
• Duration combines the length of time to maturity and bond
cash flows and measures the sensitivity of bond value to
changes in the interest rates. The higher the duration of a
bond, higher the sensitivity of the bond value to the interest
rate changes.
• Duration is defined 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
© Dr. C. Bulent Aybar
McCauley Duration: Calculation
N
McCauley Duration
=
D
n wn
n 1
n = Years until payment
w n = CF (1+r)
n
n
Price
CFn=Cash Flow at year n, r = required rate of return
Calculating the Duration of Two Bonds
Duration/Price Relationship
• While duration measures the effective maturity of a bond it also captures
the sensitivity of bond price to changes in the required return or bond
yield.
• Price change is proportional to duration:
P
P
D
(1 r )
1 r
• A more commonly used measure is “Modified Duration which is:
D* = modified duration=D/(1+r)
• With Modified Duration Bond Price –Bond Yield Relationship can
written as:
P
P
D* r
© Dr. C. Bulent Aybar
Example
• ABC Inc. has an outstanding bond with 10% coupon and 10 years to
maturity. The YTM of ABC Inc. bonds is 6%. The bond has McCauley
duration of 7.16 and the current price of the bond is $1294.40 . Calculate
the change in the bond price if the required rate of return on ABC bonds
go up to 8%.
• Solution:
• Since we have the MC Duration, we can easily calculate the change in
the value of bond by using bond price duration relationship which is
given as:
P
(1 r )
D
P
1 r
• Modified Duration=-7.16 /(1+0.06) =-6.754
• Change in Bond Price= -(MD) x (Change in Yield) x Bond Price
•
=-6.754 x (0.02) x 1294.40=$174.87
© Dr. C. Bulent Aybar
Duration Approximation to Bond Price Change
Yield
Actual Change in Bond
Actual Change in Price/Duration
Bond Price Bond Price Appoximation
Bond Price vs Bond Yield
1700.00
2.00%
1294.40
1718.61
424.20
350.20
3.00%
1597.11
302.71
262.65
4.00%
1486.65
192.25
175.10
1400.00
5.00%
1386.09
91.68
87.55
1300.00
6.00%
1294.40
0.00
0.00
1200.00
7.00%
1210.71
-83.70
-87.55
8.00%
1134.20
-160.20
-175.10
9.00%
1064.18
-230.23
-262.65
10.00%
1000.00
-294.40
-350.20
1600.00
1500.00
1100.00
1000.00
2.00% 3.00% 4.00% 5.00% 6.00% 7.00% 8.00% 9.00% 10.00%
As you see in the table, actual bond price change and the change approximated by
duration are not exactly the same. This happens because bond price yield relationship is
a convex relationship, and duration is a linear approximation. We miss the convexity in
linear approximation. We can make up for this by including another term to the
approximation called convexity.
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 5: The duration of a level perpetuity is equal to: (1+r) / r
© Dr. C. Bulent Aybar
Bond Duration versus Bond Maturity
Bond Duration versus Bond Maturity
Note that lower the coupon rate, more linear the relationship is.
Bond Durations (Yield to Maturity = 8% APR; Semiannual Coupons)
Bond durations
(yield to
maturity=8%
APR,
semiannual
coupons
CONFIRMING THE RULES 2 &3:
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