Duration
Measuring Interest Rate Sensitivity
Measuring Interest Rate Risk
• We know:
– An increase in interest rates causes bond prices
to fall, and a decrease in interest rates causes
bond prices to rise.
• We also know that longer maturity debt
securities tend to be more volatile in price.
– For a given change in interest rates, the price of
a longer term bond generally changes more
than the price of a shorter term bond.
Measuring Interest Rate Risk
• Two bonds with the same term to maturity
do not have the same interest-rate risk.
– A 10 year zero coupon bond makes all of its
payments at the end of the term.
– A 10 year coupon bond makes payments before
the maturity date.
• Which bond has the highest interest-rate
risk?
Interest Rate Risk Problem
• Calculate the rate of capital gain or loss on a
ten year zero coupon bond for which the
interest rate has increased from 10% to
20%. The bond has a face value of $1000.
– Capital gain = (Pt+1 - Pt) / P t
– - 49.7%
= ($193.81 - $385.54)/$385.54
Interest Rate Risk Problem
• The rate of capital gain or loss on a ten year
coupon bond that has a face value of $1000
for which the interest rate has increased
from 10% to 20% is -40.3%.
• The interest rate risk on a ten year coupon
bond is less than the interest rate risk on a
10 year zero coupon bond.
• Why?
Varying Coupon Rates: Coupon
Effect
• A security promising lower annual coupon
payments behaves as though it has a longer
maturity even if it is due to mature on the same
date as a security carrying a higher coupon
rate.
– Investors must wait longer to realize a
substantial return.
• The farther in the future cash payments are to be
received, the more sensitive the present value of the
stream of payments to changes in interest rates.
Coupon Effect: Definition
• When interest rates rise, the prices of low
coupon securities tend to fall faster than the
prices of high coupon securities.
• Similarly, when interest rates decline, the prices
of low coupon rate securities tend to rise faster
than the prices of high coupon rate securities.
• Therefore, the potential for capital gains and
capital losses is greater for low coupon
securities.
Duration: Introduction
• Knowledge of the impact of varying coupon
rates on security price volatility led to the
development of a new index of maturity other
than straight calendar time.
• The new measure permits analysts to construct
a linear relationship between term to maturity
and security price volatility, regardless of
differing coupon rates.
Duration
Present value of interest and
principal payments from a
security weighted by the
timing of those payments
D=
n
=
Present value of the security’s
promised stream of interest
and principal payments
CPt
t
S (1 + i)t
t=1
n
CPt
S (1 + i)t
t=1
Duration
CP represents the expected
payment of principal and
interest income.
n
t represents the time period in
which each payment is to be D =
received.
And i is the security’s yield
to maturity.
CPt
t
S (1 + i)t
t=1
n
CPt
S (1 + i)t
t=1
Duration Example
Assume there is an investor who is interested in buying a $1,000
par value bond that has a term to maturity of 10 years, a 10 percent
annual coupon rate, and a 10 percent yield to maturity based on its
current price.
D =
$100(1) $100(2) ... $100(10) $1000(10)
+
+ +
+
(1.10) (1.10)2
(1.10)10 (1.10)10
$100 + $100 + ... + $100
$1000
+
(1.10) (1.10)2
(1.10)10 (1.10)10
= 6758.9
1000
=
6.758 years
Duration: Zero Coupon Bonds
Example
• To get the effective maturity of a set of zero
coupon bonds we must:
– Sum the effective maturity of each zero coupon
bond, weighting it by the percentage of the total
value of all the bonds that it represents.
• The duration of the set is the weighted average of
the effective maturities of the individual zero
coupon bonds, with the weights equaling the
proportion of the total value represented bye each
zero coupon bond.
Duration: Example
Yield = 10%
Year Cash Payments
1
2
3
4
5
6
7
8
9
10
10
Total
100
100
100
100
100
100
100
100
100
100
1000
Present Value
of Cash Payments
90.01
82.64
75.13
68.30
62.09
56.44
51.32
46.65
42.41
38.55
385.54
1000.00
Weights
Weighted
% of Total PV Maturity
0.09001
0.08264
0.07513
0.06830
0.06209
0.0.644
0.05132
0.04665
0.04241
0.03855
0.38554
1.00
0.09091
0.16528
0.22539
0.27320
0.31045
0.33864
0.35924
0.37320
0.38550
0.38550
3.85500
6.75850
Zero Coupon Bond Example: Steps
• Calculate the present value of each of the zero coupon
bonds when the interest rate is 10% (column 3).
• Divide each of these present values by $1000 (the
total present value of the set of zero-coupon bonds) to
get the percentage of the total value of all the bonds
that each bond represents. Note that the sum equals 1
(column 4).
• Calculate the weighted maturities (column 5) by
multiplying column 1 by column 4.
• Get the effective maturity of the set of bonds by
adding column 5.
Duration: Another Example
Yield = 20%
Year Cash Payments
1
2
3
4
5
6
7
8
9
10
10
Total
100
100
100
100
100
100
100
100
100
100
1000
Present Value
Weights
of Cash Payments % of Total PV
83.33
69.44
57.87
48.23
40.19
33.49
27.91
23.26
19.38
16.15
161.15
580.76
0.14348
0.11957
0.09650
0.08305
0.06920
0.05767
0.04806
0.04005
0.03337
0.02781
0.27808
1.00
Weighted
Maturity
0.14348
0.23914
0.29895
0.33220
0.34600
0.34602
0.33642
0.32040
0.30033
0.27810
2.78100
5.72204
Things to Notice
• When the yield to maturity rises, the duration of
the coupon bond falls.
• The higher the coupon rate on the bond, the
shorter the duration of the bond.
• When the maturity of a bond lengthens, the
duration rises as well.
• Duration is additive: the duration of a portfolio of
securities is the weighted-average of the durations
of the individual securities, with the weights
equaling the proportion of the portfolio invested in
each.
Duration is Additive
• The duration of a portfolio of securities is
the weighted average of the durations of the
individual securities with the weights
reflecting the proportion invested in each.
– Example: Let 25% of a portfolio be invested in
a bond with a duration of 5 and let 75% of the
portfolio be invested in a bond with a duration
of 10.
• Dp = (0.25 x 5) + (0.75 x 10) = 8.75 years
Duration and Interest Rate Risk
• Because duration is related in linear fashion
to the price volatility of a security, there is
an approximate relationship between
changes in interest rates and percentage
changes in security prices.
Duration and Interest Rate Risk
% Change in the price
of a debt security
= -D x /\ i x 100%
1+i
D = duration
/\ i = change in interest rates
% Change in the price
of a debt security
0.02 x 100% = -11.91%
= -6.55 x 1 + 0.10
An increase in interest rates of 2% causes a decline in the bond’s
price of approximately 12%.