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Document Sample
Duration
Riccardo Colacito
1
Motivational example
• Two bonds sell today with the same
payment schedule
– Time 1: $10
– Time 2: $20
• However one has a YTM of 1% and the
other one has a YTM of 99%
• What is the temporal distribution of the
payments?
2
Example (cont’d)
Time Payment PV (YTM=1%) PV (YTM=99%)
1 10 9.9 (33.6%) 5.025 (49.9%)
2 20 19.6 (66.4%) 5.050 (50.1%)
3
Effective maturity?
• Both bonds have maturity 2 years
• But they differ in the temporal distribution
of the (present value of) payments
• Their effective maturities should differ,
i.e. the second one should have shorter
maturity
4
Duration
• A measure of the effective maturity of a
bond
• The weighted average of the times until
each payment is received, with the
weights proportional to the present value
of the payment
• Duration is always shorter or equal to
maturity for all bonds
5
Duration: Calculation
• Duration is
T
D t wt
t 1
where
CFt
(1 y ) t
wt
Bond Price
6
Duration: an example
• Consider the following bond
– Maturity: 4 years
– Par: $500
– Coupon: $80 (once per year)
– YTM: 38.5%
– Price is $287.205 (remember how to compute this?)
7
Compute the duration
Present Value
Year CF weight
CF
1 80 57.76 0.20
2 80 41.71 0.15
3 80 30.11 0.10
4 580 157.63 0.55
8
Duration is…
• Just use the formula
D 1.20 2 .15 3.10 4 .55 3
• … or three years
9
Cash flows and their present
values
700
600
500
400
CF
Dicounted CF
300
200
100
0
1 2 3 4
10
Question
• What happens if the YTM decreases?
• Would you expect the duration to increase
or decrease?
• Why?
11
Duration: another example
• Consider the following bond
– Maturity: 4 years
– Par: $500
– Coupon: $80 (once per year)
– YTM: 15.6% 38.5%
– Price is $505.641 $287.205
12
How are the weights affected?
700
600
500
400
300
200
100
0
1 2 3 4
13
What happens to the duration?
• You now attach a relatively higher weight
to cash flows that happen further in the
future
• That is: the duration goes up!
• Can verify that the duration is now 3 years
and a quarter.
14
Question
• What is the duration of a zero coupon
bond?
• The duration of a zero coupon bond is
equal to its maturity
15
A perpetuity
• Definition: a security that pays a
constant coupon forever
• Its maturity is infinite!
• Its duration is
1 y
D
y
16
Duration vs maturity
• Holding the coupon rate constant, a
bond’s duration generally increases with
time to maturity
17
Duration vs maturity
35
30 Zero-coupon bond
25
20
15
10
Par=$1,000, Coupon=$30 (yearly),
YTM=5%
5
0
0 5 10 15 20 25 30 35
18
Duration vs Maturity (cont’d)
35
30 Zero-coupon bond
25
20
Par=$1,000, Coupon=$30 (yearly),
YTM=15%
15
10
5
Perpetuity
0
0 5 10 15 20 25 30 35
19
Why does this happen?
• As the YTM increases, the present value
of the face value at high maturities gets
smaller and smaller.
• That is the bond starts looking as a
perpetuity with a very high yield to
maturity.
20
Figure 10.3 Duration as a
Function of Maturity
21
Uses of Duration
1. Summary measure of length or effective
maturity for a portfolio
2. Measure of price sensitivity for changes
in interest rate
3. Immunization of interest rate risk (next
class)
22
Duration/Price Relationship
• Take derivative of price wrt YTM
P T
C Par
t t 1
T
(1 y ) t 1 (1 y ) (1 y )T 1
1 T C Par
t (1 y )t T (1 y )T
1 y t 1
1
D P
1 y
• Hence the following holds as an approximation
P (1 y )
D
P 1 y
23
Modified Duration
• More compact formula
P
y D*
P
where
D
D
*
1 y
24
Fact #1
• Prices of long-term bonds are more
sensitive to interest rate changes than
prices of short term bonds
25
0
500
1000
1500
2000
2500
3000
3500
4000
0
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
Fact #1
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
30
10
26
Intutition
• The higher the maturity, the higher the
duration
• The higher the duration, the higher the
price sensitivity
• Higher maturity implies higher price
sensitivity to interest rate risk
27
Fact #2
• The sensitivity of bond prices to changes
in yields increases at a decreasing rate as
maturity increases.
• Equivalently, interest rate risk is less than
proportional to bond maturity.
28
0
1000
2000
3000
4000
5000
6000
0
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
Fact #2
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
50
30
10
29
Intuition
• Duration increases less than proportionally
with maturity (e.g. see slide 18)
• Hence price sensitivity to interest rate risk
increases less than proportionally with
maturity.
30
Fact #3
• Prices of high coupon bonds are less
sensitive to changes in interest rates than
prices of low coupon bonds.
• Or equivalently, interest rate risk is
inversely related to the bond’s coupon rate
31
-100
100
300
500
700
900
1100
1300
1500
0
0.01
0.01
0.02
0.02
0.03
0.03
0.04
0.04
0.05
0.05
0.06
0.06
0.07
0.07
0.08
0.08
0.09
0.09
0.1
0.1
0.11
0.11
0.12
0.12
Fact #3
0.13
0.13
0.14
0.14
0.15
0.15
0.16
0.16
0.17
0.17
0.18
0.18
0.19
0.19
10
100
32
Intuition
• The higher the coupon, the lower the
duration
• Lower duration implies lower price
sensitivity.
• The higher the coupon, the lower the price
sensitivity to interest rate risk
33
Fact #4
• The sensitivity of a bond’s price to a
change in its yield is inversely related to
the yield to maturity at which the bond
currently is selling.
34
Fact #4
2000
1800
Percentage Price change =(1341-1000)/1000=34.1%
1600
1400
1200
Percentage Price change =(769-610)/610=26%
1000
800
600
400
200
0
0.001 0.011 0.021 0.031 0.041 0.051 0.061 0.071 0.081 0.091 0.101 0.111 0.121 0.131 0.141 0.151 0.161 0.171 0.181 0.191
Busi 580 - Investments 35
Intuition
• The higher the YTM, the lower the
duration
• The lower the duration, the lower the price
sensitivity
• High YTM implies low price sensitivity
36
Convexity
• The formula
P
y D*
P
is an approximation of the more precise relation
P
y D convexity y
* 1 2
P 2
37
Figure 10.6 Bond Price
Convexity
38
The higher the convexity, the
better it is for the investor
39
Yet another fact
• An increase in a bond’s yield to maturity results
in a smaller price change than a decrease in
yield of equal magnitude
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0.001 0.011 0.021 0.031 0.041 0.051 0.061 0.071 0.081 0.091 0.101 0.111 0.121 0.131 0.141 0.151 0.161 0.171 0.181 0.191
40
