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# Duration by WG7zpP5

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```									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

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