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1

Duration

Riccardo Colacito

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Motivational example

Two bonds sell today with the samepayment schedule

Time 1: $10

Time 2: $20

However one has a YTM of 1% and theother one has a YTM of 99%

What is the temporal distribution of thepayments?

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Example (contd)

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%)

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Effective maturity?

Both bonds have maturity 2 years

But they differ in the temporal distributionof the (present value of) payments

Theireffective maturities should differ,i.e. the second one should have shortermaturity

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Duration

A measure of the effective maturity of abond

The weighted average of the times untileach payment is received, with theweights proportional to the present valueof the payment

Duration is always shorter or equal tomaturity for all bonds

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Duration: Calculation

= =

T

ttwtD 1

6

Duration is

where

PriceBond

)1( tt

yCF

tw+=

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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?)

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Compute the duration

Year CFPresent Value

CFweight

1 80 57.76 0.20

2 80 41.71 0.15

3 80 30.11 0.10

4 580 157.63 0.55

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Duration is

355.410.315.220.1 =+++=D

9

Just use the formula

or three years

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Cash flows and their present

values

0

100

200

300

400

500

600

700

1 2 3 4

CF

Dicounted CF

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Question

What happens if the YTM decreases?

Would you expect the duration to increaseor decrease?

Why?

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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

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How are the weights affected?

0

100

200

300

400

500

600

700

1 2 3 4

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What happens to the duration?

You now attach a relatively higher weightto cash flows that happen further in thefuture

That is: the duration goes up!

Can verify that the duration is now 3 yearsand a quarter.

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Question

What is the duration of a zero couponbond?

The duration of a zero coupon bond isequal to its maturity

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A perpetuity

Definition: a security that pays aconstant coupon forever

Its maturity is infinite! Its duration is

yyD

+= 1

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Duration vs maturity

Holding the coupon rate constant, abonds duration generally increases withtime to maturity

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Duration vs Maturity (contd)

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30 35

Par=$1,000, Coupon=$30 (yearly),YTM=15%

Zero-coupon bond

Perpetuity

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Why does this happen?

As the YTM increases, the present valueof the face value at high maturities getssmaller and smaller.

That is the bond starts looking as aperpetuity with a very high yield tomaturity.

20

Fi 10 3 D ti

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Figure 10.3 Duration as a

Function of Maturity

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Uses of Duration

1. Summary measure of length or effectivematurity for a portfolio

2. Measure of price sensitivity for changesin interest rate

3. Immunization of interest rate risk (nextclass)

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Duration/Price Relationship

Take derivative of price wrt YTM

Hence the following holds as an approximation

PD1

1

)1()1(1

1

)1()1()1(

1

111

+

=

++++=

+

+=

+

=

=++

y

y

Par

Ty

C

ty

y

ParT

y

Ct

y

P

T

tTt

T

tTt

D1

)1(

+

+=

y

y

P

P

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Modified Duration

*

D=

yP

P

24

More compact formula

where

yDD+= 1

*

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Fact #1

Prices of long-term bonds are moresensitive to interest rate changes thanprices of short term bonds

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Fact #1

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

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

30

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Intutition

The higher the maturity, the higher theduration

The higher the duration, the higher theprice sensitivity

Higher maturity implies higher pricesensitivity to interest rate risk

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Fact #2

The sensitivity of bond prices to changesin yields increases at a decreasing rate asmaturity increases.

Equivalently, interest rate risk is less thanproportional to bond maturity.

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Fact #2

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

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

30

50

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Intuition

Duration increases less thanproportionally with maturity (e.g. see slide18)

Hence price sensitivity to interest rate riskincreases less than proportionally withmaturity.

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Fact #3

Prices of high coupon bonds are lesssensitive to changes in interest rates thanprices of low coupon bonds.

Or equivalently, interest rate risk isinversely related to the bonds coupon rate

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Fact #3

-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

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

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Intuition

The higher the coupon, the lower theduration

Lower duration implies lower pricesensitivity.

The higher the coupon, the lower the pricesensitivity to interest rate risk

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Fact #4

The sensitivity of a bonds price to achange in its yield is inversely related tothe yield to maturity at which the bond

currently is selling.

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Fact #4

0

200

400

600

800

1000

1200

1400

1600

1800

2000

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

Percentage Price change =(1341-1000)/1000=34.1%

Percentage Price change =(769-610)/610=26%

Busi 580 - Investments

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Intuition

The higher the YTM, the lower theduration

The lower the duration, the lower the pricesensitivity

High YTM implies low price sensitivity

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Convexity

The formula

is an approximation of the more precise relation

( ) 2*21D yconvexityy

P

P+=

37

*D=

yP

P

Figure 10 6 Bond Price

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Figure 10.6 Bond Price

Convexity

38

The higher the convexity the

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The higher the convexity, the

better it is for the investor

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Yet another fact

An increase in a bonds yield to maturity resultsin a smaller price change than a decrease inyield of equal magnitude

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0. 00 1 0 .0 11 0. 021 0 .0 31 0 .04 1 0 .05 1 0 .0 61 0. 07 1 0. 08 1 0. 091 0 .10 1 0 .111 0 .1 21 0. 131 0 .14 1 0 .15 1 0 .16 1 0 .1 71 0. 18 1 0 .1 91

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