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Duration by WG7zpP5

VIEWS: 6 PAGES: 40

									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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