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             BOND DURATION

             A bond’s duration represents the weighted average time to full recovery of
             interest and principal payments. A bond with a maturity of five years
             (assuming it has a coupon rate greater than zero) will have a Macaulay du-
             ration of less than five.

                         Example:
                         Calculate the Macaulay duration of a bond with five years to
                         maturity, a coupon rate of 8% and a market rate of 10%.

             The formula to calculate the Macaulay Duration is:
                                                                    n  CF × t 
                                                                  ∑           t 
                                                                  t =1 ( + i ) 
                                                                         1
                                         Duration Macaulay   =
                                                               Current Value bond

             Thus, calculating a bond’s duration involves weighting each cash flow by
             the time period it is received, discounting the result to time zero and divid-
             ing the sum of these values by the bond’s current value.
                                           80 × 1 80 × 2 80 × 3 80 × 4 1080 × 5 
                                                    +        +         +       +
                                         =
                                            (1.10 )1 (1.10 )2 (1.10 )3 (1.10 )4 (1.10 )5  
                     Duration Macauley
                                                                924.18                    
                                                                                          
                                                                                          
                                           80         160       240      320     5400 
                                                    +        +         +       +
                                         =
                                            (1.10 ) (1.10 ) (1.10 ) (1.10 ) (1.10 )5 
                                                   1        2         3       4
                                                                                        
                                                               924 .18                 
                                                                                       
                                                                                       
                                           72.73 + 132.23 + 180.32 + 218.56 + 3,352.98 
                                         =                                               
                                                               924.18                    
                                            3,956.82 
                                         =           
                                            924.18 
                                         = 4.28
             The denominator of $924.18 is calculated using the process shown on
             pages 52-53 (Bond Values on Interest Payment Dates). Calculating a
             bond’s duration using the formula is rather time consuming, especially for
             long-term bonds. While there is no Duration function available for the cal-




Effective Use of Financi...                                                                    Monday, May 18, 1998 13:16
page 76
                                                                                                                                           77
                   culator models discussed in this handbook, the process of calculating dura-
                   tion can be made a bit easier by utilizing the NPV function to calculate the
                   numerator in the duration equation, and the basic calculation of bond
                   prices to determine the denominator. This process is shown in Bar 45.

                   Bar 45: Calculating Duration
                               TI BA II +                        HP 10B                              HP 17B II
                               Mode:          END               Mode:         END                     Mode:         END

                               2nd        CLR TVM               n       CLEAR ALL                      FIN          TVM

                                    80     PMT                          80    PMT                n     CLEAR DATA

                               1000        FV                    1000        FV                       80       PMT

                                    10     I/Y                          10   I/YR                 1000       FV

                                      5   N                              5   N                        10       I%YR
                          CPT       PV     -924.18                      PV   +/-                       5     N
                                +/-       STO     1                 n        STO                 PV    +/-       STO      1
                                         CF                         C    0    CFj                     EXIT CFLO

                               2nd        CLR Work                      80    CFj                n     CLEAR DATA

                                      0   CF ↓                  2 x 80        CFj                       YES

                                    80     ENTER      ↓ ↓       3 x 80        CFj                      0       INPUT

                              2 x 80       ENTER      ↓ ↓       4 x 80        CFj                     80       INPUT       INPUT

                              3 x 80       ENTER      ↓ ↓   5 x 1080          CFj                2 x 80        INPUT       INPUT

                              4 x 80       ENTER      ↓ ↓               10   I/YR                3 x 80        INPUT       INPUT

                        5 x 1080           ENTER                        n    NPV                 4 x 80        INPUT       INPUT

                              NPV   10     ENTER      ↓     ÷    RCL     1   =     4.2814   5 x 1080           INPUT       INPUT

                                CPT        3956.81                                                     EXIT CALC

                                     CE/C                                                             10     I%      NPV

                         ÷     RCL    1       =   4.28                                       ÷    RCL      1    =    4.2814




Effective Use of Financi...                                                                                          Monday, May 18, 1998 13:16
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