Coupon rate

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					                                                            W                      X
Coupon rate                                                 5%                   10%
Maturity                                                  5 years               7 years
Face value                                                 $100                  $100
Yield to maturity                                           6%                    6%
Value (2.6)                                               $95.73                122.59
$ Value of basis point - exact                           $0.0416               $0.0642
$ Value of +125 basis points - exact                      -$5.03                -$7.70
$ Value of –125 basis points - exact                       $5.37                 $8.36
Yield value of 1/32                                      0.0075%               0.0049%
Macaulay’s (4.5)                                        4.47 years            5.39 years
Modified (4.10)                                         4.34 years            5.23 years

2nd (4.22)                                               8,541.36             16,942.39
Convexity (6mo) (4.19)                                    89.22                138.20
Convexity (years) pg. 75                                  22.30                 34.55


From Taylor series expansion of bond price-yield to maturity relationship:

                          P               1 2P
 dP  P( y1 )  P( y0 )     ( y1  y0 )         ( y1  y0 ) 2  Rm
                          y               2 y 2

Modified duration (4.10) = MD

       C
       y   2
               1  (1  y)  n(M  C y)  (1  y)
                         n                            ( n 1)
                                                                       1 P
D*                                                               
                                 P                                     P y


(4.11) dP  D*  ( P)  (dy)

Using modified duration to predict bond price change in response to a 1 basis point
change in yield to maturity.

(W)    -4.34 (95.73) (.0001) = -$0.0415
(X)    –5.23 (122.59) (.0001) = -$0.0641

Using modified duration to predict bond price change in response to a 125 basis point
increase in yield to maturity.

(W)    -4.34 (95.73) (.0125) + = -$5.19
(X)    –5.23 (122.59) (.0125) = -$8.01


                                                 1
The larger a bond’s convexity the greater the approximation error due to a change in
yield to maturity.

Convexity captures the extent of curvature of the bond price-yield to maturity
relationship.

Convexity is determined by the second derivative of the bond price-yield to maturity
relationship (4.22):

  2P
  y   2
           
               2C
               y   3
                       1  (1  y)  2Cny
                                  n           2
                                                    (1  y )  ( n 1)  n(n  1)(M  C )  (1  y )  ( n  2)
                                                                                       y


                                   2P
Convexity (4.19) CV =        (4  P) years
                       y 2
From Taylor series expansion of bond price-yield to maturity relationship:

                                   P               1 2P
 dP  P( y1 )  P( y0 )              ( y1  y0 )         ( y1  y0 ) 2  Rm
                                   y               2 y 2

                       1
(4.20) dP                (CV )  ( P)  (dy) 2
                       2

(4.11) + (4.20)

                         1
dP   D*  ( P)  (dy)   (CV )  ( P)  (dy) 2
                         2

Using modified duration and convexity to predict bond price change in response to a 125
basis point increase in yield to maturity.

(W)        -4.34 (95.73) (.0125) + 0.50 (22.30)(95.73) (.0125)2 = -$5.03
(X)        –5.23 (122.59) (.0125) + 0.50 (34.55) (122.59) (.0125)2 = -$7.68




                                                            2
The greater the convexity of the bond price-yield to maturity relationship the larger are
changes in duration with respect to changes in yield to maturity necessitating more
frequent rebalancing to maintain the immunization property.

Holding yield to maturity and time to maturity constant, zero-coupon bonds will have
the greatest convexity.

    ytm           T         c                P                   D*              CV
    5%            5         0             $78.08               4.878            26.175
    5%            5        3%             $91.25               4.548            23.832
    5%            5        5%             $100.00              4.376            22.612



Holding yield to maturity and modified duration constant, zero-coupon bonds will
have the smallest convexity.

   ytm           T             c                 P                 D*                 CV
   5%            5             0              $78.08             4.878              26.175
   5%           5.5         3.77%             $94.16             4.878              27.637
   5%            6          7.64%             $113.54            4.878              28.956



Convexity and immunization:

Choosing a coupon bond with Macaulay’s duration equal to the investment horizon
immunizes the invested capital from a one-time parallel shift in the term structure of
interest rates.

         This is exactly true only for zero coupon bonds with maturity equal to the
          investment horizon.

         It is approximately true for bonds with coupon rates greater than zero.

         Consequently an immunization strategy is less exact the greater the convexity of
          the bond.



The convexity of a portfolio is the weighted average of the convexities of the portfolio’s
assets.




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