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**Question is at the bottom**

Suppose you have a portfolio of bonds A, B, and C with the following characteristics:

(the "Frequency" column is the # of coupon pmts per year and also the # of compounding periods)

Bond | Coupon Rate | Frequency      | Years to Maturity | YTM (%) | FV ($)
A    |   5.00      | quarterly      |  2                | 5.16    | 1,000
B    |   5.50      | semi-annually  |  3                | 5.40    | 1,000
C    |   6.25      | annually       |  4                | 5.90    | 1,000

From the information above I calculate each Bond's price and duration:

Bond A

CFt      = $1,000 × (5.00% ÷ 4) = $12.50
YTMt     = 0.0516 ÷ 4 = 0.0129 or 1.29%
Price    = $12.50 × [1 – ( 1.0129)^-8 ÷ 0.0129] + ($1,000 ÷ 1.0129)^-8
         = $94.44 + $902.54
         = $996.98
Duration = 7,638.9447 ÷ $996.98 ÷ 4 = 1.9155 years

Bond B

CFt      = $1,000 × (5.50% ÷ 2) = $27.50
YTMt     = 0.0540 ÷ 2 = 0.0270 or 2.70%
Price    = $27.50 × [1 – ( 1.0270)^-6 ÷ 0.0270] + ($1,000 ÷ 1.0270)^-6
         = $150.47 + $852.27
         = $1,002.74
Duration = 5,628.5639 ÷ $1,002.74 ÷ 2 = 2.8066 years

Bond C

CFt      = $1,000 × (6.25% ÷ 1) = $62.50
YTMt     = 0.0590 ÷ 1 = 0.0590 or 5.90%
Price    = $12.50 × [1 – ( 1.0590)^-4 ÷ 0.0590] + ($1,000 ÷ 1.0590)^-4
         = $217.07 + $795.09
         = $1,012.16
Duration = 3,707.4842 ÷ $1,012.16 ÷ 1 = 3.6630 years

Portfolio Duration

Bond    Duration(D)     Quantity (Q)    Bond Price (P)  V = Q × P        DP = D × V
A       1.9155              2            $	996.98      $   1,993.96    3,819.4304
B       2.8066              1            $	1,002.74    $   1,002.74    2,814.2901
C       3.6630              2            $	1,012.16    $   2,024.32    7,415.0842
                                                   Total    $ 5,021.02   14,048.8046
Portfolio Duration = 14,048.8046 ÷ $5,021.02 = 2.7980 or 2.80 years

Portfolio Weighted Average Discount Rate

Bond    YTM/YR  Periods/YR  YTM/Period  Quantity (Q)    Bond Price (P)  V = Q × P   YTM/Period × V
A        0.0516        4    0.0129     2          $	996.98	      $  1,993.96    25.7221 
B        0.0540        2    0.0270     1          $	1,002.74      $  1,002.74    27.0740 
C        0.0590        1    0.0590     2          $	1,012.16      $  2,024.32    19.4349 
                                                          Total       $	   5,021.02   172.2309
Weighted Average Discount Rate = 172.2309 ÷ $5,021.02 = 0.0343 or 3.43%

QUESTION

Now suppose a fourth bond D, has a coupon rate of 5.25% paid semi-annually, a maturity of two years, a face value of $1,000, a yield of 6.25%, and a duration of 1.9238 years. How could one immunize the interest rate risk of the portfolio above with bond D? I think we need to find the proportion of bond D that will make the overall duration 0, by solving the equation for the augmented-portfolio duration.

w × Dp + (1 – w ) × Dd = 0

where

Dp = duration of the portfolio of bonds A, B, and C
w = proportion of the portfolio in bonds A, B and C
1 – w = proportion of the portfolio in bond D
Dd = duration of bond D
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1 Answer 1

"I think we need to find the proportion of bond D that will make the overall duration 0" ... by doing so, you will be matching the duration of bond D, because it has the shortest one.

In order to immunize portfolio, you should have some benchmark duration you want to match. If it is the one of bond D, than your suggestion is right.

If I am wrong, please, correct me somebody.

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Can you do the calculation? –  emcor Sep 4 at 9:40

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