# Bond Portfolio Immunization - Duration Matching

**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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