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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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I may be understanding this wrong but doesn't the question of immunization inherrently have to do with "do can you cover your cash outflows with current cash inflows if there is change to interest rate?" –  Kamster Mar 4 at 6:10

2 Answers 2

up vote 2 down vote accepted

If I understand correctly the question, you wish to completely hedge the interest rate risk (defined as a parallel shift in the yield curve). If that is the case, you should use modified duration, which is the price sensitivity, rather than the MacAulay duration. They are usually close in value, but not quite the same.

Fortunately, you can easily transform your durations to modified durations:

$D_{mod} = \frac{D_{MA}}{1 + \frac{yield}{frequency}}$

This yields a vector of modified durations: $D_m = [1.8911; 2.7328; 3.4589; 1.8655]$ for bonds A, B, C and D.

Since we want to immunize against parallel shifts:

$\Delta P \approx -P * D_{mod} * \Delta y = 0$

where P is the portolio value, $\Delta P$ the change in value, $D_{mod}$ the modified duration of the portolio and $\Delta y$ the change in yield. To immunize, we want $P * D_{mod} = 0$, or, taking $P$ as a given, $D_{mod} = 0$, where $D_{mod} = w^T * D_m$ (weighted sum of the modified durations).

Given our weights (value of the bonds) $w = [1993.96; 1002.74; 2024.32; w_4]$, we only have 1 unknown, so this is easy to solve; we obtain $w_4 = -7243.81$. In other words, you would have to short 7243.81\$ worth of bond D to immunize your bond portfolio. This might be problematic, given their 1000\$ value, though.

Another way to see it is that you need to match the modified duration of your assets (your initial portfolio) and liabilities (using bond D), hence the need to short bond D (which becomes a liability). Keep in mind, this only (and only approximately) protects against parallel shifts in the curve.

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Thank you. I was expecting the amount to be a negative amount of Bond D and I understood the answer involved shorting Bond D. I just couldn't get there algebraically. Not sure I understand the modified durations but will study it more. Thanks again! –  Ben Mar 7 at 2:32
    
My pleasure. As you can see, the difference between the two is relatively small. Modified duration is actually the first derivative of the relative change in price (log price) with respect to the yield. The difference between the two only appears when using periodically compounded yields; using continuously compounded yields, the two are the same. Since in most cases the bonds use the same payment frequency and have similar yields, duration is still a good approximation. –  Olivier Chassé St-Laurent Mar 7 at 13:23

"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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1  
Can you do the calculation? –  emcor Sep 4 '14 at 9:40

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