Investing proceeds while maintaining dollar duration of portfolio

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johntavv

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Reading 21 practice problem 9 says:
Alonso has an overweight in a 5yr T-Bond ($10m par). His strategy will be to sell all 5 yr T-Bonds, and invest proceeds in 10 yr T-Bonds while maintaining the dollar duration of the portfolio.
Duration Price Yield
5yr 4.53 100.40625 4.03
10yr 8.22 109.09375 4.14
Using the above, the par value of the 10yr bonds to be purchased to execute the strategy is?
1. Sale price of the $10m 5yr bonds = $10m x 1.0040625
=10,040,625.
2. Dollar duration of 5 yr = 4.53 x 10,040,625 x 0.01
= 454,840.31
3. Divide 454,940.31 by dollar duration of 10yr = 454,940.31 / (8.22 x 1.0909375 x 0.01)
= $5,072,094
Can someone explain why are we dividing 5 yr dollar duration by 10 yr dollar duration?
 
Because we want to maintain the dollar duration of the PF..
Duration(5 Yr) * Sale Price (5 Yr) * .01 = Duration(10 Yr) * Sale Price (10 Yr) * .01
Hence Sale Price (10 Yr) = Duration(5 Yr) * Sale Price (5 Yr) / Duration(10 Yr)
Hope this was helpful.
 
The dollar duration of the 10-year is your instrument for adjusting the current duration to target duration
If this were a bond future (in place of the 10-year), it would make more sense.
 
The steps I have seen in the text are:
1. Original (target) dollar duration / New dollar duration
= 5yr dollar duration / 10yr dollar duration
= (10,040,625 x 0.01 x 4.53) / (10,909,375 x 0.01 x 8.22)
= 454,840.31 / 896,750.63
= 0.50721
2. Subtract 1 from this ratio to give the % change to make in the holding of each asset in the portfolio to restore the desired DD.
0.507 - 1 = - 0.4928
How do we then get to the answer of 5,072,094 as par value of the $10m 10yr bonds?
 
Here’s an example.
You have 10m in a -5year bond with a duration of 5. For some reason you need to buy 10-year bonds but you do not want to have the 10-year bond’s duration, you want to keep the duration of the 5-year.
You investment choice is now:
$0 in the 5-year bonds
?? in the 10-year bonds
??? as cash
Cash has zero duration, and you want to hold your portfolio duration constant at 5 but your know the 10-year has a duration of 10.
Therefore you keep 5m in cash and invest 5m in the 10-year bond. Your portfolio has now acheived all of it’s objectives. It no longer has the 5-year bond exposure but has managed to keep the same duration by keeping some $ in cash.
Just to note, this person could have also put the money into another investment that does not exibit duration.
 
I understand your concept written there Galli, but can someone please explain why the par value of 10yr bonds to be purchased is = (5yr dollar duration / 10yr dollar duration) x par value
=(454,840.31 / 896,750.63) x 10,000,000 = $5,072,094
What is the reasoning behind this?
 
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