Selling Treasuries but maintaining dollar duration

[johntavv]

CFAI reading 21 practice problem 9 says:
Alonso has discovered an overweight in a 5yr reasury bond (10million par value). Alonso’s strategy will be to sell all the 5-year Treasury bonds, and invest the proceeds in 10-year Treasury bonds and cash while maintaining the dollar duration of the portfolio. Treasury info is below:
Issue description Duration Price Yield
5yr 4.125% 4.53 100.40625 4.03%
10yr 5.25% 8.22 109.09375 4.14%
The par value of 10yr bonds to be purchased to execute this strategy is closest to?
The answer is = dollar duration of the 5-year / (duration x quoted price x 0.01 of the 10-year)
= 454,840.31 / (8.22 x 1.0909375 x 0.01)
Why do we not do: dollar duration of 5-yr / dollar duration of 10-yr?

 

[myriam2222]

Because you are comparing the dollar duration of two securities that do not have the same price (same par value not same price). So you can’t compare one to another. Careful: for duration its the value that matters, not par (duration: % change in value for % change in interest rates).
If you want to go for something intuitive like such a comparison, rather go for:
Duration of 5-yr / duration of 10-yr (not dollar duration)
That would give you the percentage of your proceeds you need to reinvest in the 10-yr bond to achieve the same duration (intuitive: if duration of 5-yr is half as much of 10-yr, you reinvest half). So 4.53/8.22 = 55.11%
How much do you have to invest?
10m x 1.0040625 = 10.0406m
How much do you reinvest in 10-yr?
10.0406 x 55.11% = 5.53339m
How much is that in par value?
5.53339 / 1.0909375 = 5.072m
Same result.

 

[zulu007]

Thanks Myriam2222 - helpful