# Reading 29 - example 10

I can’t figure out how they solved the problem, i believe they mixed duration with dollar duration too much. For those who do not have reading I present the example and give CFAI and mine solution. Please correct me if I am wrong. The manager wants to change old bonds for new issue that he belives is undervalued without changing the portfolio’s duration (dollar duration should remain the same). Old portfolio: bond price \$80, duration 4, total market value \$5.5 mn CFAI shows the dollar duration is \$5.5 mn * 4/100 = \$220,000 - I found it is OK (the dollar duration of a single bond would be 3.2 then). New bond: duration 5, price 90, which gives according to CFAI 4.5 dollar duration per bond - again that's OK but the solution (they ask for par value, so simply the market value divided by 0.9): 'The amount of the new bond reuired to keep the portfolio constant is 4.889 mn (\$220,000/4.5*100) and the required par would be \$5.432 mn (4.889/0.9)’ is wrong. If I check the dollar duration of the new portfolio in the way they calculated the old one it is: market value * duration = \$4.889 * 5 /100(as example says) is about \$244,000 not \$220,000, so the portfolio after bond change is different, more risky. They treat dollar duration of a single bond in new portfolio (4.5) as a duration (which they say is 5) So the solution should be 4.4 mn (220/5*100) market value required to buy (that is 4.4/0.9=4.889 mn par value). Am I wright?

Well, no you are not wright as they are American icons not to be trifled with. But your analysis seems correct. 220,000/4.5 says you need 48888 bonds priced at 90 not \$4.88M in bonds. But this is a pretty easy mistake to make so we still like them.

Many thanks for your answer Joey, you have boosted my morale a bit before the exam;) … and I think CFAI made sth wrong again with the dollar duration in another example: Reading 27 Example 6 (page 340). They calculate dollar duration of the portfolio as a ‘weighted average of dollar duration as the weighted dolar duration of component securities.’ Hey, the portfolio dollar duration is the SUM not WEIGHTED AVERAGE of component securities. Only a simple duration (as percantege change of value, not the dollar change of value) might be weighted on average to derive a portfolio duration. To make things worse CFAI even presents the example with this totally flawed and illogic (in my humple opinion) method of calculation. The guy who rewrited the reading 29 from Fabozzi book (Fabozzi was the author of the fixed income reading in the curriculum for 2007) made a mistake, isn’t he? There was a thread on the forum concerning the example, but focusing only on the calculation method to derive weigthed average and not on the fact that weighted average is not a proper solution here… If someone has schweser notes 2008 - do they present the same solution? (according to my schweser 2007 the answer should be the SUM not the average) Anybody supports my view on the matter?

guys, example 6 on page 340 of book 3 is not incorrect. if you look at this link you’ll see the sam formula (slide 17) http://pages.stern.nyu.edu/~jcarpen0/courses/b403333/06durh.pdf the dollar duration of the portfolio is the sum of each individual dollar durations multiplied by a given rate change. But this ends up implying a weighted average summation as bonds that have a higher dollar duration (therefore more weight in a portfolio) contribute more the the dollar duration of the entire portfolio. the CFAI’s wording might not be the easiest to follow sometimes but in this case it is accurate.

Hi, I do not see this topic in the LOS this year. It’s in the text, but not in schweser. Right?

I don’t see that in the pdf and in general I think dollar duration is a sum not a weighted average. This might be semantics, but a weighted average has a bunch of weights that sum to 1. Is there some meaningful formula for dollar duration of a portfolio that has that characteristic?

dominiko…i came up with the same answer as you did. I think the last formula on page 12, reading #29 is wrong. Instead of “New bond market value” it should be “New bond par value”, which, if you read the line above the formula, they seem to suggest should be the case. So, New bond par value = DDo/DDn x 100 = (220,000/4.5) x 100 = 4.889 million The new bond market value would be 4.889 million x 0.9 = 4.4 million