Maintain Portfolio Duration / Dollar Duration

CFA Level III
#1

Vol 4 Page 112 states

“To maintain the portfolio duration when one security is being exchanged for another, the dollar durations of the securities being exchanged must be matched”

Assuming two securities:

S1: market value $100, duration 4

S2: market value $100, duration 5

Portfolio duration = (100/200)*4 + (100/200)*5 = 4.5

If S1 is exchanged with a third security S3 having duration 2, in order to maintain $ duration

mkt value S3*2 = market value S1 * duration S1 = $100*4

hence required market value S3 = $200

taking the weighted average formula for portfolio duration

new portfolio duration = (200/300)*2 + (100/300)*5 = 3 is not equal to 4.5 (original portfolio duration)

Is the statement in the text incorrect or am I missing something here?

#2

I think 200 would be the PAR value of the bond required, not the market value.

#3

Very good point you are making here. I ll give it a try:

if you look at the porfolio dollar duration, both of them are 9.

meaning the changing of dollar amount in the portfolio is the same for both.

the duartion of 3 is only the effective duration which measures sensitivity differently than dollar duation.

#4

cpk - the par value should not figure in computing the $ duration, what matters is the market value (Example 10 on the same page validates this)

passme - agree that duration will be constant, but the text states that portfolio "duration" (not duration) is maintained

#5

text says, To maintain the portfolio duration when one security is being exchanged for another, the dollar durations of the securities being exchanged must be matched.

IMO, we would be required both the par value & the mkt value of securities being exchanged, otherwise it does not hold good. In example there are macthing the dollar duration of the bonds being exchanged. They have not given the earlier port duration & after the exchange, what is the port duration…

For examination purpose however, the replaced bonds $ dur should match with the bond which is replacing the existing bond in portfolio.

#6

and in the example - they are calculating the PAR value of the Bond being exchanged. You are calculating the Market value - and using that to recalculate the Portfolio Duration - the two steps which they do not perform, either.

#7

rahuls / cpk - portfolio duration = (V1d1+…+Vndn)/(V1+…+Vn) where Vi = mkt value, di = duration

maintaining the $ duration only maintains the numerator, but the resultant denominator changes too since the total portfolio value changes. portfolio duration cannot be constant. par value is irrelevant i think.

#8

oz

all that is understood. Are you seeing the example - that they are calculating the PAR value of the Bond required, NOT THE MARKET VALUE - like you have. That is all I am pointing out.

And yes the formula is known.

Out here - the numerator is also not maintained - as Market Value and the PAR value could be way off, and we have no means of ensuring / knowing how the two are related in teh absence of more information.

So PAR value is relevant, because that is all you can calculate.

#9

#1. I believe oz is correct in using Mkt Value. I don’t have the book, but that’s how I remember it - and what my notes say.

#2. In your original calc, oz, you are using a portfolio worth $200, and in your new portfolio calc you are using a portfolio worth $300. I wonder where the extra $100 comes from, I wish an extra $100 would appear in my portfolio when I want to buy something extra.

The fact is the extra $100 doesn’t apear, it was there all along - just invested in cash (duration = 0) if you do the original calc adjusted for the cash your portfolio would have a dollar duration of 3, which matches you ending dollar duration.

(100/300)*2 + (100/300)*5 + (100/300)*0 = 3

#10

I am not contending using Market Value for Portfolio Duration. I am saying that what you are calculating using the

If S1 is exchanged with a third security S3 having duration 2, in order to maintain $ duration

mkt value S3*2 = market value S1 * duration S1 = $100*4

S3 -> is not Market Value - but it is Par Value.

#11

I see what you are saying, it’s the part about using the price of 1 bond in calculating dollar duration if the security is not held in the portfolio, correct?

That would still be a Mkt value though, no?

#12

Maintaining Portfolio Duration in Changing Portfolio Holdings A portfolio manager wants to exchange one bond issue for another that he believes is undervalued. The existing position in the old bond has a market value of 5.5 million dollars. The bond has a price of $80 and a duration of 4. The bond’s dollar duration is therefore 5.5 million * 4/100 or $220,000. The new bond has a duration of 5 and a price of $90, resulting in a dollar duration of 4.5 ($90 * 5/100) per bond. What is the par value of the new bond needed to keep the duration of the portfolio constant? Solution: The market value of the new bond issue would be ($220,000/5)100 = $4,400,000. The bond is trading at $90 per $100 of par. The par value of this issue would be $4,400,000/0.9 = $4.889 million. This can also be calculated as $4.889 million ($220,000/4.5 * 100).

The 4.889 M$ is being calculated as the PAR value for the Bond. This is the same number being calculated above as S3. But oz0001 calls it Market Value - which it is NOT. That is all I am trying to say…

#13

from oz:

mkt value S3*duration S3 = market value S1 * duration S1

From CPK:

The market value of the new bond issue would be ($220,000/5)100 = $4,400,000

The solution goes on to find PAR Value, but these two equations are the same (both using Mkt Value).

#14

thanks for pointing that out, FinNinja.

#15

thanks FinNinja, the $100 cash that I overlooked explains it. I was thinking that an additional 100 investment is required to maintain the duration.