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#1
11-23-2010, 04:55 PM
 Final_Judgement Member Join Date: Oct 2010 College: University of Waterloo Posts: 371
Duration of a Portfolio

I could probably prove this, but can you find the duration of a portfolio using both Maccauley and modified durations?

Like, will combining Macauleys and converting to modified yield the same result as converting to modifieds and combining?

Thanks.
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#2
11-23-2010, 04:57 PM
 Final_Judgement Member Join Date: Oct 2010 College: University of Waterloo Posts: 371

Also, while we're on duration, when a question says "find the duration", is this to always be interpretted as Macauley? Seems to be the case. Thanks.
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#3
11-23-2010, 05:03 PM
 JUICE Member SOA AAA Join Date: Sep 2007 Favorite beer: Bell's Two Hearted Posts: 6,463

Quote:
 Originally Posted by Final_Judgement I could probably prove this, but can you find the duration of a portfolio using both Maccauley and modified durations? Like, will combining Macauleys and converting to modified yield the same result as converting to modifieds and combining? Thanks.
Let's find out... bear in mind I'm a bit rusty

Modified duration, MD = v * Macaulay duration

Portfolio Macaulay duration = w1 * D1 + .... + wn*Dn

where Di's are the Macaulay durations of the assets and wi's are the weights associated with said assets

Then portfolio modified duration = w1*D1*v+.... + wn*Dn*v

= v * Portfolio Macaulay duration

QED?

of course, a single interest/yield rate must be chosen when calculating duration of portfolios
#4
11-23-2010, 05:08 PM
 JUICE Member SOA AAA Join Date: Sep 2007 Favorite beer: Bell's Two Hearted Posts: 6,463

Quote:
 Originally Posted by Final_Judgement Also, while we're on duration, when a question says "find the duration", is this to always be interpretted as Macauley? Seems to be the case. Thanks.
Macaulay, I thought, referred to a SPECIFIC duration calculation where the PVs are discounted using the asset's implied yield rate. A question asking for generic 'duration' could imply several methods.... if A/L matching, you want to discount the asset CFs at the same rate as you discount the liability CFs... if a simple duration calculation on a bond, I think you would want to use the implied yield unless specifically told otherwise.

But, yes, "duration" = "Macaulay duration" and not "modified duration."

Last edited by JUICE; 11-23-2010 at 05:26 PM..
#5
11-23-2010, 06:08 PM
 bigb Member Join Date: Jul 2010 Posts: 400

In layman's terms, and I think this is correct, but you can treat the MacD of a portfolio like you calculate MacD for a bond. Instead however, if you have multiple bonds, you have multiple cash flows occurring for each one. Instead of calculating t*A(t) for one bond, where A(t) is the PV of the cashflow at time t, add all the cashflows that occur at time t from multiple bonds. Likewise your price for a bond is just P, and for a portfolio, instead of dividing by P, sum up total price of the portfolio.

Think of it as adding all of your individual bonds into one "bigger" bond with cashflows.

The only thing you need to keep in mind is that is

MacD(portfolio) does not equal MacD(1st instrument)+MacD(2nd instrument)+...+MacD(nth instrument).

I did that a few times. It is not additive.

Last edited by bigb; 11-23-2010 at 06:20 PM..
#6
11-24-2010, 12:59 PM
 dlt Join Date: Nov 2010 Posts: 2

Quote:
 Originally Posted by bigb Think of it as adding all of your individual bonds into one "bigger" bond with cashflows. The only thing you need to keep in mind is that is MacD(portfolio) does not equal MacD(1st instrument)+MacD(2nd instrument)+...+MacD(nth instrument). I did that a few times. It is not additive.

Can't MacD(portfolio) be computed as a weighted average of the duration of all of the components? Using the (PV of the component)/(PV of the portfolio) as the weight?
#7
11-24-2010, 01:16 PM
 bigb Member Join Date: Jul 2010 Posts: 400

Quote:
 Originally Posted by dlt Can't MacD(portfolio) be computed as a weighted average of the duration of all of the components? Using the (PV of the component)/(PV of the portfolio) as the weight?
It might be better to memorize it this way, as the weighted averages, but it really just depends on what information they give you in the problem. If they don't give you the individual portfolio's price and duration, then your going to have to calculate it and end up doing the same thing either way.

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