Rate of return methods -- XIRR, Dietz, Modified Dietz, BAI Iteration
Jay Petrulis
time-weighted rate of return for a total portfolio -- to be used to
validate the "black box" solution from my company's performance
system. While the program was running, I came across this old stream
of messages, and I thought I might share some insight into the
calculations.....
Both the Dietz and Modified Dietz methods are "back of the envelope"
calculations that can give a relatively quick and reasonably accurate
answer for the rate of return.
The conventions used in the formulas:
Vo = Beginning Market Value
Ve = Ending Market Value
C = Flow, can be + (contribution) or - (withdrawal)
n = number of days in the period
d = day in which the flow occurred
r = rate of return for the period
Modified Dietz r =
(Ve-Vo-SUM(C))
-------------------- = (End-Begin-Net flows)/(Begin+Weighted flows)
(Vo+SUM(weighted C))
Dietz r =
(Ve-SUM(weighted C))
-------------------- = (End-Weighted flows)/(Begin+Weighted flows)
(Vo+SUM(weighted C))
Of the two, the modified dietz give a far "truer" picture of the
"real" rate of return.
The XIRR function has been amply described elsewhere, so I won't
bother with that until the end.
Investment firms are now almost fully taking direction from the AIMR
guidelines for performance and are using a more accurate measure
developed by the Bank of America Institute (BAI). It is an iterative
formula called the BAI Iteration.
BAI Iteration
Vo(1+r) + Sum[(C)*(1+r)^((n-d)/n)] = Ve
This cannot be solved exactly, so you must use a iterative algorithm
solution:
r = Ve/Vo - [Sum(C)*(1+r)^((n-d)/n)]/Vo - 1
and solve until it converges to a single point.
I believe that if the Modified Dietz result is used as the initial
guess, it converges quickly. I leave it to others to describe the
mathematical concepts when it starts to diverge. It was a pain to
figure out in Excel how to "kick" the rate back into a converging
sequence.
This is the recommended formula to generate the rate of return. Large
flows (>10% of Vo is current threshold) distort the return, so the
recommendation is to revalue the portfolio in the interim period and
link the two (or more) periods.
The differences between the IRR (XIRR) and the BAI Iteration have to
do with the treatment of flows (contribution/withdrawals). The BAI
method seeks to eliminate their impact by explicily factoring them out
of the rate of return, to give the portfolio a fair picture of how the
investment(s) did for the period. The IRR is the opposite, where the
flows are vitally important.
The idea has to do with who has the discretion to move the money. The
typical money manager doesn't have discretion of the timing and size
of the cash flows -- the client does -- so there has to be a way to
measure performance taking that into account.
Where the manager has discretion (venture capital, real estate, etc.)
then the IRR is often used.
Please note that really simple explanation -- the academics can better
describe the appropriateness of each method.
One last thing. The BAI method is a very good approximation of the
'real world.' Ideally, every time a flow occurs, the portfolio is
revalued at the time prior to the flow, the flow is incorporated, and
the period is linked to the next one geometrically. This isn't
practical for the most part, so the BAI formula is used.
Thanks,
Jay
original message stream cc'd into new post
_____________________________________________________
The usual method for computing rate of return on a portfolio is the
algorithm used by Excel's XIRR function, at least that's what I
thought.
I recently noticed a discrepancy between the XIRR results and the
numbers
posted at www.janus.com.
I called Janus and was told they are using a modified Dietz method
which
is:
(EndingBalance - StartingBalance - Additions +
Withdrawals)/(BeginningBalance + A1 - W1)
where A1 is the time weighted additions and W1 is the time weighted
withdrawals.
They admitted the modified Dietz was not as accurate as XIRR.
Where can I find info on the details of the Deitz method?
Is there anything online that discusses such algorithms?
Is there a more appropriate newsgroup for this question?
In another newsgroup, I received the following pointer:
"Financial Planning Solutions Network provide a comparison between IRR
and the modified Dietz method at their site:
http://www.fpsn.com/irr/irrmn.htm"
Anything other than the usual XIRR seems rather misleading.
--
Please post your response to the newsgroup.
http://www.standards.com/ipusers/standards; Word macros, including
converting from WordBasic to VBA; Technical writing and reviewing;
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------------------------------------------------
Message 2 in thread
From: Ken Mintz (mi...@cup.hp.com)
Subject: Re: XIRR vis a vis Dietz vis a vis ...
Newsgroups: sci.stat.math
Date: 2000-12-24 13:08:47 PST
Howard Kaikow (kai...@standards.com) wrote:
> I called Janus and was told they are using a modified Dietz method which
> is:
> (EndingBalance - StartingBalance - Additions +
> Withdrawals)/(BeginningBalance + A1 - W1)
> where A1 is the time weighted additions and W1 is the time weighted
> withdrawals.
Do you have more specifics on the formula, specifically A1 and W1?
Since the modified Dietz is supposed to be easier to compute than
the IRR, I presume that these are not "weighted" in the same that
IRR cash flows are "weighted".
> Anything other than the usual XIRR seems rather misleading.
I agree with you, based on what little I've read about Dietz. (Just
the one or two web pages I pointed you to in misc.taxes.moderated.)
But could we have an objective discussion of the method, without
drawing conclusions? This is, after all, sci.stat.math.
First, the web pages allude to several Dietz formulas. As I recall,
they are called Dietz, modified Dietz, and modified Dietz with
weighted something-or-other.
What are the formulas for each of these?
-- Ken Mintz
Message 3 in thread
From: Howard Kaikow (kai...@standards.com)
Subject: Re: XIRR vis a vis Dietz vis a vis ...
Newsgroups: sci.stat.math
Date: 2000-12-24 16:50:58 PST
The formula I gave below is what I was able to get from Janus.
It was very difficult in getting past their front-line phone support
to talk
to somebody who even know what I was talking about.
The person I spoke to admitted that he not heard of Dietz until he got
to
Janus and that XIRR was more accurate.
The article I pointed to in Canada really worries me. If the
investment
companies in Canada adopt "Dietz", somebody in the USA, such as Janus,
could
get the same "bright" idea.
I would like to find a formal objective discussion of "Dietz" and
"modified
Dietz" that I can use as ammo to try to get the SEC to cause the right
thing
to happen.
Janus is just computing the returns overnight, they most certainly can
handle that even if they need a separate computer/drive storage to
handle
the input data. Computational convenience is a poor excuse for using
misleading numbers.
--
Please post your response to the newsgroup.
http://www.standards.com/ipusers/standards; Word macros, including
converting from WordBasic to VBA; Technical writing and reviewing;
Standards
------------------------------------------------
"Ken Mintz" <mi...@cup.hp.com> wrote in message
news:925o1a$njd$1...@web1.cup.hp.com...
> Howard Kaikow (kai...@standards.com) wrote:
> > I called Janus and was told they are using a modified Dietz method which
> > is:
> > (EndingBalance - StartingBalance - Additions +
> > Withdrawals)/(BeginningBalance + A1 - W1)
> > where A1 is the time weighted additions and W1 is the time weighted
> > withdrawals.
>
> Do you have more specifics on the formula, specifically A1 and W1?
>
> Since the modified Dietz is supposed to be easier to compute than
> the IRR, I presume that these are not "weighted" in the same that
> IRR cash flows are "weighted".
>
> > Anything other than the usual XIRR seems rather misleading.
>
> I agree with you, based on what little I've read about Dietz. (Just
> the one or two web pages I pointed you to in misc.taxes.moderated.)
>
> But could we have an objective discussion of the method, without
> drawing conclusions? This is, after all, sci.stat.math.
>
> First, the web pages allude to several Dietz formulas. As I recall,
> they are called Dietz, modified Dietz, and modified Dietz with
> weighted something-or-other.
>
> What are the formulas for each of these?
>
> -- Ken Mintz
Message 4 in thread
From: Howard Kaikow (kai...@standards.com)
Subject: Re: XIRR vis a vis Dietz vis a vis ...
Newsgroups: sci.stat.math
Date: 2000-12-24 16:53:34 PST
Oooops, I forgot to mention that Janus told me that the weights are
the
number of days.
--
Please post your response to the newsgroup.
http://www.standards.com/ipusers/standards; Word macros, including
converting from WordBasic to VBA; Technical writing and reviewing;
Standards
------------------------------------------------
"Howard Kaikow" <kai...@standards.com> wrote in message
news:9265le$9ol$1...@pyrite.mv.net...
> The formula I gave below is what I was able to get from Janus.
>
> It was very difficult in getting past their front-line phone support to talk
> to somebody who even know what I was talking about.
>
> The person I spoke to admitted that he not heard of Dietz until he got to
> Janus and that XIRR was more accurate.
>
> The article I pointed to in Canada really worries me. If the investment
> companies in Canada adopt "Dietz", somebody in the USA, such as Janus, could
> get the same "bright" idea.
>
> I would like to find a formal objective discussion of "Dietz" and "modified
> Dietz" that I can use as ammo to try to get the SEC to cause the right thing
> to happen.
>
> Janus is just computing the returns overnight, they most certainly can
> handle that even if they need a separate computer/drive storage to handle
> the input data. Computational convenience is a poor excuse for using
> misleading numbers.
>
>
>
> --
> Please post your response to the newsgroup.
>
> http://www.standards.com/ipusers/standards; Word macros, including
> converting from WordBasic to VBA; Technical writing and reviewing; Standards
> ------------------------------------------------
> "Ken Mintz" <mi...@cup.hp.com> wrote in message