Talk:Time-weighted return

Rewrite

I have substantially re-written this page, in order to clarify key points throughout.Jonathan G. G. Lewis 15:29, 29 March 2012 (UTC)

I have just corrected the sign in the formulae for a flow at the beginning of a period from a minus C for cash flow to a plus.

Jonathan G. G. Lewis 09:33, 27 November 2013 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

Confusion in Discussion, Paragraph 6

In the Discussion, paragraph (6) is confusing. The beginning and end of it refer to the approximate condition where cash flows occur within a period, not just at the end. -- Yet, the equations are exactly the same as in the "true" case at the top of the article.

I think the first paragraph should be followed by the last paragraph of that section, with the intervening material removed.

E. Bruskin June 4, 2012

I agree. The formulae in the section in question were left behind from an earlier version of the page. I will remove them.Jonathan G. G. Lewis 16:53, 1 April 2013 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

In the absence of flows

Ebruskin removed "In the absence of flows ... (unless the returns are continuous $ln({\frac {M_{2}}{M_{1}}})$)." because (a) it's wrong (e.g. simple interest is different from compound interest), and (b) the parenthetical doesn't make any sense.

It would have been nice if this change had been proposed first on this discussion page, but I am just as guilty of going ahead with making changes without any discussion first.

The point I was trying to make (but failing, apparently) was that the only difference between the methods (time-weighted, internal rate of return, modified Dietz, simple Dietz, linked internal rate of return, etc.) is the handling of flows, i.e. if there are no flows, then there is no difference between the results of the various methods.

Does anyone have any counter-evidence?

This point has nothing to do with simple or compound interest. Have I confused anyone by explaining the link between IRR and the modified Dietz method in terms of simple and compound interest? I hope not.

The subsidiary point I was attempting to make was to draw attention to the fact that all these methods give 'raw' returns, as opposed to continuous or logarithmic returns, i.e. those which take the form $ln({\frac {M_{2}}{M_{1}}})$. I'm sorry that Ebruskin thinks this makes no sense - clearly I am to blame for not making the point clearly enough.

I propose replacing these points in the article thus:

If there are no external flows, then all these methods give identical results. They are not to be confused with continuous or logarithmic returns.

Jonathan G. G. Lewis 10:21, 27 November 2013 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

Duration of Sub-Periods

This article does not directly mention anything about the duration of the sub-periods, indeed it could be implied from the text that the sub-periods could be of any duration.

All formulae for time-weighted returns that I have seen assume that the sub-periods are of equal duration. I think this should be clearly stated in this article.

Further, the text as it stands ("Suppose that the portfolio is valued immediately after each external flow") implies that the sub-periods can be taken respective to external flows. It would be fair to assume that external flows are not regularly spaced, and thus assume that the sub-periods do not need to be of the same duration.

Mgwalker (talk) 08:49, 8 September 2015 (UTC)

In response to the first two paragraph immediately above, the implication that the sub-periods could be of any duration is correct. The duration of the sub-periods makes no difference. There is no assumption in the time-weighted method that the sub-periods are of equal duration. The factors in the formulae are growth factors related to holding-period returns, where the holding periods are the sub-periods. These holding period returns over the sub-periods compound together to give the holding period return over the overall period. Deriving a rate of return, such as annual return, is a separate step, as mentioned towards the end of the article.

The point in the third paragraph is correct. External flows are not in general regularly spaced, and the sub-periods do not need to be of the same duration.

Jonathan G. G. Lewis 05:04, 19 October 2015 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

I have just added the following clarification to the article:

"In general, these sub-periods will be of unequal lengths."

Jonathan G. G. Lewis 06:14, 19 October 2015 (UTC)

Oct 2016 Changes Are Incorrect

Forgive me: This "new section" was appended to an existing section. Newbie mistake. I will see later if there is a way for me to correct my mistake. Later: I hope I figured it out. — Preceding unsigned comment added by JoeU2004 (talk • contribs) 23:45, 17 November 2016 (UTC)

Much of the changes apparently made in Oct 2016 are completely incorrect.

(Note: For simplicity for me, I will use Excel notation.)

Let's start with the subsection "General Formula for Ordinary Time-Weighted Return".

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1. "If an annualized rate of return r1 applies over a period of t1 years" should read "if the rate of return r1 applies for t1 periods". In other words, (1+r1)*(1+r1)*...*(1+r1) for t1 times can be written (1+r1)^t1.

See the correct "Example 1" under "Why It's Called Time Weighted": (1+0.10)*(1+0.10)*(1-0.03)*(1-0.03)*(1-0.03) - 1 = (1+0.01)^2 * (1-0.03)^3 - 1. There, r1 = 1+0.01, t1 = 2, r2 = 1-0.03, and t2 = 3.

Note the subtle difference between "for t1 periods" and "over t1 periods". r1 is a periodic rate of return, not necessarily an annual rate, much less an annualized rate. And t1 is simply a number of periods, not necessarily a number of years.

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2. The statement "the powers t1/T, t2/T [...] can be thought of weights" is misleading. Although that generalized form is correct mathematically, they are not "weights" at all, in a sense similar to "weighted average". Instead, it is the commutative presentation of the geometric mean, to wit:

( (1+r1)^t1 * (1+r2)^t1 * ... * (1+rk)^tk ) ^ n - 1, for n years covered by t1+t2+...+tk periods.

There really is no relationship between n years (T in the article) as the factors t1, t2,..., tk.

In fact, I think this entire subsection can be simplified to something like the following:

For the rates of return r1, r2,..., rk for each of k periods that cover n years, the annualized time-weighted return is the geometric mean of the product of 1+r1, 1+r2,..., (1+rk), to wit:

( (1+r1)*(1+r2)*...*(1+rk) ) ^ (1/n) - 1

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3. Example 2 is completely incorrect.

a.First, it says that T and ti "are not necessarily whole years". I think "are not necessarily in years" is meant; if not, it should be.

b. Second, on the one hand, it says the example is for "the overall period T is [...] 674 days". Then it says the calculation is "over the whole 309 day period". Note: 674 v. 309. Arguably, a simple typo.

c. Third and most importantly, the calculations are completely wrong. The notation 1.1025^(337/674) = (1.1025^(1/674))^337. The first part, 1.1025^(1/674), might represent the daily rate that compounds to 1.1025 over 674 days. That has nothing to do with a rate of over 337 days, even if we compounded that 337 times (...^337).

The fact is, if the rate of return is 10.25% for 337 days and -19% for another 337 days, the time-weighted return over 674 days is simply:

(1+10.25%)*(1-19%) - 1

That results in about -10.70% (-10.6975%). But does not result in -10.70% "p.a", much less "-5.5% p.a" (correct calculation), where "p.a" typically means "per annum". Instead, it is simply -10.70% over 674 days.

We can annualize that using:

( (1+10.25%)*(1-19%) )^(365/674) - 1

assuming 365 days per year. I suspect that is the calculation that was intended by the incorrect use of exponential powers like 337/674.

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4. The subsection "Continuous Time-Weighted Rate of Return" is incorrect, misleading and, arguably, not very useful.

First, there is nothing more "continuous" in nature about it than the formulas presented in the previous subsection. It is just the log of (1+r1)^t1 * (1+r2)^t2 *...* (1+rk)^tk

Second, in this context, r1 must be the log return, not the same rate of return r1 used in the previous subsection. In other words, "log r1" = log(1 + r1).

Third, it perpetuates the misunderstandings noted above about the terminology "time-weighted". That is, ti/T is not a "weight" at all.

Finally, all of these mistakes are compounded (pun intended) in Example 3.

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I hope this helpful. It should lead to a revocation of most, if not all, of the Oct 2016 changes. If you wish, I can offer to write a prototype of corrected text. But really, I think the wiki article should be restored to the way it was "long ago". If I recall correctly, the wiki page was correct when I first looked at it circa 2014.

For references, simple do a browser search for "time-weighted return". You might narrow your search to schwab.com. But many other sites echo the same details. — Preceding unsigned comment added by JoeU2004 (talk • contribs) 23:43, 17 November 2016 (UTC)

Apologies to my fellow editor, if the edits he or she refers to are unclear. I myself find the comments above unclear, unfortunately. Jonathan G. G. Lewis 06:16, 9 September 2017 (UTC)

Can anyone explain these criticisms above? Jonathan G. G. Lewis 06:16, 9 September 2017 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

I think the steps in this section may have been too big for some readers, so I have just added points about the relationships between continuous rate of return, period length and continuous holding-period return. I hope these additional steps help the reader to recognize more easily the general formula for the time-weighted average continuous rate of return. Jonathan G. G. Lewis 08:28, 16 September 2017 (UTC) — Preceding unsigned comment added by Jonazo (talk • contribs)

I just removed the following content on why it is called time-weighted:

General formula for continuous time-weighted average rate of return

Note that:

1. The continuous holding-period return over a period of $t$ years, where the continuous annual rate of return is a constant $r$, is $rt$,

and conversely that:

2. The continuous annualized rate of return over a period $T$ years, where the continuous overall holding-period return is $R$, is ${\frac {R}{T}}$.

The general formula for the continuous rate of return is:

${\frac {1}{T}}\sum _{i=1}^{n}{r_{i}t_{i}}=\sum _{i=1}^{n}{r_{i}\times {\frac {t_{i}}{T}}}=r_{1}\times {\frac {t_{1}}{T}}+r_{2}\times {\frac {t_{2}}{T}}+r_{3}\times {\frac {t_{3}}{T}}+...+r_{n}\times {\frac {t_{n}}{T}}$

The continuous time-weighted rate of return is the weighted average of the sub-period continuous rates of return. The weight ${\frac {t_{i}}{T}}$ assigned to the continuous rate of return $r_{i}$ in sub-period $i$ is the duration of the respective sub-periods, as a proportion of the overall period $T$.

General formula for ordinary time-weighted return

More generally, if an annualized rate of return $r_{1}$ applies over a period of $t_{1}$ measured in years, $r_{2}$ over another period of $t_{2}$ years, etc. then the time-weighted return over the overall period of $T=t_{1}+t_{2}+t_{3}+...+t_{n}$ years is:

$(1+r_{1})^{t_{1}}(1+r_{2})^{t_{2}}(1+r_{3})^{t_{3}}...(1+r_{n})^{t_{n}}-1$

and the annualized time-weighted rate of return is:

$(1+r_{1})^{\frac {t_{1}}{T}}(1+r_{2})^{\frac {t_{2}}{T}}(1+r_{3})^{\frac {t_{3}}{T}}...(1+r_{n})^{\frac {t_{n}}{T}}-1$

The powers ${\frac {t_{1}}{T}},{\frac {t_{2}}{T}},{\frac {t_{3}}{T}}...{\frac {t_{n}}{T}}$ can be thought of as weights.

Example 4

The overall period $T$ and sub-periods $t_{i}$ where $i=1,2,3,...n$ are not necessarily whole years.

For example, suppose the overall period $T$ is the 674 days from the year-end of 2014, to the end of the day on 4 November 2016. Over the sub-period of $t_{1}=$ 337 days between the end of 2014 and the end of the day on 3 December 2015, the rate of return on a portfolio was 10.25% p.a., and over the remaining $t_{2}=$ 337 days of the period, it fell to -19% p.a.

The rate of return over the whole 674 day period was:

$1.1025^{337/674}\times 0.81^{(674-337)/674}-1$

$=1.05\times 0.9-1$

$=0.945-1$

$=-5.5\%{\text{ p.a.}}$

I wrote this a while back, but I think it is overly mathematical, and I don't think it is effective to get the basic point across to the average reader. I think what is left now does the job.

--Jonathan G. G. Lewis 08:05, 1 April 2018 (UTC)

Duration of Sub-Periods (redux)

Ostensibly, change the last sentence of the first paragraph from

``The rate of return over each different sub-period is weighted according to the duration of the sub-period.``

to

``To calculate the average rate of return per unit time, the rate of return for each sub-period is geometrically weighted by the duration of the sub-period.``

But in my opinion, that is too much detail so early in the article. It is a mathematical nitpick. It would be better to relegate such detail to the section titled ``Why is it called "time-weighted"``.

The important take-away is in the second sentence of the first paragraph. I think the first paragraph can be clarified as follows:

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The time-weighted return (TWR)^[1][2] is a method of calculating investment return. The TWR is calculated by the product of the rates of return (plus 1) for each sub-period, regardless of their duration. The result is the cumulative TWR for the entire time covered by the sum of the sub-periods.

The average TWR per unit of time can be calculated by compounding the cumulative TWR by an appropriate factor. For example, if the cumulative TWR ("cumlTWR") covers a period of "t" months, the monthly TWR is (1 + cumlTWR)^{(1 / t)} - 1. The annual TWR is (1 + cumlTWR)^{(12 / t)} - 1.

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--JoeU2004 (talk) 06:49, 3 April 2022 (UTC)