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Suppose I am invested in the same fund for the first 200 days in 2013, some combination of 150 days in 2014, and the last 100 days in 2015. Further suppose that geometrically linking the daily returns over every day invested gives a cumulative return of 10% over the total 450 days invested during this 3 year time frame.

What is the annualized return in this case?

A couple possibilities seem reasonable:

  1. Annualize only over days invested: $$(1+10\%)^{365/450} - 1 \approx 8.037\%$$

  2. Annualize over the entire time frame: $$(1+10\%)^{1/3} - 1 \approx 3.228\%$$

Is one of these very different answers the standard or is it entirely context dependent?

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4 Answers 4

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Your second suggestion can be interpreted as an actual return, while the first one would be a hypothetical return, given you could keep making the same gains you did for those days over the entire 3 year period.

Everything else depends on your assumptions. You can take (actual days)/365, often used is actual/360 or even actual/252 if you only consider the number of trading days.

You know what you had and what you ended up with. Your interpretation of how well you did, or how high the return was, is up to the definition.

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  • $\begingroup$ While I agree with your last sentence if he wants to do apple to apples comparisons he needs to fix a day count and report the years separate or, very roughly, use the second method. $\endgroup$
    – HerbN
    Sep 27, 2016 at 22:20
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Let's assume that you are manager of a fund and you have made the aforementioned investments and gains (and only those). When I ask you what was the annualized return of your fund during 2013-2015, the answer most certainly is 3.228% and not 8.037%.

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  • $\begingroup$ That's definitely true on the portfolio/fund level, but the capital from that one deal is moved to another investment when it's divested. Choosing that option doesn't really represent the performance of how the deal itself did while invested. I guess it depends what you are trying to show? $\endgroup$
    – Alexis Olson
    Sep 27, 2016 at 22:40
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The industry standard is to display annualized returns on a yearly basis because it is requested by most of the authorities (SEC...) and you shouldn't mix years. So you should present your returns as:

2013 : X %

2014 : X %

2015 : X %

Also you should use formula 2) because formula 1) is misleading :

If you tell someone that you have a 8% of yearly returns, and that this person lends you x dollars during 100 days and that after this period this investors come back to get its money, you can't tell him to come back later on because up to now you didn't invest its money...

Time is money and so you should use the entire time frame.

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  • $\begingroup$ I understand that, but I need to show returns on a trailing 3/5/10 year basis, which means annualization is necessary. $\endgroup$
    – Alexis Olson
    Sep 27, 2016 at 22:23
  • $\begingroup$ ok sorry, but still you should use formula 2. $\endgroup$
    – Malick
    Sep 27, 2016 at 22:26
  • $\begingroup$ Do you know of any reference that I could cite as a guideline or is this your personal opinion? $\endgroup$
    – Alexis Olson
    Sep 27, 2016 at 22:27
  • $\begingroup$ you can have a look item 13 here : sec.gov/about/forms/formn-1a.pdf $\endgroup$
    – Malick
    Sep 27, 2016 at 22:34
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    $\begingroup$ Item 13 - Instruction 3a : "calculated on the last business day before the first day of each period shown." So you should think in period (3/5/10) - ( usually by fiscal period but you can use a different period). It is detailled that it should be in period , not in time invested $\endgroup$
    – Malick
    Sep 27, 2016 at 22:40
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What happened to the cash in the intervening periods? Was it in a mattress somewhere or in the money market in some form? The problem to me appears less you have these gaps but you only want to count these return periods. That would make anyone you are dealing with extremely nervous.

Your capital exists in between these investments and the returns on them should be added to get continuous values for the entire timeframe. If that is, "I put it in a mattress" you need to own that and accept 0% return for those days.

As for industry standards I am used to seeing all rates specified as annualized although when applying them to non-year aligned periods the day count convention needs to be specified.

Thus, your second calculation is closest to correct.

You could declare a day count convention (probably actual/252) and report each year individually based on that convention by taking the return for held days and annualizing it out but that would only be acceptable, I think, if you report each year individually with the day count convention and number of days held.

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  • $\begingroup$ The cash would have been invested elsewhere. I'm not looking to calculate a total portfolio return, but how to show an annualized return for a single deal inside of a portfolio. $\endgroup$
    – Alexis Olson
    Sep 27, 2016 at 22:20
  • $\begingroup$ Why? That's a very odd measure to my mind. Calculating a specific holding while held is one thing but moving in and out of something and only wanting the value of that something annualized over three years when you were never in it continuously just sets off my spidey senses. Can you elaborate on how you intend to use this measure? What kinds of comparisons are you interested in? $\endgroup$
    – HerbN
    Sep 27, 2016 at 22:22
  • $\begingroup$ It's certainly not intended to be a measure used in reporting to clients. For some the internal reports I run, performance is broken out on the deal level and I was curious about the best approach for this odd situation (which only occurs in a few deals out of thousands). $\endgroup$
    – Alexis Olson
    Sep 27, 2016 at 22:58
  • $\begingroup$ 1. Internal reports have no industry standard although they may follow one. 2. I have never seen things broken out. I have see all kinds of reporting but I never seen something along the lines of "We held FOO for 178 days in 2013, 192 days in 2014, and 122 in 2015 and here is the performance of just those swaps over 3 years but limited to the days we held them". I'm not even sure of the value of such a report. A P&L for the period would work but that would sum up income, loses, and gains. Any return would be on that P&L. $\endgroup$
    – HerbN
    Sep 28, 2016 at 14:00

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