Annualizing Returns Across Investment Gaps

# What is the industry standard for annualizing returns over non-contiguous time periods?


# What is the industry standard for annualizing returns over non-contiguous time periods?












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:

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

- 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?

## Answer by Andreas (score 2, accepted)

https://quant.stackexchange.com/a/30277

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.

## Answer by Juha Lipponen (score 1)

https://quant.stackexchange.com/a/30332

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%.

## Answer by Malick (score 1)

https://quant.stackexchange.com/a/30333

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.

## Answer by HerbN (score 1)

https://quant.stackexchange.com/a/30334

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.