Rolling Returns Calculation

An end of reading question asked to calculate the average 9 month rolling return given a set of 12 one-month returns.

My approach (which was wrong) was to compound the monthly returns (geometric sum) for 9 months using monthly data 1-9, then again using 2-10, 3-11, 4-12. I then averaged the 4, 9-month returns.

The “correct answer” was to take an arithmetic average of 9 monthly returns, using months 1-9, 2-10, 3-11, and 4-12. The monthly averages using the 4 9-month sets were then averaged.

My questions:

  1. Would we always use monthly data in this calculation. In other words, the specification of “9-month rolling returns” means we used 9 individual months to get an average 1-month return of the past 9 months. So, are we always calculating month average returns over the specified period?

  2. If we are not always using 1 month as the period average, what indication would we have as to the time period we are using in the average? In other words, I would expect a 9 month rolling return to be the average return over a 9 month holding period of time. But the text has averaged the 1-month return over a 9 month period - which gives the average return over a 1 month holding period of time. If we did the same calculation using daily data, the 9 month rolling return would be very low and one would need to know the holding period used in calculating that average.

yes you are trying to figure out the “average return” earned over the 9 month period.

a 9 month rolling return is what it is - you calculate the average monthly return over a 9 month period.

if you were provided daily returns - you would use that to first calculate the monthly HPR - and then further do the same average over the 9 month period.

your quote" But the text has averaged the 1-month return over a 9 month period - which gives the average return over a 1 month holding period of time." is incorrect. Averaging the 1 month return over a 9 month period - gives you the average 1 month return over the 9 month holding period.

And “average return” = average monthly return? Always and without exception? So if, in real life, I see a 10-year rolling return for a security it is the monthly returns, averaged over 10 years? Because there is plenty of data that contradicts that - e.g.

It seems so arbitrary that rolling returns are ALWAYS monthly returns. But this is consistent with the text. It is inconsistent with how I’ve seen it used in the industry but I guess I’ll memorize it for the test.

Sorry if my statement was ambiguous and thanks for rephrasing it. What I meant to say was that if I am providing statistics on an investment and I give the 3-month, 6-month, 9-month, 12-month, 2-year, 5-year, and 10-year rolling returns for that investment all these measures are based on a 1-month return?

3 month return would be rolling over 3 months at a time 1-3, 2-4, 3-5, 4-6 months etc.

6 month => 1-6, 2-7, 3-8 months etc.

9 month = 1-9, 2-10, 3-11 etc. months

12 month 1-12, 2-13, 3-14 etc/

2 year return => 0-2 years, 1-3 years, 2-4

you get the drift?

DJS: My understanding is that you’re asking why you’re calculating the monthly arithmetic average for these rolling periods instead of the geometrically linked cumulative returns.

I would expect these returns to be calculated as the cumulative return as well (annualized for periods over a year). This is what I’ve seen to be the common interpretation of ‘return’ with no other qualifier, with common rollback periods being monthly or quarterly.

Just to be clear, obviously we’d be taking the arithmetic average of these rolling returns to get the final ‘average rolling return’.

But when you “roll over” the first 3 months, are you computing the total return over those 3 months? No. You are taking the return in month 1, month 2, and month3 and computing an arithmetic mean. This is SO inconsistent with the way I’ve seen rolling returns applied in real life.

YES. This. I have always thought that the X-month rolling return was the return for an X-month holding period rolling forward.

So, let’s say we are calculating average 3 month rolling returns for 1 calendar year…

What I would do:

  1. Apr 1 is the first time we have 3 months of returns. Calculate the total return from Jan 1 to Apr 1.

  2. Next, move forward in time - it doesn’t need to be in any specific increment. For example, on Apr 2, the 3 mo. return is calculated Jan 2 - Apr 2.

  3. For the entire year, calculate the 3 month returns by rolling forward through the year in some specified increment (days, months, whatever)

  4. Average the set of 3 month returns that have been generated in step 3 - and presto, average 3 month rolling return.

Rather, the way it is defined in the CBOK, is that the magic holding period is 1 month and if we are evaluating a 3 month rolling return we do the following:

  1. For each month, calculate the return for that month.

  2. Calculate the arithmetic average for months 1-3, 2-4, 3-5, … 9-12

  3. Finally, we take the arithmetic average of the set of 9 arithmetic averages each for 3 1-month periods.

Does anyone other than me and Katalepsis think the CBOK’s method for rolling returns is super bizarre and in conflict with how the value is defined everywhere else? I was really hoping someone would chime in and say, ‘o no that is CBOK errata.’

for one, they did not give you daily returns.

they gave you a monthly return. so jan, feb, mar, apr was provided.

for a 2 month rolling return - jan-feb, feb-mar, mar-apr would be the means you calculate.

it is the same thing. you are rolling over a month at a time

idea is to get a picture of how consistent the returns have been across the “rolling return period”.

I understand what you are saying - it just doesn’t seem logical to me.

If I say to someone, this fund’s average 6-month rolling return over the last 10 years has been 5% I would expect that to mean that the return for a 6-month holding period, rolled over a 10 year period, averages to 5%.

The way it is defined in the CBOK, it would mean the average monthly return averaged over a 6-month period and then the average of the averages for 10 years. How does THIS provide more insight than simply averaging all the monthly returns over 10 years?

It sounds like you are suggesting regardless of the data provided, that 1-month returns are always the value that is used in the arithmetic average for rolling return.

you are attributing a lot of stuff to me, which I really did not say, since you are taking stuff from 2 different posts to put them together.

your first question was - they asked you for monthly rolling return but provided you with daily return data – in that case - you first need to go from daily returns to monthly returns - and then roll the monthly returns to arrive at a monthly rolling return.

you would not go and perform days take returns 1-30, 2-31 etc. to get a monthly rolling return, in my mind.

Sorry cpk - I didn’t mean to put words in your mouth. I genuinely appreciate your assistance on this.

FYI I have emailed CFA institute’s cirriculum department on this issue. I believe it has been introduced in an ambiguous way, to say the least. Rather than comment further here, I will wait until they respond or post errata and then follow up on this thread.

Hi - did you get an answer from CFAI? I agree with you, the question is ambiguous. 9-month average rolling return does not necessarily mean return per month, it can be for the whole 9-month period. The 4 rolling returns (sums) were 6.8, 7, 3.4 and 2.2, therefore, average 9-month rolling return would be 4.85% But the EOC took the extra step of dividing that by 9 for each rolling return.

Does anyone have a clear way of describing the answer, I for one tried to do it geometrically and got it wrong.

anyone :slight_smile:

Say you are presented with returns for Jan-Dec of 1 year

and asked to calc a 9 month rolling return

R1 = sum(Jan-Sep)/9

R2 = sum(Feb-Oct)/9

R3 = sum(Mar-Nov)/9

R4 = sum(Apr-Dec)/9

and finally average 9 month rolling return = (R1+R2+r3+r4)/4