# cumulative return calculation, disagreement

A friend of mine and myself are having an argument on how to correctly determine cumulative return.

The dataset has monthly return data and we are trying to determine the 6-month cumulative return.

I proposed the following: Cumulative return for 6 months is a product of monthly returns:

(1) Ri=(1+ri_1)∗ ... ∗(1+ri_6)−1

He mentioned just adding up the different return values:

(2) Ri=(ri_1)+ ... (ri_6)

Now I am pretty sure my way of calculating this is correct, although differences can be small.

Question Am I correct that when academic papers talk of cumulative return they calculate this with equation (1)

Thank you in advance. S Gontscharoff

To expand on my comment, consider the following R code:

set.seed(1)
returns <- runif(1000, 0.95,1.055) #Extremely simple return generation with a slight drift.
plot(cumprod(returns), type = "l")
lines(cumsum(returns-1)+1, col = "blue")


Which gives the following result: As you can see the effect is not linear, as the difference nearly disappears around index 300 and is greatly reduced at 900, but increases again as the compounding effect increases. So, you will most likely not see major differences in 6 month data, but as soon as you go to larger time scales the effect become noticeable.

• Thank you! Was thinking that as well, but this illustrates it very nicely. I think I will have to switch from stata to R soon, since it seems everyone is using R!. But thank you both @dm63 ,Forgottenscience – S. Gontscharoff Jun 2 '16 at 13:11

If the dataset contains arithmetic returns where 1+r(i)= S(i)/S(i-1) then you are correct. If the dataset contains logarithmically defined returns where r(i) = log (S(i)/S(i-1)) then your friend is correct.

• Thank you for your quick response. The data is indeed in the first form you indicated, it is the return data from CRSP database. One final thing; could using one method or the other cause results to vary widely? – S. Gontscharoff Jun 2 '16 at 12:40
• Over time the differences accumulate, as compounding becomes a force. – Forgottenscience Jun 2 '16 at 12:53
• Consider just (the first) two months. Then your first method gives $r_1+r_2+r_1r_2$. This shows that the difference is due to sceond order effects. These might indeed sum up to a larger difference over time. – Dr_Be Jun 3 '16 at 14:49
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