# Get Annualized return from a time series of daily returns

I have a time series of data which has daily returns over 15-year time period. I understand that you get multi-period simple return using the following formula.

$$1 + R_{t}(k) = \prod_{j=0}^{k-1}(1 + R_{t-j})$$

I want to know how to get annualized return from this data.

Thanks.

• Hi: you need to make reinvestment assumptions when doing something like that. For instance, is the return made in the first year, re-invested during the second year or are the returns just left on the side and not re-invested. For that long of a time period, the difference matters. Oct 2 '19 at 2:12
• Hi. There are no reinvestments in my case. If possible could you please explain the solution in both the cases. Thanks. Oct 2 '19 at 9:06
• So, in that case, you can calculate the returns for each of the 15 years ( just link the daily returns so $( 1+ r_1 )(1+ r_2) \ldots, (1+ r_{365})$ ) and then take the arithmetic average of the 15 yearly returns. if you were re-investing, you'd need to link all of them and then take the 15th root of that and subtract 1.0. Oct 2 '19 at 12:56
• Thanks. That information answered my question. Oct 5 '19 at 5:36
• Note that my suggestion, linking the daily returns for a year, is still assuming that re-investment is done within the year. But that makes some sense because you have to assume that you hold the position for some length of time (which is equivalent to re-investing ). Oct 6 '19 at 13:12