Wikipedia:Reference desk/Archives/Mathematics/2022 June 9

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June 9

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Annualized rate of return

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Okay, simple question, but I can't figure out the magic formula for this. $100 is invested. After 3 years, with no further contributions being made, and interest having been accrued daily, the balance stands at $130. So what is the annualized rate of return? Working backwards I can conclude the answer is about 9.139% but it's not exact and I don't know what formula to use. Many Thanks! Uhooep (talk) 06:37, 9 June 2022 (UTC)[reply]

We work in fractions. Let   (a fraction) stand for the annualized rate. If   stands for the value in year  , we have:
 
We can "solve" this equation for  :
 .
Then   and so on. In general:
 .
We can solve this for  :
 
In this specific case,  ,  ,   So we get:
 
The decimal fraction for the cube root of 1.3 does not terminate, but for practical purposes 0.09139 (or, for that matter, 0.0914) is good enough.  --Lambiam 07:23, 9 June 2022 (UTC)[reply]