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 ${\displaystyle r}$(a fraction) stand for the annualized rate. If ${\displaystyle V_{y}}$stands for the value in year ${\displaystyle y}$, we have:

${\displaystyle r={\frac {V_{y+1}-V_{y}}{V_{y}}}.}$

We can "solve" this equation for ${\displaystyle V_{y+1}}$:

${\displaystyle V_{y+1}=(1+r)V_{y}}$.

Then ${\displaystyle V_{y+2}=(1+r)V_{y+1}=(1+r)^{2}V_{y},}$and so on. In general:

${\displaystyle V_{y+n}=(1+r)^{n}V_{y}}$.

We can solve this for ${\displaystyle r}$:

${\displaystyle r=\left({\frac {V_{y+n}}{V_{y}}}\right)^{\frac {1}{n}}-1.}$

In this specific case, ${\displaystyle n=3}$, ${\displaystyle V_{y}=100}$, ${\displaystyle V_{y+3}=130.}$So we get:

${\displaystyle r=\left({\frac {130}{100}}\right)^{\frac {1}{3}}-1=1.091392883...-1=0.091392883...\,.}$

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]