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

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June 9

Annualized rate of return

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)

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

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

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

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

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

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

We can solve this for $r$:

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

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

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

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)