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

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