Testwiki: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)