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I'm trying to boil this scenario down to an equation:
A token is worth $1.50. There is a total of 100 tokens invested. Last week, this investment earned $2 in interest.
Given this, how would I calculate the daily/weekly/yearly average APR for holding the token?
APR = I/(PY), where I is the interest, P is the principal, and Y is the number of years. Your interest is $2, and your total principal is 100*$1.50, and the portion of the year is 7/365 (since a week is less than a year, Y is fractional). So you take $2/(100 * $1.50*7/365). We can move the 365 to the top (we're dividing by something divided by 365, which is the same as multiplying by 365), and the dollars cancel out, giving 365*2/(150*7) = 0.69523809523, or 69.52%. If this were to compound over a year, then, using the approximation 7/365 ~= 1/52, we have 52 compoundings of an interest of 2/(100*1.50), so that's (1+0.01333333333)^52-1 = 0.99122851206, or an APY of about 100%. That is, your return corresponds to an APR of 69% compounded weekly, or 100% compounded yearly. If you want to know what APR compounded daily would correspond to this return, the interest rate per day is the total interest rate divided by 365 (because that's how many days are in a year), the multiplication factor is 1 more than that, and it's being applied 7 times, so you have that (1+i/365)^7 = 1.01333333333, so you can take the seventh root of both sides, subtract 1 from both sides, then multiply both sides by 365, which gives 0.6912977881, or 69.13%, which is only slightly lower than the 69.52% from before.
