> If y = 10x where y = ARR and x = time in months, then after two months ARR = $20. After 12 months, ARR = $120 (assuming we start from $0 of ARR)<p>I don't think this makes sense, because the time is in months yet you're calculating a yearly value. Time should be in years, because otherwise you're calculating forecasted values.<p>> So at the end of a year, the business has grown from $0 to $120 in ARR. But what is the recognized revenue? The complex answer is that it's the integral of 10x from 0 to 12 months<p>(Recognized revenue = $720 in this example)<p>Again, this doesn't make sense due to the fact that your calculated ARR values are forecasted.<p>If your projected ARR has grown throughout the year, it's intuitive that your recognized revenue <i>must</i> be lower than your ARR.<p>It's not possible to recognize $720 in revenue when your ARR is only $120, it should be $60.<p>I think the issue here is using ARR, since ARR grows quadratically given a linear increase in MRR. This math would make sense if you used MRR instead, or modelled actual ARR instead of forecasted ARR.