Doesn't work like that. Return rates on assets average about 6% or so. i.e., if you invest $100, you expect a $6 annual income. In other words, a $100b investment would expect a $6b annual income.
Take average PE ratios for example, about 15 (so 6.6% earnings as a percentage of market cap).
That would require about $6 per user per year in profit. Facebook is at $6 average revenue per user (ARPU) right now, but >$25 for US users for example, and growing.
Instagrams revenues are currently around $7b with a fraction of that profit, and at current growth rates, the valuation isn't crazy.
If the average user spends 1 hour per day, that's 365 hours per year at $5 or about 1 penny per hour of use. It's not hard to squeeze in a penny worth of profit from ads per hour of use. This would get you at $6b net income. Given that's in the cards for the future, the valuation isn't crazy.
The only flaw investors can make is that growth either isn't sustainable (I think it is for quite some time), or that Instagram users will eventually move on to other things (e.g. Myspace) such that the years of profit will be shortlived (more likely). Nobody invests $100 to gain $5 per year for a short period, that's based on perpetual income expectations which would require Instagram to stay around for quite a long time.
Nice back-of-the-envelope analysis. But just wanted to let you know, the ARPU numbers are per quarter, not per year! For Q1’2018, from the official earnings report, has worldwide ARPU at $5.53 for that quarter. In US, for Q1, it was $23.59 ARPU.
