11.6% a year for 10 years compounds out to 3x. If it was 10 years to 4x, that would be 15%. Which would be a very respectable risk premium over BH,
the greatest publicly-traded large hedge fund/conglomerate of all time.
But, the "percent" returns, as opposed to cash-on-cash, aren't actually calculated that way. Most funds eventually call something close-ish to 100% of capital (1.0x paid-in), so you can meaningfully-ish talk about the "times money" on the entire fund. But it doesn't work for "percent" returns, which are necessarily annualized.
Funds call capital over time, not all at once. They also pay back distributions as they become liquid, not all at the end. Investors measure this with IRR, taking into account the timing and magnitudes of the cash flows. So a fund that posts up a 4x "times money" within ~ 10 years is going to have a substantially better IRR than 15%.
WRT LP expectations: institutional investors aren't just weighing VC against "safer investments." They look at long-term historical data about different asset classes, project out their best opinions about likely risk and return AND correlations among classes, and make a top-down decision about where to allocate. Then, within each bucket, they try to select a group of managers who can meet (or ideally safely beat) those assumptions. (If rather small, they may delegate this to a fund-of-funds; if rather large, they may further differentiate into various stages or strategies of venture, and the process is more or less fractal.)
Partly we may be talking past each other because "expect" here could mean various things. One thing that's clear is no major LP / institution is going to be invested only in one VC firm or only one vintage year. They think about these things in aggregate and over time.
If by "expect" you mean, "in aggregate and over time, some LPs project 20% IRR for the venture asset class (subject to a very high, like say 25%+ std dev)" I think that's very fair and you'd be right.
But if by "expect" you mean "LPs look at each venture investment and rely on it to consistently return 20% IRR, or that will have been deemed a failure" then you're not giving them enough credit.
#1 driver of venture performance: fund vintage year.
#2 driver of venture performance: manager persistence (manager skill/alpha).
But to bubble back up to the higher level: LPs who see 10x, 9-figure exits in 2 years coming out of the portfolio of a ~$400M fund are not going to say "why not $3B?" They are going to understand that it's a rare fund in a rare vintage year that is going to return 7x+ on the whole fund, and they are going to be exceedingly happy if the "off years" consistently return 3x.