The problem when you invest your emergency fund in a stock index is that the time when you are most likely to lose your job is also the time when your stocks are most likely to dive. ie, during difficult economic situations (recessions). You don't want to need cash when everybody else is doing the same.
Yeah, but even in major crash that starts a recession what's the worst that'll happen? Maybe a 50% loss if you bought it all near the peak and had to sell it all in the month or so at the very bottom? What are the chances you both buy stock such that it's at it's peak, that the stock crashes 50%, that you lose your job AND that you need all of that money at once? An enormous 50% crash happens maybe once every decade or two. If you lose your job, you just need money for rent/mortage/utilities every month and get a 30-day interest free loan for your groceries (and possibly rent and utilities if you pay a small fee)- you will not need $15000 immediately. It's an expected positive return, so especially if you intend on keeping that much in the market/savings account for more than a few years, the odds that you'll actually encounter a worse off-situation than a savings account are slim, and the odds that you'll actual end up experiencing something catastrophic because you chose an index fund are near nil.
In the meantime, if you don't have a massive emergency, in 9 years, you'll be expected (within bounds) to have nearly doubled your money. So keeping $15000 in a savings account for 9 years will expect to cost you $15000. That's a lot!
I guess everyone has their own comfort level here, but considering the massive opportunity costs here and size of the account I would just double check how much you're willing to pay for this insurance. Would you really pay $1000 a year to an insurance company for coverage of your utilities, groceries and rent/mortgage in the event of a job loss?