Cash for a down payment on the apartment would be a good thing. That would lead to keeping the funds liquid. If you could get a line of credit on your paid off house and the rentals, I would definitely consider that. That would give you a decent 4% return and you would have the liquidity available to make an offer when an opportunity came by.
I assume you are hoping that prices/values of apartments go down since you state you are looking to get a cap rate of 10. If you invest in the stock market, you are hoping that stocks return more than the interest rates of the mortgages you could have paid off. It seems that if you invested in stocks you would be hoping stocks continue to rise at a steady pace but that, at the same time, hoping real estate takes a considerable drop. Not that it can't or won't happen, but I don't see that combination happening together as a high probability.