The fundamental financial question is supposed to be "do I need this money soon?" It does not apply in your situation, because you don't need exactly $370K. The admonishment to keep the money in cash applies to people who for whatever reason need the principle value to be stable. For example, if you invested all of the money in SPY, the fear is that the market would crash and you'd have to sell a great deal of SPY for not very much money. However, since you don't need exactly $370K you can allocate the money you don't need to any sort of investment you think is prudent. Ideally you'd want to put the needed money in an asset that was 100% correlated with the value of the home you will end up buying. If you expect the price of this home to be relatively stable (ie. no moves greater than 10%) cash is good enough, and it is why people suggest keeping your money in it. The correct way to treat the problem is by asking yourself this question: "If this transaction had already occurred, what should the allocation of my money be." After the house is bought you will be long 1 to 1.3 million in real estate and have a mortgage that is a significant fraction of a million dollars. You can treat the mortgage as being short bonds, with the benefit that the principal value can never go up. So the real question is would you be willing to short bonds in order to go long on some other asset class?
Without knowing exactly what your financial situation is I would suspect that the best option is to put an estimated 20% down payment in cash in a savings account, CD or something with a similar level of safety. Take the rest of the money and invest it according to your long term asset allocation strategy assuming your expected return on this strategy exceeds 4% and assuming that if things go to pot in the stock market/bond market etc, you'll be able to leave the money in there. I realize this sounds like market timing, but the principle at work is not that you're trying to avoid "selling low" but rather that you realize that the assets you purchase have intrinsic value and you don't want to be in a position to sell your assets at a discount to that value because you need the money. You should have a plan that doesn't involve raiding these assets if for example you lose your job and still have to make this new mortgage payment. If you can support your fixed payments in some fashion even if you and your spouse are fired, then you can afford to be much riskier with the excess cash. This all holds even if you plan on using that excess money to pay off the mortgage debt if you can make two assumptions: first you're willing to hold your assets if the best price you can get from them is below your estimate of their intrinsic value, and second you believe the expected return on these assets is greater than the expected cost of the mortgage debt.