First, congrats. What an empowering position to be in. I was in a similar position in NYC several years ago.
To me, there are several questions that you need to ask yourself. 1) Do you see yourself in LA for a long time (say 5 years or more)? 2) Can you find a building in the LA area in a neighborhood that you want to live and that will at least allow you to lock in your housing expenses at or close to your current housing expenses? I would look for something that you can turn into a 3 unit or 4 unit but a duplex may work at least initially — a shell is totally OK (and may be preferred) if you have the budget to renovate (note that you may be able to find a construction loan with as little as 3.5% down although you’ll need to run the numbers to see whether it will work for you).
If the answer to the first question is no (or the answer to the first question is yes and the answer to the second question is no), then don’t invest in LA because you are likely able to get better cash flow elsewhere to offset your current housing costs. If the answer to the first question is yes and the second question is yes, I think you may be in a better spot by buying in LA.
I initially was not sure that I was going to stay in NYC so invested outside of NYC. When I determined that I would be in NYC for a long time, I bought a duplex in a neighborhood that I liked that locked in my housing costs (including taxes, insurance and maintenance) at roughly the cost of renting. I didn’t plan for it but rents have gone up as have prices, which has put me in a better cash flow and equity position. I’m now in the process of expanding the duplex into a four unit. When this is done, the cash flow off the building will more than cover my housing expenses.
One final observation - if you think it is going to take you time to watch the markets before making a decision, consider becoming a lender over the short run (either by investing in hard money loans or re-performing notes). You can get good returns over the short run to increase your “war chest” without sacrificing the same level of liquidity as if you bought real estate equity. If you’re curious about the debt side, read articles by David Van Horn.
PM me if you want further thoughts. Good luck.