As others have already pointed out, there is no monetary sensible reason to borrow at 5% cost to invest at 1% return.
However, just because it doesn't make perfect sense financially doesn't mean it can't make sense for peace of mind. And you should not dismiss the peace of mind argument out of hand.
Ignoring tax effects, credit score effects, cost of higher levels of insurance required, etc., and assuming a five year repayment plan, borrowing $15,000 at 5% will cost you about $283/month for a total cost of $16,980. 1% interest on the same $15,000 would give you about $12/month. In other words, your "loan premium" is $21/month (interest expense about $33/month on the car loan, reduced by interest earned $12/month on the retained savings) plus the capital repayment amount. If you were to take the money out of savings you would probably want to replenish that over a similar time period (ignoring interest, saving $15,000 in five years means $250/month), so this boils down to the $21/month interest premium.
Now consider that the times when an emergency fund is most often needed are very often the times when banks will be reluctant to extend a loan (a job loss being a common example). While foreclosing on an existing loan can still happen, as long as you keep making payments, I suspect that most banks are far more willing to overlook the fact that you would not have qualified for the loan after the job loss.
If a loss of income situation develops after you pay the car with your savings without a loan, you start out with $15,000 in the bank plus whatever "car payments to yourself" you have been able to save afterwards. Depending on when things turn bad for you, this could mean that you having only half of the savings that you used to, but of course you also have no car payment expense (which is the same as you do now).
If a loss of income situation develops while you are still paying off the car, you start out with $30,000 in the bank instead of $15,000, but run the risk of having to make the car payments with money out of your savings. The net result of that is that your savings are potentially effectively reduced by whatever the remaining debt outstanding on the car is, which in turn is reduced over time. Even if you were not to actively save, your net financial situation becomes better over time.
If a loss of income situation develops after you have paid off the car, you now own the car free and clear and still have $30,000 in the bank. Assuming that you would repay yourself on a schedule similar to that of a car loan if you took the $15,000 out of the bank instead, this is a very similar situation.
Consequently, the important consideration becomes: Is it worth it to you to pay $21/month extra to have an extra $15,000 on hand if something happens to your financial situation?
I have been in pretty much exactly the same situation, albeit with smaller amounts, and determined that having the cash on hand was worth the small additional interest expense, not the least of which because I was able to secure a loan at a pretty good interest rate and with no early repayment penalties. You may reach a different conclusion, and that's okay. But do consider it.
