Can you find a formula for the following situation?

What is more cost effective: Buy a car with cash or invest the cash and take a loan?

Loan: $50,000 at 5% APR for 5 years. 60 payments
Investment: $50,000 at 12% annual return
The idea is to account for the payments each month which reduce the investment as time increases. I am looking for a formula I could use to reproduce similar situations so please show your work. Thank You.

PS This is strictly curiousity so please do not add advice or advertisements, just a solution.

If the investment returns more than the loan repayments then that is the better way, as you are making money.
The loan will decrease more rapidly (as interest is building slower) than the investment.
Assumptions: There are no penalties for drawing on the investment account. The investment interest rate is always higher thann the loan.
Reality: Loans generally have a higher interest rate than guaranteed investments (like a high interest savings account). Higher interest means than there is higher risk in investment.

Formula: (assuming one year, one payment)
gain = (50,000 x 1.12) - (50,000 x 1.05)
gain = 56,000 - 52,500
gain = $3,500 + a $50,000 car!

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