Rent or Buy a House

You are staying in Raleigh! With a post$-$graduation career in the Triangle, you need a place to live. A move$-$in ready townhouse is $$300,000$ using a $20-$year mortgage and putting $10\%$ as a cash down payment $($i$.$e$.,$ you only need to finance $90\%$ of the purchase price$).$ The current interest rate you have available is a $4\mathrm{.}25\%$ APR compounded monthly. "Closing costs," that is$,$ transaction costs associated with purchasing the house, are $2\%$ of the purchase price. Suppose, you expect home values in Raleigh to continue to grow at $2\mathrm{.}5\%$ per year. Finally, if you were to sell the home, you would realize $5\%$ in fees $($costs of a realtor and associated transaction costs$).$ Alternatively, you can rent an apartment for $$1,900$ per month. There is a $$3,000$ security deposit, due at the time you move in$.$ Assume that the cost of the rental increases $15\%$ every five years.

Mortgage and rental payments are made at the end of every month. Assuming you will live in this location for $10$ years $($i$.$e$.,$ you sell the house or stop renting at that time$),$ find the implied cost of the rental. Do you rent or do you buy? $($You can find NPV at the rental rate or the implied interest cost of the lease. For both of these, you need some cash flows!$)$

Pay attention to formatting and do not hardcode numbers into your formulas.