After graduation, you moved to Lower East Manhattan, and now you are deciding between renting an apartment and purchasing a condominium. Both units have the same design, location, and quality. If you rent, you will pay $$4,000$ each month for the first year. You expect the monthly rent to increase by $10$ percent each year and the rental lease is for $12$ months when you sign the contract $($so you will pay $$4,000$ monthly rent for the first $12$ months, $$4,400$ monthly rent for the next $12$ months, and so on$.).$ If you decide to buy, imagine the same unit costs $$800,000.$ Assume you make a down payment of $20\%($$$160,000)$ and pay closing costs of $3\%($$$24,000).$ You will need to make a monthly mortgage payment of $$3,055$ and pay property taxes of $1\mathrm{.}2\%$ of the home's value at the end of each year. Assume $30-$year fixed rate mortgage of $4\%.$ Your home value will be increasing $10\%$ a year $($For simplicity, assume your home value stays $$800,000$ for the first $12$ months, $$880,000$ for the next $12$ months, and so on$..)$ You will be asked to compare the present value of renting vs purchasing. You will be using monthly discount rate of $0\mathrm{.}8\%.$ What is the present value of the rent you will pay over the next four years?