Melinda and Milan both need a place to live. Melinda has decided to rent a house for $1200 per month. Milan has decided to buy a house for $210000 which he will finance with a 18 year mortgage at 3.9%, compounded semi-annually. Milan must make a down payment of $17000 and he will pay off the mortgage with regular monthly payments. The house appreciates at a rate of 2% Melinda and Milan both move out after 7 years.

a) What are Melinda's costs?
b) What are Milan's costs?

A lot

To calculate Melinda's costs, we need to determine the total amount she spends on rent over the 7-year period.

Melinda's monthly rent is $1200, so her annual rent expenses are $1200 x 12 = $14,400.
Over 7 years, her total rent costs would be $14,400 x 7 = $100,800.

To calculate Milan's costs, we need to consider several factors:
1. Down payment: Milan makes a down payment of $17,000.
2. Mortgage principal: The mortgage principal is calculated by subtracting the down payment from the purchase price, i.e., $210,000 - $17,000 = $193,000.
3. Interest rate: The interest rate is 3.9% compounded semi-annually. To calculate the effective interest rate per compounding period, we divide the annual rate by the number of compounding periods in a year, i.e., 3.9% / 2 = 1.95%.
4. Mortgage duration: Milan has an 18-year mortgage, which means there are 18 x 2 = 36 compounding periods.

To calculate monthly mortgage payments, we'll use the formula for a fixed-rate mortgage:

M = P [ i(1 + i)^n ] / [ (1 + i)^n - 1],

Where:
M = Monthly payment
P = Principal (mortgage amount)
i = Monthly interest rate
n = Total number of payments (36)

First, we need to calculate the monthly interest rate. We take the annual interest rate and divide it by 12, i.e., 1.95% / 12 = 0.1625%.

Now, substitute the known values into the equation:

M = $193,000 * [0.001625 * (1 + 0.001625)^36] / [(1 + 0.001625)^36 - 1]

Using a calculator, the monthly mortgage payment comes out to be approximately $1,153.79.

To calculate Milan's costs, we multiply the monthly mortgage payment by the number of months (7 years = 7 x 12 = 84 months):

Milan's costs = $1,153.79 * 84 = $96,970.36.

Lastly, we need to consider the house appreciation. The house appreciates at a rate of 2% annually. To calculate the appreciation value over 7 years, we calculate the compound interest on the original house price ($210,000) over the period:

Appreciation value = $210,000 * (1 + 0.02)^7 - $210,000 = $31,066.68.

Therefore, Milan's total costs would be the sum of his down payment, mortgage payments, and appreciation value:

Milan's total costs = $17,000 + $96,970.36 + $31,066.68 = $144,036.04.

In summary:
a) Melinda's costs: $100,800 (rent over 7 years)
b) Milan's costs: $144,036.04 (down payment, mortgage payments, and appreciation value)