# Acquisition + Rent payment calculations

I was reading some documentation which had an example scenario. In the scenario, they gave the following details

Price = 150000

ownership
bank = 80% = 120000

period = 30 years

rent = 4.99%


Using the above, apparently the payments for the first 3 months will be:

month 1
acquisition payment = 144.45
rent payment = 499.00
total monthly payment = 643.45

month 2
acquisition payment = 145.05
rent payment = 498.40
total monthly payment = 643.45

month 3
acquisition payment = 145.65
rent payment = 498.40
total monthly payment = 643.45


How have they used the details above to work out the monthly payments? I can't seem to work it out.

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Let's get some mathematical notation down. We'll call the amount that you pay each month $X$, which is a fixed amount, ie it doesn't vary month on month. The total value of the house is $V$ and the monthly rental rate is $r$ (which is 0.416% in your example, or 4.99% divided by 12). Your ownership in month $n$ is $E_n$ (E for equity) and the bank's ownership is $L_n$ (L for liability). The amount of rent paid in month $n$ is $R_n$, and the acquisition payment is $A_n$.

Then we have the following relationships between these quantities:

(1) $A_n + R_n = X$ (the acquisition and rent sum to the monthly payment)

(2) $E_n + L_n = V$ (your equity plus your liability give the total value)

(3) $R_n = rL_n$ (your rent is the rental rate times your liability)

(4) $L_{n+1} = L_n - A_n = L_n - (X - R_n) = (1+r)L_n - X$ (your liability next month is your liability this month, minus your acquisition capital)

The first month your liability is $L_0 = 120,000$. Therefore your rent is $0.00416 \times 120,000 = 499$. The remainder of your monthly payment goes towards acquisition capital, so it is calculated as $643.45 - 499 = 144.45$. Therefore your liability goes down to $119855.6$ for the second month, and the rent you pay in that month is $0.00416 \times 119855.6 = 498.4$, so you pay slightly more in acquisition that month.

The trick is in how we calculate the monthly payment, $X$. Notice that using the equations (1)-(4) above, if we know $X$ we can calculate by how much your liability decreases from one month to the next. As long as your monthly payment exceeds your rent, your liability will decrease every month. At some point, it will decrease to zero. The monthly payment $X$ is decided by choosing the value that means your liability decreases to zero after 30 years, or 360 months, which is the period of the loan.

You can either solve equation (4) to give an explicit formula for your remaining liability in month $n$ and then solve for $X$ by setting $L_{360} = 0$, or you can plug these equations into a spreadsheet and fiddle with the value for $X$ until the liability comes out to be zero after 30 years.

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Thanks for the very detailed comprehensive answer. I now understand the rent calculations. What I don't understand is how 144.45 is worked out for the first month if the total 643.45 is not known? –  oshirowanen Sep 7 '11 at 12:13
You can determine the 643.45 by trial and error. For example, you might first try paying 600 per month, and hence only 101 in acquisition in the first month. This would leave you with a positive liability after 30 years, so you might try 700 per month, meaning you pay 201 in acquisition in the first month. This leaves you with negative liability after 30 years. You want your liability to be zero, so you now know that the correct amount to pay per month lies between 600 and 700. –  Chris Taylor Sep 7 '11 at 12:51