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## Formula and table methods

A condominium is purchased for \$80,000 with a down payment of \$12,000 at an annual interest rate of 11% for 15 years. Calculate the unpaid balance after 12 years of making payments.   Use the formula method and the table method.

The first step is to determine the periodic interest rate. The payments are monthly so we need a monthly rate. If the 10% is a nominal annual rate, then the monthly rate is 11%/12 = 0.9167%. If 11% is an annual effective rate, the monthly rate is 1.1^(1/12)-1=0.8735%. I will assume 11% is a nominal annual rate.

Let's define some parameters:

L: loan amount = \$80,000-\$12,000 = \$68,000
i: periodic interest rate = 0.9167% per month
n: number of payments = 15*12 = 180
P: level monthly payment

The next step is to calculate the level monthly payment using this formula:

P=Li/(1-(1+i)^-n)

Plugging in the values above we get \$772.89

To build a table:
The interest portion of the payment is the prior balance times i.
The principal portion is the difference between the payment and the interest portion.

You can see an example of this here.

For the formula method:

Let D: the balance after kth payment (k=12*12=144)

D = P[1-(1+i)^-(n-k)]/i

Plugging in we get \$23,607.70