Time Value of Money – Payback of a Loan with Fixed Monthly Payments

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One of the interesting applications is the payback of a loan or mortgage. Typically, people will take a mortgage to buy a house and pay it over a longer period like 10, 15, 20, 25, 30 years, or even longer in some cases. The lenders will pay back the loan in fixed monthly installments that doe not change over the duration of the loan.

In this first example, we will be looking into fixed monthly payments that are composed of payback of the capital + the interest on the remaining outstanding amount.
In our example, we have a loan of $150,000 with a yearly interest rate of 6% and a period of 30 years.

We can use two different formulas:
1. The PV formula with fixed future payments (PV) which we deducted in one of the previous videos. When we transform it the fixed monthly payments (FV) then become the PMT or monthly fixed payments.
2 Using the integrated formula in spreadsheet programs like Excel:

When using the PMT-formula with the following parameters:
rate = 6%/12 because we have monthly payments
nper = 30 x 12 = 360
pv = -150,000
fv = 0
type = 0

When we enter all in the formula:

PMT(0.05,360,-150000,0,0) = $899.33
Remark = the separates may differ from program to program!

The final result is that you will pay 360 x $899.33 = $323.758.8

When we look deeper into the calculations, there are two elements in the payment: interest on the outstanding capital and interest:

First payment
Now, the outstanding amount is $150,000 on which we have to pay 0.5% interest = $750
The difference 899.33 – 750 = $149.33 which is used to reduce the outstanding capital to $149,850.67

Second payment
The outstanding amount has been reduced to $149,850.67, hence the interest payment on this amount then becomes equal to $149,850.6 x 0.005 = $749.25 which is a little lower than the interest paid at the end of the first period of $750.
The new amount of capital paid at the end of period 2 is now: 899.33 – 749.25 = $150,08

We see that the interest to pay is reduced slowly every month while the part of capital is increasing. At the end of the loan, we see that we paid a total of interest equal to $173,758.8 which is more than the total amount we lend at the beginning of the period!

This is completely normal because of the time value of money.
In the next video, we will look into an alternative way to pay a loan: we will pay a fixed proportion of the capital each period.

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