Lesson Amortizing a debt on a credit card

Amortizing a debt on a credit card

Problem 1

Use the same formula as for the monthly payment for a loan

    M = 


where P is the loan amount; r =  is the nominal interest rate per month;
n is the number of payments (same as the number of months); M is the monthly paymemt.


In this problem  P = $2000;  r = .


Substitute these values into the formula and get for monthly payment

    M =  = $58.75.


ANSWER.  The monthly payment is $58.75.


In total, you will pay  4*12*58.75 = 2,820 dollars in 4 years.


The difference $2,820 - $25,000 = $2,000 = $820  is the interest you pay to the financial company.

Problem 2

In this problem, the debt is numerically equal to the future value of the ordinary annuity 
with the monthly deposits of $600 and the interest of 19.8% per year compounded monthly.


To calculate the future value of an ordinary annuity, use the general formula 


    FV = ,    (1)


where  FV is the future value of the account;  P is the monthly payment (deposit); 
r is the monthly percentage yield presented as a decimal; 
n is the number of deposits (19 in this case).


Under the given conditions, P = 600;  r = 0.198/12;  n = 19.  
So, according to the formula (1), you get at the end of the 20-th year


    FV =  = $13262.17.


ANSWER.  $13,262.17.

Problem 3

Use the formula which connects the loan amount with the monthly payment

    PMT = .


Here L is the loan amount, 'r' is the monthly effective interest rate, 
PMT is the monthly payments, n is the number of monthly payments (same as the number of months).


Substitute the given numbers into this equation.  You will get

    60 = .


         This equation is to find 'n'.


Simplify it step by step

     = 

    1.7142857 =     (*)

     = 1 - 1.011666667^(-n)

    0.583333333 = 1 - 1.011666667^(-n)

    1.011666667^(-n) = 1 - 0.583333333

    1.011666667^(-n) = 0.416666667

    1.011666667^n = 1/0.416666667

    1.011666667^n = 2.4

    n*log(1.011666667^n) = log(2.4)

    n =  = 75.47706349.


We must round this decimal value to the closest greater integer.


ANSWER.  It requires 76 months, or 6 years and 4 months.

As an alternative to these long calculations, you can solve equation (*) numerically,
using online solver/solvers for non-linear equations.


I did it using online calculator https://comnuan.com/cmnn03/cmnn03007/cmnn03007.php

It produced an output 75.4771, which is quite close to my answer.


You may consider it as a confirming check.