Question A (10 marks) - Credit Card Mathematics
Introduction
On a monthly credit card balance of $1000, a typical credit card company will only ask for a minimum payment of $20. Why do credit card companies do that?
Mathematics of Credit Card Debt
Suppose we do what the company wants and make only the minimum payment p every month against an initial balance of b. If the company charges monthly interest rate r, what is the balance after n months?
See if we can notice a pattern.
Balance after n months
n=1 (b-p)(1+r)=b(1+r)-p(1+r)
n=2 b(1+r)^2-p(1+r)^2-p(1+r)
n=3 b(1+r)^3-p(1+r)^3-p(1+r)^2-p(1+r)
n=4 b(1+r)^4-p(1+r)^4-p(1+r)^3-p(1+r)^2-p(1+r)
A1. (2 marks) Looking at the pattern above, derive a general function, f(n,r,p,b), for the balance after n months. Hint: use summation notation ∑_(k=1)^nwhere applicable when deriving the function.
A2. (1 mark) If your credit card company charges a monthly interest rate of 2% (annually 24%) on an initial balance of $1000, and you make a monthly payment of $30, what is your balance after one year? That is, find the value of f(12,0.02,$30,$1000).
A3. (1 marks) Based on your answer in A2, how much did you end up paying in interest rate charges over a year?
A4. (2 marks) Use geometric progression properties to convert the general formula in A1 above to a functional form that excludes the summation notation. Hint: You want to replace the summation notation ∑_(i=1)^n with a ratio; see https://en.wikipedia.org/wiki/Geometric_progression, subsection titled Related Formulas.
A5. (2 marks) How many months would it take to pay off a balance of $1000 if you made $30 monthly payments while being charged 2% monthly interest?What if we double the payment to $60, do we cut the time in half?Hint: equate the function for the balance after n month to zero and solve for n.
A6. (2 marks) Plot the function derived in A5 in a two-dimensional coordinate system with n on the y-axis and p on the x-axis. Assume the initial balance of b=$1000, and monthly interest of r=0.02. Find the vertical asymptote of this function, that is, find the value p (monthly minimum payment on your credit card) such that the number of months required to pay off your credit card debt is equals to infinity (that is a monthly minimum payment that makes you forever indebted to your credit card provider!).
Attachment:- QBA-part-A.docx