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Credit card balance transfer fee problem

• To: mathgroup at smc.vnet.net
• Subject: [mg103184] Credit card balance transfer fee problem
• From: Kelly Jones <kelly.terry.jones at gmail.com>
• Date: Thu, 10 Sep 2009 07:18:52 -0400 (EDT)

I want to use Mathematica to solve this problem.

My credit card company loans me $10000 for a cash advance fee of 3%
($300), and an interest rate of 2% per year. I have to pay off the
loan in 1 year, but my monthly minimum payment is only 3% of my
outstanding balance. In other words, I can pay 3% of my balance for
the first 11 months, and then pay off the remaining balance in the
12th month.

Assuming I do this, how does this loan compare to a regular, amortized loan?

At first glance, this looks like a 5% loan: 3% upfront fee, and 2%
interest for 1 year.

Using Mathematica, I found this is really a ~6.4% loan: if I invested
all the money I got at ~6.4%, I'd break even after one year.

What's the general solution here? Is there a well-known formula?

My take: let f[t] be the amount I have after t years. This starts at
$10000, and decreases by 36% each year (3% per month), but increases
because I'm investing at p% annualized. In other words:

DSolve[{f'[t] == f[t]*Log[1+p]-36/100*(f[t]+300), f[0] == 10000},f[t],t]

Note that I pay 36% of my balance per year, which is $300 higher than
the amount I actually have.

Let g[t] be the amount I owe. This starts at $10300, and decreases 36%
per year from my payments, but increases by 2% annualized. In other words:

DSolve[{g'[t] == -36/100*g[t] + g[t]*Log[1+2/100], g[0]==10300},g[t],t]

These are the equations I used to come up w/ the 6.4% number.

I realize I'd really be paying monthly, not constantly, but I prefer
using differential equations, as they seem cleaner/purer.

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• Follow-Ups:
  • Re: Credit card balance transfer fee problem
    • From: Kelly Jones <kelly.terry.jones@gmail.com>
  • Re: Credit card balance transfer fee problem
    • From: Kelly Jones <kelly.terry.jones@gmail.com>
  • Re: Credit card balance transfer fee problem
    • From: Kelly Jones <kelly.terry.jones@gmail.com>
  • Re: Credit card balance transfer fee problem
    • From: Kelly Jones <kelly.terry.jones@gmail.com>
  • Re: Credit card balance transfer fee problem
    • From: "Benedetto Bongiorno" <bongiob@sbcglobal.net>