I am trying to calculate a loan amount where the total principle depends on the payment amount.
Here's a simple example, leaving aside that a) no bank would actually do this, and b) my income is not specified:
$0 in the bank right now. I want to take out a loan for a
$100 car, for which I will be making a regular annual payment, starting at the end of year 0 with an interest-only payment. My loan rate is 5%. In order to qualify for the loan, the bank wants to make sure that I have a balance equal to twice my annual payment amount throughout the life of the loan.
If I borrow exactly
$100, my payment at the end of year 0 would be
5%). However, I need to borrow a little extra so that I have a
$10 balance in the account (to satisfy the requirement that I maintain a balance of twice my annual payment). This means I actually need a loan of
$110. However, a loan of
$110 actually raises my payment amount to
$5.50, which means I need to maintain a balance of
$11 in my account.
This means that I actually need to borrow
$111 to begin with, which raises my payment to
$5.55, which means I actually need to borrow
I have worked it out to a summation that resembles the following, although I can't be certain this is accurate:
Where $r$ is the interest rate
I believe this would allow for me to iterate $x$ number of times over a formula that sums the additional principle needed for each incremental amount of interest. However, I'm not sure how to carry this out to infinity.
Ultimately, I need to plug this into Excel as part of a cash flow, but I wanted to think through the math first.