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I am attempting to determine for formula for "n" in an annuity due (aka annuity in advance) where there is a balloon value involved. I have been unable to determine the formula myself.

Here are the formulas for n as an Ordinary Annuity, Advance Annuity, and n as an Ordinary Annuity with a balloon value. FV represents the balloon value.

  • PV = Present Value
  • PMT = Payment
  • i = interest rate
  • n = number of periods
  • ln = natural log
  • FV = Future Value (aka balloon payment)

Here are the formulas that I have so far:

$n$ (arrears): $ n = \frac{\ln \left[ \left( 1-\frac{\text{PV}(i)}{\text{PMT}}\right)^{-1}\right]}{\ln (1+i)} $

$n$ (arrears w/ balloon): $ n = \frac{ \ln \left[ \frac{\text{PMT}-\text{FV}(i)}{\text{PMT}-\text{PV}(i)} \right]}{\ln (1+i)}$

$n$ (advance):

$ n = \frac{ \ln \left[ \left(1+i\left(1-\frac{\text{PV}}{\text{PMT}}\right)\right)^{-1}\right]}{\ln (1+i)} + 1 $

(original link: http://danieljlong.com/tvm/n.png)

As a potential starting point, here is the formula for the balloon value (in advance):

$FV$ (advance as balloon):

$ FV = PV(1 + i)^n -\left[\text{(1 + i)} \times \text{PMT}\left[\frac{(1 + i)^n - 1}{i}\right]\right] $

As another potential starting point, here is the formula for PMT (advance) w/ a balloon:

$PMT$ (advance w/balloon):

$ PMT = \left(PV - \frac{FV}{(1 + i)^n} \right) \times \frac{i}{1 - (1 + i)^{-n}} \times \frac{1}{(1 + i)} $

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