Compute Loan Balance for a specific month (period) on an amortized loan

Question

I'm looking for is a method that will allow me to compute the Loan Balance value for a specific period (month) in the amortization table without having to compute all of the values before it.

Let's say: your bank loan of 100000$ at the rate of 10% with annual rate compounded monthly and for a maturity of a 10 years.

So we have following information:

PV = 100000$, Time (or n) = 10, Frequency (#) = 12 (monthly), Rate (r) = 10% or 0.1.

We should find 66th month Loan Balance value without establishing whole table. Please do not suggest to calculate with Future Value (FV of annuity) formula replacing Time (n) = by 66 (month that we want to find value)

Answer 1

Given

PV = principal
i  = periodic rate
m  = number of periods
d  = periodic payment

d  = PV i (1 + 1/((1 + i)^m - 1))

the balance b remaining in month x is

b  = (d + (1 + i)^x (i PV - d))/i

Applying your figures

  PV = 100000
  i  = 0.1/12
  m  = 10*12

∴ d  = 1321.51

  x  =  66
∴ b  =  57276.53

Check: final balance

  x  =  120
∴ b  =  0

Formulae

Formula for periodic payment - loan payment formula

Formula for loan balance - inhomogeneous difference equation (Arne Jensen, Aalborg Uni.)