I Monthly Payment and Loan Balance
Many readers, for
reasons of their own, request the algebraic formulas used to calculate the
monthly payment and loan balance on amortized mortgages. Here they are:
The following
formula is used to calculate the fixed monthly payment (P) required to
fully amortize a loan of L dollars over a term of n months at a monthly
interest rate of c. [If the quoted rate is 6%, for example, c is .06/12 or
.005].
P = L[c(1 + c)n]/[(1
+ c)n - 1]
The
next formula is used to calculate the remaining loan balance (B) of a
fixed payment loan after p months.
B
= L[(1 + c)n - (1 + c)p]/[(1 + c)n - 1]
II
Annual
Percentage Rate (APR)
Other readers
ask about the formula used to calculate the APR. The APR is what
economists call an "internal rate of return" (IRR), or the
discount rate that equates a future stream of dollars with the present
value of that stream. In the case of a home mortgage, the formula is
L - F = P1/(1 + i) + P2/(1
+ i)2 +… (Pn + Bn)/(1 + i)n
Where:
i = IRR
L = Loan amount
F = Points and all other lender
fees
P = Monthly payment
n = Month when the balance is
paid in full
Bn = Balance in month
n
This equation can be solved for i
only through a series of successive approximations, which must be done by
computer. Many calculators will also do it provided that all the
values of P are the same.
The APR is a special case of the
IRR, because it assumes that the loan runs to term. In the equation,
this means that n is equal to the term, and Bn
is zero.
Note that on ARMs, the payments
used to calculate the APR are those that would occur under the assumption
that the index rate does not change over the life of the loan.
On a cash-out refinance, the APR ignores the existing mortgage
that is paid off, which makes it a poor guide to the decision (see
The APR
on a Cash-Out Refinance). The better guide is a "net-cash APR", in
which the balance of the existing loan (including interest accrued to
the day of payoff) is subtracted from the left side of the equation, and
the "Ps" represent the difference in payment between the old and new
mortgage.
III Future Values
Many of my calculators
measure financial results in terms of "future values" -- the
borrower's net wealth at the end of a specified period.
The future value of a
single sum today is:
FVn =
S(1+c)n
Where:
FVn is the
value of the single sum after n periods
S is the amount of the
single sum now
c is the applicable
interest rate
n is the length of the
period
The future value of a
series of payments of equal size, beginning after one period, is:
FVn =
P[(1+c)n - 1]/c
Where P is the periodic
payment, and the other terms are as defined above.
Copyright Jack Guttentag
2007
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