Amortization calculator-Derivation of the method ( Simple Quick Loan)

Amortization calculator-Derivation of the method

Derivation of the method

The formula for the actual periodic payment amount A comes as follows. For a good
amortization schedule, we may define a function p(t) which represents the
principal quantity remaining at time capital t. We can then derive a formula with this
function given an unfamiliar payment amount A as well as r = 1 + we.

p(0) = P
p(1) = p(0) ur - A = G r - A
p(2) = p(1) ur - A = G r² - A ur - A
p(3) = p(2) ur - A = G r³ - A r² -- A r - The

This may be generalized in order to

p(t) = P r^t - A S_k=0 ^t-1 r^k

Using the substitution (see geometric progressions)

S_k=0 ^t-1 r^k = 1 + ur + r^2 +... + r^t-1 = \fracr^t-1 r-1

This leads to

p(t) = P r^t - A \fracr^t-1 r-1

With regard to n payment periods, we expect the main amount will be totally
paid off at the final payment period, or

p(n) = G r^n - A \fracrn-1 r-1 = 0


Solving for any, we get

A = P \fracrn ?r-1) rn-1 = P \frac(i+1)^n ((i+\cancel1 )-\cancel1
) (i+1)n-1 = P \fraci (1 + i)n (1 + i)n-1

or even

\fracA P = \fraci 1 - (1+i)^-n

Following substitution and simplification all of us get

\fracp(t) P = 1 - \frac(1+i)^t-1 (1+i)n-1

Additional uses

While often employed for mortgage-related purposes, an amortization loan calculator
can also be accustomed to analyze other debt, such as short-term loans, student
loans and charge cards.