Linemo

How it works

Mathematical deep-dive: how it works

This walkthrough mirrors the writing style of the About page while spelling out how a fixed monthly payment m , duration N , and rate r translate into the amortization engine behind the simulator.

Key inputs

  • Payment: lock a monthly cash outflow m .
  • Duration: decide on the number of months N .
  • Rate: use the nominal monthly rate r .

Outputs

  • Borrowed amount b implied by those parameters.
  • Monthly interest terms i_k through time.
  • Cumulative interest I_k and final balance logic.

Derivation

Step-by-step amortization logic

The goal is to show how the interest portion evolves when payments stay constant. Each paragraph below mirrors the bullet points from the About page but dives into the algebra.

1. Express the monthly interest

We consider fixed monthly payments m over N months with rate r . The unknowns are the borrowed amount b and the monthly interest terms i_k . The interest at month k follows the subtraction of all previous principal repayments.

\begin{aligned} i_0 &= rb \\ i_1 &= r (b - (m - i_0)) \\ i_2 &= r [b - (m - i_0) - (m - i_1)] \\ \vdots \\ i_k &= r(b - km + \sum_{i=0}^{k-1}i_k) \\ \end{aligned}

2. Look at the decrement

Instead of expanding every term, track how the interest drops from one month to the next. This gives a compact recurrence for i_k as a function of the previous interest and the fixed payment.

i_{k+1} - i_k = -r (m - i_k)
i_{k+1} = (1 + r) i_k - rm

3. Turn it into a geometric sequence

The recurrence is arithmetico-geometric, so shift the sequence by defining a_k . This lets us write the evolution as a pure geometric sequence with ratio (1 + r) .

a_k := i_k - m
a_{k + 1} = (1 + r) a_k
a_k = (1 + r)^k a_0

Substituting back provides the closed form for i_k , which decays exponentially from the initial interest rb .

i_k = m - (1 + r)^k (m - rb)

4. Sum the interests

Let I_k be the cumulative interest paid through month k . Using the geometric sum formula yields the compact expression below.

I_k = \sum_{j = 0}^k i_j
I_k = km - (m - rb) \frac{(1 + r)^k - 1}{r}

5. Link payments to balances

The sum of all monthly payments equals the principal plus the cumulative interest. Rearranging isolates b as a function of m , r , and N .

b + I_N = Nm
b = Nm - I_N

A final substitution reveals the closed-form borrowing capacity for any fixed payment plan.

b = m \frac{(1 + r)^N - 1}{r(1 + r)^N}

QED \blacksquare

This proof keeps the conversational tone of the About page yet exposes every algebraic step, so you can reconcile the simulator outputs with pen-and-paper math.

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