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Outstanding loan balance

Two ways to find what is still owed: look forward at the payments left, or look back at what has been paid. They must agree.

The formulas

Prospective: $B_{t} = P a_{\overline{n - t} ∣ i}$
Retrospective: $B_{t} = L (1 + i)^{t} - P s_{\overline{t} ∣ i}$
Recursion: $B_{t} = B_{t - 1} (1 + i) - P$

Where it comes from

Prospective: the balance is by definition the present value of the remaining $n - t$ payments.
Retrospective: the loan accumulates at $i$ , and the payments made accumulate too; the difference is what is still owed.
They agree for a loan running on schedule — which is exactly the identity `verify-answers` asserts numerically.

Worked example

For the 250,000 twenty-year loan at 5.5%, find the outstanding balance immediately after the 8th payment.

$P = 20, 919.83$ as before, and 12 payments remain.
Prospective: $B_{8} = P a_{\overline{12} ∣0.055} = 20, 919.83 \times 8.618518$ .
$B_{8} = 180, 297.95$ .
Retrospective gives the same figure: $250, 000 (1.055)^{8} - 20, 919.83 s_{\overline{8} ∣0.055}$ .

Answer: 180,297.95

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

Prospective when you know the payment; retrospective when you know the original loan and have to prove where the money went.
Any answer for a balance can be checked with the other method in under a minute.

Traps

Using a_{n−t+1} (one payment too many) after payment t.
Forgetting that a payment made at time t is already gone when you value the balance 'immediately after' t.

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Outstanding loan balance

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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