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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
Bt=P an−t‾∣iB_t = P\,a_{\overline{n-t}|i}
Retrospective
Bt=L(1+i)t−P st‾∣iB_t = L(1+i)^{t} - P\,s_{\overline{t}|i}
Recursion
Bt=Bt−1(1+i)−PB_t = B_{t-1}(1+i) - P

Where it comes from

  1. Prospective: the balance is by definition the present value of the remaining n−tn-t payments.
  2. Retrospective: the loan accumulates at ii, and the payments made accumulate too; the difference is what is still owed.
  3. 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.

  1. P=20,919.83P = 20{,}919.83 as before, and 12 payments remain.
  2. Prospective: B8=P a12‾∣0.055=20,919.83×8.618518B_8 = P\,a_{\overline{12}|0.055} = 20{,}919.83 \times 8.618518.
  3. B8=180,297.95B_8 = 180{,}297.95.
  4. Retrospective gives the same figure: 250,000(1.055)8−20,919.83 s8‾∣0.055250{,}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.

Related

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