Loans, Annuities and Level Payment Cashflows Explained

Understand Loans, Annuities and Level Payment Cashflows

Key rules to remember

Annuity-immediate present value

a_n̄| = (1 − vⁿ) ÷ i, where v = 1 ÷ (1 + i)

Payments of 1 at the end of each period for n periods. i is the effective rate per payment period.

Annuity-immediate accumulation

s_n̄| = ((1 + i)ⁿ − 1) ÷ i = (1 + i)ⁿ × a_n̄|

Value of the payments just after the last one.

Annuity-due present value

ä_n̄| = (1 − vⁿ) ÷ d = (1 + i) × a_n̄|

Payments at the start of each period. d = i ÷ (1 + i).

Level loan repayment

R = L ÷ a_n̄|

Loan L repaid by n payments of R at the end of each period.

Interest in a payment

Iₜ = i × B_{t−1}

B_{t−1} is the balance just after the previous payment.

Capital in a payment

Cₜ = R − Iₜ

For a level loan, Cₜ = R × v^(n−t+1), so capital repaid grows by a factor (1 + i) each period.

Balance (prospective)

Bₜ = R × a_{n−t}̄|

Balance just after the t-th payment, with n − t payments remaining.

Balance (retrospective)

Bₜ = L(1 + i)ᵗ − R × s_t̄|

Gives the same value as the prospective method.

Perpetuity

a_∞ = 1 ÷ i

Payments of 1 at the end of each period, forever.

How to solve Loans, Annuities and Level Payment Cashflows questions

1. 1Draw a timeline. Mark the loan date, each payment date and the number of payments n.
2. 2Find the effective interest rate per payment period. Convert nominal or annual rates if needed.
3. 3Decide whether payments are at the end (immediate) or start (due) of each period.
4. 4Write the equation of value: loan = PV of payments. Solve for the unknown, usually R, n or i.
5. 5For the balance after t payments, use the prospective method: PV of remaining payments at the same rate.
6. 6For the interest part of a payment, multiply the previous balance by i. Capital part = payment − interest.
7. 7For a full schedule, track: opening balance, interest, payment, capital, closing balance. Check the final balance is zero.
8. 8Check reasonableness: total payments must exceed the loan, and the balance must fall each period.

Quickest way: Prospective balance and capital shortcut

Common mistakes in Loans, Annuities and Level Payment Cashflows

• Using the annual rate when payments are monthly.

  The quoted rate is annual, so students plug it straight into the formula.

  Fix: Convert to the effective rate per payment period first. If the rate is a nominal 12% convertible monthly, use i = 1% per month.

• Mixing up annuity-immediate and annuity-due.

  The wording about when the first payment falls is easy to skim past.

  Fix: Mark the first payment on a timeline. If it is one period after the loan, use a_n̄|. If it is at the loan date, use ä_n̄|.

• Calculating interest on the original loan every period.

  Students confuse it with simple interest on the starting amount.

  Fix: Charge interest on the outstanding balance after the previous payment: Iₜ = i × B_{t−1}.

• Counting the wrong number of remaining payments.

  Off-by-one errors between balance after payment t and before payment t.

  Fix: After the t-th payment, n − t payments remain. Write this on the timeline before you compute.

• Treating the capital part as constant.

  The payment is level, so students assume both parts are level.

  Fix: Only the total is level. Interest falls each period and capital rises by a factor of (1 + i).

• Rounding early and ending with a non-zero final balance.

  Rounded intermediate values accumulate error.

  Fix: Keep full calculator precision until the final step. In a schedule, tolerate small rounding in the last row.

Worked examples