Actuarial Mathematics for Modelling · Practical applications of the equation of value (loans and APR)

Loans with Varying Instalments and Early Repayment

Updated 11 October 2026 · Fact-checked

Set up an equation of value: the present value of all repayments, at the loan rate, equals the amount borrowed. Include every instalment, any balloon payment and any deferment. For early settlement, the amount due is the outstanding balance, which is the present value of the remaining payments at the loan rate, plus any stated fee.

Understand Loans with Varying Instalments and Early Repayment

A loan is a deal where the lender pays out an amount now and gets repayments later. The deal is fair, at the loan's interest rate, when the two sides have equal value. That is the equation of value: PV of repayments = amount borrowed, with every payment discounted at the loan rate.

The repayments need not be level. They can rise by a fixed amount, rise by a fixed percentage, or be a mix of regular payments and one large final payment, called a balloon payment. The method does not change. You write each payment, attach its time, discount it, and set the total equal to the loan.

A loan can also be deferred. Nothing is paid for the first few years, although interest still builds up on the loan. Deferment only shifts the timing of the payments. You discount the payments by extra years, or equivalently you roll the loan forward to the first payment date.

If the borrower settles early, the outstanding balance is what they owe at that moment. You can find it two ways. The prospective method is the PV of future payments at the loan rate. The retrospective method is the accumulated loan less the accumulated payments made so far. Both give the same answer when the same rate is used throughout. Any penalty or fee is added after you find the balance.

Key rules to remember

Equation of value for a loan: L = Σ Pt × v^t, where v = 1 ÷ (1 + i); L is the amount borrowed and Pt is the payment at time t. Use the effective rate per payment period.
Level annuity-immediate: aₙ = (1 − vⁿ) ÷ i; Use for n level payments of 1 at the end of each period, valued one period before the first payment.
Increasing annuity-immediate (payments 1, 2, ..., n): (Ia)ₙ = (äₙ − n × vⁿ) ÷ i; Payment at time t is t. For a different start and step, split into a level part plus an increasing part.
Payments rising in geometric progression: PV = P × Σ for t = 1 to n of (1 + g)^(t−1) × v^t = P × [1 − ((1 + g) ÷ (1 + i))ⁿ] ÷ (i − g); Valid when i ≠ g. If i = g, PV = n × P ÷ (1 + i).
Balloon payment: B = (L − PV of regular payments) × (1 + i)ⁿ; n is the time of the balloon. The balloon is paid on top of any regular payment due at that time.
Deferred loan: PV at time 0 = v^d × (PV of payments valued at time d); d is the number of periods the payment stream is shifted back to reach time 0.
Outstanding balance (prospective): Bt = PV at time t of all future payments, at the loan rate; Right after the payment at time t, exclude that payment.
Outstanding balance (retrospective): Bt = L × (1 + i)^t − accumulated value of payments made up to t; Gives the same result as the prospective method at the same rate.

How to solve Loans with Varying Instalments and Early Repayment questions

Use the same method for every question on varying instalments, balloons, deferment and early settlement.

1Draw a timeline. Mark the loan date and every payment, including any balloon and any deferred period.
2Convert the rate to an effective rate for the payment period. Check whether the rate is nominal, a force of interest or a discount rate.
3Write the payments in terms of the unknown. Split complicated patterns into a level annuity plus an increasing annuity, or a level annuity plus a single payment.
4Choose a valuation date, usually time 0 or the date just before the first payment. Write: PV of payments = loan.
5Solve for the unknown, which could be the instalment, balloon, loan amount, term or rate.
6For early settlement, find the date and the balance just after that date's payment. Use the prospective method unless the past payments are easier.
7Add any penalty or fee exactly as the question states it, for example a percentage of the balance.
8Check the answer by sense: the balance should be below the sum of the remaining payments and, early in the loan, not far above the original loan.

Quickest way: Prospective balance and one-line equation

When to use it: Use when you are short of time and the payments are level, or level plus one extra payment.

Value everything at one date with one equation. Do not build a full loan schedule unless the question asks for it.
Treat the balloon as a separate single payment: B × vⁿ. Treat a level part as a × annuity factor.
For settlement, use only the future payments: balance = PV of those payments. This avoids the past entirely.
Keep at least six decimal places in factors and round only the final answer.
Substitute your answer back into the equation of value as a quick check.

Common mistakes in Loans with Varying Instalments and Early Repayment

Valuing an annuity at the wrong date, for example using aₙ for a deferred loan without discounting further
The annuity factor values payments one period before the first payment, and students forget where that date is.
Fix: State the valuation date of the annuity factor on your timeline. Then discount or accumulate it to the date you need.
Forgetting that the balloon is paid in addition to the last regular instalment
The wording 'final payment of ...' is read as replacing the regular payment.
Fix: Read carefully. If the question says 'in addition', include both at that time. If it says 'the final payment is', it replaces the regular one.
Including the payment made on the settlement date in the outstanding balance
Confusion between the balance just before and just after the payment.
Fix: For 'immediately after the t-th payment', count only payments after time t. For 'just before', include the payment.
Using the nominal rate directly as the rate per payment period
Students rush to the formula without converting.
Fix: Convert first. A nominal rate i(m) convertible m-thly gives i(m) ÷ m per period. Match the period to the payment frequency.
Applying the early settlement fee to the wrong amount
Questions vary: the fee may be a percentage of the balance or of the instalments missed.
Fix: Quote the base of the fee in your answer. Compute the balance first, then the fee, then add them.
Using (Ia)ₙ when the payments do not start at 1 or do not step by 1
The formula is memorised for the pattern 1, 2, 3, ... only.
Fix: Rewrite the payments as a level payment plus a multiple of the pattern 1, 2, 3, ... Check the first and last terms against the formula.

Worked examples

Example 1

A loan of ₹5,00,000 is repaid by three annual instalments in arrears. The first is X, the second is 2X and the third is 3X. The effective annual interest rate is 10%. Find X.

Show the solution

Equation of value at time 0: 5,00,000 = X × (v + 2v² + 3v³) = X × (Ia)₃ at 10%.
v = 1 ÷ 1.1 = 0.909091, v² = 0.826446, v³ = 0.751315.
(Ia)₃ = 0.909091 + 2 × 0.826446 + 3 × 0.751315 = 0.909091 + 1.652893 + 2.253944 = 4.815928.
X = 5,00,000 ÷ 4.815928 = 1,03,822 (to the nearest rupee).
Check: 4.815928 × 1,03,822 ≈ 5,00,000.

Answer: X ≈ ₹1,03,822, so the instalments are about ₹1,03,822, ₹2,07,644 and ₹3,11,466.

Example 2

A loan of ₹10,00,000 is repaid by level annual payments of ₹1,50,000 in arrears for 5 years, together with a balloon payment B at the end of year 5, in addition to the fifth regular payment. The effective annual rate is 10%. (a) Find B. (b) Separately, a different loan is repaid by 20 level annual payments of ₹1,00,000 in arrears at 8% effective. The borrower settles it immediately after the 12th payment and is charged a fee of 2% of the balance. Find the total amount paid at settlement.

Show the solution

(a) Equation of value at time 0: 10,00,000 = 1,50,000 × a₅ + B × v⁵.
a₅ at 10% = (1 − 1.1⁻⁵) ÷ 0.1 = 3.790787, so the PV of the regular payments = 1,50,000 × 3.790787 = 5,68,618.
Then B × v⁵ = 10,00,000 − 5,68,618 = 4,31,382.
v⁵ = 0.620921, so B = 4,31,382 × 1.1⁵ = 4,31,382 × 1.61051 = 6,94,745 (to the nearest rupee).
(b) Immediately after the 12th payment, 8 payments remain, at times 1 to 8 from then.
Balance = 1,00,000 × a₈ at 8%. v⁸ = 0.540269, so a₈ = (1 − 0.540269) ÷ 0.08 = 5.746639.
Balance = 1,00,000 × 5.746639 = 5,74,664.
Fee = 2% × 5,74,664 = 11,493. Total = 5,74,664 + 11,493 = 5,86,157.

Answer: (a) B ≈ ₹6,94,745. (b) The balance is about ₹5,74,664 and the total paid at settlement is about ₹5,86,157.

Exam tips

Draw the timeline before you write any algebra. Most lost marks come from the wrong payment date.
Write the equation of value in words first, for example 'PV of payments = loan', then in symbols. Written questions give method marks for this.
In computer-based Paper B questions, build the cashflows in a column against time, discount each one, and solve for the unknown with a solver or algebra. State the rate and the timing assumptions.
Read the wording on the balance: 'immediately after', 'just before' and 'at the start of year' mean different things.
In MCQs, test an answer by substituting it into the equation of value. It is often faster than solving.

Practice questions from Practical applications of the equation of value (loans and APR)

Loans with Varying Instalments and Early Repayment: frequently asked questions

How do I calculate a balloon payment on a loan?

Value the loan and the regular payments at the same date. The balloon is the shortfall: take the loan minus the PV of the regular payments, then accumulate this to the date of the balloon at the loan rate. Check whether the balloon is in addition to the last regular payment.

What is the outstanding balance on early settlement?

It is the amount needed to clear the loan at that date. Using the prospective method, it is the present value of the remaining payments at the loan rate. The retrospective method gives the same value: the accumulated loan minus the accumulated payments made.

How do I handle a loan with increasing repayments?

Write the payments as a pattern such as P, P + d, P + 2d and so on. Split them into a level annuity of P plus d times an increasing annuity. If they rise by a percentage, use the geometric series formula and check whether i equals g.

Does a deferred start change the equation of value?

The method is the same. Only the timing changes, because you discount the payments for the extra deferred years. Interest still builds up on the loan during the deferment.