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

Equation of Value and Loan Schedules Explained

Updated 11 October 2026 · Fact-checked

The equation of value sets the present value of money you receive equal to the present value of money you pay, at a stated interest rate. For a loan, solve it for the level instalment. Then build the schedule: interest = rate × opening balance, capital = instalment − interest, closing balance = opening − capital.

Understand Equation of Value and Loan Schedules

A loan is a cashflow exchange. The lender pays you an amount now. You pay it back later in instalments. At the loan rate, both sides must have the same value at the same date. That statement is the equation of value.

Choose a valuation date, usually the date the loan is made. Discount every payment back to that date at the loan rate. Set the present value of repayments equal to the amount borrowed. For level instalments paid at the end of each period, the repayments form an annuity-immediate, so the loan L = X × a_n at the rate per period i.

Each instalment does two jobs. Part pays the interest that has built up since the last instalment. The rest repays capital. Interest is always the periodic rate times the balance outstanding at the start of the period. Early on the balance is large, so interest is large and capital is small. As the balance falls, interest falls and the capital share grows.

The outstanding balance just after a payment can be found two ways. The prospective method is the present value of the remaining instalments. The retrospective method is the accumulated loan less the accumulated payments made so far. With the same interest rate, both give the same answer.

In exams, always match the rate to the payment frequency. If payments are monthly, use the monthly effective rate, not the annual one.

Key rules to remember

Loan from level instalments (arrears): L = X × a_n = X × (1 − v^n) ÷ i; i is the effective rate per payment period. v = 1 ÷ (1 + i). First payment one period after the loan.
Level instalment: X = L ÷ a_n; Rearrangement of the equation of value at the loan date.
Interest in an instalment: I_t = i × B_(t−1); B_(t−1) is the balance just after the previous payment.
Capital in an instalment: C_t = X − I_t; Capital repaid equals the fall in the balance.
Closing balance: B_t = B_(t−1) − C_t = B_(t−1)(1 + i) − X; Use this to roll the schedule forward.
Prospective balance: B_t = X × a_(n−t); For level instalments in arrears after t payments, with the same i throughout.
Retrospective balance: B_t = L(1 + i)^t − X × s_t; s_t = ((1 + i)^t − 1) ÷ i. Gives the same value as the prospective method.
Capital in the first instalment and growth: C_(t+1) = C_t × (1 + i); For level instalments in arrears, capital repayments grow geometrically at rate i.
Total interest paid: Total interest = n × X − L; For level instalments with no other charges.

How to solve Equation of Value and Loan Schedules questions

Use this order for any loan question, whether it asks for an instalment, a balance or a split.

1Write down the loan amount, term, payment frequency, payment timing (arrears or advance) and the rate quoted.
2Convert the rate to an effective rate per payment period. For a nominal rate i^(m) compounded m times a year, the period rate is i^(m) ÷ m.
3Choose the valuation date, normally the loan date, and write the equation of value: loan = PV of instalments.
4Solve for the unknown, usually the instalment X = L ÷ a_n.
5For the balance after t payments, use the prospective method: X × a_(n−t). Check with the retrospective method if time allows.
6For interest and capital in a given instalment, find the opening balance, then interest = i × balance and capital = X − interest.
7For a full schedule, roll forward row by row: opening balance, interest, instalment, capital, closing balance. Check the final balance is zero.
8State the answer with units and to a sensible accuracy. Round only at the end.

Quickest way: Direct balance and capital shortcuts

When to use it: Use when the question asks for one row of the schedule, such as the interest in the 7th instalment, not the whole table.

Find X = L ÷ a_n with the period rate.
Find the balance before the required instalment as X × a_(n−t+1), where t is the instalment number.
Interest in instalment t = X × (1 − v^(n−t+1)). Capital = X × v^(n−t+1).
Check: interest + capital = X.

Common mistakes in Equation of Value and Loan Schedules

Using the annual rate for monthly or half-yearly instalments.
The rate is quoted annually and students plug it straight into the annuity.
Fix: Convert first. For a nominal rate compounded monthly, use i^(12) ÷ 12 per month and count n in months.
Calculating interest on the original loan every period.
Students confuse the schedule with simple interest.
Fix: Interest is always the period rate times the current outstanding balance.
Using a_n for payments made in advance.
The annuity-immediate is the default in memory.
Fix: If the first payment is at the loan date, use the annuity-due: L = X × ä_n, with ä_n = (1 + i) × a_n.
Counting the wrong number of remaining payments in the prospective balance.
Off-by-one errors between n, t and n − t.
Fix: After t payments with n in total, n − t remain. Draw a timeline and count the dots.
Rounding the instalment early and finding a non-zero final balance.
Rounded X is reused in later rows.
Fix: Keep full calculator accuracy in X. Quote a rounded figure only in the final answer, unless told to use a rounded instalment.
Mixing prospective and retrospective methods with different rates.
The rate changed during the loan and the formulas are mixed.
Fix: If the rate varies, roll the balance forward period by period, or value the remaining payments at the rate that applies to them.

Worked examples

Example 1

A loan of ₹10,00,000 is repaid by level annual instalments in arrears over 5 years at 8% effective per year. Find the instalment, then the interest and capital in the second instalment.

Show the solution

Rate i = 0.08, n = 5. v = 1 ÷ 1.08 = 0.925926.
v^5 = 0.680583. a_5 = (1 − 0.680583) ÷ 0.08 = 3.992710.
X = 10,00,000 ÷ 3.992710 = ₹2,50,456 (nearest rupee).
Balance after instalment 1: 10,00,000 × 1.08 − 2,50,456 = 10,80,000 − 2,50,456 = ₹8,29,544.
Interest in instalment 2 = 0.08 × 8,29,544 = ₹66,364.
Capital in instalment 2 = 2,50,456 − 66,364 = ₹1,84,092.

Answer: Instalment ≈ ₹2,50,456. In the second instalment, interest ≈ ₹66,364 and capital ≈ ₹1,84,092.

Example 2

A loan is repaid by 10 level annual instalments of ₹1,00,000 in arrears at 6% effective per year. Find the loan outstanding just after the 4th instalment by the prospective method, and check it by the retrospective method.

Show the solution

i = 0.06, n = 10, t = 4. Six payments remain.
Prospective: B_4 = 1,00,000 × a_6.
v^6 = 1.06^(−6) = 0.704961. a_6 = (1 − 0.704961) ÷ 0.06 = 4.917324.
B_4 = ₹4,91,732.
Retrospective: loan L = 1,00,000 × a_10. 1.06^(−10) = 0.558395, so a_10 = 0.441605 ÷ 0.06 = 7.360087, L = ₹7,36,009.
Accumulate: 1.06^4 = 1.262477. L × 1.262477 ≈ 9,29,194.
s_4 = (1.262477 − 1) ÷ 0.06 = 4.374616. Payments accumulated = ₹4,37,462.
B_4 = 9,29,194 − 4,37,462 = ₹4,91,732, which agrees with the prospective value.

Answer: Outstanding balance after the 4th instalment ≈ ₹4,91,732 (both methods agree).

Exam tips

Always state the valuation date and write the equation of value in words before the numbers. Examiners give method marks for it.
Use the prospective method as your main route. It needs fewer steps. Use the retrospective method only as a check or when asked.
Show a short schedule table in written answers and confirm the final balance is zero. This catches errors fast.
In computer-based Paper B questions, set the rate per period in a cell, use a formula for X, and fill the schedule down with formulas rather than typed numbers.
Read the timing carefully. Arrears versus advance, and a deferred first payment, are the usual traps in MCQs.

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

Equation of Value and Loan Schedules: frequently asked questions

What is the equation of value for a loan?

It says the loan amount equals the present value of all repayments at the loan rate. You choose a date, usually the loan date, and discount each payment to it. Solve the equation for the unknown, such as the instalment.

How do I split an instalment into interest and capital?

Multiply the opening balance by the rate per period to get the interest. Subtract that from the instalment to get the capital. The closing balance is the opening balance less the capital.

Do the prospective and retrospective methods give the same balance?

Yes, when the same interest rate applies throughout the loan and the loan is fully consistent with the payments. The prospective method values future payments. The retrospective method accumulates past cashflows. Rounding may cause tiny differences.

Why is the capital repaid small at the start of a loan?

Interest is charged on the full outstanding balance, which is largest at the start. With a level instalment, more of it goes to interest early. Capital repayments then grow at the rate i each period.