Accumulation - Irregular Payments and Withdrawals

Video coming soon3 worked examples

Theory

Real-world finances are rarely as perfectly structured as a regular monthly deposit. People receive unexpected bonuses, make emergency withdrawals, and borrow additional funds.

1. The Chronological Method (Payment Tracking)

When dealing with irregular payments and withdrawals, the "Individual Accumulation Method" from the previous section becomes highly complicated and prone to error.

Instead, you should always use the Chronological Method (Payment Tracking).

Strategy: Treat the timeline like a stepping-stone path. You must calculate the interest to move the balance forward from one specific transaction date to the exact date of the next transaction, stop, adjust the balance, and then calculate the next "step".

2. Handling Withdrawals and Repayments

Savings: A deposit is added to the balance. A withdrawal is subtracted from the balance.
Loans: Borrowing more money is added to the total debt. A repayment is subtracted from the total debt.

Crucial Rule:

Never add or subtract a transaction amount until you have completely finished calculating the interest up to that specific date!

3. The Rate Change Trap

Sometimes, an interest rate will change on a date when no transaction takes place.

You must still treat this rate change as a "stepping-stone" on your timeline. Calculate the interest up to the date of the change, note the new balance, and then continue forward using the new rate.

Worked examples

Example 1

Example 1: Irregular Deposits and Withdrawals (Consistent Rate)

Amira has a savings account that offers an effective rate of interest of 0.4% per month. She makes the following transactions:

1 February: Deposits £800
1 June: Deposits £300
1 September: Withdraws £250

Calculate the exact balance of Amira's account on 1 November of the same year.

We will track the balance chronologically from transaction to transaction. The multiplier is $100\% + 0.4\% = 1.004$ .

1 Feb to 1 June (4 months): $\text{\pounds}800 \times 1.004^4 = \text{\pounds}812.87...$
Add 1 June deposit: $\text{\pounds}812.87... + \text{\pounds}300 = \text{\pounds}1,112.87...$
1 June to 1 Sept (3 months): $\text{\pounds}1112.87... \times 1.004^3 = \text{\pounds}1126.28...$
Subtract 1 Sept withdrawal: $\text{\pounds}1126.28... - \text{\pounds}250 = \text{\pounds}876.28...$
1 Sept to 1 Nov (2 months): $\text{\pounds}876.28... \times 1.004^2 = \text{\pounds}883.31$ (rounded to nearest penny).

Final Balance = £883.31

Example 2

Example 2: Loan Repayments and Additional Borrowing

A small business takes out a commercial loan. The lender charges an effective rate of interest of 1.2% per quarter. The business makes the following transactions:

1 January 2024: Borrows £5,000
1 July 2024: Repays £1,500
1 April 2025: Borrows an additional £2,000

Calculate the total amount the business owes on 1 October 2025.

The multiplier is $100\% + 1.2\% = 1.012$ . We must count the quarters carefully.

1 Jan 2024 to 1 July 2024 (2 quarters): $\text{\pounds}5000 \times 1.012^2 = \text{\pounds}5120.72$
Subtract 1 July repayment: $\text{\pounds}5120.72 - \text{\pounds}1500 = \text{\pounds}3,620.72$
1 July 2024 to 1 April 2025 (3 quarters): (July->Oct, Oct->Jan, Jan->Apr)
$\text{\pounds}3620.72 \times 1.012^3 = \text{\pounds}3752.54...$
Add 1 April borrowing: $\text{\pounds}3752.54... + \text{\pounds}2000 = \text{\pounds}5,752.54...$
1 April 2025 to 1 October 2025 (2 quarters):
$\text{\pounds}5752.54... \times 1.012^2 = \text{\pounds}5,891.41$ .

Amount Owed = £5,891.41

Example 3

Example 3: Interest Rate Changes (Exam Style)

Marcus pays £2,000 into a new savings account on 1 May 2024. The bank offers an effective rate of interest of 0.3% per month. On 1 September 2024, the bank changes the interest rate to 1.5% per quarter. Marcus deposits a further £500 into the account on 1 December 2024.

Calculate the balance of Marcus's account on 1 March 2025.

Even though no money is deposited or withdrawn on 1 September, we must stop and calculate the balance on that date because the interest rate changes.

1 May to 1 Sept (4 months at 0.3%):
Multiplier: 1.003. Time: $n = 4$ .
Balance on 1 Sept: $\text{\pounds}2000 \times 1.003^4 = \text{\pounds}2,024.108...$
1 Sept to 1 Dec (1 quarter at 1.5%):
Multiplier: 1.015. Time: $n = 1$ .
$\text{\pounds}2024.108... \times 1.015^1 = \text{\pounds}2054.469...$
Add 1 Dec deposit: $\text{\pounds}2054.469... + \text{\pounds}500 = \text{\pounds}2,554.469...$
1 Dec to 1 March (1 quarter at 1.5%):
Multiplier: 1.015. Time: $n = 1$ .
Final Balance: $\text{\pounds}2554.469... \times 1.015^1 = \text{\pounds}2,592.79$ .

Final Balance = £2,592.79

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5 Accumulation with Irregular Payments questions with answers and video solutions