Accumulation Calculations (Regular Payments)

Video coming soon3 worked examples

Theory

When a person makes regular payments into a savings account (e.g., depositing £100 on the 1st of every month), you cannot simply add the payments together and calculate the interest at the end. Because the deposits are made at different times, each deposit sits in the account earning interest for a different length of time.

To calculate the final accumulated balance, you can choose between two valid mathematical methods. Qualifications Scotland awards full marks for both, so you should use the one that makes the most logical sense to you.

Method 1: The Chronological Method (Payment Tracking)

This is often the safest method to avoid getting confused by dates. You track the balance of the account sequentially from one payment to the next.

Take the first deposit.
Calculate the interest it earns up until the exact moment the next deposit is made.
Add the new deposit to the running total.
Calculate the interest on this new combined balance up to the next deposit, and repeat.

Method 2: The Individual Accumulation Method

This method treats every single deposit as its own separate "mini-investment".

Calculate exactly how much time Deposit 1 spends in the account, and calculate its final value.
Calculate exactly how much time Deposit 2 spends in the account, and calculate its final value.
Repeat for all deposits, and then add all the final values together at the very end.

Crucial Exam Trap:

Always read the final sentence of the question carefully! You must check whether you are calculating the balance immediately after the final payment is made (meaning the final payment earns zero interest), or some time after the final payment (meaning the final payment also earns interest).

Worked examples

Example 1

Example 1: Basic Regular Payments (Chronological Method)

Liam deposits £400 into a savings account on 1 January 2024, 1 January 2025, and 1 January 2026. The account pays an effective rate of interest of 3.5% per year.

Calculate the balance of Liam's account on 1 January 2026, immediately after he makes his final deposit.

We will use the Chronological Method to track the balance year by year.

1 Jan 2024: Balance = £400.
1 Jan 2025 (Before deposit): The £400 earns one year of interest. $\text{\pounds}400 \times 1.035 = \text{\pounds}414$ .
1 Jan 2025 (After deposit): Liam deposits another £400. $\text{\pounds}414 + \text{\pounds}400 = \text{\pounds}814$ .
1 Jan 2026 (Before deposit): The £814 earns one year of interest. $\text{\pounds}814 \times 1.035 = \text{\pounds}842.49$ .
1 Jan 2026 (After deposit): Liam makes his final £400 deposit. $\text{\pounds}842.49 + \text{\pounds}400 = \text{\pounds}1,242.49$ .

Final Balance = £1,242.49

Example 2

Example 2: Varying Interest Rates (Individual Accumulation Method)

Nadia deposits £150 into a new savings account on 1 March, 1 April, and 1 May. The account has the following effective rates of interest:

0.5% per month for March and April.
0.8% per month from 1 May onwards.

Calculate the balance of Nadia's account on 1 June.

We will use the Individual Accumulation Method, tracking how long each specific £150 deposit sits in the account before 1 June.

The 1 March Deposit: Sits for 2 months at 0.5% (March, April) and 1 month at 0.8% (May).
Value = $\text{\pounds}150 \times 1.005^2 \times 1.008 = \text{\pounds}152.715...$
The 1 April Deposit: Sits for 1 month at 0.5% (April) and 1 month at 0.8% (May).
Value = $\text{\pounds}150 \times 1.005 \times 1.008 = \text{\pounds}151.956$
The 1 May Deposit: Sits for 1 month at 0.8% (May).
Value = $\text{\pounds}150 \times 1.008 = \text{\pounds}151.20$
Total Balance: Add the three individual values together.
$\text{\pounds}152.715... + \text{\pounds}151.956 + \text{\pounds}151.20 = \text{\pounds}455.87$ (rounded to nearest penny).

Example 3

Example 3: Mixed Time Units (Exam Style)

A charity sets up a fund, depositing £1,000 on 1 January 2023, 1 January 2024, and 1 January 2025. The effective rate of interest on the fund is:

4.2% per year during 2023.
1.1% per quarter during 2024 and 2025.

Calculate the accumulated value of the fund on 1 January 2026.

We will use the Chronological Method. Keep the long decimals on your calculator screen and do not round prematurely!

1 Jan 2023: Balance = £1,000.
1 Jan 2024 (After 1 year at 4.2%): ( $\text{\pounds}1000 \times 1.042$ ) = £1042.
Add the new deposit: $\text{\pounds}1042 + \text{\pounds}1000 = \text{\pounds}2042$ .
1 Jan 2025 (After 4 quarters at 1.1%): ( $\text{\pounds}2042 \times 1.011^4$ ) = £2133.3503...
Add the new deposit: $\text{\pounds}2133.3503... + \text{\pounds}1000 = \text{\pounds}3133.3503...$
1 Jan 2026 (After 4 quarters at 1.1%): ( $\text{\pounds}3133.3503... \times 1.011^4$ ) = £3,273.50.

Final Balance = £3,273.50

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5 Accumulation with Regular Payments questions with answers and video solutions