Combining Interest Rates

Video coming soon3 worked examples

Theory

In the real world, interest rates rarely stay exactly the same for years at a time. The Bank of England regularly changes the central interest rate, meaning the effective rates of interest on savings accounts and loans will fluctuate over long periods.

1. The "Chain" Calculation

To calculate the final balance of an account when the interest rate changes, you do not need to calculate each time period separately.

Instead, you can find the accumulated value by multiplying the original principal by a continuous chain of multipliers, each raised to the power of its own specific time period.

\text{Final Balance} = \text{Capital} \times (m_1)^{n_1} \times (m_2)^{n_2} \times \dots

2. The "Counting Time" Trap

The Classic Exam Trap:

According to your course notes, miscounting the number of months between two dates is one of the most common errors made in these types of questions.

You must take your time to physically count the months or years to ensure your powers ( $n$ ) are absolutely correct. For example, the time between 1 March and 1 September is exactly 6 months.
Just like in the previous section, the time period you use for your power ( $n$ ) must always perfectly match the time units of the specific interest rate.

Worked examples

Example 1

Example 1: Combining Years and Half-Years

Oliver deposits £2,500 into a long-term savings bond. The effective rates of interest on the bond are as follows:

3.4% per year for the first 2 years.
1.6% per half-year for the next 18 months.

Calculate the accumulated value (balance) of Oliver's bond at the end of the 3.5 year period.

First Period (Years):

Multiplier: $100\% + 3.4\% = 103.4\% \rightarrow 1.034$ .
Time ( $n$ ): The rate is per year, and the duration is 2 years, so $n = 2$ .

Second Period (Half-Years):

Multiplier: $100\% + 1.6\% = 101.6\% \rightarrow 1.016$ .
Time ( $n$ ): The rate is per half-year. 18 months is exactly 1.5 years, which contains 3 half-years, so $n = 3$ .

Chain Calculation:

$\text{Final Balance} = \text{\pounds}2500 \times 1.034^2 \times 1.016^3$

$\text{Final Balance} = \text{\pounds}2500 \times 1.069156 \times 1.04877... = \text{\pounds}2,802.73$ (rounded to the nearest penny).

Example 2

Example 2: Counting Months and Quarters

Aisha takes out a business loan of £4,200 on 1 May 2023. She makes no repayments. The lender applies the following effective rates of interest:

0.8% per month between 1 May 2023 and 1 November 2023.
2.5% per quarter from 1 November 2023 to 1 August 2024.

Calculate the amount Aisha owes on 1 August 2024.

First Period (Months):

Multiplier: $100\% + 0.8\% = 100.8\% \rightarrow 1.008$ .
Time ( $n$ ): Counting carefully from 1 May to 1 November is exactly 6 months. So, $n = 6$ .

Second Period (Quarters):

Multiplier: $100\% + 2.5\% = 102.5\% \rightarrow 1.025$ .
Time ( $n$ ): Counting carefully from 1 Nov 2023 to 1 Aug 2024 is exactly 9 months. Since there are 3 months in a quarter, 9 months is exactly 3 quarters. So, $n = 3$ .

Chain Calculation:

$\text{Amount Owed} = \text{\pounds}4200 \times 1.008^6 \times 1.025^3$

$\text{Amount Owed} = \text{\pounds}4200 \times 1.04897... \times 1.07689... = \text{\pounds}4,744.40$ .

Example 3

Example 3: Calculating an Overall Effective Rate

The effective rate of interest on a financial product is:

5.2% per year for the first year.
0.45% per month for the second year.

Calculate the overall effective rate of interest for the entire two-year period, giving your answer as a percentage to two decimal places.

To find the overall effective rate, we multiply the multipliers for the entire period together. (Tip: You can imagine you are investing exactly £1 to make this easier to visualise).

Year 1 Multiplier: 1.052 ( $n = 1$ ).
Year 2 Multiplier: 1.0045 ( $n = 12$ , because it is a monthly rate for a full year).

Combine Multipliers:

$\text{Overall Multiplier} = 1.052 \times 1.0045^{12}$

$\text{Overall Multiplier} = 1.052 \times 1.055356... = 1.11023...$

Convert Back to Percentage:

$1.11023... - 1 = 0.11023...$

$0.11023... \times 100 = 11.02\%$ .

The overall effective rate of interest for the two years is 11.02%.

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7 Combining Interest Rates questions with answers and video solutions