Working with Start and End Values

Video coming soon3 worked examples

Theory

Sometimes in finance, you will know exactly how much money you started with (the Capital) and exactly how much you ended up with (the Final Balance), but you will not know the interest rate that was applied.

To find the interest rate, you must use your algebra skills to work backwards and find the hidden multiplier.

1. Rearranging the Formula

We start with our standard compound interest formula: $\text{Final Balance} = \text{Start Value} \times (\text{Multiplier})^n$

To find the multiplier, we reverse the process in two steps:

Step 1 (Divide): $\text{Multiplier}^n = \frac{\text{Final Balance}}{\text{Start Value}}$
Step 2 (Root/Fractional Power): $\text{Multiplier} = \left(\frac{\text{Final Balance}}{\text{Start Value}}\right)^{\frac{1}{n}}$

2. The Calculator Syntax Trap (Again!)

Just like in Section 11, when you calculate the fractional power on your calculator, you must put brackets around the fraction (e.g., ^(1/n)).

Once you have the decimal multiplier, do not forget the final step: subtract 1 and multiply by 100 to convert it back into a percentage interest rate.

3. Matching the Time Period ( $n$ )

The power you use for $n$ dictates the specific time period of the interest rate you will find.

If you want to find an annual interest rate, $n$ must be the number of years. If the duration is 18 months, you must use $n = 1.5$ years.
If you want to find a monthly interest rate, $n$ must be the number of months.

Worked examples

Example 1

Example 1: Basic Start and End Values (Whole Years)

An investment of £3,500 grows to a final balance of £3,950 over a period of exactly 4 years.

Calculate the effective annual rate of interest. Give your answer as a percentage to two decimal places.

Set up the formula: $n = 4$ years.
$\text{Multiplier}^4 = \frac{3950}{3500}$
Calculate the multiplier:
$\text{Multiplier} = \left(\frac{3950}{3500}\right)^{\frac{1}{4}}$
$\text{Multiplier} = 1.12857...^{\frac{1}{4}} = 1.0307...$
Convert to a percentage:
$1.0307... - 1 = 0.0307...$
$0.0307... \times 100 = 3.07\%$ .

Example 2

Example 2: Fractional Years

A small business deposits £8,200 into a savings bond. After exactly 30 months, the bond matures and returns a final balance of £9,150.

Calculate the effective annual rate of interest on the bond. Give your answer as a percentage to two decimal places.

Match the time period: Because we want an annual rate, $n$ must be in years. 30 months ÷ 12 = 2.5 years.
Set up and solve:
$\text{Multiplier}^{2.5} = \frac{9150}{8200}$
$\text{Multiplier} = \left(\frac{9150}{8200}\right)^{\frac{1}{2.5}}$
$\text{Multiplier} = 1.11585...^{\frac{1}{2.5}} = 1.0448...$
Convert to a percentage:
$1.0448... - 1 = 0.0448...$
$0.0448... \times 100 = 4.48\%$ .

Example 3

Example 3: Short-Term Monthly Rates

A student borrows £600 to buy a new laptop. They make no repayments. After 7 months, their total debt has increased to £612.50.

Calculate the effective monthly rate of interest charged on the debt. Give your answer as a percentage to two decimal places.

Match the time period: Because we want a monthly rate, $n$ must be in months. $n = 7$ .
Set up and solve:
$\text{Multiplier}^7 = \frac{612.50}{600}$
$\text{Multiplier} = \left(\frac{612.50}{600}\right)^{\frac{1}{7}}$
$\text{Multiplier} = 1.02083...^{\frac{1}{7}} = 1.00294...$
Convert to a percentage:
$1.00294... - 1 = 0.00294...$
$0.00294... \times 100 = 0.29\%$ .

Ready to practise?

4 Start and End Values questions with answers and video solutions