Present Value (Variable Interest Rates)

Video coming soon3 worked examples

Theory

Calculating the Present Value of an investment becomes slightly more complex when the interest rate changes over the term of the investment. Just as we used a chain of multipliers to accumulate money forward, we can use a chain of division to work backwards.

1. The Reverse Chain Calculation

To find the original deposit when there are multiple interest rates, you take the final balance and divide it by the product of all the multipliers.

\text{Present Value} = \frac{\text{Future Balance}}{(m_1)^{n_1} \times (m_2)^{n_2} \times \dots}

Exam Tip:

It is often safer to calculate the bottom line (the chain of multipliers) first, keep the long decimal on your calculator screen, and then perform the final division to avoid rounding errors.

2. Present Value with Intermediate Transactions

This is one of the most challenging question types in the Higher Applications course. If an account has a variable interest rate and the person makes deposits or withdrawals halfway through, you cannot use a single formula.

The "Reverse Chronological" Method: You must work backwards step-by-step from the end date to the start date.

The Ultimate Trap:

Because you are travelling backwards in time, the effect of transactions is reversed.

If someone withdrew money in the past, you must add it back onto your running total.
If someone deposited money, you must subtract it to find the balance before the deposit occurred!

Worked examples

Example 1

Example 1: Single Deposit with a Variable Rate

A single deposit was placed into a 5-year savings bond. After exactly 5 years, the bond matured and returned a final balance of £12,000. The effective rates of interest for the bond were:

4.0% per year for the first 3 years.
1.5% per half-year for the final 2 years.

Calculate the value of the original deposit.

First Period (Years): Multiplier = 1.04. Time = 3 years.
Second Period (Half-years): Multiplier = 1.015. Time = 4 half-years (since 2 years = 4 half-years).
Apply the Reverse Chain Formula:
$\text{Present Value} = \frac{\text{\pounds}12000}{1.04^3 \times 1.015^4}$
$\text{Present Value} = \frac{\text{\pounds}12000}{1.124864 \times 1.06136...}$
$\text{Present Value} = \frac{\text{\pounds}12000}{1.19389...} = \text{\pounds}10,051.14$ .

Example 2

Example 2: Mismatched Time Units

After exactly two years, a business has accumulated £4,500 in a corporate savings account from a single initial deposit. The account had an effective rate of interest of:

0.6% per month during the first year.
2.1% per quarter during the second year.

Calculate the exact amount of money the business originally deposited.

First Period (Months): Multiplier = 1.006. Time = 12 months (for the first full year).
Second Period (Quarters): Multiplier = 1.021. Time = 4 quarters (for the second full year).
Apply the Reverse Chain Formula:
$\text{Present Value} = \frac{\text{\pounds}4500}{1.006^{12} \times 1.021^4}$
$\text{Present Value} = \frac{\text{\pounds}4500}{1.07442... \times 1.08672...}$
$\text{Present Value} = \frac{\text{\pounds}4500}{1.16763...} = \text{\pounds}3,853.96$ .

Example 3

Example 3: The "Reverse Chronological" Method (Transactions)

Darren holds a savings account. He makes the following transactions:

1 January 2022: Opens the account with an initial deposit.
1 January 2023: Withdraws £500 to pay for car repairs.

On 1 January 2024, Darren checks his account and sees a final balance of £8,000. The effective rate of interest was 4.0% per year in 2022, and 6.0% per year in 2023.

Calculate the Present Value (the initial deposit) of Darren's account on 1 January 2022.

We must work backwards from 2024, undoing the interest and reversing the transactions as we travel back in time.

Reverse the 2023 Interest (1 Jan 2024 back to 1 Jan 2023):
Balance after withdrawal = $\frac{\text{\pounds}8000}{1.06} = \text{\pounds}7547.169...$
Reverse the Transaction on 1 Jan 2023:
Because Darren withdrew £500, we must add it back to find the balance before he took the money out.
Balance before withdrawal = $\text{\pounds}7547.169... + \text{\pounds}500 = \text{\pounds}8047.169...$
Reverse the 2022 Interest (1 Jan 2023 back to 1 Jan 2022):
Initial Deposit (PV) = $\frac{\text{\pounds}8047.169...}{1.04} = \text{\pounds}7,737.66$ .

Initial Deposit = £7,737.66

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4 Present Value with Variable Rates questions with answers and video solutions