Present Value (Fixed Interest Rates)

Video coming soon3 worked examples

Theory

Often in financial planning, a person will have a specific future savings goal in mind (e.g., saving a deposit for a house or paying for university fees). When we need to find out exactly how much money must be invested today to reach that future target, we are calculating the Present Value.

1. Reversing the Formula

Because we are working backwards from the end of the timeline to the start, we must perform the exact opposite mathematical operation to our standard accumulation formula.

Instead of multiplying the capital, we take the final future balance and divide it by the compound multiplier.

\text{Present Value} = \frac{\text{Future Balance}}{(\text{Multiplier})^n}

2. The Golden Rule Still Applies

Even though we are working backwards, the "Golden Rule of Time Units" still applies.

The power you use for your time period ( $n$ ) must always perfectly match the time units of the interest rate you are given.
If the rate is per month, $n$ must be the total number of months.

Worked examples

Example 1

Example 1: Basic Present Value (Years)

Alistair wants to buy a new car in exactly 3 years' time. He estimates that he will need £8,000. He opens a savings account that offers a fixed effective rate of interest of 4.5% per year.

Calculate the exact minimum amount Alistair must deposit into the account today to ensure he reaches his £8,000 target.

Find the multiplier: $100\% + 4.5\% = 104.5\% \rightarrow 1.045$ .
Match the time period: The rate is annual, and the time is 3 years. $n = 3$ .
Apply the Present Value formula:
$\text{Present Value} = \frac{\text{\pounds}8000}{1.045^3}$
$\text{Present Value} = \frac{\text{\pounds}8000}{1.141166...} = \text{\pounds}7,010.37$ .

Alistair needs to deposit £7,010.37 today.

Example 2

Example 2: Present Value with Mismatched Time Units

Bianca wants to save £15,000 to pay for a wedding in 2 years' time. She finds a high-yield savings bond that offers a fixed effective rate of interest of 1.2% per quarter.

Calculate the amount of money she needs to invest in the bond today to reach her target.

Find the multiplier: $100\% + 1.2\% = 101.2\% \rightarrow 1.012$ .
Match the time period: The rate is per quarter. There are 4 quarters in a year, so over 2 years, $n = 2 \times 4 = 8$ quarters.
Apply the Present Value formula:
$\text{Present Value} = \frac{\text{\pounds}15000}{1.012^8}$
$\text{Present Value} = \frac{\text{\pounds}15000}{1.100127...} = \text{\pounds}13,634.78$ .

Example 3

Example 3: Working Backwards from Maturity

A long-term investment bond matured after exactly 42 months, returning a final accumulated balance of £5,420. The bond had a fixed effective rate of interest of 2.15% per half-year.

Calculate the original amount of money that was invested in the bond.

Find the multiplier: $100\% + 2.15\% = 102.15\% \rightarrow 1.0215$ .
Match the time period: The rate is per half-year (every 6 months). To find how many half-years are in 42 months, we divide by 6. $n = 42 \div 6 = 7$ half-years.
Apply the Present Value formula:
$\text{Present Value} = \frac{\text{\pounds}5420}{1.0215^7}$
$\text{Present Value} = \frac{\text{\pounds}5420}{1.160677...} = \text{\pounds}4,669.69$ .

Ready to practise?

4 Present Value questions with answers and video solutions