What do we mean by the **time value of money**?

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In simple terms, it is about how we calculate the **future value** of an investment, or

how we work back from a future required value to calculate the **present value**.

Compounding is where we take the known **present value** and use this to calculate a the potential **future value**.

For example, John has invested £50,000 into his investment and wants to know, assuming 5% net, what this will be worth in 10 years time when he retires.

What would the formula look like?

The formula:

FV = PV (1+r)^{n}

Where:

FV = Future value

PV = Present value

r = Rate of return

n = Number of years

Using John as an example, what would his investment be worth in 10 years time?

FV = PV x (1 + 0.05)^{10}

FV = £50,000 x (1.05)^{10}

FV= £50,00 x 1.6289

FV = £81,445

In AF4, you are recommended to have a financial /scientific calculator. You should have this before your course and be familiar with undertaking a calculation such as this.

Discounting is working from a **known future value** to determine the **present value**.

For example, Janet wants to provide university fees of £15,000 to her grandaughter in 12 years time. Assuming a 3% net return, how much must she invest now?

What would the formula look like? I'll give you a clue, it uses all of the same elements as the compounding formula but in a different order.

Discounting formula:

PV = __ FV __

(1+r)^{n}

Where:

PV = Present value

FV = Future value

r= Rate of retiurn

n = Number of years

Example. How much does Janet need to invest today for her grandchild's university fees?

If you don't have a calculator, work your way through the steps to the process required.

PV = __ FV __

(1+r)^{n}

PV = __£15,000__

(1+03)^{12 }

PV =__ £15,000__

1.4258

PV= £10,520

How do we find the annual compound interest rate when we know the present and future value along with the time frame?

r = [(FV / PV)^{1/n} - 1] x 100

•FV = Future value

•PV = Present value

•r = Interest rate as a decimal e.g. 4% = 0.04

•n = Number of years

Kate requires £35,000 in 12 years time for little Hugo’s university fees. She has £21,000 available to invest now. What annual rate of return will be required to reach her goal?

If you don't have a calculator, work your way through the steps to the process required.

r = [(FV / PV)^{1/n} - 1] x 100

r = [(£35,000 / £21,000)1/12 - 1] x 100

r = [1.6671/12 - 1] x 100

r = [1.0435 - 1] x 100

r = 4.35%

How do we find the Annual Effective Rate (AER) when there are more than one payments in a year?

AER = [(1 + r/n)^{n} - 1] x 100

FV = Future value

PV = Present value

r = Nominal interest rate

n = Number of payments made in a year

Best Bank pays a nominal interest of 3.2% gross per annum, paid monthly. Calculate, showing all your workings, the Annual Effective Rate (AER).

If you don't have a calculator, work your way through the steps to the process required.

Best Bank:

AER = [(1 + r/n)^{n} - 1] x 100

(1 + 0.032 / 12)^{12} – 1 x 100

(1.03247)^{12} – 1 x 100

= 3.25%