CIMA BA2 Syllabus D. DECISION MAKING - Time Value of Money, Annuities and Perpetuities - Notes ^{1 / 3}

Money now is worth more than money in the future.

This is because we can get interest on the money now.

For example, if we have $100 in the bank, you can receive (10%) interest on it, therefore in one year, it will be worth $110 with the interest.

One way of calculating this is $100 x 1.10 = $110

The $100 value is known as the present value
The $110 value is known as the future value

This formula can work the other way round also, going from the future value to the present value like this:

$110/1.10 = $100

Illustration 1 - Time Value of Money

What is $100 worth in 2 years time? (At 10% interest rate)

Solution

$100 x 1.10 (year 1) x 1.10 (year 2) = $121

Let's do it the other way now.

What is $121 in 2 years time worth now?

$121 /1.10 (year 1) / 1.10 (year 2) = $100

This can also be written as: $121 / 1.10^2 = $100

Illustration 2 - Time Value of Money

What is the future value of $200,000 in 8 years time, with a growth rate of 1.2% per annum

Solution

$200,000 x 1.012^8 = $220,026

Illustration 3 - Compounding period is 6 months

What is the future value of $1,000 in 1 years time with a 6 month interest rate of 6%

Solution

Here, we need to compound in the period of 6 months, therefore:

$1,000 x 1.06^2 (because interest would be paid every 6 months, so 2 times in the year) = $1,123.6

Equivalent annual interest rate

If compounding interest is more than once a year, we might want to turn this into an annual interest rate, and the way to do this is:

1 + EAR = 1 + R (interest rate given)^n (number of periods in the year)

Illustration - EAR

Quarterly interest rate is 3.9%

What is the EAR?

1 + EAR = 1 + 0.039^4 (4 because it was compounded 4 times in the year)

1 + EAR = 1.165

Therefore, the EAR is 16.5% (Remove the 1)

Single future discount tables

If we have one single figure in the future, and we want to go to the present value, we can use a table to calculate it (this table will be given to you in the exam).

So, if we are told that we have $5,000 in 3 years time at an interest rate of 8%.

We can find the present value by using the table below (Column of 8% and row of 3 years) = 0.794

5,000 x 0.794 = $3,970 is the present value

Annuities

An annuity is a fixed (constant) periodic payment or receipt which continues either for a specified time or until the occurrence of a specified event, e.g. ground rent.

For example, $100 every year, for 3 years (paid at the end of each year, at an interest rate of 10%)

If we want to know the present value of this amount, we can use the annuity tables (Column 10%, Row 3)

= $100 x 2.487 = $248.7 is the present value

Illustration - Annuity with flow starting NOW

PV invested now $1,500
Then, we will get $1,500 for the next 4 years at 6%. 

What is the present value of the whole amount?

$1,500 + ($1,500 x 3.465) = $6,697.5

The reason that the $1,500 invested now is not discounted is because $1,500 today is worth $1,500 today - there is no need to discount that amount, the other amounts are being received later, and that is why we discount those.

Perpetuities

Perpetuity is a periodic payment or receipt continuing for a limitless period.

Calculating the PV of a perpetuity:

Cash flow
---------------
Interest rate (r)

For example if we have $1,000 in perpetuity starting at the end of the year at an interest rate of 10%.

 What is the present value?

$1,000 / 0.1 = $10,000