Time Value of Money, Annuities and Perpetuities

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What is the time value of money?

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

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