Inter-Quartile Range
2. Financial Mathematics
2.3 Present Value
Learning Objective
5.2.1 Be able to calculate the present value of: lump sums; regular payments; annuities; perpetuities
The other formula that advisers should be able to do readily is one called present value. This is the reverse of the future value formula and is expressed as:
Present value = Terminal value
(1+r)n
2.3.1 Present Value of a Future Lump Sum
So going back to the same example, we saw that $100 invested for five years at 5% per annum would be worth $127.63 at the end of the term. The present value formula can be used to answer the question:
how much needs to be invested today to produce $127.63 if the funds can be invested for five years and are expected to earn 5% per annum?
The amount that needs to be invested today = $127.63 = $100
(1 + 0.05)5
Using the Microsoft Windows calculator that you will need to use in the exam, you enter this as: 127.63
÷ 1.05 followed by pressing ‘x^y’ (or xy), then 5 and then the equals sign.
As an example of a practical use of this formula, consider the question: how much does an investor need to invest today if they need a lump sum of $25,000 in seven years’ time and the rate that can be earned is 6% per annum? The formula would be:
$25,000
(1+0.06)7 = $16,626.43
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A further useful tool is to know how to simply calculate how long it would take for an investment to double in value. Something known as the ‘Rule of 70’ gives a shorthand way of working this out. If the investment is expected to grow at 5% per annum, then you divide 70 by the rate of interest (5%) which gives 14 – in other words it will take about 14 years for the investment to double in value at a compound rate of growth of 5%.
This is an approximate figure only, but you can see how reasonably close the result is by using the future value formula: $100 invested today at 5% per annum for 14 years is:
$100 * (1.05)14 = $197.99
Being readily able to calculate how much a client’s investment might grow by or what sum is needed to achieve a desired objective is clearly a key skill in providing the correct financial advice.
As the present value formula expresses future cash flows in today’s terms, it allows a comparison to be made of competing investments of equal risk which have the same start date but have different payment timings or amounts.
2.3.2 Present Value of an Annuity or Regular Payment
Present value of an annuity refers to a series of equal cash payments that will be received over a specified period of time.
As before, the present value of an annuity is calculated by discounting the cash flows to today’s value.
The same formula can be used for regular payments.
We will first consider where payments are made in arrears. If we expect to receive payments of $100 over the next three years, we can calculate their present value (assuming interest rates are 5%) using the above formula:
Year Cash Flow Discount Rate Formula Discount Factor Present Value
1 100.00 5% 100.00 ÷1.05 0.9524 95.24
2 100.00 5% 100.00 ÷1.05^2 0.9070 90.70
3 100.00 5% 100.00 ÷1.05^3 0.8638 86.38
Total 272.32
So the present value of those future payments is $272.32. The table also shows how this converts into a discount factor, that is, how much is the future value discounted by in decimal terms.
Instead of calculating each present value, this can be calculated by using another formula:
Present value of an annuity = $ x x 1 1 – 1 r (1 + r)n
where:
• $x is the amount of the annuity paid each year;
• r is the rate of interest over the life of the annuity;
• n is the number of periods that the annuity will run for.
Taking the example above, the formula would become:
Present value of an annuity = $100 x 1 0.05 (1 + 0.05)
[
1 – 1 3]
To calculate this using the Microsoft Windows calculator that you will need to use in the exam, start with the figures inside the square brackets and enter 1 – 1 ÷ 1.05 ^ 3 followed by = to give 0.136162401.
Then multiply this by the values outside the square bracket, in other words, 0.136162401 x 100 x 1 ÷ 0.05 followed by = which will give the answer of 272.32.
An alternative method of calculation and one that can also be used to find out the present value of a bond is:
Present value of an annuity = Annuity x
[
1 – (1+r)r –n]
You should note that (1+r)–n is more simply calculated as 1
(1+r)n
So, using the same figures as above, the formula becomes:
£100 x
[
1 –(
1) ]
or[
1 –(
1) ]
1.053 1.157625 or
[
1 – 0.8638376]
=0.05 0.05 0.05
100 x 0.13616 = 272.32 0.05
The present value of an annuity can be used for calculating such things as an annuity or the monthly repayments on a mortgage. It can also be used in investment appraisal.
2.3.3 Present Value of Perpetuities
Perpetuity is a series of regular cash flows that are due to be paid or received indefinitely, which in practice is defined as a period beyond 50 years.
It is simply calculated using the following formula:
Present value of a perpetuity = annuity x 1
r
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So, for example, if $1,000 is to be received each year in perpetuity, what is its present value at an interest rate of 5%?
$1,000 x 1 = $20,000 0.05
Although a perpetuity really exists only as a mathematical model, it can be used to approximate the value of a long-term stream of equal payments by treating it as an indefinite perpetuity. So, for example, if you have a commercial property that generates $10,000 of rental income and the discount rate is 8%, then the formula can be used to calculate its present value by capitalising those future payments into its present value, which would be $125,000.