If we invest a Single Payment Present Sum P (at Time Zero), for n interest periods, at an interest rate i per interest period, this will yield a Final Sum S at the end of the n periods.
The formulae, expressing the relationship between the parameters involved in this investment situation, are referred to as Single Payment Formulae and are as follows:
(F1)S=
(F2)P=

The following diagrammatic illustration will facilitate your understanding of the interrelationship between the Single Payment Formulae parameters.
Single Payment / Investment (P)
at Time Zero
i = interest rate per interest period
Final Sum (S) achieved
We will now illustrate the use and meaning of these formulae, by example. (For an understanding of the derivation of these formulae, see Appendix 5/1.)
Example 5.1
If we invest a Present Sum (P) of £10,000 for 5 years at 10% per annum, what will be the Final Sum (S) realised by this investment at the end of the 5 years?
If we invest £10,000 for 1 year at 10% p.a., the Sum at the end of the year
=£10,000 + £10,000 x 0.1=£11,000
We tabulate the investment sequence over the 5 year period as follows:
Interest Period. Year No.

Sum at Start of Interest Period

Sum at End of Interest Period

1

£10,000

£10,000 (1.1) = £ 11,000

2

£11,000

£11,000 (1.1) = £ 12,100

3

£12,100

£12,100 (1.1) = £ 13,310

4

£13,310

£13,310 (1.1) = £ 14,641

5

£14,641

£14,641 (1.1) = £ 16,105.10

So, if we invest a Present Sum of £10,000 (at Time Zero) for a time period of 5 years at an interest rate of 10% per annum, the Final Sum at the end of the 5 year period will be £16,105.10.
Using the Single Payment Formula (F1)

S 
= 
P(1 + i)n 
where 
P 
= 
Present Sum, in this case £10,000 

i 
= 
interest rate per interest period, in this case 10% per annum 

n 
= 
number of interest periods, in this case 5 years 

S 
= 
Final Sum at the end of the n interest periods, in this case at the end of 5 years 
So, 
S 
= 
£10,000(1.1)5 
Using the xy function on a scientific calculator we compute (1.1)5 = 1.61051
Note!(1.1)5=1.1 x 1.1 x 1.1 x 1.1 x 1.1= 1.61051
Therefore 
S 
= 
£10,000(1.61051) 


= 
£16,105.10 as above 
Using the parameters of Formulae (F1) and (F2), and the data of Example 5.1 above, the following logically equivalent statements can be made:
1. 
The Final Value (FV) of a Present Sum (P) of £10,000 invested for 5 years at 10% p.a. is £16,105.10.

Note! Final Value (FV) = Final Sum value (S)
2. 
If we borrow a Present Sum (P) of £10,000 for 5 years at 10% p.a. and this borrowing is to be repaid by a Single Repayment (S) at the end of the 5 year period, then that repayment amount will be £16,105.10

3. 
If we know that we are to receive a Future Sum (S) of £16,105.10 from some source (an inheritance, a gift, an endowment or some other source) at the end of a 5 year period, what is the Present Value (PV) (i.e. the value at Time Zero), in terms of present borrowing potential, of this Future Sum if the cost of money is 10%p.a.?

Note! Present Value (PV) = Present Sum value (P)

The Present Value (PV) is obtained from Formula (F2).
P=
i.e. PV===£10,000
In other words, a Future Sum of £16,105.10 that we will receive in 5 years time affords us a Present Value (PV) borrowing potential of £10,000, when the cost of borrowing is 10% p.a.

Note! 
It is very important to grasp the concept of Present Value (PV) as being the Value at Time Zero.

4. 
If we invest a Present Sum (P) of £10,000 for a period of 5 years and we will receive a Final Value (FV) of £16,105.10 at the end of that 5 year period, what (Internal) Rate of Return will we have received on our investment?
From the Formula (F1) S=P(1 + i)^{n} i.e. FV=P(1 + i)^{n}
Therefore£16,105.10=£10,000 (1 + i)5
We know from the above Example that i computes at 10%p.a.
NOTE ! Because we are dealing here with a simple expression with a single dependent unknown, i, its value could be computed, directly, using a logarithmic expression. (For the mathematical purists, this logarithmic solution method, using the parameters of Example 5.2 above, is illustrated in Appendix 5/1.) But a main purpose of these early Examples is to introduce the mathematical process of ‘trial and error’ by which an unknown parameter can be computed. We will need to have knowledge of this process to solve for parameters under more complicated investment scenarios in later Chapters.
Note! It is clear, from the above, that Formulae (F1) and (F2) could also be written as:
FV= (F1) equivalent
PV=(F2) equivalent
where PV is the Present Value and FV is the Final Value.

The repetitive process of ‘trial and error’ used to solve the value of a variable in a mathematical expression is known as ‘Iteration’, and it is a key mathematical tool of financial analysis.
We will now illustrate this process of ‘Iteration’, by example.
Example 5.2
If we invest a Present Sum (P) of £10,000 for a period of 5 years and we will receive a Final Sum (S) of £17,860.71 at the end of the 5 year period, what (Internal) Rate of Return (i) will we have earned on our investment?
Using Formula (F1) S=P(1 + i)n
We have£17,860.71=£10,000 (1 + i)5
We want to compute the value of i that makes this expression valid.
We first assume a trial value for i.
Let i = 10% p.a. Thence Trial No.1.
We tabulate the iterative ‘trial and error’ process as follows:
Trial No.

Present Sum

Trial Value of i

Computed Value of ( 1+ i)5

Corresponding Final Sum

1

£10,000

10 % p.a.

1.61051

£16,105.10

2

£10,000

15 % p.a.

2.011357

£20,113.57

3

£10,000

12.5 % p.a.

1.802032

£18,020.32

4

£10,000

12.4 % p.a.

1.794038

£17,940.38

5

£10,000

12.3 % p.a.

1.786071

£17,860.71

From Trial No. 1,i = 10%p.a. yields S = £16,105.10. This is lower than the Final Sum of £17,860.71, so we must try a higher value of i.
From Trial No. 2,i = 15% p.a. yields S = £20,113.57. This is higher than the Final Sum of £17,860.71, so we must try a lower value of i.
We now know that the value of i is between 10% p.a. and 15% p.a.
Thence Trial No. 3,using a value of i = 12.5%p.a. ––– and so on, until we arrive at a value of i that satisfies our expression.
i computes at 12.3% p.a.,i.e. £17,860.71= £10,000 (1.123)5
Therefore the (Internal) Rate of Return received on our investment is 12.3%p.a.