**Derivation of Single Payment Formulae**

Note! The formulae derivations of Appendix 5/1 and Appendix 5/2 are included for ‘first principles’ purists like myself, and to pre-empt **any** credulity being given to the customary denial position adopted by the Financial Services Institutions.

**SINGLE PAYMENT FORMULAE**

If we invest a Present Sum P for n interest periods at an interest rate i per interest period, this will yield a Final Sum S at the end of n periods.

The following diagrammatic illustration will facilitate your understanding of the inter-relationship between the Single Payment Formulae parameters.

**Single Payment/Investment (P)**

at **Time Zero**

**i** = interest rate per interest period

If we invest **P** for **1** interest period at the interest rate i per interest period, then

**S at the end of 1 interest period**=P + Pi=**P( 1 + i )**

If we invest **P** for **2** interest periods at the interest rate i per interest period, then

**S at the end of 2 interest period**=P( 1 + i ) ( 1 + i )=**P( 1 + i )2**

If we invest **P** for **3** interest periods at the interest rate i per interest period, then

**S at the end of 3 interest period**=P( 1 + i ) ( 1 + i ) (1 + i )=**P( 1 + i )3**

If we invest **P** for **n** interest periods at the interest rate i per interest period, then

**S at the end of n interest periods**=**P( 1 + i )n**

Hence the following formulae :-

**(F1)S=**

**(F2)P=**

If we borrow the sum **P** over **n** interest periods at interest rate **i** per interest period, then we will be required to repay the sum **S** at the end of the **n** periods.

**Illustration of Logarithmic Solution using 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?

**From Formula (F1) we have:** S=P( 1 + i )^{n}

=( 1 + i )^{n }

**Note!** If you have a scientific calculator with a function on it, you could work out the value of ** i** immediately, at this stage.

= = **12.3%p.a. **

**The value of i is computed using Logarithms as follows:**

=log ( 1 + i )

^{n}=n log ( 1 + i )

=log ( 1 + i )

log ( 1 + i ) = 0.0504

( 1 + i ) = 10^{0.0504}

** **
**Giving: i = 10**^{0.0504} - 1=12.3%p.a.