CHAPTER 5 HOW MONEY WORKS


If we invest an equal amount R at the End of each period for a duration of n interest periods, at an interest rate i per interest period, these investments 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 Uniform Series Payment Formulae and are as follows: (F3)S=R (F4)R=S  The following diagrammatic illustration will facilitate your understanding of the interrelationship between the Uniform Series Payment Formulae parameters. (For an understanding of the derivation of these formulae, see Appendix 5/2.)
i = interest rate per Final Sum (S) achieved
In the above Uniform Series of Payments investment situation we see, from Formula (F3), that we can realise a Final Sum S at the end of n interest periods, by investing an equal amount R at the End of each interest period (over the n periods) at an interest rate i per interest period. But from Formula (F1) (see Section 5.1) we know that a Final Sum S, at the end of n interest periods, can also be realised by investing a present sum P (at Time Zero) at an interest i per interest period, over the n periods. i.e. S = P(1 + i)n
(F5)R=P (F6)P=R
Single Payment / Investment (P) Final Sum (S) achieved
It is important to understand that financial analysis evaluation using these basic Uniform Series Payment Formulae is on the basis that the series of equal payments are End of interest period payments. The mathematics of the financial analysis evaluation using these basic formulae is on the basis that the first payment / investment is made at the End of the first interest period. For example: ——— if the interest period applicable is 1 month, then the first payment / investment is to be made at the end of the first month, and Time Zero will therefore be at the beginning of the first month; ——— if the interest period applicable is 1 year, then the first payment / investment is to be made at the end of the first year, and Time Zero will therefore be at the beginning of the first year. Note! For situations where payments are made from the start of the first interest period (which is generally the case with investments), the use of the general Geometric Series format would be employed (as in the derivation of the Uniform Series Payment Formulae in Appendix 5/2) or a timevalue adjustment could be made to the above formulae. But these situations are not of relevance to the subject matter of this websitebook. We will now illustrate the use and meaning of the Uniform Series Payment Formulae, by example.
Example 5.3 If we invest an equal amount (R) of £2,000 at the end of each year for 5 years, at an interest rate of 10% p.a., what will be the Final Value (FV) of our investment at the end of the 5 year period? From Formula (F3)S=R R = £2,000n = 5 yearsi = 10% p.a. ThereforeS=£2,000 S=£2,000 =£12,210.20 i.e. the Final Value of our investment at the end of the 5 year period will be £12,210.20.
So, our Final Value (FV) at the end of year 5 is ( £9,282 x 1.1 ) + £2,000 = £12,210.20, as we have already computed using the Uniform Series Formula (F3).
Using the parameters of Formulae (F1), (F2), (F3), (F4), (F5) and (F6), and the data of Example 5.3 above, the following logically equivalent statements can be made:
We know from the above Example that the value of n computes at 5 years. NOTE ! In the above case the value of n could be computed directly, using logarithms. (For the mathematical purists, this logarithmic solution method, using the parameters of Example 5.3 above, is illustrated in Appendix 5/2.) But, again, a main purpose of these early Examples is to introduce the mathematical Iteration process of ‘trial and error’ by which an unknown parameter can be computed.
Example 5.4
If we invest £3,000 at the end of each year for a period of 5 years and we receive a Final Sum of £19,518.13 at the end of the 5 year investment period, what (Internal) Rate of Return will we have achieved on our investment? We know:R=£3,000 n=5 years S=£19,518.13 i=?
S = R
From Trial No. 1,i = 10% p.a. yields S = £18,315.30. This is lower than the Final Value of £19,518.13 required, so we must try a higher value of i. From Trial No. 2,i = 15% p.a. yields S = £20,227.14. This is higher than the Final Value of £19,518.13 required, so we must try a lower value of i. We now know that the value of i lies between 10% p.a. and 15% p.a.
Trial No. 2 i2=15%p.a. yields S2=£20,227.14
i = 10% + = 13.143%
Copyright © 2013, 2014 John O'Meara. All Rights Reserved. 