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# Compounding and discounting

The time value of money is accounted for by the simple concept of compounding interest.

There is a required rate, which since the arithmetic is being used in an exercise of 'discounting' will be called the discount rate. Since the required rate is so critical as to whether the investment proceeds or not, setting required rates for investment is a key strategic issue and is discussed further below.

Compounded amounts increase in a geometric progression: at the end of a period (normally a year), 1 becomes 1 + the interest rate. With a rate of 12 per cent, 1 becomes 1.12. This is then the base sum on which interest is calculated for the second year. The sum at the end of the second year = 1.12 x 1.12 = 1.2544 and so on.

Discounted future amounts are decreased in a similar progression. Discounting is the inverse of compounding. Table 11.1 shows the progression in compounding or discounting at 12 per cent over five years.

TABLE 11.1 Progression in compounding or discounting at 12 per cent over five years

Tables of compound and discount factors exist, but it is easier to obtain the factors from a spreadsheet calculation as above. These factors form an integral part of an appraisal model. The formulae and notation are as follows:

Required/interest/discount rate r or i

Number of years or periods y or n

Future amount, worth or value F

Present amount, worth or value P The compound factor for a future amount F F = (1+r) An

(A is the symbol for to the power, upper

case on the key for 6 on the keyboard)

The discount factor to arrive at a present P = l/(l+r) An or (1+r)A_n

amount P

# Time base

The convention used when dealing with cash flows is to assume that costs will be incurred at the earliest time and that income will not arise until the end of a period. There is a time zero, year 0, and the factor at this time is 1, as the present value is the same as the future value (ie now!) and a dollar is worth a dollar. Normally the first outflows of expense or capital cost will be incurred at time zero, year 0, and the first inflows recorded at year 1, at the end of the first year. This is the concept of prudence in operation and is adopted in most appraisals.

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