# Compounding and Discounting of FV

Our objective is to calculate and compare returns on an investment in a capital project with an alternative equal risk investment in securities traded in the financial markets. This comparison is made using a technique called discounted cash flow (DCF) analysis.
Because a DCF analysis is the opposite of the concept of compounding interest, we shall initially focus on compound interest calculations, Suppose you are investing \$100 000 in a risk-free security yielding a return of 10 percent
payable at the end of each year. Below photo shows that if the interest is reinvested, your investment will accumulate to \$146,410 by the end of year 4. Period 0 in the first column of photo means that no time has elapsed or the time is now, period 1 means one year later, and so on The values in photo can also be obtained by using the formula:

FVn= Vo(1 + K)n where FVn denotes the future value of an investment in n years, ( denotes the amount invested at the beginning of the period (year O), K denotes the rate of return on the investment and n denotes the number of years for which the money is invested.

The calculation for \$100 000 invested at 10 percent for two years is: FV2= \$100 000 (1 + 0.1O)2= \$121,000

In photo, all of the year-end values are equal as far as the time value of money is concerned. For example, \$121,000 received at the end of year 2 is equivalent to \$100 000 received today and invested at 10 percent. Similarly, \$133,100 received at the end of year 3 is equivalent to \$121,000 received at the end of year 2, because \$121,000 can be
invested at the end of year 2 to accumulate to \$133,100. Unfortunately, none of the amounts are directly comparable at any singlie moment in time, because each amount is expressed at a different point in time.

When making capital investment decisions, we must convert cash inflows and outflows for different years into a common value. This is achieved by converting the cash f lows into their respective values at the same point in time Mathematically any point in time can be chosen, since all four figures in Exhibit 6.1 are equal to \$100,000 at year 0,  \$110,O0O at year 1 , \$121,000 at year 2, and so on However, it is preferable to choose the point in time at which the decision is taken, and this is the present time or year O. All of the values in below photo can therefore be expressed in values at the present time (i e. present value) of \$100,000.
The process of converting cash to be received in the future into a value at the present time by the use of an interest rate is termed discounting and the resulting present value is the discounted present value. Compounding is the opposite of discounting, because it is the future value of present value cash flows. Equation for calculating future values can be rearranged to produce the present value formula:

Vo (present vatue) = FVn / (1+ k)n

By applying this equation, the calculation for e 121 000 received at the end of year 2 can be expressed as

Present value = \$121,000 / (1 + 0.10)2 = \$100,000

You should now be aware that \$1 received today is not equal to \$1 received one year from today. No rational person will be equally satisfied with receiving \$1 a year from now as opposed to receiving it today, because money received today can be used to earn interest over the ensuing year. Thus one year from now an investor can have the original
\$1 plus one year’s interest on it. For example, if the interest rate rs 10 per cent each \$1 invested now will yield \$1.10 one year from now. That is \$1 received today is equal to \$1.10 one year from today at 10 percent interest Alternatively,\$1 one year from today is equal to 10.9091 today, its present value because \$0.9091, plus 10 per cent interest for one year amounts to \$1. The concept that \$1 received in the future is not equal to \$1 received today is known as the time value of money.

We shall now consider four different methods of appraising capital investments: the net present value, internal rate of return, accounting rate of return and payback methods. We shall see that the first two methods take into account the time value of money whereas the accounting rate of return and payback methods ignore this factor.