# Mathematics of Finance

To keep pace with the competition, companies are always gong in for new ideas implemented through new projects whether it be by expansion, diversification or modernization. All these involve raising of borrowed funds, issuing of equity and purchasing of fixed assets etc., all of which affect the firm’s cash inflows and outflows over a period of time. For evaluating the project and making a sound decision the inflows and outflows are to be logically comparable. Absolute cash flows which differ in timing and risk are not directly comparable. These inflows and outflows of cash are logically comparable when they are adjusted for their differences in timing and risk i.e. by considering the time value of money.

We all know that a rupee today is more valuable than a rupee a year hence. The reasons for this are

1. Time preference for money i.e. most individuals value the opportunity to receive money now than waiting for one or more years to receive the same amount.
2. Subjective preference for present consumption over future consumption of goods and services. This may be because of urgency of their present wants or may be to avoid risk as the future is uncertain.
3. Investment opportunities in which an investor can invest present cash to earn additional cash in future.

We intuitively know that Rs 5,000 on hand is more valuable than Rs 5,000 receivable after a year. In other words, we do not invest Rs 5,000 now to be repaid the same sum after a year. But we may invest Rs 5,000 now if we are assured that some thing more than Rs 5,000 will be received at the end of the first year. This additional compensation required for investing Rs 5,000 is called interest to compensate for the time value of money or the decrease in value of money over time. Normally interest is expressed in terms of percentage per annum, For example 12% or 24% p.a. and so on.

In this article we will discuss two techniques which consider the time value of money.

1. Compounding
2. Discounting

To understand these two techniques let us consider an example where a firm is making an investment for purchase of an asset. This involves an immediate out flow of Rs 2,000 and the pattern of inflows are Rs 250, Rs 500, Rs750 and Rs 1,000 for years 1, 2, 3, and 4 respectively. This inflow and outflow can be represented on a time line as shown below

Year Rupees

0 –2000
1 250
2 500
3 750
4 1,000

Compounding: Compounding is a technique of finding the future values of all the cash flows at the time horizon at a particular rate of interest in the process of compounding all the inflows and outflows of cash are compounded and compared at the end of the life of the project. (In the above example it is year four). The project will be accepted if the future value of inflows is greater than future value of outflows, otherwise the project will be rejected. Considering the above mentioned project the process of compounding is shown as:

Year

0 — 2000 F V (2000)
1 250 F V (250) Compared with
2 500 F V (500) +
3 750 F V (750) +
4 1000 —

When doing the above, we are bringing cash flows at various points of time to a common point in time (at the end of 4 years) to enable comparison. In above representation the outflow of cash is indicated with negative sign.

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