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# Determining the rate of interest

Topic 1.The Instruments of Management of Financial Resources

The time value of money

When people undertake to set aside money for investment something has to be given up now. For instance, if someone buys shares in a firm or lends to a business there is a sacrifice of consumption. One of the incentives to save is the possibility of gaining a higher level of future consumption by sacrificing some present consumption. Therefore, it is apparent that compensation is required to induce people to make a consumption sacrifice. Compensation will be required for at least three things:

· Time. That is, individuals generally prefer to have \$1 today than \$1 in five years’ time. To put this formally: the utility of \$1 now is greater than \$1 received five years hence. Individuals are predisposed towards impatience to consume, thus they need an appropriate reward to begin the saving process. The rate of exchange between certain future consumption and certain current consumption is the pure rate of interest – this occurs even in a world of no inflation and no risk. If you lived in such a world you might be willing to sacrifice \$100 of consumption now if you were compensated with \$104 to be received in one year. This would mean that your pure rate of interest is 4%.

· Inflation. The price of time (or the interest rate needed to compensate for time preference) exists even when there is no inflation, simply because people generally prefer consumption now to consumption later. If there is no inflation then the providers of finance will have to be compensated for that loss in purchasing power as well as for time.

· Risk. The promise of the receipt of a sum of money some years hence generally carries with it an element of risk; the payout may not take place or the amount may be less than expected. Risk simply means that the future return has a variety of possible values. Thus, the issuer of a security, whether it is a share, a bond or a bank account, otherwise no one will be willing to buy the security.

Take the case of Mr. Investor who is considering a \$1000 one-year investment and requires compensation for three elements of time value. First, a return of 4% is required for the pure time value of money. Second, inflation is anticipated to be 10 % over the year. Thus, at time zero (n0) \$1000 buys one basket of goods and services. To buy the same basket of goods and services at time n1 (one year later) \$1100 is needed. To compensate the investor for impatience to consume and inflation the investments needs to generate a return of 14.4 %, that is:

(1+0.04) (1+0.1) – 1 = 0.144

The figure 14.4 % may be regarded here as the risk-free return (RFR), the interest rate which is sufficient to induce investment assuming no uncertainty about cash flows.

However, different investment categories carry different degrees of uncertainty about the outcome of the investment. Investors require different risk premiums on top of the RFR to reflect the perceived level of extra risk. Thus,

Required return (time value of money) = RFR + Risk Premium

Simple Interest is interest paid only on the original principal.

Example:

Suppose that a sum of \$100 is deposited in a bank account that pays 12 % per annum. At the end of year 1 the investor has \$112 in an account. That is:

F=P(1+r)

112=100(1+0.12)

At the end of 5 years:

F=P(1+rn)

160=100(1+0.12 x 5)

Compound Interest is interest paid on the accumulated interest and principal.

Example:

An investment of \$100 is made at an interest rate of 12 % with the interest being compounded. In one year the capital will grow by 12 % to 112\$. In the second year the capital will grow by 12 %, but this time the growth will be on the accumulated value of \$112 and thus will amount to an extra 13.44 at the end of two years:

F=P(1+r) (1+r)

F=112 (1+r)

F= 125.44

Present values

There are many occasions in financial management when you are given the future sums and need to find out what those future sums are worth in present-value terms today. For example, you wish to know how much you would have to put aside today which will accumulate, with compound interest, to a defined sum in the future; or you are given the choice between receiving \$200 in five years or \$100 now and wish to know which is the better option, given anticipated interest rates; or a project gives a return of \$1 mln. in three years for an outlay of \$800 000 now and you need to establish if this is the best use of the \$800 000 by the process of discounting a sum of money to be received in the future is given a monetary value today.

Determining the rate of interest

Sometimes you wish to calculate the rate of return that a project is earning. For instance, a savings company may offer to pay you \$10 000 in five years if you deposit \$8 000 now, when interest rates on accounts elsewhere are offering 6% per annum. In order to make a comparison you need to know the annual rate being offered by the savings company. Thus, we need to find r in the discounting equation.

To be able to calculate r it is necessary to rearrange the compounding formula.

The investment period

Rearranging the standard equation so that we can find n, we create the following equation:

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