Sunil K. Parameswaran

Fundamentals of Financial Instruments


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      The Simple Interest Approach

      EXAMPLE 2.22

      Michael has borrowed $8,000 from a bank for a year. The bank charges simple interest at the rate of 10% per annum. If the loan is repaid in one lump sum at the end of the year, the amount payable will be:

1.10 times 8 comma 000 equals dollar-sign 8 comma 800

      This consists of $8,000 by way of principal repayment and an interest payment of $800.

      Now let us consider a case where Michael repays the principal in two equal semiannual installments. For the first six months, interest will be computed on the entire principal. So the first installment will be:

8 comma 000 times 0.10 times 0.5 plus 4 comma 000 equals dollar-sign 4 comma 400

      The second installment will be lower, for it will include interest only on the remaining principal, which in this case is $4,000. So the amount repayable will be:

4 comma 000 times 0.10 times 0.5 plus 4 comma 000 equals dollar-sign 4 comma 200

      The sum of the two payments is $8,600. In the first case the interest payable was $800, whereas in the second case it is only $600. Quite obviously, the more frequently principal is repaid, the lower will be the amount of interest.

      The Add-on Rate Approach

      This approach entails the calculation of interest on the entire principal. The sum total of principal and interest is then divided by the number of installments in which the loan is sought to be repaid. As should be obvious, if the loan is repaid in a single annual installment, the total interest payable will be $800 and the effective rate of interest will be 10%. However, if Michael were to repay in two equal semiannual installments of $4,400 each, the effective rate of interest may be computed as follows:

StartLayout 1st Row 8 comma 000 equals StartStartFraction 4 comma 400 OverOver left-parenthesis 1 plus StartFraction i Over 2 EndFraction right-parenthesis EndEndFraction plus StartFraction 4 comma 400 Over left-parenthesis 1 plus StartFraction i Over 2 EndFraction right-parenthesis squared EndFraction 2nd Row right double arrow StartFraction i Over 2 EndFraction equals 6.5965 percent-sign 3rd Row right double arrow i equals 13.1930 percent-sign EndLayout

      The Discount Technique

      The interest for the loan amount of $8,000 is $800. So the lender will give him $7,200 and ask him to repay $8,000 after a year. The effective rate of interest is:

i equals StartFraction 8 comma 000 minus 7 comma 200 Over 7 comma 200 EndFraction times 100 equals 11.11 percent-sign

      Many banks require borrowers to keep a percentage of the loan amount as a deposit with them. Such deposits, referred to as compensating balances, earn little or no interest. Obviously such requirements will increase the effective rate of interest, and the higher the required balance, the greater will be the rate of interest that is paid by the borrower.

      Assume that in Michael's case, the bank required a compensating balance of 12.50%. So while he will have to pay interest on the entire loan amount of $8,000, the usable amount is only $7,000.

      The effective rate of interest is:

i equals StartFraction 800 Over 7 comma 000 EndFraction times 100 equals 11.4286 percent-sign

      We will first demonstrate how to compute effective rates given nominal rates, and vice versa.

      EXAMPLE 2.23

      Mary has borrowed money from a bank, which is quoting a rate of 6.4% per annum compounded quarterly. To calculate the effective annual rate, we use an Excel function called EFFECT. The parameters are:

       Nominal_rate: This is the nominal rate of interest per annum.

       Npery: This is the frequency of compounding per annum.

      The nominal rate is 6.40% or 0.064 in this case. The frequency of compounding per annum is 4. Using the function, we get the effective annual rate of 6.5552% per annum.

EFFECT left-parenthesis 0.064 comma 4 right-parenthesis equals 6.5552 percent-sign

      If we are given the effective rate, we can compute the equivalent nominal rate using the NOMINAL function in Excel. The parameters are

       Effect_rate: This is the effective rate of interest per annum.

       Npery: This is the frequency of compounding per annum.

      Assume that the bank is quoting an effective annual rate of 7.2% per annum with quarterly compounding. What is the equivalent nominal annual rate? In this case the effective