The Art of Mathematics in Business - Dr Jae K Shim

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      $10,000 × 0.10 × 3/12 month = $250

      The maturity value of the loan is

      S = P + I

      $10,000 + $250 = $10,250

      How is it used and applied?

      The owner must know how to determine the cost of money in a debt or lease agreement. By knowing the cost of borrowing, the owner can better plan a business strategy to obtain an adequate return on his or her money and to provide sufficient funds to pay principal and interest.

      The business owner who seeks a loan should shop around to obtain the best interest terms possible. Excessive interest rates, although they are tax deductible, are a drain on an owner’s profits. Large outstanding loans, coupled with high interest rates, can substantially reduce an owner’s profits and impair a business’ credit rating. A balance sheet showing high interest-bearing loans can also result in lenders charging even higher rates of interest because of the possibility of default and even bankruptcy if the loans are not paid at their maturity date.

      See Sec. 2, Real (Effective) Interest Rate.

      Introduction

      The stated rate of interest does not tell the whole story: you need to be sure you understand all the fees and charges that might affect the real interest rate. The real rate of interest on a loan is expressed as an annual percentage applicable for the life of the loan.

      How is it computed?

      For a discounted loan, which is a common form of loan, interest is deducted immediately in arriving at the net proceeds - which increases the effective interest rate. A bank may require a compensating balance, i.e., a deposit that offsets the unpaid loan. In this case, no interest is earned on the compensating balance, which is stated as a percentage of the loan. A compensating balance also increases the effective interest rate.

      Example

      A business takes out a $10,000, 1-year, 10% discounted loan. The compensating balance is 5 percent. The effective interest rate is:

      How is it used and applied?

      Business owners compute the effective interest rate to determine the true cost of borrowing.

      See Sec. 3, The Cost of Credit: Annual Percentage Rate.

      Introduction

      The annual percentage rate (APR) is a true measure of the effective cost of credit. It is the ratio of the finance charge to the average amount of credit in use during the life of the loan, and is expressed as a percentage rate per year.

      How is it computed?

      We present below a discussion of the way the effective APR is calculated for various types of loans.

      Single-Payment Loans

      The single-payment loan is paid in full on a given date. There are two ways of calculating APR on single-payment loans: the simple interest method and the discount method.

      1.Simple interest method

      Recall from Sec. 1 that, under the simple interest method, interest is calculated only on the amount borrowed (proceeds):

      Example 1

      A business owner took out a single-payment loan of $1000 for 2 years at a simple interest rate of 15 percent. The interest charge is: $300 ($1,000 × 0.15 × 2 years). Hence the APR is:

      Under the simple interest method, the stated simple interest rate and the APR are always the same for single-payment loans.

      Under the discount method, interest is determined and then deducted from the amount of the loan. The difference is the actual amount the borrower receives. In other words, the borrower prepays the finance charges.

      Example 2

      Using the same figures as in Example 1, the actual amount received $700 ($1,000 - $300), not $1,000. The APR is:

      The rate the lender must quote on the loan is 21.43 percent, not 15 percent.

      The discount method always gives a higher APR than the simple interest method for single-payment loans at the same interest rates because the proceeds received are less.

      Most consumer loans are the add-on method. One popular method of calculating the APR for add-on loans is the constant-ratio method. The constant-ratio formula is:

      Example 3

      Assume that a business owner borrows $1,000 to be repaid in 12 equal monthly installments of $93 each for a finance charge of $116. The APR under the constant-ratio method is computed as follows:

      Note that some lenders charge fees for a credit investigation, a loan application, or for life insurance. When these fees are required, the lender must include them in addition to the finance charge in dollars as part of the APR calculations.

      How is it used and applied?

      The lender is required by the Truth in Lending Act (Consumer Credit Protection Act) to disclose to a borrower the effective annual percentage rate (APR) as well as the finance charge in dollars.

      Banks often quote their interest rates in terms of dollars of interest per hundred dollars. Other lenders quote in terms of dollars per payment. This leads to confusion on the part of borrowers. Fortunately, APR can eliminate this confusion. By comparing the APRs of different loans, a borrower can determine the best deal.

      Example 4

      Bank A offers a 7 percent car loan if a business owner puts down 25 percent. That is, if the owner buys a $4,000 auto, she will finance $3,000 over a 3-year period with carrying charges that amount to $630 (0.07 × $3,000 × 3 years). The owner will make equal monthly payments of $100.83 for 36 months.

      Bank B will lend $3,500 on the same car. In this case the business owner must pay $90 per month for 48 months.

      Which of the two quotes offers the best deal?

      The APR calculations (using the constant-ratio formula) follow.

      In the case of Bank B, it is necessary to multiply $90 × 48 months to arrive at a total cost of $4320. Therefore, the total credit cost is $920 ($4,320 - $3,500).

      Based on the APR, the business owner should choose Bank B over Bank A.

      Introduction

      The due date, also called maturity date, is the exact date when a loan must be repaid.

      How is it computed?

      Some

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