THE FRIENDLY FINANCE COMPANY:

The Friendly Finance Company: An Illustrative Problem Bridging the Gap Between the Logic of Time Value of Money and The Application of Financial Calculators

William B. Joyce – Eastern Illinois University

Appearing in: Journal of Accounting Education: May 1, 2003

Abstract: Application of the “friendly finance company” illustrative problem forces students to demonstrate use of a financial calculator while also demonstrating an understanding of the underlying logic of the time value of money.

INTRODUCTION

The concept of the time value of money is used throughout a student's entire academic program in accounting as well as their subsequent careers. It is vital for students to understand both the underlying logic and the practical application of the time value of money.

Because the cost of financial calculators is falling rapidly many students have them. As a result, procedures for obtaining financial calculator solutions will be used in each of the major sections, along with the procedures for obtaining solutions by using regular, non-financial calculators. Still, it is highly desirable for each student to obtain a financial calculator and learn how to use it. Calculators (and not clumsy, rounded, and incomplete tables) are used exclusively in well-run, efficient businesses.

Even though financial calculators are efficient, they do pose a danger: students sometimes learn how to use them in a “cookbook” fashion without understanding the logic process that underlie the calculators. Then, when confronted with a new type of problem, the students can have difficulty in determining how to set up the problem. The “friendly finance company” problem bridges the gap between the underlying logic and the practical use of a financial calculator.

TIME VALUE OF MONEY BASICS

Accounting decisions often involve situations in which someone pays money at one point in time and receives money at some later time. Dollar amounts that are paid or received at two different points in time are different; this difference is the concept of the time value of money.

Compounding

To introduce students to the concept of time value of money, a simple compounding problem is used. The compounded amount, or future value(FV), is equal to the beginning amount (present value or PV) plus the interest earned. The equation for future value (of a single payment) is given by:

FV = PV*[1 + R]ⁿ, (1)
where R is the interest rate and n the number of periods.

A simple example is depositing $1,000 in a Certificate of Deposit (CD) that pays 5 percent for one year. When asked what the future value of the CD will be in 1-year, most students can perform the calculation in their head and provide the answer of $1,050. When asked how the answer was derived, a students generally provides some variation of equation (1). Next, equation (1) is applied to answer the example:

FV = PV*[l + R]ⁿ

FV = $1,000*[1.05]¹

FV = $1,050

With this simple problem easily solved, the students have overcome a basic aversion that many have towards the application of algebraic equations. Moreover, students have a concrete, practical application as part of their theoretical foundation.

Discounting

The concept of discounting can be more intellectually challenging than compounding for students. Determining present values (or discounting as it is commonly called) is simply the reverse of compounding, and equation (1) can be transformed into a present value formula by solving equation (1) for the present value (PV):

FV = PV*(1 + R)ⁿ (1)

which, when solved for PV, yields

PV = FV / (1 + R)ⁿ. (2)

The previous example can be worked “backwards” to illustrate discounting. Using a regular four-function calculator to find the present value, equation (2) is applied:

PV = FV / (1 + R)ⁿ (2)

PV = $1,050 / (1.05)1

PV = $1,000.

Alternatively, a financial calculator could be used to find the present value of $1,050. With a financial calculator (for example the Texas Instruments BA-35), one enters the following key strokes:

$1,050 FV

1 N

5 %i

CPT PV

The present value of $1,000 is the result. With the concepts of compounding and discounting understood, students have all of the necessary underlying logic from the concept of time value of money.

Annuities

An annuity is a series of payments of an equal amount for a specified number of periods. An ordinary (deferred) annuity is an annuity whose payment occurs at the end of each period. The concept of annuities builds upon and expands the basic skill the students have acquired. Determining the present value of annuity is simply a matter of summing across the present value of the series of payments. The present values of the series of payments is determined by modifying equation (2). Payment, PMT, is substituted for future value, FV; and the sum is taken as follows:

PV = FV / (1 + R)ⁿ. (2)

Substituting payment, PMT, for future value, FV, yields:

PV = PMT¹/(1+R)¹ + PMT² /(1+R)² + ... + PMTⁿ /(1+R)ⁿ,

PV = PMT*[ 1/(1 + R)¹ + 1/(1 + R)² +... + 1/(1 + R)ⁿ].

Then, the sum of these factors is taken:

PV = PMT*S [1 / (1 + R)ⁿ]. (3):

As an example, one could use a regular calculator to find the present value interest factor of the annuity of $1,000 per year for 3 years at a discount rate of 10 percent. The calculation and analysis are next:

Time (in years) 0 1 2 3

Cash Flows $1,000 $1,000 $1,000

Discount Factors 1 / (1.1) 1 / (1.21) ( / (1.331)

Present Values (flows*factor) $909.09 $826.45 $751.31

The sum is $2,486.85: $909.09 + $826.45 + $751.31. A financial calculator (again for example, Texas Instruments BA-35) could be used to find the present value of the $1,000 payment stream. With the Texas Instruments BA-35, the key strokes are:

$1,000 PMT

3 N

10 %i

CPT PV.

The result, once again, is $2,486.85. The present value of an annuity concept is used next to illustrate the practical application of loan amortization.

Loan Amortization:

An amortized loan is one that is paid off in equal payments over a specified period. To illustrate, suppose a mobile home buyer borrows $10,000 to be repaid in 3 equal payments at the end of the next three years. The lender is to receive 10 percent interest on the loan balance that is outstanding at the beginning of. each period. The first task is to determine the amount the borrower must repay each year, or the annual payment. To find this amount, one needs to recognize that the $10,000 represents the present value of an annuity of payment (PMT) dollars per year for 3 years, discounted at 10 percent:

PV = PMT/(1 + R)¹ + PMT/(1 + R)² + PMT/(1 + R)³

$10, 000 = PMT / (1.10)¹ + PMT / (1.10)² + PMT / (1.10)³

$10,000 = PMT / (1.10) + PMT / (1.21) + PMT / (1. 331)

$10,000 = PMT *[1/1.10+1/1.21 +1/1.331]

$10,000 = PMT * [ .90909 + .82645 + .75131 ]

$10,000 = PMT * [2.48685]

$10,000 / 2.48685 = PMT

$4,021.15 = PMT

The payment can also be determined by using a financial calculator. One simply enters:

$10,000 PV

3 N

10 %I

CPT PMT.

The solution, PMT, $4,021.15 is the result. Each payment consists partly of interest and partly of a repayment of principal. This breakdown is given in the amortization schedule shown below:

Year Beginning Payment Interest Repayment Remaining

Amount of Principal Balance

(1) (2) (3) (4) (5) (6)

1 $10,000.00 $4,021.15 $1,000.00 $3,021.15 $6,978.85

2 $ 6,978.85 $4,021.15 $ 697.89 $3,323.26 $3,655.59

3 $ 3,655.59 $4,021.15 $365.60 $3,655.55 $ 0.04 (rounding).

Now, students possess all of the skills to determine payment amount with either a four-function calculator or a financial calculator. However, the “friendly finance company” illustrates and tests the use of both!

THE FRIENDLY FINANCE COMPANY

It is not uncommon for “friendly” finance companies to provide their customers with the opportunity to “skip a payment” after a major holiday. From a teaching perspective, the “friendly finance company” provides an excellent opportunity to demonstrate the practical application of a financial calculator while also requiring an understanding of the underlying logic involved in the time value of money.

In order to make the previous problem more realistic, monthly compounding will be applied to the loan amortization problem. Then, the “Memorial Day” payment or the 5th payment of each year will be “skipped” to bridge the gap between the efficiency of the financial calculator and the underlying logic that the time value of money make uses. First, the amount of monthly payments must be determined. Then, these payments needs to be “adjusted” to take into account the “skipped” payments.

Determining Monthly Payments

The present value of the loan balance remains the same at $10,000. However, the annual rate of interest (10 percent) and the number of payments (3) need to be converted into monthly amounts. The monthly rate is determined by dividing the annual interest rates by the number of months in a year, 12: 10%/12 = .8333%. The number of periods is determined by taking the number of years (3) and multiplying by the number of months per year (12): 3*12=36. Now, a financial calculator can be used to determine the monthly payments. The key strokes are:

$10,000 PV

36 N

.8333 %i

CPT PMT

The monthly payment is $322.67. This is illustrated by the following adaptation of a derivative of equation (3), the present value of an annuity:

PV = PMT*[1/(1 + R)¹ + 1/ (1 + R)²+ ... +1/ (1 + R)³⁶]

$10,000 = PMT*[1/(1.0083)¹]+[1/(1.0083)²+...+1/(1.0083)³⁶]

$10,000 = PMT*[30.9912]

$10,000 / 30.9912 = PMT

$322.67 = PMT

The payment solution of $322.67.

Due to the shear number of calculations involved, the students are forced to use a financial calculator to complete the task in a efficient and accurate manner. Consequently, a the calculation provides an excellent means of testing comprehension and application of a financial calculator.

Next, the value of the “skipped” payments can be determined by applying the logic of the present value of an annuity equation (3) for the “skipped” payments. Since most financial calculators lack the capability to determine the present value of the “skipped payments” directly, the students are forced to determine the present value of the “skipped payments” by hand. Thus, the students must demonstrate an understanding and application of the underlying logic.

Assuming the loan was granted in January, the “Memorial Day” payments would be numbers 5, 17, and 29, respectively. Thus, the equation for determining the present value of these “skipped” payments is an application of equation (3) as follows:

PV = PMT* S [1 / (1 + R)ⁿ]. (3)

which in expanded form for this problem becomes:

“skipped” = PMT*[1/(1.0083)⁵ + 1/(1.0083)¹² + 1/(1.0083)²⁹]

“skipped” = PMT *[1/(1.00833)⁵ +1 / (1.0083)¹² +1/(1.0083)²⁹]

“skipped” = PMT*[.9593+.8684+.7861 ]

“skipped” = PMT * [2.6139]

Now, the value of these “skipped” payments are deducted from the value of the annuity without skipping payments to determine the payment amount with skipping the payments in May for Memorial Day:

$10,000 = PMT*[30.99124 - 2.61389]

$10,000 = PMT*[28.377371].

$10,000 / 28.377371 = PMT

$352.39 = PMT

Thus, the new payment amount with “skipping” the payments in May is $352.39. This result makes use of both the efficiency of the financial calculator and the underlying logic of the financial concepts. The “friendly finance company” is an ideal question for examinations because it bridges the gap between a financial calculator and the underlying logic (time value of money).

CONCLUDING REMARKS

Financial calculators have built-in programs which perform most of the operations in this paper. It is very important for students to get such a calculator and learn how to use it. However, it is essential that students also understand the logic process involved. Application of the “friendly finance company” illustrative problem forces students to demonstrate use of a financial calculator while also demonstrating an understanding of the underlying logic of the time value of money. Consequently, the “friendly finance company” is an excellent teaching and testing application of the time value of money.