Teaching Business Mathematics II without formulas

Oded Tal

School of Business

Conestoga College

2

Outline

Rationale and goal

Challenges and solutions

Solving compound interest questions

Solving for effective and equivalent interest rates

Solving annuities

Solving perpetuities

Solving amortization schedules

Measuring students’ success

Conclusions

3

Rationale

The best way of solving compound interest based problems is using pre-programmed functions on a financial calculator. Why?

Sometimes it is inevitable, e.g.: Solving for an annuity’s interest rate Solving for the number of payments required to

accumulate a certain amount in an annuity, starting with a given initial amount.

4

Rationale (cont.)

Easier and less error-prone - no need to memorize and/or to manipulate numerous complex formulas

Simpler and quicker- in many cases the number of required steps/calculations is smaller

This is also how professionals do it on the job!

5

The Goal

Using the “calculator method” for solving every question involving compound interest, without any* formulas.

6

Challenges

Jerome’s textbook does not provide calculator-based solutions to several types of questions:

– Effective rates– Equivalent interest rates– Perpetuities*– The final payment in an amortization schedule

Some students prefer using formulas The Sign Convention.

7

Solutions

Emphasizing the three keys to success in the course

Finding ways of using the calculator method for every type of question

Early introduction of the calculator method Demonstrating the formula method and the

calculator method Facilitating using the calculator method: charts,

tips, sanity checks, extended sign convention.

8

Tips for compound interest questions

N=number of compounding periods N= #years * (C/Y) or N= #years * m

Sign Convention part 1: PV and FV must have opposite signs

Sanity check no. 1: N and I/Y must be positive.

N I/Y P/Y C/Y PV PMT FV BGN

9

Using ICONV for effective rates

Effective rate: f=(1+i)m-1 ICONV can be used to calculate any of the following

three parameters:– NOM: Nominal interest rate (I/Y or j)– EFF: Effective interest rate (f)– C/Y: Compoundings per year (m).

NOM EFF C/Y

10

USING ICONV for equivalent interest rates

Equivalent rate: i2=(1+i1)m1/m2-1 ICONV can be used in two steps to calculate an

equivalent rate for any given interest rate:– Step 1: Find the effective rate (f) of the given rate– Step 2: Find the nominal rate (j) corresponding to f.

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Tips for annuities

N is the total number of payments, calculated by – N= #years * P/Y

P/Y is different from C/Y for General Annuities END/BGN for Ordinary annuities/annuities Due Sign Convention part 2: PMT gets the same sign

as either PV or FV, depending on which of the two can be considered as a lump payment, serving the same purpose as PMT.

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Solving perpetuities

The present value of a simple perpetuity is PV=PMT/i Calculating the present value of a general perpetuity

requires two additional formulas: – i2=(1+i)c-1– c= (C/Y)/(P/Y)

Can it be done using the financial calculator? How do you enter an infinite number of payments? N must be large enough to reflect a perpetuity, but not

too large for the calculator to handle (in some cases)!

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Mini literature survey

Lyryx suggests using N=1000 Hummelbrunner’s textbook (8th edition, 2008) suggests

using 300 years as the term, and hence N=300*P/Y, with a note regarding inaccuracies due to rounding errors

Jerome’s textbook (6th edition, 2008) does not mention the calculator method. The 5th edition suggests using N=9999

None of three methods is universally accurate .

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Example no. 1

What is the present value of a perpetuity paying $1,000 every month, if money earns 3% annually compounded?

The exact solution is: $405,470.65 Using N=1,000: $370,941.11 (Lyryx) Using N=300*12=3,600: $405,413.53 (Hummelb.) Using N=9999: $405,470.65 (Jerome) The minimum required value for N is close to 8,000.

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Example no. 2

What is the maximum semiannual payment that can be made in perpetuity if the initial investment is $186,828.49 and money earns 5% annually compounded?

The exact solution is: $4,613.74 Using N=1,000: $4,613.74 (Lyryx) Using N=300*2=600: $4,613.75 (Hummelb.) Using N=9999: Error 1 (Jerome) The minimum required value for N is close to 625 The maximum value for N is 9438.

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Research objective

Finding an empirical rule for N in order to accurately estimate the PV of any perpetuity (simple or general, ordinary or due):

P/Y: 1 to 12 C/Y: 1 to 365 I/Y: 1% to 20% Required accuracy: no difference between the

exact value and the estimated value, rounded to the nearest cent.

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Research approach

Calculating the exact present value of perpetuities with a variety of possible combinations of I/Y, P/Y and C/Y

Using ever-increasing values of N to estimate the present value on the calculator, and stopping when the required accuracy level has been reached (and PV doesn’t change anymore)

Looking for patterns.

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Results

0100020003000400050006000700080009000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

i2

N

N vs. i2

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Results (cont.)

y = 19.999x + 0.2518

R2 = 10

2000

4000

6000

8000

10000

0 100 200 300 400 500

(P/Y)/j

N

N vs. (P/Y)/j

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The 20-year Rule

The present value of any perpetuity is identical to the present value of an equivalent annuity, whose term is 20 years divided by the perpetuity's nominal interest rate (as a decimal fraction)

In other words: N=20*(P/Y)/(j) This is the minimum number of payments required

to represent a perpetuity Larger values are fine, but sometimes only up to a

certain point.

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Example 3

What is the present value of an annuity paying $1,000 once a year, if money earns 12% daily compounded?

The exact solution is: $7844.70 The minimum required value for N is

20*1/0.12=166.67, or 167 The estimated solution is $7844.70.

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Tips for amortization schedules

The AMORT function accurately calculates all the rows in any amortization schedule except for the last one

The formula-based solution is based on:

Final payment=(1+i2)* previous balance.

P1 P2 BAL PRN INT

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Amortization schedules (cont.)

The last row can also be calculated using AMORT as follows:

– Final INT= INT of the last payment– Final PRN= BAL after the previous payment– Final payment= Final INT + Final PRN

The final payment can also be very accurately approximated by PMT + final BAL (positive or negative)

Sanity check no. 2: Last payment is approximately (non-integer portion of N) * PMT.

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Measuring students’ success

Two Winter 2008 sections from two 3-year business programs at Conestoga College:

– Accounting (44 students)– Materials and Operations Management (33 students)

Test 1 covered Simple Interest and was entirely formula-based

Test 2 covered Compound Interest; students could use formulas, the calculator method or both

Very similar concepts: equivalent payment streams, unknown loan payments, promissory notes, T-bills/ Strip bonds, etc.

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Test results

0

10

20

30

40

50

60

70

80

90

100

Test 2Test 1

68.9 88.078.7 86.3

Average improvement: MOM- 19.1%, Accounting- 7.6%.

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Failure rates

0%

5%

10%

15%

20%

25%

30%

35%

30% 9%14% 7%

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Major improvements (15% or more)

0

2

4

6

8

10

12

14

16

18

MOM Accounting

Formulas Formulas

Calc. Calc.

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The largest improvements

Materials and Operations Management– 27% to 99%– 27% to 97%– 50% to 100%– 52% to 97% – 40% to 82%– 42% to 82%

Accounting– 42% to 84%– 39% to 80%– 39% to 79%

Was using the calculator method the only reason?!

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The evolution of “die hard” formula fans

Stronger students tend to initially prefer the formula method, until one of the following “break points”:

Solving for i in an annuity Solving for N in an annuity with a lump initial

payment Solving general annuities Solving general perpetuities Constructing amortization schedules.

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Conclusions

It is possible to solve every compound interest based question in Business Mathematics II using the pre-programmed functions of the BAII Plus, without memorizing any formulas

Eventually, most students prefer the calculator method to the formula method

Using the calculator method tends to improve students’ marks and to reduce failure rates

It also tends to “level the field”- both at the student level and at the program level.

32

Questions