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Teaching Business Mathematics II without formulas

- Oded Tal
- School of Business
- Conestoga College

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

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 annuitys interest rate
- Solving for the number of payments required to

accumulate a certain amount in an annuity,

starting with a given initial amount.

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!

The Goal

- Using the calculator method for solving every

question involving compound interest, without

any formulas.

Challenges

- Jeromes 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.

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.

Tips for compound interest questions

- Nnumber 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.

Using ICONV for effective rates

- Effective rate f(1i)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).

USING ICONV for equivalent interest rates

- Equivalent rate i2(1i1)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.

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.

Solving perpetuities

- The present value of a simple perpetuity is

PVPMT/i - Calculating the present value of a general

perpetuity requires two additional formulas - i2(1i)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)!

Mini literature survey

- Lyryx suggests using N1000
- Hummelbrunners textbook (8th edition, 2008)

suggests using 300 years as the term, and hence

N300P/Y, with a note regarding inaccuracies due

to rounding errors - Jeromes textbook (6th edition, 2008) does not

mention the calculator method. The 5th edition

suggests using N9999 - None of three methods is universally accurate .

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 N1,000 370,941.11 (Lyryx)
- Using N300123,600 405,413.53 (Hummelb.)
- Using N9999 405,470.65 (Jerome)
- The minimum required value for N is close to

8,000.

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 N1,000 4,613.74 (Lyryx)
- Using N3002600 4,613.75 (Hummelb.)
- Using N9999 Error 1 (Jerome)
- The minimum required value for N is close to 625
- The maximum value for N is 9438.

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.

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 doesnt change anymore) - Looking for patterns.

Results

N vs. i2

Results (cont.)

N vs. (P/Y)/j

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

perpetuitys nominal interest rate (as a decimal

fraction) - In other words N20(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.

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

201/0.12166.67, or 167 - The estimated solution is 7844.70.

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(1i2) previous balance.

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.

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.

Test results

68.9

88.0

78.7

86.3

Average improvement MOM- 19.1, Accounting- 7.6.

Failure rates

30

14

9

7

Major improvements (15 or more)

Calc.

Calc.

Formulas

Formulas

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!

(No Transcript)

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.

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.

Questions

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