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Teaching Business Mathematics II without formulas

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... 'calculator method' for solving every question involving compound interest, ... Test 1 covered Simple Interest and was entirely formula-based ... – PowerPoint PPT presentation

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Title: Teaching Business Mathematics II without formulas


1
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 annuitys 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
  • 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.

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
  • 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.

9
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).

10
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.

11
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.

12
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)!

13
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 .

14
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.

15
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.

16
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.

17
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.

18
Results
N vs. i2
19
Results (cont.)
N vs. (P/Y)/j
20
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 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.

21
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.

22
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.

23
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.

24
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.

25
Test results
68.9
88.0
78.7
86.3
Average improvement MOM- 19.1, Accounting- 7.6.
26
Failure rates
30
14
9
7
27
Major improvements (15 or more)
Calc.
Calc.
Formulas
Formulas
28
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?!

29
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30
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.

31
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
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