Teaching Business Mathematics II without formulas

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Teaching Business Mathematics II without formulas

• Oded Tal
• School of Business
• Conestoga College

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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

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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.

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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!

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The Goal

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

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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.

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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.

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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.

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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).

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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.

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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
  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)!

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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 .

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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 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.

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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 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.

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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 doesnt change anymore)
• Looking for patterns.

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Results
N vs. i2
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Results (cont.)
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 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.

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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
  201/0.12166.67, 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(1i2) previous balance.

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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
68.9
88.0
78.7
86.3
Average improvement MOM- 19.1, Accounting- 7.6.
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Failure rates
30
14
9
7
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Major improvements (15 or more)
Calc.
Calc.
Formulas
Formulas
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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.

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Questions