Teaching Business Mathematics II
without formulas
Oded Tal
School of Business
Conestoga College
Outline
2
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Rationale and goal
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Challenges and solutions
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Solving compound interest questions
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Solving for effective and equivalent interest rates
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Solving annuities
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Solving perpetuities
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Solving amortization schedules
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Measuring students’ success
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Conclusions
Rationale
The best way of solving compound
interest based problems is using preprogrammed functions on a financial
calculator. Why?
 Sometimes it is inevitable, e.g.:
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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.

3
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!

4
The Goal
the “calculator method” for
solving every question involving
compound interest, without any*
formulas.
 Using
5
Challenges
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Jerome’s textbook does not provide calculator-based
solutions to several types of questions:
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Effective rates
Equivalent interest rates
Perpetuities*
The final payment in an amortization schedule
Some students prefer using formulas
The Sign Convention.
Solutions
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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
N
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8
P/Y
C/Y
PV
PMT FV
BGN
N=number of compounding periods
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I/Y
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
NOM
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C/Y
Effective rate: f=(1+i)m-1
ICONV can be used to calculate any of the following
three parameters:
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EFF
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
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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:
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Step 1: Find the effective rate (f) of the given rate
Step 2: Find the nominal rate (j) corresponding to f.
Tips for annuities
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N is the total number of payments, calculated by
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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
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The present value of a simple perpetuity is PV=PMT/i
Calculating the present value of a general perpetuity
requires two additional formulas:
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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)!
12
Mini literature survey
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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 .
Example no. 1
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14
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.
Example no. 2
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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.
Research objective
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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
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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.
Results
N vs. i2
9000
8000
7000
6000
5000
N
4000
3000
2000
1000
0
0
0.02
0.04
0.06
0.08
i2
18
0.1
0.12
0.14
Results (cont.)
N vs. (P/Y)/j
10000
8000
6000
N
4000
y = 19.999x + 0.2518
R2 = 1
2000
0
0
100
200
300
(P/Y)/j
19
400
500
The 20-year Rule
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20
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.
Example 3
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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.
Tips for amortization schedules
P1
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P2
BAL
PRN INT
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.
Amortization schedules (cont.)
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The last row can also be calculated using AMORT as
follows:
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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
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Two Winter 2008 sections from two 3-year business
programs at Conestoga College:
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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
100
90
80
70
60
50
40
30
20
68.9
78.7
88.0
86.3
10
0
25
Test 1
Test 2
Average improvement: MOM- 19.1%, Accounting- 7.6%.
Failure rates
35%
30%
25%
20%
15%
10%
5%
0%
26
30%
14%
9%
7%
Major improvements (15% or more)
18
16
14
12
10
8
6
Calc.
Calc.
Formulas
Formulas
MOM
Accounting
4
2
0
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The largest improvements
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Materials and Operations Management
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Accounting
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27% to 99%
27% to 97%
50% to 100%
52% to 97%
40% to 82%
42% to 82%
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
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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
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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
32