New Zealand Applied Business Journal
Volume 1, Number 1, 2002
A BACK-TO-BASICS APPROACH TO FINANCIAL MODELLING
THE USE OF MATHEMATICAL MODELS AND SPREADSHEETS IN THE TEACHING OF FINANCIAL
MATHEMATICS CAN LEAD TO A DEEPER UNDERSTANDING OF THE TIME VALUE OF MONEY AND
ENHANCE CRITICAL THINKING
Peter Watson
Department of Applied Mathematics
Auckland University of Technology
Auckland, New Zealand
Abstract: Financial mathematics provides many examples of arithmetic, geometric
and arithmetico-geometric progressions in action. In this paper a fragment of the
material will be discussed and this will indicate the potential of the subject to explore
mathematical models
The examples explored and compared will be the table mortgage and the reducing
mortgage. These are two popular methods used to construct a repayment schedule
for a home mortgage so the real life context of the examples is immediately apparent.
Some popular misconceptions regarding the way to compare them will be exposed
and a more correct methodology proposed.
INTRODUCTION
To allow a comparison to be made the following example will be used throughout this paper.
Imagine you have applied for a loan of $120 000 to complete the purchase of a property and
you are offered the choice of a table mortgage or a reducing mortgage. Each loan will have
an interest rate of 9% pa compounded semi-annually and the loan is for 25 years.
The table mortgage has a timeline with the following features.
$120 000
R
R
R
R
0
1
2
3…
50
Where i =
j2
0.09
=
= 0.045
2
2
The formula to calculate the regular payment can be found in any financial mathematics
textbook
1
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
R
=
P
an i
Where P is the principal (P = $120 000), n is the number of periods (n = 50), and i is the
period interest rate (i = 0.045).
And
an i
So
R
=
1  (1  i )  n
i
=
1  1.04550
0.045
=
19.76200778
=
$6072.257503
=
$6072.26
1  (1  i )  n
is surprisingly straightforward so
i
the mathematics involved will not obscure the model that is being solved. Appendix 1 gives
an explanation of the proof from first principles based on the concept of a geometric
progression (GP). A finance course tends to focus on the use of a formula to solve problems
whereas in a financial mathematics course the derivation of the formulae is also a feature.
The way the derivation is developed, it is submitted, leads towards a deeper understanding of
the time value of money and makes the concept of replacing a stream of payments by a single
payment on a particular day more obvious.
The proof of the formula
an i
=
Students need to come to realise the effect of this formula. One important feature is the 50
payments of $6072.26 in the example are replaced with the single payment of $120 000 on
day 1 (period 0).
The spreadsheet provides an excellent tool to show the calculations for a table mortgage.
Appendix 2 shows an example of a spreadsheet that is designed to show this. In an
introductory course this might be as far as you would go. A usual question that arises is to
enquire as to how much interest is paid. You often find the following procedure followed.
To find the total interest paid it is a simple matter to autosum the column of interest
payments. Similarly it is easy to verify this is right by finding the sum of the payments and
subtracting the original principal. This is so easy to do that the verification might fool some
people. However maybe the solution of the equation of value (see appendix 1) might alert
students to the fallacy of what has just been done.
2
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
Before this is taken any further the reducing mortgage is now introduced and developed to
the same point.
The reducing loan is a loan where the principal is reduced by the same amount each period
and the interest owed is paid with each principal payment. You either state the size of each
principal payment which determines the number of payments or you state the number of
payments that will be made and calculate the size of each payment. The loan is called a
reducing loan because the amount paid each period reduces over time.
A reducing mortgage is a reducing loan where the loan is secured with a mortgage over a
property. This is a popular method of financing property deals in New Zealand.
For a loan with a principal of P the size of each principal payment is found by calculating
P
n
where n is the number of payments.
Again a spreadsheet can show the calculations for a reducing mortgage. Appendix 3 shows
an example of an Excel spreadsheet that is designed to show this. It can be seen that both this
spreadsheet and the spreadsheet in appendix 2 are created to be as similar in shape as possible
to show the way the calculations produce different answers.
The autosum button can in the same way be used to find
t
r
where tr is the interest at step
r.
A useful exercise in summing series can be developed here because it can be shown that tr is
 Pi
an arithmetic progression (AP) with a first term of Pi and a common difference of
.
n
The result is
P
( n  1) i
2
(See appendix 4 for a proof.)
This is very easy to calculate and to verify that it gives the same value as
t
r
from the
spreadsheet.
Again the result can be verified as
 Payments  P
Comparing the Table Mortgage and the Reducing Mortgage
Looking at the two spreadsheets now it is easy to see that the sum of the interest payments for
the table mortgage is greater than the sum of the interest payments for the reducing mortgage.
Does this lead to the conclusion that the table mortgage is more expensive than the reducing
mortgage when the interest rate is the same? This calculation is often performed to show the
table mortgage is more expensive and an accompanying argument might go like this.
Because the table mortgage results in a smaller quantum of the principal being paid in the
earlier stages of the loan’s repayment the principal will diminish more slowly in the
3
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
beginning and more rapidly towards the end of the loan period hence the higher interest.
While the argument is correct the addition of the interest payments is not appropriate. The
addition of different sums at different time periods shows a lack of appreciation of the time
value of money. It is therefore more appropriate to find the present value (PV) of the stream
of interest payments and to compare those.
In appendix 5 the table mortgage schedule in appendix 2 has been extended to provide 3 extra
columns. At each step the present value of the interest, the principal outstanding and the
payment has been calculated. This is achieved by multiplying the corresponding value in row
r by (1  i )  r .
The patterns in the data are immediately apparent however the most interesting pattern is the
value in the principal column. The value is clearly independent of r.
The Net Present Value (NPV) of the interest in cell I2 ($87 834.57) is obtained by summing
the present values in column G (G11:G60). Similarly I3 = Sum (H11:H60) and I4 = Sum
(I11:I60).
In appendix 6 the reducing mortgage schedule in appendix 3 has been similarly extended and
it is immediately apparent that the NPV of the interest payments ($72 571.18) is less than the
NPV for the table mortgage and this is the appropriate comparison to make.
While it might be obvious that the NPV of the payments in column I is equal to the principal
($120 000) in each case the spreadsheets provide a useful and spectacular verification of this
fact.
Deriving the formulae for the Net Present Value
The derivation of the results for the various NPV values can verify the numbers obtained and
lead to an algebraic comparison. The analysis will also lead to a deeper understanding of the
concepts involved.
The following table summarises the results of the various calculations.
4
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
At step r
Outstanding Principal
Table Mortgage
Reducing Mortgage
P (1  i )  Rsr i
Pr
PV of the Principal Repaid
( R  Pi)(1  i)1
PV of the Payment
R(1  i) r
PV of the Interest
R(1  i) r  ( R  Pi)(1  i)1
For n Payments
NPV of the
Repaid
r
Principal
NPV of the Repayments
NPV of the Interest
Where sr i
=
n( R  Pi)(1  i)1
P
(1  i )  r
n
 P Pi

r
 n  n 1  n  r   (1  i )
P

r
 P  (r  1) n  i  (1  i )
P
a
n ni
P
P
P  n( R  Pi)(1  i)
(1  i ) r  1
i
P
n
and
ar i
1
P
=
P
a
n ni
1  (1  i)  r
i
Proofs of the key formulae in this table can be found in appendices 7 and 8. The proofs of
the NPV formulae generally involve the summation of a GP. This type of summation might
be found in any introductory algebra course. The same cannot be said for the arithmeticogeometric progression, which is a progression that is simultaneously arithmetic and
geometric. The way this is treated can be seen in appendix 8 and to find a need to solve a
problem of this type in a real life context provides a worthwhile extension to the material.
Scientific calculators are now a basic requirement for students so all have access to the
functions required in a finance course. While all students should develop facility with the use
of their calculator a financial mathematics course will provide many opportunities for them to
do so. Many finance textbooks still provide compound interest tables and annuity tables and
show students how to use them. [Adams et al, Brealey and Myers, Croucher, Francis and
Taylor, Gitman, McLean and Stephens, Shim and Siegel, Van Horne]. A financial
mathematics course on the other hand can treat the generation of these tables as an exercise.
A smaller number of textbook show the formulae in action with worked examples and
problems [Knox et al, McLean and Stephens, Shannon, Waters, Zima and Brown]. Some
show the formulae and give worked examples based on the tables [Brealey and Myers,
Croucher, Gitman, Shim and Siegel, Van Horne]. Burton et al take a different approach,
neither tables nor formulae are mentioned with every problem being tackled from first
principles. A text with a financial mathematics approach often includes the tables for
completeness but more often than not shows worked examples that focus on the formulae
[Adams et al, McCutcheon and Scott].
5
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
CONCLUSION
A financial mathematics course provides the opportunity to explore not only the practical
applications of financial problems in a business context but also the underlying mathematical
formulae and how they are derived. The development of the mathematical models and the
building of the spreadsheets using an algorithmic approach lead to a deeper understanding of
the processes involved. During the earlier stages of a financial mathematics course
deterministic models such as those touched on in this paper provide a practical application of
the algebraic processes under the general category of “progressions”. An appreciation of the
concepts involved can lead to an ability to develop proofs outside those found in standard
discussions of the subject.
6
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
REFERENCES
Adams, A.T., Bloomfield, D.S.F., Booth, P.M., England, P.D. (1993). Investment Mathematics and Statistics.
London: Graham & Trotman
Brealey, R. and Myers, S. (1984). Principles of Corporate Finance (2nd ed.). Singapore: McGraw-Hill
Burton, G., Carrol, G. and Wall, S. (1999). Quantitative Methods for Business and Economics. Harlow: Addison
Wesley Longman
Croucher, J.S. (1998). Introductory Mathematics and Statistics for Business (3rd ed.). Australia: McGraw-Hill
Francis, J. C., Taylor, R. W. (2000). Investments. New York: McGraw-Hill
Gitman, L. J. (1997). Principles of Managerial Finance (8th ed.). Massachusetts: Addison-Wesley Longman
Knox, D.M., Zima, P., Brown, R.L. (1997). Mathematics of Finance. Australia: McGraw-Hill
McCutcheon, J.J., Scott, W.F. (1986). An Introduction to the Mathematics of Finance. Oxford: ButterworthHeinemann
McLean, A., Stephens, B. (1996). Business Mathematics and Statistics. Melbourne: Addison Wesley Longman
Shannon, J. (1995). Mathematics for Business, Economics and Finance. Brisbane: John Wiley & Sons
Shim, J. K., Siegel, J. G. (1998). Financial Management (2nd ed.). New York: McGraw-Hill
Van Horne, J.C. (1989). Financial Management and Policy (8th ed.). London: Prentice Hall International
Waters, D. (1997). Quantitative Methods for Business (2nd ed.). Harlow: Addison Wesley Longman
Zima, P., Brown, R.L. (1996). Mathematics of Finance. New York: McGraw-Hill
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Volume 1, Number 1, 2002
New Zealand Applied Business Journal
APPENDIX 1: DERIVING THE PRESENT VALUE ANNUITY FACTOR FROM FIRST
PRINCIPLES
1  (1  i )  n
from first principles provides a useful
i
insight into the use of geometric progressions (GPs) in financial mathematics.
A proof of the formula an i
=
The present value (PV) of the payment R at period r = R(1  i ) r . There are n payments so
n
PV of the n payments
=
 R(1  i )
r
r 1
=
R(1  i )1  R(1  i )2  ...  R(1  i ) n
=
R (1  i )1  (1  i )2  ...  (1  i ) n 
The series in the square brackets is a GP with t1 = (1  i )1 and common ratio r = (1  i )1
From the formula Sn = t1
Sn
1 rn
1 r
1  (1  i ) n
1  (1  i )1
=
(1  i )1
=
1 1  (1  i ) n

1  i 1  (1  i )1
=
1  (1  i ) n
(1  i )  (1  i )(1  i )1
=
1  (1  i ) n
1 i 1
=
1  (1  i ) n
i
So PV of the n payments is R
1  (1  i ) n
i
8
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
APPENDIX 2: THE TABLE MORTGAGE CALCULATOR
Table Loan and Table Mortgage Calculator
Principal
$120,000.00
Number of Payments
Interest
50
Monthly Payment
$6,072.26
Period Interest Rate
Added
$183,612.88
Calculated
$183,612.88
4.50%
Number
Principal
of Periods
Outstanding
Principal
n
at Beginning
Payment
at End
1
$120,000.00
$5,400.00
$672.26
$6,072.26
$119,327.74
2
$119,327.74
$5,369.75
$702.51
$6,072.26
$118,625.23
3
$118,625.23
$5,338.14
$734.12
$6,072.26
$117,891.11
4
$117,891.11
$5,305.10
$767.16
$6,072.26
$117,123.95
5
$117,123.95
$5,270.58
$801.68
$6,072.26
$116,322.27
6
$116,322.27
$5,234.50
$837.76
$6,072.26
$115,484.52
7
$115,484.52
$5,196.80
$875.45
$6,072.26
$114,609.07
8
$114,609.07
$5,157.41
$914.85
$6,072.26
$113,694.22
9
$113,694.22
$5,116.24
$956.02
$6,072.26
$112,738.20
10
$112,738.20
$5,073.22
$999.04
$6,072.26
$111,739.16
11
$111,739.16
$5,028.26
$1,044.00
$6,072.26
$110,695.16
12
$110,695.16
$4,981.28
$1,090.98
$6,072.26
$109,604.19
13
$109,604.19
$4,932.19
$1,140.07
$6,072.26
$108,464.12
14
$108,464.12
$4,880.89
$1,191.37
$6,072.26
$107,272.75
15
$107,272.75
$4,827.27
$1,244.98
$6,072.26
$106,027.76
16
$106,027.76
$4,771.25
$1,301.01
$6,072.26
$104,726.76
17
$104,726.76
$4,712.70
$1,359.55
$6,072.26
$103,367.20
18
$103,367.20
$4,651.52
$1,420.73
$6,072.26
$101,946.47
19
$101,946.47
$4,587.59
$1,484.67
$6,072.26
$100,461.80
20
$100,461.80
$4,520.78
$1,551.48
$6,072.26
$98,910.33
21
$98,910.33
$4,450.96
$1,621.29
$6,072.26
$97,289.03
22
$97,289.03
$4,378.01
$1,694.25
$6,072.26
$95,594.78
23
$95,594.78
$4,301.77
$1,770.49
$6,072.26
$93,824.29
24
$93,824.29
$4,222.09
$1,850.16
$6,072.26
$91,974.13
25
$91,974.13
$4,138.84
$1,933.42
$6,072.26
$90,040.70
26
$90,040.70
$4,051.83
$2,020.43
$6,072.26
$88,020.28
27
$88,020.28
$3,960.91
$2,111.35
$6,072.26
$85,908.93
28
$85,908.93
$3,865.90
$2,206.36
$6,072.26
$83,702.58
29
$83,702.58
$3,766.62
$2,305.64
$6,072.26
$81,396.94
30
$81,396.94
$3,662.86
$2,409.40
$6,072.26
$78,987.54
31
$78,987.54
$3,554.44
$2,517.82
$6,072.26
$76,469.72
32
$76,469.72
$3,441.14
$2,631.12
$6,072.26
$73,838.60
33
$73,838.60
$3,322.74
$2,749.52
$6,072.26
$71,089.08
34
$71,089.08
$3,199.01
$2,873.25
$6,072.26
$68,215.83
35
$68,215.83
$3,069.71
$3,002.55
$6,072.26
$65,213.29
36
$65,213.29
$2,934.60
$3,137.66
$6,072.26
$62,075.63
37
$62,075.63
$2,793.40
$3,278.85
$6,072.26
$58,796.77
38
$58,796.77
$2,645.85
$3,426.40
$6,072.26
$55,370.37
39
$55,370.37
$2,491.67
$3,580.59
$6,072.26
$51,789.78
40
$51,789.78
$2,330.54
$3,741.72
$6,072.26
$48,048.06
41
$48,048.06
$2,162.16
$3,910.09
$6,072.26
$44,137.97
42
$44,137.97
$1,986.21
$4,086.05
$6,072.26
$40,051.92
43
$40,051.92
$1,802.34
$4,269.92
$6,072.26
$35,782.00
44
$35,782.00
$1,610.19
$4,462.07
$6,072.26
$31,319.93
45
$31,319.93
$1,409.40
$4,662.86
$6,072.26
$26,657.07
46
$26,657.07
$1,199.57
$4,872.69
$6,072.26
$21,784.38
47
$21,784.38
$980.30
$5,091.96
$6,072.26
$16,692.42
48
$16,692.42
$751.16
$5,321.10
$6,072.26
$11,371.32
49
$11,371.32
$511.71
$5,560.55
$6,072.26
$5,810.77
50
$5,810.77
$261.48
$5,810.77
$6,072.26
-$0.00
Principal
Monthly
Interest Repayment
9
Outstanding
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
APPENDIX 3: THE REDUCING MORTGAGE CALCULATOR
Reducing Loan and Reducing Mortgage Calculator
Principal
$120,000.00
Number of Payments
50
Principal Repayment
$2,400.00
Period Interest Rate
Interest
Added
$137,700.00
Calculated
$137,700.00
4.50%
Number
Principal
of Periods
Outstanding
Principal
n
at Beginning
Payment
at End
1
$120,000
$5,400.00
$2,400.00
$7,800.00
$117,600
2
$117,600
$5,292.00
$2,400.00
$7,692.00
$115,200
3
$115,200
$5,184.00
$2,400.00
$7,584.00
$112,800
4
$112,800
$5,076.00
$2,400.00
$7,476.00
$110,400
5
$110,400
$4,968.00
$2,400.00
$7,368.00
$108,000
6
$108,000
$4,860.00
$2,400.00
$7,260.00
$105,600
7
$105,600
$4,752.00
$2,400.00
$7,152.00
$103,200
8
$103,200
$4,644.00
$2,400.00
$7,044.00
$100,800
9
$100,800
$4,536.00
$2,400.00
$6,936.00
$98,400
10
$98,400
$4,428.00
$2,400.00
$6,828.00
$96,000
11
$96,000
$4,320.00
$2,400.00
$6,720.00
$93,600
12
$93,600
$4,212.00
$2,400.00
$6,612.00
$91,200
13
$91,200
$4,104.00
$2,400.00
$6,504.00
$88,800
14
$88,800
$3,996.00
$2,400.00
$6,396.00
$86,400
15
$86,400
$3,888.00
$2,400.00
$6,288.00
$84,000
16
$84,000
$3,780.00
$2,400.00
$6,180.00
$81,600
17
$81,600
$3,672.00
$2,400.00
$6,072.00
$79,200
18
$79,200
$3,564.00
$2,400.00
$5,964.00
$76,800
19
$76,800
$3,456.00
$2,400.00
$5,856.00
$74,400
20
$74,400
$3,348.00
$2,400.00
$5,748.00
$72,000
21
$72,000
$3,240.00
$2,400.00
$5,640.00
$69,600
22
$69,600
$3,132.00
$2,400.00
$5,532.00
$67,200
23
$67,200
$3,024.00
$2,400.00
$5,424.00
$64,800
24
$64,800
$2,916.00
$2,400.00
$5,316.00
$62,400
25
$62,400
$2,808.00
$2,400.00
$5,208.00
$60,000
26
$60,000
$2,700.00
$2,400.00
$5,100.00
$57,600
27
$57,600
$2,592.00
$2,400.00
$4,992.00
$55,200
28
$55,200
$2,484.00
$2,400.00
$4,884.00
$52,800
29
$52,800
$2,376.00
$2,400.00
$4,776.00
$50,400
30
$50,400
$2,268.00
$2,400.00
$4,668.00
$48,000
31
$48,000
$2,160.00
$2,400.00
$4,560.00
$45,600
32
$45,600
$2,052.00
$2,400.00
$4,452.00
$43,200
33
$43,200
$1,944.00
$2,400.00
$4,344.00
$40,800
34
$40,800
$1,836.00
$2,400.00
$4,236.00
$38,400
35
$38,400
$1,728.00
$2,400.00
$4,128.00
$36,000
36
$36,000
$1,620.00
$2,400.00
$4,020.00
$33,600
37
$33,600
$1,512.00
$2,400.00
$3,912.00
$31,200
38
$31,200
$1,404.00
$2,400.00
$3,804.00
$28,800
39
$28,800
$1,296.00
$2,400.00
$3,696.00
$26,400
40
$26,400
$1,188.00
$2,400.00
$3,588.00
$24,000
41
$24,000
$1,080.00
$2,400.00
$3,480.00
$21,600
42
$21,600
$972.00
$2,400.00
$3,372.00
$19,200
43
$19,200
$864.00
$2,400.00
$3,264.00
$16,800
44
$16,800
$756.00
$2,400.00
$3,156.00
$14,400
45
$14,400
$648.00
$2,400.00
$3,048.00
$12,000
46
$12,000
$540.00
$2,400.00
$2,940.00
$9,600
47
$9,600
$432.00
$2,400.00
$2,832.00
$7,200
48
$7,200
$324.00
$2,400.00
$2,724.00
$4,800
49
$4,800
$216.00
$2,400.00
$2,616.00
$2,400
50
$2,400
$108.00
$2,400.00
$2,508.00
$0
Principal
Monthly Outstanding
Interest Repayment
10
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
APPENDIX 4: FINDING THE SUM OF THE INTEREST PAYMENTS FOR A REDUCING
MORTGAGE
The formula to calculate the total interest is
P
n  1 i
2
.......... (1)
Where P is the principal, n is the number of periods and i is the period interest rate
The derivation of this formula from first principles follows
Let the principal be P, the number of periods be n and the period interest rate be i.
Let the regular equal payment by which the principal is reduced each period be
P
n
Let tr be the interest paid in period r.
t1
=
Pi
t2
=
P

 P  n   i
t3
=
P

 P  2 n   i
.......... (2)
.....................................
.....................................
.....................................
tr
=
P

 P  (r  1) n   i
.....................................
.....................................
.....................................
tn
=
P

 P  (n  1) n   I
This is an arithmetic progression (AP) with a common difference of 
11
P
i
n
…….. (3)
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
To find the total interest use sn
Where
=
n
2t1  n  1d 
2
.......... (4)
n is the number of periods
t1 is P  i
d=
from (2)
P
xi
n
from (3)
Substitute these into (4)
sn
Or
=
n
P 
2P  i  n  1
i
2 
n 
=
n
 P
2P  n  1
i

2
n 
=
n
P P
2P  n     i

2
n n
=
n
P
2P  P    i

2
n
=
n
P
P  i
2 
n
=
n  P  n  1
i
2  n 
=
P
n  1 i
2
=
P
n  1i
2
12
which is (1)
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
APPENDIX 5: THE COMPLETED TABLE MORTGAGE CALCULATOR WITH NPV
CALCULATIONS
Table Loan and Table Mortgage Calculator
NPV
Principal
$120,000.00
Number of Payments
50
Monthly Payment
$6,072.26
Period Interest Rate
Interest
Interest
4.50%
Added
$183,612.88
Calculated
$183,612.88
NPV
Check
$87,834.57
Principal
$32,165.43
Payment
$120,000.00
I+P=
$120,000.00
$87,834.57
Present
Present
Value
Value of
Value of
Outstanding of Interest @ Principal @
Payment @
Principal
Present
Number
Principal
of Periods
Outstanding
n
at Beginning
Payment
at End
4.50%
4.50%
4.50%
1
$120,000.00
$5,400.00
$672.26
$6,072.26
$119,327.74
$5,167.46
$643.31
$5,810.77
2
$119,327.74
$5,369.75
$702.51
$6,072.26
$118,625.23
$4,917.24
$643.31
$5,560.55
3
$118,625.23
$5,338.14
$734.12
$6,072.26
$117,891.11
$4,677.79
$643.31
$5,321.10
4
$117,891.11
$5,305.10
$767.16
$6,072.26
$117,123.95
$4,448.65
$643.31
$5,091.96
5
$117,123.95
$5,270.58
$801.68
$6,072.26
$116,322.27
$4,229.38
$643.31
$4,872.69
6
$116,322.27
$5,234.50
$837.76
$6,072.26
$115,484.52
$4,019.55
$643.31
$4,662.86
7
$115,484.52
$5,196.80
$875.45
$6,072.26
$114,609.07
$3,818.76
$643.31
$4,462.07
8
$114,609.07
$5,157.41
$914.85
$6,072.26
$113,694.22
$3,626.61
$643.31
$4,269.92
9
$113,694.22
$5,116.24
$956.02
$6,072.26
$112,738.20
$3,442.74
$643.31
$4,086.05
10
$112,738.20
$5,073.22
$999.04
$6,072.26
$111,739.16
$3,266.79
$643.31
$3,910.09
11
$111,739.16
$5,028.26
$1,044.00
$6,072.26
$110,695.16
$3,098.41
$643.31
$3,741.72
12
$110,695.16
$4,981.28
$1,090.98
$6,072.26
$109,604.19
$2,937.28
$643.31
$3,580.59
13
$109,604.19
$4,932.19
$1,140.07
$6,072.26
$108,464.12
$2,783.09
$643.31
$3,426.40
14
$108,464.12
$4,880.89
$1,191.37
$6,072.26
$107,272.75
$2,635.55
$643.31
$3,278.85
15
$107,272.75
$4,827.27
$1,244.98
$6,072.26
$106,027.76
$2,494.35
$643.31
$3,137.66
16
$106,027.76
$4,771.25
$1,301.01
$6,072.26
$104,726.76
$2,359.24
$643.31
$3,002.55
17
$104,726.76
$4,712.70
$1,359.55
$6,072.26
$103,367.20
$2,229.94
$643.31
$2,873.25
18
$103,367.20
$4,651.52
$1,420.73
$6,072.26
$101,946.47
$2,106.21
$643.31
$2,749.52
19
$101,946.47
$4,587.59
$1,484.67
$6,072.26
$100,461.80
$1,987.81
$643.31
$2,631.12
20
$100,461.80
$4,520.78
$1,551.48
$6,072.26
$98,910.33
$1,874.51
$643.31
$2,517.82
21
$98,910.33
$4,450.96
$1,621.29
$6,072.26
$97,289.03
$1,766.09
$643.31
$2,409.40
22
$97,289.03
$4,378.01
$1,694.25
$6,072.26
$95,594.78
$1,662.33
$643.31
$2,305.64
23
$95,594.78
$4,301.77
$1,770.49
$6,072.26
$93,824.29
$1,563.05
$643.31
$2,206.36
24
$93,824.29
$4,222.09
$1,850.16
$6,072.26
$91,974.13
$1,468.04
$643.31
$2,111.35
25
$91,974.13
$4,138.84
$1,933.42
$6,072.26
$90,040.70
$1,377.12
$643.31
$2,020.43
26
$90,040.70
$4,051.83
$2,020.43
$6,072.26
$88,020.28
$1,290.11
$643.31
$1,933.42
27
$88,020.28
$3,960.91
$2,111.35
$6,072.26
$85,908.93
$1,206.86
$643.31
$1,850.16
28
$85,908.93
$3,865.90
$2,206.36
$6,072.26
$83,702.58
$1,127.18
$643.31
$1,770.49
29
$83,702.58
$3,766.62
$2,305.64
$6,072.26
$81,396.94
$1,050.94
$643.31
$1,694.25
30
$81,396.94
$3,662.86
$2,409.40
$6,072.26
$78,987.54
$977.98
$643.31
$1,621.29
31
$78,987.54
$3,554.44
$2,517.82
$6,072.26
$76,469.72
$908.17
$643.31
$1,551.48
32
$76,469.72
$3,441.14
$2,631.12
$6,072.26
$73,838.60
$841.36
$643.31
$1,484.67
33
$73,838.60
$3,322.74
$2,749.52
$6,072.26
$71,089.08
$777.42
$643.31
$1,420.73
34
$71,089.08
$3,199.01
$2,873.25
$6,072.26
$68,215.83
$716.24
$643.31
$1,359.55
35
$68,215.83
$3,069.71
$3,002.55
$6,072.26
$65,213.29
$657.70
$643.31
$1,301.01
36
$65,213.29
$2,934.60
$3,137.66
$6,072.26
$62,075.63
$601.68
$643.31
$1,244.98
37
$62,075.63
$2,793.40
$3,278.85
$6,072.26
$58,796.77
$548.06
$643.31
$1,191.37
38
$58,796.77
$2,645.85
$3,426.40
$6,072.26
$55,370.37
$496.76
$643.31
$1,140.07
39
$55,370.37
$2,491.67
$3,580.59
$6,072.26
$51,789.78
$447.67
$643.31
$1,090.98
40
$51,789.78
$2,330.54
$3,741.72
$6,072.26
$48,048.06
$400.69
$643.31
$1,044.00
41
$48,048.06
$2,162.16
$3,910.09
$6,072.26
$44,137.97
$355.73
$643.31
$999.04
42
$44,137.97
$1,986.21
$4,086.05
$6,072.26
$40,051.92
$312.71
$643.31
$956.02
43
$40,051.92
$1,802.34
$4,269.92
$6,072.26
$35,782.00
$271.54
$643.31
$914.85
44
$35,782.00
$1,610.19
$4,462.07
$6,072.26
$31,319.93
$232.15
$643.31
$875.45
45
$31,319.93
$1,409.40
$4,662.86
$6,072.26
$26,657.07
$194.45
$643.31
$837.76
46
$26,657.07
$1,199.57
$4,872.69
$6,072.26
$21,784.38
$158.37
$643.31
$801.68
47
$21,784.38
$980.30
$5,091.96
$6,072.26
$16,692.42
$123.85
$643.31
$767.16
48
$16,692.42
$751.16
$5,321.10
$6,072.26
$11,371.32
$90.81
$643.31
$734.12
49
$11,371.32
$511.71
$5,560.55
$6,072.26
$5,810.77
$59.20
$643.31
$702.51
50
$5,810.77
$261.48
$5,810.77
$6,072.26
-$0.00
$28.95
$643.31
$672.26
Principal
Interest Repayment
Periodic
13
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
APPENDIX 6: THE COMPLETED REDUCING MORTGAGE CALCULATOR WITH NPV
CALCULATIONS
Reducing Loan and Reducing Mortgage Calculator
NPV
Principal
$120,000.00
Number of Payments
50
Principal Repayment
$2,400.00
Period Interest Rate
Interest
Interest
4.50%
Added
$137,700.00
Calculated
$137,700.00
NPV
Check
$72,571.18
Principal
$47,428.82
Payment
$120,000.00
I+P=
$120,000.00
$72,571.18
Present
Present
Principal
Value of
Value of
Present
Value of
Outstanding
Interest @
Principal @
Payment @
Number
Principal
of Periods
Outstanding
n
at Beginning
Payment
at End
4.50%
4.50%
4.50%
1
$120,000
$5,400.00
$2,400.00
$7,800.00
$117,600
$5,167.46
$2,296.65
$7,464.11
2
$117,600
$5,292.00
$2,400.00
$7,692.00
$115,200
$4,846.04
$2,197.75
$7,043.79
3
$115,200
$5,184.00
$2,400.00
$7,584.00
$112,800
$4,542.72
$2,103.11
$6,645.83
4
$112,800
$5,076.00
$2,400.00
$7,476.00
$110,400
$4,256.54
$2,012.55
$6,269.08
5
$110,400
$4,968.00
$2,400.00
$7,368.00
$108,000
$3,986.58
$1,925.88
$5,912.46
6
$108,000
$4,860.00
$2,400.00
$7,260.00
$105,600
$3,731.97
$1,842.95
$5,574.92
7
$105,600
$4,752.00
$2,400.00
$7,152.00
$103,200
$3,491.90
$1,763.59
$5,255.49
8
$103,200
$4,644.00
$2,400.00
$7,044.00
$100,800
$3,265.59
$1,687.64
$4,953.24
9
$100,800
$4,536.00
$2,400.00
$6,936.00
$98,400
$3,052.29
$1,614.97
$4,667.27
10
$98,400
$4,428.00
$2,400.00
$6,828.00
$96,000
$2,851.31
$1,545.43
$4,396.74
11
$96,000
$4,320.00
$2,400.00
$6,720.00
$93,600
$2,661.98
$1,478.88
$4,140.86
12
$93,600
$4,212.00
$2,400.00
$6,612.00
$91,200
$2,483.66
$1,415.19
$3,898.86
13
$91,200
$4,104.00
$2,400.00
$6,504.00
$88,800
$2,315.77
$1,354.25
$3,670.02
14
$88,800
$3,996.00
$2,400.00
$6,396.00
$86,400
$2,157.73
$1,295.93
$3,453.67
15
$86,400
$3,888.00
$2,400.00
$6,288.00
$84,000
$2,009.01
$1,240.13
$3,249.14
16
$84,000
$3,780.00
$2,400.00
$6,180.00
$81,600
$1,869.09
$1,186.73
$3,055.82
17
$81,600
$3,672.00
$2,400.00
$6,072.00
$79,200
$1,737.50
$1,135.62
$2,873.13
18
$79,200
$3,564.00
$2,400.00
$5,964.00
$76,800
$1,613.78
$1,086.72
$2,700.50
19
$76,800
$3,456.00
$2,400.00
$5,856.00
$74,400
$1,497.49
$1,039.92
$2,537.42
20
$74,400
$3,348.00
$2,400.00
$5,748.00
$72,000
$1,388.22
$995.14
$2,383.37
21
$72,000
$3,240.00
$2,400.00
$5,640.00
$69,600
$1,285.59
$952.29
$2,237.88
22
$69,600
$3,132.00
$2,400.00
$5,532.00
$67,200
$1,189.22
$911.28
$2,100.51
23
$67,200
$3,024.00
$2,400.00
$5,424.00
$64,800
$1,098.77
$872.04
$1,970.81
24
$64,800
$2,916.00
$2,400.00
$5,316.00
$62,400
$1,013.90
$834.49
$1,848.39
25
$62,400
$2,808.00
$2,400.00
$5,208.00
$60,000
$934.31
$798.55
$1,732.86
26
$60,000
$2,700.00
$2,400.00
$5,100.00
$57,600
$859.69
$764.17
$1,623.85
27
$57,600
$2,592.00
$2,400.00
$4,992.00
$55,200
$789.76
$731.26
$1,521.02
28
$55,200
$2,484.00
$2,400.00
$4,884.00
$52,800
$724.26
$699.77
$1,424.03
29
$52,800
$2,376.00
$2,400.00
$4,776.00
$50,400
$662.94
$669.64
$1,332.58
30
$50,400
$2,268.00
$2,400.00
$4,668.00
$48,000
$605.56
$640.80
$1,246.36
31
$48,000
$2,160.00
$2,400.00
$4,560.00
$45,600
$551.89
$613.21
$1,165.09
32
$45,600
$2,052.00
$2,400.00
$4,452.00
$43,200
$501.71
$586.80
$1,088.51
33
$43,200
$1,944.00
$2,400.00
$4,344.00
$40,800
$454.84
$561.53
$1,016.37
34
$40,800
$1,836.00
$2,400.00
$4,236.00
$38,400
$411.07
$537.35
$948.42
35
$38,400
$1,728.00
$2,400.00
$4,128.00
$36,000
$370.23
$514.21
$884.44
36
$36,000
$1,620.00
$2,400.00
$4,020.00
$33,600
$332.15
$492.07
$824.21
37
$33,600
$1,512.00
$2,400.00
$3,912.00
$31,200
$296.65
$470.88
$767.53
38
$31,200
$1,404.00
$2,400.00
$3,804.00
$28,800
$263.60
$450.60
$714.20
39
$28,800
$1,296.00
$2,400.00
$3,696.00
$26,400
$232.85
$431.20
$664.04
40
$26,400
$1,188.00
$2,400.00
$3,588.00
$24,000
$204.25
$412.63
$616.88
41
$24,000
$1,080.00
$2,400.00
$3,480.00
$21,600
$177.69
$394.86
$572.55
42
$21,600
$972.00
$2,400.00
$3,372.00
$19,200
$153.03
$377.86
$530.89
43
$19,200
$864.00
$2,400.00
$3,264.00
$16,800
$130.17
$361.59
$491.76
44
$16,800
$756.00
$2,400.00
$3,156.00
$14,400
$108.99
$346.01
$455.01
45
$14,400
$648.00
$2,400.00
$3,048.00
$12,000
$89.40
$331.11
$420.52
46
$12,000
$540.00
$2,400.00
$2,940.00
$9,600
$71.29
$316.86
$388.15
47
$9,600
$432.00
$2,400.00
$2,832.00
$7,200
$54.58
$303.21
$357.79
48
$7,200
$324.00
$2,400.00
$2,724.00
$4,800
$39.17
$290.15
$329.33
49
$4,800
$216.00
$2,400.00
$2,616.00
$2,400
$24.99
$277.66
$302.65
50
$2,400
$108.00
$2,400.00
$2,508.00
$0
$11.96
$265.70
$277.66
Principal
Interest Repayment
Periodic
14
New Zealand Applied Business Journal
Volume 1, Number 1, 2002
APPENDIX 7: ALGEBRAIC VERIFICATION OF THE TABLE MORTGAGE FORMULAE
To prove that the present value of the principal repaid at step r+1 for a table mortgage
is ( R  Pi)(1  i)1 , which is independent of r. Hence the NPV of all principal repayments
is n( R  Pi)(1  i)1 .
At step r the outstanding principal is P (1  i ) r  Rsr i i.e. the accumulated value of the
principal at the end of period r minus the accumulated value of the r payments of R made at
the end of each period.
At step r+1 the interest charged
=
 P(1  i)r  Rs   i
r i

The principal repaid at step r+1
=
R   P(1  i)r  Rsr i   i


The present value of the principal repaid at step r+1
=
 R   P(1  i)r  Rs   i   (1  i) r 1
r i



=
 R 1  is  Pi(1  i)r   (1  i) r 1
r i


=
 

(1  i ) r  1 
r
 r 1
R
1

i

 
  Pi (1  i )   (1  i )
i

 

=
 R 1  (1  i)r  1  Pi(1  i)r   (1  i)  r 1


=
 R(1  i ) r  Pi (1  i ) r   (1  i )  r 1
=
( R  Pi )(1  i ) r   (1  i )  r 1
=
 R  Pi 1  i 
=
 R  Pi 1  i 


r  r 1
1
There are n such payments so the NPV of the n payments
=
n  R  Pi 1  i 
1
For the model n = 50 so R = 6072.257503 (for verification purposes), P = 120 000, i = 0.045
So
NPV
=
50   6072.257503  120000  0.045   1.045 
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Volume 1, Number 1, 2002
New Zealand Applied Business Journal
=
32165.43077
=
$32 165.43
As shown on the spreadsheet
To show the NPV of the payments is P
This is true by definition of an ordinary simple annuity
At step r
PV of R
=
R 1  i 
=
 R 1  i 
r
n
The sum of the n payments
r
r 1
n
=
R  1  i 
r
r 1
=
Ran i
=
P
To prove the NPV of the interest payments is P  n  R  Pi 1  i 
1
It is clear that this formula is simply the difference between the NPV of the repayments
minus the NPV of the principal repaid
Looking at step r+1 the interest
=
 P 1  i r  Rs   i
r i

And the PV of this interest
=
 P 1  i r  Rs   i  1  i  r 1
r i

Following the simplification above it becomes
=
r

1  i   1

r
 r 1
 Pi 1  i   Ri
  1  i 
i


=
 Pi 1  i r  R 1  i r  R   1  i  r 1


=
Pi 1  i 
=
R 1  i 
r  r 1
 r 1
 R 1  i 
r  r 1
 R 1  i 
  R  Pi 1  i 
 r 1
1
Close inspection will show this has been rearranged to resemble the previous two present
value calculations
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New Zealand Applied Business Journal
So
PV interest
Volume 1, Number 1, 2002
=
n
So
 PV interest
=
r 1
For our example
PV payments – PV principal repaid
n
n
r 1
r 1
 PV payments   PV principal repaid
=
P  n  R  Pi 1  i 
=
120000 - 32165.43077
=
$87 834.57
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Volume 1, Number 1, 2002
New Zealand Applied Business Journal
APPENDIX 8: ALGEBRAIC VERIFICATION OF THE REDUCING MORTGAGE
FORMULAE
To show that the present value of the principal repaid at step r is
The principal repaid at each step is the same and is equal to
P
r
1  i 
n
P
. This is simply discounted to
n
produce the necessary result
PV of principal repaid
=
NPV of the n principal repayments
=
P
r
1  i 
n
n
P
 n 1  i 
r
r 1
=
P n
r
1  i 

n r 1
=
P
a
n ni
By definition
The net present value of the repayments (including principal repayment and interest incurred)
is equal to P. Whereas in the case of the table mortgage the result follows directly from the
definition it might not be so apparent in this case.
At step r
Repayment
=
Principal repaid + Interest at step r
Principal outstanding at the beginning of step r
P
n
=
P   r  1
=
P

 P   r  1   i
n

=
Pi
1  n  r 
n
Repayment
=
P 
P
  P   r  1   i
n 
n
PV of the repayment
=
P 
P 
r
 n   P   r  1 n   i   1  i 

 

Interest incurred at step r
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New Zealand Applied Business Journal
Volume 1, Number 1, 2002
=
r
 P Pi

 n  n 1  n  r    1  i 
=
P
Pi
r
r
1  i   1  n  r 1  i 
n
n
=
PV principal repaid + PV interest incurred
n
So NPV
 PV principal repaid + PV interest incurred
=
r=1
=
n
n
r=1
r=1
 PV principal repaid +  PV interest incurred
n
n
P
Pi
r
r
1

i



1  n  r 1  i 


r 1 n
r 1 n
=
The first sum has already been evaluated as
n
the interest incurred. 
r 1
Pi
r
1  n  r 1  i 
n
P
a so it only remains to simplify the NPV of
n ni
Pi n
r
=
1  n  r 1  i 

n r 1
This is an example of an arithmetico-geometric progression. Here is one way this sum can be
evaluated.
n
Let S =
 1  n  r 1  i 
r
r 1
=
1  n  11  i 
1
 1  n  2 1  i   1  n  31  i  
2
3
 1  n   n  1  1  i 
=
n 1  i    n  11  i    n  2 1  i  
1
2
3
 n 1
 1  n  n 1  i 
 2 1  i 
 n 1
n
 11  i 
n
(1)
Multiply by 1  i 
1  i  S =
n 1  i    n  11  i    n  2 1  i  
 2 1  i 
=
n  1  i   1  i  

 n 1
=
n  an i
1
0
2
 n 2
 11  i 
(2) – (1)
iS
1
2
 1  i 
19
 n 2
 1  i 
n
 1  i  

 n 1
(2)
Volume 1, Number 1, 2002
New Zealand Applied Business Journal
So from above NPV interest incurred =
P
 iS
n

=
P
n  an i
n
=
P

P
a
n ni
By proving this independently it can be seen that the results are interconnected.
So NPV repayments =
P
P
an i  P  an i
n
n
=
P
It can be seen that these proofs can be verified immediately by referring to the spreadsheet.
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