CHAPTER 7
FINANCIAL MATHEMATICS
Page
Contents
7.1
Compound Value
115
7.2
Compound Value of an Annuity
116
7.3
Sinking Funds
117
7.4
Present Value
120
7.5
Present Value of an Annuity
120
7.6
Term Loans and Amortization
121
Exercise
Objectives:
7.1
125
After working through this chapter, you should be able to:
(i)
explain the term compound interest;
(ii)
explain the meaning of an annuity;
(iii)
calculate the compound amount and the present value of an annuity;
(iv)
set up an amortization schedule for the amortization of debt;
(v)
set up a sinking fund schedule and explain some of the applications
of sinking funds.
Compound Value
Chapter 7: Financial Mathematics
Define the following terms :
P0 = principal, or beginning amount at time 0.
i
= interest rate
I
= total amount of interest earned.
Pn = principal value at the end of n periods
Then Pn may be calculated as follows :
P1 = P0 + I = P0 + P0i = P0(1 + i)
P2 = P1 + P1i = P1(1 + i) = P0(1 + i)2
P3 = P2 + P2i + P2(1 + i) = P0(1 + i)3


Pn = P0(1 + i)n
where (1 + i)n is called the Compound Value Interest Factor.
Example 1
A loan of $35,000 made today is to be repaid by a single payment of 42,000 two years
from now. Find the annual interest rate.
42,000 = 35,000 (1 + i)2
(1 + i)2 = 1.2
(1 + i) = 1.095
i = 9.5%
When compounding periods are more frequent than once a year, then
i

Pn  P0  1  

m
mn
where m is the numbers of times per year compounding occurs.
Example 2
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Chapter 7: Financial Mathematics
Find the amount to which $10,000 will grow after five years if semiannual compounding is
applied to a stated 5% interest rate.
i = 5%
m=2
5% 

P5  10,000 1 


2 
7.2
25
 10,000(1.025)10  $12,800
Compound Value of an Annuity
Def : An annuity is defined as a series of payments of a fixed amount for a specified
number of years. Each payment occurs at the end of the year.
Example 3
Suppose you are to receive a three-year annuity of $10,000 and deposit each annual
payment in a saving account paying 8% interest. How much will you have at the end of
the third year?
End of Year
0
1
2
3
10,000
10,000
10,000
 1.08
10,800
 (1.08)2
11,664
Compound Sum :
$32,464
======
In general, we have :
0
1
2
3· · ·
R
R
R · · ·
n1
R
n
R
R(1 + i)
·
·
·
R(1 + i)n2
R(1 + i)n1
Sn
Define
Sn = compound sum
R = periodic receipt
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Chapter 7: Financial Mathematics
n = length of annuity
Sn  R  R(1  i ) R(1  i )n  2  R(1  i )n 1

 R 1  (1  i ) (1  i ) n  2  (1  i ) n 1

 (1  i ) n  1
 R

i


Compound Value Factor of an annuity.
7.3
Sinking Funds
-
When a sum of money will be needed at some future date, a good practice is to
accumulate systematically a fund that will equal the sum desired at the time it is
needed. Money accumulated in this way is called a sinking fund.
Example 4
A machine acquired at the beginning of this year is expected to last 10 years and its
replacement price is estimated to be $8,000. What annual provision must be made to
ensure sufficient fund is available if money can be invested at 8% per annum?
(i)
If the sinking fund a is to be set aside at the end of each year,
0
1
2
3 · · · · ·
10
a
a
a · · · · ·
a
then
a + a(1.08) + a(1.08)2 +  + a(1.08)9 = 8,000
 (1.08)10  1
a
  8,000
 0.08 
a = $552.23
117
replace the machine
Chapter 7: Financial Mathematics
(ii)
Suppose the firm wish to start the fund now (i.e., set aside a sum of money a at the
beginning of each year)
0
1
2 · · · · ·
a
a
a · · · · ·
9
10
replace the machine
a
then
a (1  i)  a (1  i)2 a (1  i)10  8,000


a (1  i ) 1  (1  i ) 2  (1  i ) 9  8,000
 (1  i )10  1
a (1  i )
  8,000
i


a = 511.33
Example 5
Sinking Fund Schedules
The View Royal Fire District needs a new fire truck which will cost $300,000. The district
is able to arrange the necessary financing provided a sinking fund is established to provide
for repayment of the debt. The loan must be repaid in four years. Monies for repayment
will come from a tax increase on the land owners who are part of the fire district. The
interest cost on the loan must be paid every six months as per the loan agreement with the
province. If the district earns 8% compounded annually on the sinking fund, if the interest
rate on the loan is 12% compounded semi-annually and if the payment must be made
annually to the sinking fund, answer the following questions.
a)
Set up a sinking fund schedule to show the accumulation of the fund and the book
value of the debt for each year.
b)
Show the total annual expense associated with the interest payment and the sinking
fund payment.
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Chapter 7: Financial Mathematics
A
B
C
D
E
F
Year
Regular
Payment
Payment
Of
Interest
Increase
In
Fund
Accumulated
Balance
In the Fund
Accumulated
Book Value
Of the Debt
0
$300,000.00
1
$66,576.24
0
$66,576.24
$66,576.24
$233,423.76
2
$66,576.24
$5,326.10
$71,902.34
$138,478.58
$161,521.42
3
$66,576.24
$11,078.29
$77,654.53
$216,133.11
$83,866.89
4
$66,576.24
$17,290.65
$83,866.89
$300,000.00
$00000.00
TOTALS
$266,304.96
$33,695.04
$300,000.00
N/A
N/A
Example 6
Partial Sinking Fund Schedules
A small municipality is setting up a sinking fund with annual payments to repay a debt of
$520,000. The sinking fund will earn 12.55088%, compounded annually. If the fund is to
accumulate the desired sum of $520,000 over 28 years, construct a partial sinking fund
schedule which shows the fund at periods 3, 27 and 28.
A
B
C
D
E
F
Year
Regular
Payment
Payment
Of
Interest
Increase
In
Fund
Accumulated
Balance
In the Fund
Accumulated
Book Value
Of the Debt
2
$2,472.03
$5,254.33
$514,745.71
3
$2,472.03
$8,385.83
$511,614.18
$659.46
$3,131.50
26
7.4
$406,344.99
27
$2,472.03
$50,999.87
$53,471.91
$459,816.90
$60,183.10
28
$2,472.03
$57,711.07
$60,183.10
520,000.00
0
TOTALS
$69,216.84
N/A
N/A
N/A
N/A
Present Value
Example 7
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Chapter 7: Financial Mathematics
Suppose you are offered the alternative of either $1,500 at the end of three years
today. If your saving account pays a 10% interest p.a. Find x.
or
$x
x(110
. )3  1,500
x
1,500
(11
. )3
 1127
,
$x is defined as the present value (PV) of $1,500 due in 3 years when the applicable
interest rate is 10%.
Finding present values (discounting) is simply the reverse of compounding.
Recall that
Pn  P0 (1  i ) n
Therefore
P0 
Pn
(1  i ) n
 1 
 Pn 
n
 (1  i ) 
PV factor
When discounting periods are more frequent than once a year, then


1
P0  Pn 
i mn 
 (1  m ) 
7.5
Present Value of an Annuity
Define :
An = present value of an annuity of n years
R = periodic receipt
n = length of annuity
i = interest rate
End of year
0
1
2
120
3· · · · · · ·
n
Chapter 7: Financial Mathematics
R
R· · · · · · ·
R
R
 1 
R

 1 i 
 1 
R

 1 i 
2
 1 
R

 1 i 
n
An

An  R (1  i ) 1  (1  i ) 2  (1  i )  n

1  (1  i ) n 
 R

i


PV factor of an annuity
7.5.1 Present Value in Perpetuity
A 
7.6
R
i
Term Loans and Amortization
A term loan is a business loan with a maturity of more than one year. Ordinarily, term
loans are retired by systematic repayments (often called amortization repayments) over the
life of the loan.
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Chapter 7: Financial Mathematics
7.6.1 Determination of Repayment Schedule
Example 8
Assume that a firm borrows $1,000,000 on a ten-year loan, that interest is
computed at 5% on the declining balance, and that the principal and interest are to
be paid in ten installments. Find the amount of each of the ten annual repayments
and produce a repayment schedule.
Let R be the amount of repayment
R
1  (1  i)  n 
from An  R 

i


An
1  (1  i )  n 


i



1,000
7.722
 $130
i.e.
Ten installments of $130,000 will have retired the one million loan and
provided the lender a 5% return on his investment.
Year
Total
Payment
Interest
Amortization
Repayment
Remaining
Balance
1
$130
$50
$ 80
$920
2
130
46
84
836
3
130
42
88
748
4
130
38
92
656
5
130
34
96
560
6
130
28
102
458
7
130
23
107
351
8
130
18
112
239
9
130
13
117
122
10
130
$1,300
=====
8
$300
====
122
$1,000
=====
0
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Chapter 7: Financial Mathematics
Example 9
Setting Up an Amortization Schedule
Janice and Phil Grantham have a mortgage of $61,818.44 which is to be amortized
over six months with an annual interest rate of 12%. Set up an amortization
schedule under the following conditions :
a)
Assume the interest rate is compounded monthly.
b)
Assume the interest rate is compounded semi-annually.
A
B
C
D
E
Payment
period
Regular
Payment
Payment
to
Interest
Payment
to
Principal
OutStanding
Balance
0
$61,818.44
1
$10,666.67
$618.18
$10,048.49
$51,769.95
2
$10,666.67
$517.70
$10,148.97
$41,620.99
3
$10,666.67
$416.21
$10,250.46
$31,370.52
4
$10,666.67
$313.71
$10,352.96
$21,017.56
5
$10,666.67
$210.18
$10,456.50
$10,561.06
6
$10,666.67
$105.61
$10,561.06
$00000.00
TOTALS
$64,000.02
$2,181.59
$61,818.44
N/A
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Chapter 7: Financial Mathematics
A
B
C
D
E
Payment
period
Regular
Payment
Payment
to
Interest
Payment
to
Principal
OutStanding
Balance
0
$61,818.40
1
$10,657.83
$603.27
$10,054.56
$51,763.88
2
$10,657.83
$505.15
$10,152.68
$41,611.21
3
$10,657.83
$406.08
$10,251.76
$31,359.45
4
$10,657.83
$306.03
$10,351.80
$21,007.65
5
$10,657.83
$205.01
$10,452.82
$10,554.83
6
$10,657.83
$103.00
$10,554.83
$00000.00
TOTALS
$63,946.98
$2,128.54
$61,818.44
N/A
Example 10
Partial Amortization Tables
Omega Holdings has arranged a $200,000 mortgage on a piece of property. The
arrangements are that the mortgage will be amortized over 20 years with monthly
payments based on an interest rate of 10%, compounded monthly. Show the
amortization schedule entries for the 36th and 37th payments.
A
B
C
D
E
Payment
period
Regular
Payment
Payment
to
Interest
Payment
to
Principal
OutStanding
Balance
35
$189,347.51
36
$1,930.04
$1,577.90
$352.14
$188,995.37
37
$1,930.04
$1,574.96
$355.08
$188,640.29
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Chapter 7: Financial Mathematics
EXERCISE:
FINANCIAL MATHEMATICS
1.
A sum of money is deposited now at 10% per annum. How long will it take for the
sum invested to double?
2.
We require $10,000 in 15 years time and we can deposit money at 10% per annum.
How much must be invested now to achieve this sum?
3.
Suppose $9,500 is invested on the 1st of Jan. of a certain year at 12% compound
and $800 is withdrawn at the end of each year. How much would remain after 12
years.
4.
A Co. decide to invest $10,000 at the beginning of 1999 in a fund earning 12% per
annum. A Co. will add a further $3,000 to the fund at the beginning of each year,
commencing in 2000.
(a)
What will be the value of total investment in the fund at the end of 2007?
(b)
Suppose A Co. now decide to make equal annual instalments starting at the
beginning of 1999 at the same interest rate of 12%, calculate the annual
instalments necessary for the fund to have the same value at the end of 2007
as in (a).
5.
Suppose we deposit $10,000 now, and we withdraw X at the end of each year for 5
years so that nothing is left on deposit. What is X if money can be invested at 10%
p.a. compound?
6.
At what rate of interest will money double its value in three years? Assume that
the rate is compounded semi-annually.
7.
Mr. Wong bought a flat on January 1, 1998 at three million dollars with down
payments of 30% of the purchase price. The remaining 70% is to be repaid by 180
monthly instalments at 6% p.a. compounded monthly. The first instalment due on
February 1, 1998. Calculate the amount (rounded to the nearest dollar) of each
instalment.
8.
ABC Holdings has just borrowed $300,000 to finance a new land development
project. The repayment requires 20 quarterly payments with interest 18% p.a.
compounded monthly, the first payment due 3 years from now. What is the size of
each quarterly payment?
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Chapter 7: Financial Mathematics
9.
Mr. Leung invests $1,000 at the end of each year for 20 years in an investment fund
which pays interest at 13% p.a. compounded annually. The fund pays the interest
at the end of each year and at the same time Mr. Leung receives each interest
payment, he deposits it into his bank account which pays interest at 10% p.a.
compounded annually. How much money does Mr. Leung have at the end of 20
years?
10.
Mr. Cheung buys a house and borrows $95,000 from the ABC Financial Company.
The loan is to be repaid with monthly payments over 30 years at 15% p.a.
compounded semiannually. The interest rate is guaranteed for 5 years. After
exactly two years of making payments, Mr. Cheung sees that interest rates have
dropped to 10.5% p.a. compounded semiannually in the market place. He asks to
be allowed to repay the loan in full so he can refinance. ABC agrees to renegotiate
but sets a penalty exactly equal to the money the company will lose over the next 3
years. Find the value of the penalty.
11.
A debt of $80,000 is to be amortized with $2,500 payable every month. The
interest rate is 11% compounded monthly. Construct an amortization schedule
showing the last three entries to complete the repayment of the debt.
12.
The ABC company has a mortgage of $65,000 on a property and has been making
regular payments of $3,500 every three months. If the interest rate is 8%,
compounded quarterly, what would be the first three entries in the amortization
schedule which shows repayment of the loan?
13.
A debt of $180,000 is to be amortized by using a sinking fund. The payments are
to be made monthly over the next five years and the interest rate is 12%,
compounded monthly. Construct a sinking fund schedule showing the first and last
two entries. Show the book value for each of the three periods in the table.
14.
A city has just borrowed $7.5 million for twenty years through the sale of bonds.
The money is to assist in undertaking major sewer reconstruction. The city has
started a sinking fund to handle repayment of the bond issue when it becomes due.
The fund is to accumulate money at 8%, compounded annually. The interest rate
on the bond is 10% annually, payable every 6 months. Rounding all calculations to
the nearest dollar, find:
(a)
the annual payment to the sinking fund. Construct a partial schedule for
periods 2, 17 and 18;
(b)
the annual budget expenditure to handle the interest on the debt and the
sinking fund;
(c)
the book value of the debt after 15 years,
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Chapter 7: Financial Mathematics
(d)
the entries for the schedule in part (a) if the interest rate on the sinking fund
had been 8%, compounded semi-annually
127