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5
Mathematics of Finance
• Compound Interest
• Annuities
• Amortization and Sinking Funds
• Arithmetic and Geometric Progressions
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Simple Interest
Simple Interest - interest that is compounded on
the original principal only.
Interest:
I = Prt
Accumulated amount: A = P(1 + rt)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. $800 is invested for 9 years in an account that
pays 12% annual simple interest. How much
interest is earned? What is the accumulated
amount in the account?
P = $800, r = 12%, and t = 9 years
Interest:
I = Prt
= (800)(0.12)(9)
or $864
= 864
Accumulated amount = principal + interest
= 800 + 864 = 1664
or $1664
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Compound Interest
Compound Interest – interest is added to the original
principal and then earns interest at the same rate.
A  P (1  i )
n
where i 
r
and n  mt
m
A = Accumulated amount after n periods
P = Principal
r = Nominal interest rate per year
m = Number of conversion periods per year
t = Term (number of years)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Find the accumulated amount A, if $4000 is
invested at 3% for 6 years, compounded monthly.
P = $4000, r = 3%, t = 6, and m = 12
r .03
So i  
 .0025 and n  mt  12(6)  72
m 12
A  P (1  i ) n
 4000(1  .0025)
72
 4787.79
or $4787.79
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Effective Rate of Interest
Effective Rate – the simple interest rate that would
produce the same accumulated amount in 1 year as
the nominal rate compounded m times per year.
m
reff
r

 1    1
 m
where
reff = Effective rate of interest
r = Nominal interest rate per year
m = Number of conversion periods per year
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Find the effective rate that corresponds to a
nominal rate of 6% compounded quarterly.
r = 6% and m = 4
m
reff
r

 1    1
 m
4
 .06 
 1 
 1
4 

 .06136
So about 6.136% per year.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Present Value (Compound Interest)
Present Value (principal) – the amount required
now to reach the desired future value.
P  A(1  i )
n
r
where i 
and n  mt
m
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Jackson invested a sum of money 10 years ago
in an account that paid interest at a rate of 8%
compounded monthly. His investment has grown to
$5682.28. How much was his original investment?
A = $5682.28, r = 8%, t = 10, and m = 12
r .08
i 
and n  mt  12(10)  120
m 12
 .08 
P  5682.28 1 

 12 
 2560.00
120
or $2560
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Annuity
Annuity – a sequence of payments made at regular
time intervals.
Ordinary Annuity – payments made at the end of
each payment period.
Simple Annuity – payment period coincides with
the interest conversion period.
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Future Value of an Annuity
The future value S of an annuity of n payments
of R dollars each, paid at the end of each
investment period into an account that earns
interest at the rate of i per period is
 (1  i ) n  1 
S  R

i


Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Find the amount of an ordinary annuity of 36
monthly payments of $250 that earns interest at a
rate of 9% per year compounded monthly.
.09
R = 250, n = 36 and i 
12
 (1  i ) n  1 
S  R

i




36

.09
1

1

12
S  250 
.09

12
S  10288.18






or $10,288.18
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Present Value of an Annuity
The present value P of an annuity of n payments
of R dollars each, paid at the end of each
investment period into an account that earns
interest at the rate of i per period is
1  (1  i )  n 
P  R

i


Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Paige’s parents loaned her the money to buy a
car. They required that she pay $150 per month, for
60 months, with interest charged at 2% per year
compounded monthly on the unpaid balance. What
was the original amount that Paige borrowed?
1  (1  i )  n 
P  R

i


 

.02
1

1


12
 150 
.02

12


 8557.85


60





or $8557.85
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Amortization Formula
The periodic payment R on a loan of P dollars to
be amortized over n periods with interest charged
at a rate of i per period is
Pi
R
n
1  (1  i)
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. The Kastners borrowed $83,000 from a credit
union to finance the purchase of a house. The credit
union charges interest at a rate of 7.75% per year on
the unpaid balance, with interest computations made
at the end of each month. The Kastners have agreed
to repay the loan in equal monthly installments over
30 years. How much should each payment be if the
loan is to be amortized at the end of the term?
0.0775
P = 83000, n = (30)(12) = 360, and i 
12
Continued
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Pi
R
n
1  (1  i )


83000 .0775
12
 
1  1  .0775
12


360
 594.62
So a monthly installment of
$594.62
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. A bank has determined that the Radlers can
afford monthly house payments of at most $750.
The bank charges interest at a rate of 8% per year on
the unpaid balance, with interest computations made
at the end of each month. If the loan is to be
amortized in equal monthly installments over 15
years, what is the maximum amount that the Radlers
can borrow from the bank?
R = 750, n = (15)(12) = 180, and i 
0.08
12
Continued
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
1  1  i  n 

P  R
i


 

.08
1

1


12
 750 
.08

12




180





 78480.44
So they can borrow up to
about $78480.44
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Sinking Fund Payment
The periodic payment R required to accumulate S
dollars over n periods with interest charged at a
rate of i per period is
iS
R
n
(1  i )  1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Max has decided to set up a sinking fund for
the purpose of purchasing a new car in 4 years.
He estimates that he will need $25,000. If the
fund earns 8.5% interest per year compounded
semi-annually, determine the size of each (equal)
semi-annual installment that Max should pay into
the fund.
0.085
S = 25000, n = 4(2) = 8, and i 
4
Continued
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
iS
R
n
(1  i )  1
.085  25000

4

.085
1

  4  1
8
 2899.91
So semi-annual payments of
about $2899.91
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Arithmetic Progressions
Arithmetic progression – a sequence of numbers in
which each term after the first is obtained by
adding a constant d (common difference) to the
preceding term.
Ex. 1, 8, 15, 22, 29, …
First term
Common difference: d = 7
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Arithmetic Progression
The nth term with the first term a and common
difference d is given by
an  a   n  1 d
The sum of the first n terms with the first term
a and common difference d is given by
n
Sn   2a  (n  1)d 
2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Given the arithmetic progression
1, 8, 15, 22, 29, …
find the 10th term and the sum of the
first 10 terms.
a = 1, d = 7, and n = 10.
10th term: an  a   n  1 d
a10  1  10 1 7  = 64
Sum:
n
Sn   2a  (n  1)d 
2
10
S10   2(1)  (10  1)(7)  = 325
2
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Geometric Progressions
Geometric progression – a sequence of numbers in
which each term after the first is obtained by
multiplying the preceding term by constant r
(common ratio).
Ex. 9, 3, 1, 1/3,…
Common ratio: r = 1/3
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Geometric Progression
The nth term with the first term a and common
ratio r is given by
an  ar
n1
The sum of the first n terms with the first term
a and common ratio r is given by

 a 1 rn

Sn   1  r

na

if r  1
if r  1
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc
Ex. Given the geometric progression
4, 12, 36, 108, …
find the 8th term and the sum of the first 8
terms.
a = 4, r = 3, and n = 8.
8th term:
an  ar
n1
 
81
a8  4 3
Sum:
Sn 
S8 

a 1 rn
1 r
4 1  38

1 3
= 8748


= 13120
Copyright (c) 2003 Brooks/Cole, a division of Thomson Learning, Inc