Chapter 1. The Growth of Money
ACTSC231 — Mathematics of Finance
Department of Statistics and Actuarial Science
University of Waterloo
Fall 2010
Instructor: Chengguo Weng
C. Weng ([email protected])
– p. 1/3

Interest
•
(p10) Monday has time value
XBox
investment opportunities theory.
XBox
time preference theory.
•
(p10)
Interest
is the payment by a borrower to a lender (investor) in return for
the use of the capital.
XBox
Interest may be paid annually, monthly, weekly, or daily
......
=
⇒
continuously.
XBox
The original sum invested (or borrowed) is the
capital
or
principal
.
XBox
The
total repaid
is the principal plus interest.
C. Weng ([email protected])
– p. 2/3

•
Example 1.1.
John lends Tom $10,000 for 5 years. Tom offers the following
three options of repayments:
Opt1
: Pay $10,500 at the end of 5 years.
Opt2
: Pay $100 at the end of each year, and repay the principal after 5 years.
Opt3
: Make payment of principal and interest of $2,100 at the end of each year.
Which option is most favorable for John and which one is best for Tom?
C. Weng ([email protected])
– p. 3/3

Accumulation Function
•
(p11) Consider a single investment of $1 at time 0 without any
withdrawal up to time
t >
0
XBox
At time
t
, the
accumulated value
(or
accumulation
) of the
investment is the total of the principal plus interest added.
XBox
Notation:
a
(
t
)
is used to denote the accumulation at time
t
of
$1 invested at time 0.
RHD
a
(
t
)
as a function of
t
is called accumulation function.
RHD
a
(0) = 1
RHD
if interest is positive,
a
(
t
)
is an increasing function of
t
C. Weng ([email protected])
– p. 4/3

Amount Function
•
(p11) We use
A
K
(
t
)
to denote the accumulated value at time
t
of
$
K
invested at time 0.
XBox
Is it true:
A
K
(
t
) =
K
·
a
(
t
)
?
RHD
It may not be true
e.g. A bank account earns 3% if
K
≥
5000
, and 2.5%
otherwise.
XBox
Unless stated otherwise, we generally assume
A
K
(
t
) =
K
·
a
(
t
)
.
C. Weng ([email protected])
– p. 5/3

Example 1 of accumulation functions (p12-13)
•
e.g.1.
a
(
t
) = 1 + 0
.
05
t
(simple interest). 5% of original capital
is added each year, paid continuously.
•
Graph of
a
(
t
)
:
C. Weng ([email protected])
– p. 6/3

Example 2 of accumulation functions (p12-13)
•
e.g.2.
a
(
t
) = 1 + 0
.
05
⌊
t
⌋
,
XBox
where
⌊
t
⌋
denotes the floor of
t
, i.e. the integer part of
t
.
XBox
interest is added only at the end of year
•
Graph of
a
(
t
)
:
C. Weng ([email protected])
– p. 7/3

Example 3 of accumulation functions (p12-13)
•
e.g.3.
a
(
t
) = 1
.
05
t
(compound interest).
XBox
5% of the accumulation (capital+interest earned) is added
continually.
•
Graph of
a
(
t
)
:
C. Weng ([email protected])
– p. 8/3