Math 618 Notes
Chapter 1
•
Interest rate
j
for a period:
X
= value at the beginning,
Y
= value at the end. Then
Y

X
X
=
j
or
Y
= (1 +
j
)
X.
Example
The interest rates are
i
1
for the ﬁrst year,
i
2
the second year, and
i
3
the third
year. If $100 is deposited in the beginning of the ﬁrst year, what is the accumulated amount
A
at the end of the third year?
Answer:
A
= 100(1 +
i
1
)(1 +
i
2
)(1 +
i
3
)
If
i
1
=
i
2
=
i
3
=
i
, then
A
= 100(1 +
i
)
3
.
•
Notation: We will use
α
(
t
) as the factor for accumulated amount after
t
periods. For example,
for 5 years ubder a period rate of
i
, the factor is
α
(5) = (1 +
i
)
5
.
•
Eﬀective annual rate
Example
If the monthly rate is 0.5%, the original amount
P
becomes
P
(1 +
.
005)
12
=
P
(1
.
0616778) at the end of the year. This is equivalent to an annual rate of 6.16778%, the
eﬀective annual rate.
•
Simple (annual) rate
i
:
α
(
t
) = 1 +
ti
,
A
(
t
) =
A
(0)(1 +
ti
)
where
t
is usually
m/
12 (in months) or
d/
365 (in days).
•
Example
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