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Lecture 9

# Actuarial Science 2053 Lecture 9: Sep 30, 2015 Premium

24 Pages
128 Views

Department
Actuarial Science
Course Code
Actuarial Science 2053
Professor
Steven Kopp

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Equivalent Rates (section 2.2)
Recall
j
m
= nominal rate of interest,
compounded m-times a year
i = interest rate per period
We now define:
j = annual effective rate of interest
that is equivalent to j
m
Meaning of Equivalent Interest Rates
If j
m
and j are equivalent rates of interest,
they must produce the same amount of
interest over a one-year period
Relationship
To develop a relationship between j
m
and j,
letâ€™s accumulate \$1 for one year under both
interest rates
Investing at j, \$1 will accumulate to
(1 + j) at the end of 1-year
Investing at j
m
, \$1 will accumulate to
(1 +
m
j
m
)
m
at the end of 1-year
(Actually, they must produced he same amount of interest over any period
of time)!
!
OR!
!
They must produce the same accumulated values over any period of time.
If j and j
m
are equivalent, then you must
have the same amount of money at the end
of the year under both rates:
(1 + j) = (1 +
m
j
m
)
m
Solving this equation, we obtain:

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Description
Equivalent Rates (section 2.2) Recall j = nominal rate of interest, m compounded mtimes a year i = interest rate per period We now define: j = annual effective rate of interest t h a t i s e q u i v a l e n t t o j m Meaning of Equivalent Interest Rates If j and j are equivalent rates of interest, m they must produce the same amount of interest over a oneyear period (Actually, they must produced he same amount of interest over any period of time) OR They must produce the same accumulated values over any period of time. Relationship To develop a relationship between j and j, m lets accumulate 1 for one year under both interest rates Investing at j, 1 will accumulate to (1 + j) at the end of 1year Investing at j, 1 will accumulate to m j m m (1 + ) at the end of 1year m
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