MATH 471: Actuarial Theory I
Final Exam: Formula Summary (Chapter 6 and Chapter 7 Part)
This review sheet ONLY contains the key formulas from Chapters 6 and 7
of the text. You should refer to the review sheets for Midterms 1 and 2 for
other key formulas. You should also refer to the text, notes, homework, etc.
BENEFIT PREMIUMS:
TwoStep Process for Pricing Premiums:
1) Find an expression for the lossatissue random variable (at time 0):
L
= PV(future benefits)  PV(future premiums)
2) Using the above
L
, calculate each premium using either:
•
Equivalence Principle:
E
(
L
) = 0
=
⇒
APV(future benefits) =
APV(future premiums) at time 0, or
•
Percentile Principle:
Pr
(
L >
0) =
α
, where 0
< α <
1.
Premiums calculated via the equivalence principle are called
benefit pre
miums
. Premiums calculated via the percentile method are called
100
α
th
percentile premiums
.
Fully Continuous Premiums
:
Fully Continuous Life Insurance:
a continuous life insurance on (x).
The insurance is purchased with a continuous life annuity of annual premi
ums.
Examples of fully continuous life insurances: Refer to Table 6.2.1.
Note that in Table 6.2.1, benefit premiums are generally of the form:
¯
P
(
¯
A
)
=
¯
A
¯
a
. This is for a standard “full payment insurance,” where premiums are
payable for the duration of the insurance coverage. Note the difference for
h
payment insurance.
1
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For a fully continuous whole life insurance of 1 on (x):
var
(
L
) = (1 +
¯
P
δ
)
2
[
2
¯
A
x

(
¯
A
x
)
2
]
A similar formula for
var
(
L
) applies to a fully continuous
n
year endow
ment insurance of 1 on (x).
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 Spring '08
 Staff
 Formulas, ax, Endowment policy

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