© ACTEX 2012 SOA Exam MLC – Six Practice Exams

**TABLE OF CONTENTS **

### Practice Exam 1

### 3

### Practice Exam 2

### 15

### Practice Exam 3

### 27

### Practice Exam 4

### 41

### Practice Exam 5

### 53

### Practice Exam 6

### 65

### Solutions to Practice Exam 1

### 79

### Solutions to Practice Exam 2

### 89

### Solutions to Practice Exam 3

### 99

### Solutions to Practice Exam 4

### 113

### Solutions to Practice Exam 5

### 125

### Practice Exam 4

### 41

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© ACTEX 2012 SOA Exam MLC - Six Practice Exams

**Practice Exam 4**

**Questions__________________________________________**

4-1. The survival function for the lifetime of a newborn in a population is

= B œ " Þ!"Ba b a bÞ(&. If the force of mortality at all ages is decreased by "!!5% , the increase in the life expectancy of a b%! is 1.217 years.

Calculate 5.

(A) .02 (B) .04 (C) .06 (D) .08 (E) .10

4-2. At age 60, George purchases a 5-year deferred life annuity. You are given:

(i) Contract premiums of 10,000 are paid annually at the beginning of each of the next 5 years while George is surviving.

(ii) The life annuity pays 12,000 per year continuously while George is surviving. (iii) $ œ Þ!&

(iv) P is the loss at issue random variable.

Calculate the value of if death occurs at age 67.4.P

### 42

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### Practice Exam 4

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© ACTEX 2012 SOA Exam MLC - Six Practice Exams

4-3. ‡For a fully discrete 20-year term life insurance on (40), you are given: (i) The death benefit is 10,000.

(ii) The death benefit is payable at the end of the year of death. (iii) Values in year 4:

Anticipated Actual

Gross annual premium

Expenses as a percent of premium Annual effective rate of i

*! *!

!Þ!$! !Þ!#&

;%$ !Þ!!$ !Þ!!#

nterest !Þ!& !Þ!%

(iv) Expected value of the present value of future losses random variable based on assumed values in years 3 and later.

End of year Expected value

$ "!!

% "#&

A company issued the 20-year term life insurance to 1000 lives age 40 with independent future lifetimes. At the end of the 3-rd year 990 insurances remain in force. Calculate the total profit from mortality, interest and expenses in year 4 from the 990 insurances.

A 8,345 B 8,385 C 8,425 D 8,465 E 8,505

Ð Ñ Ð Ñ Ð Ñ Ð Ñ Ð Ñ

Study Note MLC-09-11, Problem No. 300 ‡

4-4. *For a discrete whole life insurance of 1,000 on (x), you are given:*
(i) 1,000 _{# B}Z œ ")Þ)"Þ

(ii) 1,000 39.10% BZ œ Þ

(iii) ;B# œ Þ!!$%%and ;B$ œ Þ!!$("Þ (iv) 3 œ Þ!'Þ

Calculate the annual benefit premium 1,000TB.

Practice Exam 4 - Solutions 113

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© ACTEX 201 # SOA Exam MLC - Six Practice Exams

**Solutions to Practice Exam 4**

4-1. If =ÐBÑ œ " BÎa =b<, then ‰/ œ_{B} a= B Î < "b a b and .a bB œ <Îa= Bb. If .a bB is
decreased to .a bB œ.a baB " 5 œ < " 5 Îb a a bb a= Bb, then , since the force has the
same parametric form with the shifted to < < " 5a b, the increase in life expectancy at age
B is

=B =B < "5 "a b <"

So when B œ %! we have

"Þ#"( œ _{Þ(& "5 "}_{a}'!_{b} _{Þ(&"}'! Ê 5 œ Þ!(** **ANSWER D**

4-2. The value of the annuity payments of 12,000 per year between ages 65 and 67.4 is
12,000Ð+_{(Þ%}_{l} + Ñ œ "#ß !!! 'Þ")&$"$ %Þ%#$*)% œ #"ß "$'_{&l} a b .

The value of the premiums coming in to the insurer is

(note "!ß !!! + œ "!ß !!! %Þ&$&&!' œ %&ß $&&ÞÞ_{&l} a b that . œ " @ œ " /Þ!&ÑÞ
Hence the loss value is P œ #"ß "$' %&ß $&& œ #%ß #"*. **ANSWER B**

4-3. Recall that profit from mortality, interest and expenses emerges as a result of the

experience being better than expected (and losses arise from experience being worse than expected). Thus if we evaluate total excess of experienced profit over the assumed profit, such profit must have emerged from these three sources, combined. Note that we do this calculation as of year 3, for then existing policies, so that even if 990 is not the expected number of policies in force as of time 3, that's irrelevant, because we are doing the calculation prospectively, for year 4. The expected profits for year 4 are

_{î}**! † _{Š} _{í}"!! *!_{í} !Þ!$! † *!_{ðóñóò}_{‹} † Ð" !Þ!&Ñ_{í}
Number of
policies in
force
Reserve at the
beginning of

the year Expectedpremium

Expected amount of expenses as a

percent of premium interestExpected

!Þ!!$ † "!!!! Ð" !Þ!!$Ñ † "#& ¸ %"ß '"*Þ'!ßî î í

43

Expected *q* Expected *q*43

Expected reserve

114 Practice Exam 4 - Solutio

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© ACTEX 201 # SOA Exam MLC - Six Practice Exams

4-3. (cont.)

while the actual profits are

_{í}**! † _{Š}_{í}"!! *! !Þ!#& † *! † Ð" !Þ!%Ñ_{ì} _{ðóñóò}_{‹} _{í}
Number of
policies in
force
Reserve at the
beginning of

the year Actualpremium

Actual amount of expenses as a

percent of premium Actual_{interest}

!Þ!!# † "!!! Ð" !Þ!!#Ñ † "#& ¸ &!ß !!%Þ*!ßî î í

43

Actual *q* Actual *q*43

Expected reserve

so that the total profit from mortality, interest and expenses equals approximately

50,004.90 - 41,619.60 = 8385.30 **ANSWER B**

4-4. Using the idea that assets and liabilities match in net premium accounting, the reserve at duration four is a survivors share of the assets accumulated by time four. Thus

")Þ)" I "ß !!! T = I œ $*Þ"! ÞÞ "ß !!! E # B# B B#À#l # B# B#À#l " a b

The first two terms are a survivor's share of the accumulation of assets at time 2 plus the premiums at times 2 and 3.The term subtracted is a survivors share of the accumulated cost of benefits paid out at times 3 and 4.

To solve for the premium, the following values are needed :
+ œ " œ "Þ*%!"&"
ÞÞ " Þ!!$%%
"Þ!'
B#À#l
E œ Þ!!$%% Ð" Þ!!$%%ÑÞ!!$(" œ Þ!!'&$'
"Þ!' "Þ!'
B#À#l
"
#
#IB# œ _{#} œ Þ))$'%%
" Þ!!$%% " Þ!!$("
"Þ!'
a ba b

Substituting these results into the equation above and solving for the premium results in