1. A single premium 10-year term life insurance policy with benefits payable at the end of the yearof death is issued to (30). You are given:

(i) Mortality follows the Illustrative Life Table.

(ii) i = 0.06

(iii) Sales commissions are 18% of expenses-loaded premium

(iv) Taxes are 2% of expense-loaded premium

(v) Per policy expenses

First year = 40

Renewal = 5 per year

Calculate the policy fee that should be charged.

A. 9.00 B. 11.90 C. 73.75 D. 79.30 E. 92.15

2. For a life table with a one-year select period, you are given:

(i)

x ![x] d[x] !x+1e[x]

80 1000 90 - 8.581 920 90 - -82 850 90 - -

(ii) Deaths are uniformly distributed over each year of age.

Calculatee[82].

A. 7.9 B. 8.0 C. 8.1 D. 8.2 E. 8.3

7.736842

8.836842

7.86

3. You are given the following information for a person aged x:

(i) (1)x (t) = 0.05 for 0 t(ii) (2)x (t) =

1

20 t for 0 t < 20

where the index (1) indicated death due to accidental causes and the index (2) indicates deathdue to non-accidental causes.

Determine the probability that (x) will die due to accidental causes.

A. e1 B. 1 e1 C. e1.5 D. 1 e1.5 E. e2

4. You are given:

(i) Bill and Ted are both age 30.

(ii) Bills force of mortality function is (x) = 4100x .

(iii) Teds force of mortality function is (y) = 3100x .

Find the probability that Bill dies within 10 years and after Ted.

A. 0.083 B. 0.093 C. 0.103 D. 0.113 E. 0.123

5. For a special fully-discrete whole life whole life insurance issued to a (x) you are given:

(i) The death benefit is paid at the end of the year of death.

(ii) The death benefit paid at the end of the hth year is bh.

(iii) The terminal reserve at time h is Vh+1.

(iv) The level benefit premium paid at the beginning of each year is P .

(v) The annual effective rate of interest is i.

Which of the following is an expression for the tth terminal reserve for this special policy?

A. P st t1h=0

qx+h (bh hV ) (1 + i)th

B. Pst t1h=0

qx+h (bh hV ) (1 + i)th

C. P st t

h=0

vqx+h (bh+1 h+1V ) (1 + i)th

D. Pst t

h=0

vqx+h (bh+1 h+1V ) (1 + i)th

E. P st t1h=0

vqx+h (bh+1 h+1V ) (1 + i)th

6. For fully discrete 3-year endowment insurance of 1000 on (x), you are given:

(i) i = 0.10

(ii) Expenses, which occur at the beginning of the policy year, are as follows:

First Year Renewal YearsPercentage of premium 20% 6%

Per policy 8 2

(iii) The annual contract premium is equal to 314.

(iv) The following double-decrement table:

k p()x+k q(d)x+k q

(w)x+k

0 0.54 0.08 0.381 0.62 0.09 0.292 0.50 0.50 0.00

(v) The following table of cash values and asset shares:

k k+1CV kAS0 247 01 571 -

Calculate 2AS.

A. 257 B. 326 C. 415 D. 423 E. 452

7. The random variable kL is the prospective loss at time k for a fully discrete 3-year endowmentinsurance of 3 on (x).

You are given:

(i) qx = 0.009

(ii) qx+1 = 0.011

(iii) The premium is 3Px:3 = 0.834

(iv)

k 3 kVx:31 0.8982 -3 3.000

Determine 3 2Vx:3 .

A. 1.692 B. 1.792 C. 1.892 D. 1.992 E. 2.092

8. For a last-survivor 20-year term insurance of 1000 on (x) and (y) you are given:

(i) The death benefit is payable at the moment of the second death if the second death occursduring the next 20 years.

(ii) The independent random variables T (x), T (y), and Z are the components of a commonshock model.

(iii) T (x) has an exponential distribution with T(x)

x (t) = 0.03, t 0(iv) T (y) has an exponential distribution with T

(y)y (t) = 0.05, t 0

(v) Z, the common shock random variable, has an exponential distribution with Z(t) =0.02, t 0

(vi) = 0.06

Calculate the actuarial present value of this insurance.

A. 0.198 B. 0.216 C. 0.271 D. 0.303 E. 0.326

9. You are given the following forms of decrement:

(d) death

(w) withdrawal

(i) early retirement

(r) normal retirement

You are also given:

(i) Decrements follow the Illustrative Service Table.

(ii) For partial years all decrements are uniformly distributed in the multiple-decrement table.

Calculate q(d)35 .

A. 0.00134 B. 0.00144 C. 0.00154 D. 0.00164 E. 0.00174

10. You are given:

(x) =2x

10,000 x2 for 0 x < 100Determine the probability that a life aged 30 dies between ages 40 and 50, 10|10q30 .

A. 0.05 B. 0.10 C. 0.15 D. 0.20 E. 0.25

11. The total dental claims for an insured are classifed each year as full (F), partial (P) or no claims(N). The transition from year to year is as follows

Given that an insured has no claims in 2009, what is the probability of no claims in 2012?

A. 0.17 B. 0.18 C. 0.19 D. 0.20 E. 0.21

12. On average James Washer receives 13.8 emails per day, which arrive according to a Poissonprocess. 50% of all his emails are MLC-related. For a given day, starting at midnight, what isthe median time until James receives his first MLC-related email for the day?

A. 2:20 am B. 2:25 am C. 2:30 am D. 2:35 am E. 2:40 am

13. For a 20-year endowment insurance of 100 on (40) with death benefit payable at the end of theyear of death, you are given:

(i) i = 6%

(ii) Mortality follows the Illustrative Life Table.

(iii) Z is the present value random variable for this insurance.

Calculate Var(Z).

A. 48 B. 72 C. 106 D. 183 E. 299

14. Your best friend, Amy, won the lottery and is given 5 options for her winnings:

1. A lump-sum of $1,000,000 today.

2. A lump-sum of $500,000 today and another lump-sum of $1,000,000 in 20 years if she isstill alive.

3. $70,000 at the beginning of each year for the rest of her life starting today.

4. $150,000 at the beginning of each year for the rest of her life starting today, but no morethan 10 payments.

5. $50,000 at the beginning of each year for the rest of her life starting today, but no less than10 payments.

Your friend is not an actuary and turns to you for help. You decide to advise her to take theoption with the largest actuarial present value. You make the following assumptions:

(i) i = 6%

(ii) Amy just turned 41 today.

(iii) Amys mortality follows the Illustrative Life Table.

Which option do you advise your friend to take?

A. 1 B. 2 C. 3 D. 4 E. 5

15. For a fully discrete whole life insurance of $100,000 on each of 10,000 lives age 60, you are given:

(i) The future lifetimes are independent.

(ii) Mortality follows the Illustrative Life Table.

(iii) i = 0.06

(iv) $3380 is the premium for each insurance of $100,000.

Using the normal approximation, calculate the probability of a positive total loss.

A. 0.0091 B. 0.0096 C. 0.0102 D. 0.0110 E. 0.0118

16. You are given the following:

(i) (x) = 0.04

(ii) = 0.06

(iii) Two lives (30) and (40) are independent.

Calculate Cov[vT (30:40), vT (30:40)

].

A. 0.00 B. 0.01 C. 0.02 D. 0.03 E. 0.04

17. You are given the following:

(i) Mortality follows the Illustrative Life Table.

(ii) i = 0.06

Calculate 1000 (10V30:20 ) .

A. 356 B. 366 C. 376 D. 386 E. 396

18. Rainfall is modeled by a homogeneous Markov Chain. If it rained on a given day then the probthat it rains the next day is 0.7. If it didnt rain on a given day then the probability that rainsthe next day is 0.2. When it rains the total rainfall for the day is always 2 inches of rain. Giventhat it rained today, what is the variance of the total rainfall for the next 3 days?

A. 5.08 B. 5.18 C. 5.28 D. 5.38 E. 5.48

19. A worker has become disabled as a result of an occupational disease. The workers compensationlaws of state X specify that, in addition to disability benefits, the workers estate will receive alump sum payment at the time of her death if the worker dies as a result of the occupationaldisease.

You are given:

(i) The lump sum payment is $50,000 if the death occurs within the next 20 years.

(ii) Otherwise, the lump sum payment is $25,000.

(iii) The force of mortality for death as a result of the disease, (1) = 0.02

(iv) The force of mortality for death from all other causes, (2) = 0.015

(v) The force of interest, = 0.05

Determine the actuarial present value of this death benefit.

A. $6,630 B. $10,315 C. $10,690 D. $11,765 E. $18,700

20. You are given:

(i)e30 is the expected future lifetime for (30) under the assumption of constant force with = 0.10.

(ii)e30 is the expected future lifetime for (30) if a life table is constructed at annual points in

time using a constant force of mortality model with = 0.10 and a uniform distributionof deaths for partial years is assumed.

Calculate 1000(e30 e30

).

A. 0.0 B. 2.3 C. 4.3 D. 6.3 E. 8.3

21. You are given:

(i) Mortality follows the Illustrative Life Table.

(ii) i = 0.05

(iii) 1000A 132:8

= 13.31

Calculate 1000A 130:10

.

A. 14.20 B. 14.35 C. 14.50 D. 14.75 E. 14.95

22. Which of the following expressions equals Var (aT ), where2ax is based on the force of interest

2?

A.1

(ax 2ax

) a2xB.

2

(ax 2ax

) a2xC.

1

(ax 2ax

)2 a2xD.

2

(ax 2ax

)2 a2xE.

1

(ax 2ax

)2 ax

23. You are given:

(i) Ax = 0.6

(ii) n|Ax = 0.4

(iii) Px = 0.1

(iv) Px+n = 0.2

Calculate P 1x:n .

A. 0.05 B. 0.06 C. 0.07 D. 0.08 E. 0.09

24. The status of an insured can be classified as one of three states: Healthy (1), Disabled (2) orDead (3). You are given then following:

(i) Transitions happen at the end of each year.

(ii) A death benefit of $1000 is paid at the end of the year in which the insured transitions to#3 from either #1 or #2.

(iii) Benefit premiums of P are paid by the insured at the beginning of every year if in State#1.

(iv) If the insured is disabled at the start of a year then the premium is waived and the insurancecompany pays the amount of the benefit premium to the insured.

(v) Policies are only issued to people currently in state #1.

(vi) d = 0.05%

(vii) The one-year transition matrix between states is 0.7 0.2 0.10.0 0.8 0.20.0 0.0 1.0

Using the equivalence principle calculate P for a 3-year term insurance policy.

A. 135 B. 140 C. 145 D. 150 E. 155

0.49

174.75

25. You are given two independent lives (x) and (y) and the following information:

(i) x = 20

(ii) y = 40

(iii) The lifetimes of (x) and (y) follow DeMoivres law, with = 100.

Calculate the probability that exactly one life will survive for at least 20 years.

A.1

12B.

1

4C.

5

12D.

7

12E.

3

4

26. You are given the following information about a triple decrement insurance model:

(i) (1)(x) = 180x for x < 80

(ii) (2)(x) = 280x for x < 80

(iii) (3)(x) = 380x for x < 80

(iv) i = 0

Calculate the actuarial present value of a continuous 10 year temporary annuity that pays 1per year as long as no decrement has occurred issued to a life age 30.

A. 4.6 B. 5.6 C. 6.6 D. 7.6 E. 8.6

27. Harry the Hustler runs a 3 card monte game on a street corner in New York City. Players arriveaccording to a Poisson process at a rate of 10 per day. Half of the players arriving are Foolsand the other half are Suckers.

Fools can expect to lose $100 with variance $1000. Suckers can expect to lose twice as muchas Fools, but their variance is half of the Fools variance. The police show up 20% of the time,but only stop the Suckers from playing since a Fool and his money are soon parted anyways.

Find the standard deviation of Henrys profit for any given month. Assume a month is 30 days.

A. $2550 B. $2610 C. $2670 D. $2730 E. $2780

28. You are given the following:

x Ax 15P(Ax)

20P(Ax)

40 0.15 0.015 0.013

45 0.20 0.020 0.018

60 0.40 0.040 0.039

Calculate 205 V(A40

).

A. 0.02 B. 0.03 C. 0.05 D. 0.07 E. 0.14

29. For a population of individuals age x, you are given:

(i) Each individual has a constant force of mortality.

(ii) The forces of mortality are uniformly distributed over the interval (0,1).

(iii) The constant force of interest is .

Calculate the actuarial present value of a continuously increasing, at a rate of 1 per year,continuous whole life annuity drawn at random from this population,

(I a)x.

A. 1 + B. ln(1 + ) C.1

(1 + )D. ln

(1

(1 + )

)E. Not enough information

30. After passing MLC an actuarial student decides to celebrate by drinking wine. She has twochoices - cheap and expensive. Since she is celebrating she decides her first glass will be theexpensive wine.

After her first glass there is 10% chance she will stop drinking. This probability will doubleafter each glass of wine she drinks.

After her first glass there is an 80% chance she will order the expensive wine again for her 2ndglass (assuming she doesnt stop drinking). This probability will be cut in half with each glassof expensive wine she drinks. Once she switches to cheap wine she will never order expensivewine again.

What is the probability that she has a 3rd glass of wine and it is cheap?

A. 0.45 B. 0.47 C. 0.49 D. 0.51 E. 0.53