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Question 1Actuarial Mathematics

An insurer issues a whole life policy with sum assured 100,000 payable at the end of the year of death. Mortality follows a table with q_x = 0.01 for all ages and the interest rate is 4% per year. Using the standard whole life assurance formula A_x = q_x / (q_x + i) under this constant-force assumption, calculate the net single premium.

  1. 20,000 Correct answer.
  2. 19,231
  3. 24,000
  4. 25,000

Explanation

With constant q and constant interest the whole life assurance factor is A_x = q/(q+i) = 0.01/(0.01+0.04) = 0.2. Multiplying by the sum assured of 100,000 gives a net single premium of 20,000. This shortcut is valid only when q_x is constant across all ages and the benefit is paid at the end of the year of death.

Question 2Actuarial Mathematics

Which quantity is obtained by evaluating the expression (1 - A_x)/d, where A_x is the whole life assurance factor and d is the effective rate of discount?

  1. The expected present value of a temporary annuity-immediate
  2. The expected present value of a whole life annuity-due Correct answer.
  3. The net premium reserve at duration zero
  4. The expected present value of a deferred whole life annuity

Explanation

The identity a-double-dot_x = (1 - A_x)/d links the whole life assurance factor to the expected present value of a whole life annuity-due paying one unit at the start of each year while alive. The d in the denominator is the discount rate, reflecting payment at the beginning of the year. Rearranging the same identity gives A_x = 1 - d times a-double-dot_x.

Question 3Actuarial Mathematics

A life aged 40 has force of mortality mu = 0.008 per year and the force of interest delta = 0.05 per year, both constant. Calculate the expected present value of a continuous whole life annuity of 1 per year payable while the life survives.

  1. 17.24 Correct answer.
  2. 12.50
  3. 20.00
  4. 19.23

Explanation

For constant force of mortality and constant force of interest, the continuous whole life annuity value is 1/(mu + delta) = 1/(0.008 + 0.05) = 1/0.058, which equals approximately 17.24. The sum of the two forces acts like a combined discount rate in the integral of survival times discounting.

Question 4Actuarial Mathematics

Under the equivalence principle, how is the annual net premium for a whole life assurance with sum assured S and annual premiums payable in advance determined?

  1. It equals S times A_x divided by a-double-dot_x Correct answer.
  2. It equals S times A_x multiplied by a-double-dot_x
  3. It equals S times a-double-dot_x divided by A_x
  4. It equals S divided by the product of A_x and a-double-dot_x

Explanation

The equivalence principle sets the expected present value of premiums equal to the expected present value of benefits. For premiums payable in advance the premium P satisfies P times a-double-dot_x = S times A_x, so P = S A_x / a-double-dot_x. Using the identity a-double-dot_x = (1 - A_x)/d this can also be written as S d A_x/(1 - A_x).

Question 5Actuarial Mathematics

A term assurance pays 50,000 at the end of the year of death if death occurs within 10 years. Given A^1_{x:10} = 0.045 and a-double-dot_{x:10} = 7.8, calculate the level annual net premium payable in advance for the 10-year term.

  1. 351.00
  2. 225.00
  3. 288.46 Correct answer.
  4. 576.92

Explanation

The equivalence principle gives P times a-double-dot_{x:10} = 50,000 times A^1_{x:10}. Hence P = 50,000 times 0.045 / 7.8 = 2,250/7.8, which is approximately 288.46. The temporary annuity factor is used because premiums cease at the end of the term.

Question 6Actuarial Mathematics

Which statement correctly describes the net premium reserve at duration t for a whole life policy issued to a life aged x?

  1. It is the accumulated value of all premiums paid to date with no allowance for benefits
  2. It is the difference between gross premiums collected and expenses incurred to date
  3. It is the expected present value of future benefits minus the expected present value of future net premiums Correct answer.
  4. It is always equal to the sum assured less the accumulated premiums

Explanation

The prospective net premium reserve is the expected present value of future benefits less the expected present value of future net premiums, both evaluated at duration t. Equivalently, by the retrospective method, it equals the accumulated value of past premiums less the accumulated cost of past benefits. Gross premiums and expenses belong to gross premium reserves rather than net premium reserves.

Question 7Actuarial Mathematics

For a fully discrete whole life policy on (x) with net premium P_x, which recursion holds between reserves at successive durations?

  1. (V_x + P_x)(1+i) = p_{x+t}(S) + q_{x+t} times (t+1 V_x)
  2. (V_x + P_x) = q_{x+t}(S) + p_{x+t} times (t+1 V_x)
  3. (t+1 V_x)(1+i) = q_{x+t}(S) + p_{x+t} times (t V_x)
  4. (V_x + P_x)(1+i) = q_{x+t}(S) + p_{x+t} times (t+1 V_x) Correct answer.

Explanation

The recursion accumulates the current reserve plus the premium at interest, then splits the fund between the death benefit paid with probability q and the reserve needed for survivors with probability p. This gives (t V_x + P_x)(1+i) = q_{x+t} S + p_{x+t} times (t+1 V_x). It is the standard Fackler-style recursion for a fully discrete whole life policy.

Question 8Actuarial Mathematics

A life table gives l_40 = 90,000 and l_41 = 89,100. Calculate q_40, the probability that a life aged 40 dies before age 41, assuming deaths are uniformly distributed between integer ages.

  1. 0.0111
  2. 0.0090
  3. 0.0100 Correct answer.
  4. 0.1000

Explanation

The one-year death probability is q_40 = (l_40 - l_41)/l_40 = (90,000 - 89,100)/90,000 = 900/90,000 = 0.01. The uniform distribution assumption is not needed for the annual probability itself but would matter for fractional-age calculations.

Question 9Actuarial Mathematics

Under the uniform distribution of deaths assumption between integer ages, which expression gives the probability that a life aged x dies within t years, where t lies between 0 and 1?

  1. 1 minus (1 - t) times q_x
  2. t times q_x Correct answer.
  3. q_x divided by t
  4. t times p_x

Explanation

Uniform distribution of deaths means the deaths in the year of age are spread evenly, so the probability of dying within a fraction t of the year is t q_x. Equivalently the survival probability to age x+t is 1 - t q_x. This linear interpolation of l_{x+t} is the defining feature of the UDD assumption.

Question 10Actuarial Mathematics

An annuity-immediate of 1 per year is payable annually in arrears while (x) survives, for at most 20 years. Which symbol denotes its expected present value?

  1. a-double-dot_{x:20}
  2. A^1_{x:20}
  3. a_{x:20} Correct answer.
  4. a_{20}

Explanation

The symbol a_{x:20} denotes a temporary life annuity-immediate with payments in arrears, contingent on survival and limited to 20 years. The double-dot version would indicate payments in advance, and the assurance symbol would denote a death benefit rather than an annuity.

Question 11Actuarial Mathematics

A deferred whole life annuity of 1 per year payable annually in advance begins at age 65 for a life now aged 50. Which factor converts the whole life annuity-due value at age 65 into its expected present value at age 50?

  1. The 15-year pure endowment factor 15 E_50 Correct answer.
  2. The 15-year term assurance factor A^1_{50:15}
  3. The 15-year certain discount factor v^15 alone
  4. The 15-year temporary annuity factor a-double-dot_{50:15}

Explanation

The deferred annuity value equals the annuity-due value at age 65 multiplied by the pure endowment factor 15 E_50, which discounts for 15 years and weights by the probability of survival to 65. Writing it as a-double-dot_50 minus a-double-dot_{50:15} gives the same result. The pure endowment captures both interest and survival.

Question 12Actuarial Mathematics

Which of the following best explains why the net premium reserve for a whole life policy increases over time under a level premium arrangement?

  1. Mortality rates decline with age, so the insurer needs larger reserves to cover cheaper later claims
  2. Expenses are higher in later years, and the net premium reserve must cover them
  3. The level premium exceeds the cost of insurance in early years, so a fund accumulates to meet rising mortality costs later Correct answer.
  4. Interest rates are assumed to fall over the policy term, requiring larger reserves

Explanation

Under a level net premium the amount charged exceeds the true cost of insurance in the early years when mortality is low, and falls short later when mortality is high. The reserve is the accumulated excess that funds the later shortfall. This is why reserves rise over the duration of a whole life policy.

Question 13Actuarial Mathematics

Given that the force of interest is delta = 0.06 and the force of mortality is mu = 0.02, both constant, calculate the expected present value of a whole life assurance paying 1 immediately on death.

  1. 0.25 Correct answer.
  2. 0.33
  3. 0.50
  4. 0.75

Explanation

For a continuous whole life assurance with constant forces, the value is mu/(mu + delta) = 0.02/(0.02 + 0.06) = 0.25. The numerator is the mortality force and the denominator combines mortality and interest, reflecting the chance the benefit is paid and its discounting.

Question 14Actuarial Mathematics

A life aged 60 purchases a whole life assurance with sum assured 200,000 payable at the end of the year of death. Given A_60 = 0.35 and i = 0.05, calculate the net single premium.

  1. 70,000 Correct answer.
  2. 140,000
  3. 35,000
  4. 130,000

Explanation

The net single premium equals the sum assured multiplied by the assurance factor: 200,000 times 0.35 = 70,000. The factor A_60 already incorporates both mortality and interest, so no further discounting is needed.

Question 15Actuarial Mathematics

Which relationship holds between the net premium reserve at duration t and the gross premium reserve for the same policy?

  1. The gross premium reserve equals the net premium reserve minus the expected present value of all future gross premiums
  2. The gross premium reserve equals the net premium reserve exactly, since expenses are paid from surplus
  3. The gross premium reserve equals the net premium reserve multiplied by the expense ratio for the policy
  4. The gross premium reserve equals the net premium reserve plus the expected present value of future expenses less the expected present value of future expense loadings Correct answer.

Explanation

The gross premium reserve adjusts the net premium reserve for future expenses and for the expense loading contained in future gross premiums. Specifically it adds the present value of future expenses and subtracts the present value of future expense loadings. When expense assumptions match the loadings the two reserves can coincide, but in general they differ.

Question 16Actuarial Mathematics

A life aged 30 takes out a 20-year endowment assurance with sum assured 100,000 payable at the end of the year of death or on survival to age 50. Which combination of factors gives the expected present value of the benefit?

  1. 100,000 times (A^1_{30:20} plus 20 E_30) Correct answer.
  2. 100,000 times (A_30 plus 20 E_30)
  3. 100,000 times (A^1_{30:20} plus a-double-dot_{30:20})
  4. 100,000 times (A^1_{30:20} minus 20 E_30)

Explanation

An endowment assurance pays on death within the term or on survival to the end of the term, so its value is the term assurance factor plus the pure endowment factor. Multiplying the sum of A^1_{30:20} and 20 E_30 by the sum assured gives the expected present value. The whole life factor A_30 would include deaths after the term and is therefore incorrect.

Question 17Actuarial Mathematics

In a life table, which expression equals the expected number of years of life remaining at age x under the curtate future lifetime random variable, often denoted e_x?

  1. The sum from k=1 to infinity of k p_x Correct answer.
  2. The sum from k=0 to infinity of k q_x
  3. The product from k=1 to infinity of k p_x
  4. l_x divided by the sum of l_{x+k}

Explanation

The curtate expected future lifetime is the sum over k of the probability of surviving k more years, that is the sum of k p_x for k from 1 to infinity. Each term represents the probability that the life completes at least k further years. This identity underlies the relationship between annuity values and expected lifetime.

Question 18Actuarial Mathematics

An insurer uses the equivalence principle to set premiums but then discovers that actual mortality is 20% higher than assumed at every age, with all other assumptions unchanged. What is the most likely effect on the adequacy of the net premium reserve?

  1. Reserves become inadequate only if interest rates also fall
  2. Reserves become excessive because higher mortality reduces the number of premiums received
  3. Reserves become inadequate because benefits are paid sooner and more often than assumed Correct answer.
  4. Reserves are unaffected because the equivalence principle guarantees adequacy for any mortality

Explanation

Higher mortality means more claims and earlier payment, so the present value of benefits rises while fewer premiums are collected. The reserves set on the original assumptions will therefore be insufficient to meet the actual cost. This is a mortality risk that the equivalence principle does not protect against, since it only ensures balance under the pricing basis.

Question 19Actuarial Mathematics

For a whole life annuity-due of 1 per year on (x), which identity correctly relates it to the whole life assurance factor A_x and the discount rate d?

  1. a-double-dot_x equals (1 minus A_x) multiplied by d
  2. a-double-dot_x equals (1 plus A_x) divided by d
  3. a-double-dot_x equals (1 minus A_x) divided by d Correct answer.
  4. a-double-dot_x equals A_x divided by (1 minus d)

Explanation

The standard identity is a-double-dot_x = (1 - A_x)/d, derived by equating the present value of a unit paid now plus a unit paid at the end of each year of survival. The discount rate d equals i/(1+i) and arises because annuity-due payments occur at the start of each year. Rearranging gives A_x = 1 - d times a-double-dot_x.

Question 20Actuarial Mathematics

A portfolio consists of two independent lives aged 45 and 50 with independent future lifetimes. Given q_45 = 0.004 and q_50 = 0.006 for the coming year, calculate the probability that exactly one of the two lives dies during the year.

  1. 0.010000
  2. 0.000024
  3. 0.009976 Correct answer.
  4. 0.005000

Explanation

The probability that exactly one dies is q_45 p_50 plus p_45 q_50, where p = 1 - q. This equals 0.004 times 0.994 plus 0.996 times 0.006, giving 0.003976 plus 0.005976, which totals approximately 0.009976. Independence allows the joint probabilities to be multiplied.

Question 21Actuarial Mathematics

Which of the following is the correct expression for the variance of the present value of a whole life assurance of 1 payable at the end of the year of death, in terms of assurance factors at interest rate i?

  1. The quantity (2 A_x evaluated at double the force of interest) minus the square of A_x Correct answer.
  2. The quantity 2 A_x minus A_x squared
  3. The quantity (1 minus A_x) squared divided by d squared
  4. The quantity A_x minus the square of A_x

Explanation

The second moment of the present value is found by revaluing the benefit at twice the force of interest, giving 2A_x computed with v squared, and the variance is that second moment minus the square of the mean. The result is therefore the double-force assurance factor less A_x squared. This is the standard variance formula for a whole life assurance.

Question 22Actuarial Mathematics

A life aged 55 pays a single premium for a whole life annuity-due of 10,000 per year. Given a-double-dot_55 = 15.0, calculate the single premium ignoring expenses.

  1. 15,000
  2. 150,000 Correct answer.
  3. 666.67
  4. 10,000

Explanation

The single premium for an annuity-due equals the annual payment multiplied by the annuity factor: 10,000 times 15.0 = 150,000. The factor already reflects survival probabilities and interest discounting for payments in advance.

Question 23Actuarial Mathematics

Under a select mortality table, which feature distinguishes the mortality of recently underwritten lives from that of the same lives several years after selection?

  1. Mortality is lighter in the years immediately following underwriting and converges towards ultimate rates over time Correct answer.
  2. Mortality follows the ultimate table from the outset and the select period is purely administrative
  3. Mortality is identical in all years, since selection affects only the premium charged
  4. Mortality is heavier immediately after underwriting and improves steadily thereafter

Explanation

Select mortality reflects the favourable health of lives accepted after underwriting, so death rates are lower in the first years after selection. As the effect of selection wears off, mortality converges to the ultimate rates for the attained age. Select tables therefore show mortality depending on both age and duration since selection.

Question 24Actuarial Mathematics

A 10-year term assurance on (x) has sum assured 100,000 payable at the end of the year of death, with level annual premiums in advance. If the reserve at duration 9 is 40,000 and the premium is 500, calculate the expected fund at the start of the final year after one year's interest at 4%, before any claim is paid.

  1. 40,520
  2. 42,120 Correct answer.
  3. 41,600
  4. 42,000

Explanation

The fund at the start of the final year is the duration-9 reserve plus the premium, accumulated one year at 4%: (40,000 + 500) times 1.04 = 42,120. This fund must then cover the death benefit if the life dies during the year and the terminal reserve of zero on survival.

Question 25Actuarial Mathematics

Which statement about the relationship between gross and net premiums is correct?

  1. The gross premium exceeds the net premium by a loading intended to cover expenses, profit and contingencies Correct answer.
  2. The gross premium equals the net premium when expenses are zero, and is otherwise smaller
  3. The gross premium excludes any allowance for profit or contingencies
  4. The gross premium is always exactly twice the net premium

Explanation

The gross premium is the net premium plus a loading for expenses, profit margins and contingencies. The net premium covers only the expected cost of benefits and is derived under the equivalence principle. The difference between the two is the expense loading used in gross premium valuations.

Question 26Actuarial Mathematics

A life table has l_30 = 95,000 and l_31 = 94,525. Calculate p_30, the probability that a life aged 30 survives to age 31.

  1. 0.99500 Correct answer.
  2. 0.95000
  3. 0.99474
  4. 0.00500

Explanation

Survival probability is the ratio of survivors: p_30 = l_31 / l_30 = 94,525 / 95,000 = 0.995. The death probability q_30 is the complement, 0.005, and the two must sum to one.

Question 27Actuarial Mathematics

Under the assumption of a constant force of mortality mu between integer ages, which expression gives the probability that a life aged x dies within one year?

  1. exp(-mu)
  2. mu / (1 + mu)
  3. mu
  4. 1 - exp(-mu) Correct answer.

Explanation

With a constant force of mortality over the year, survival for the full year is exp(-mu), so the probability of death within the year is 1 - exp(-mu). Note this differs from q under uniform distribution of deaths, where q equals mu/(1+mu) approximately.

Question 28Actuarial Mathematics

A whole life assurance pays 1 at the moment of death. The force of interest is delta and the force of mortality is mu, both constant. Which expression gives the expected present value of the benefit?

  1. delta / (mu + delta)
  2. exp(-mu - delta)
  3. mu / (mu + delta) Correct answer.
  4. mu / delta

Explanation

Under constant forces, the present value density integrates to mu/(mu+delta), the standard constant-force whole life assurance value. The result lies between 0 and 1, as any probability-weighted discounted benefit must.

Question 29Actuarial Mathematics

For a whole life annuity-due of 1 per year on (x), which identity correctly relates it to the whole life assurance factor A_x, given an effective annual interest rate i?

  1. a-double-dot_x = (1 - A_x) / i
  2. a-double-dot_x = A_x / d
  3. a-double-dot_x = (1 - A_x) / d, where d = i/(1+i) Correct answer.
  4. a-double-dot_x = (1 + A_x) / d

Explanation

The actuarial identity a-double-dot_x = (1 - A_x)/d uses the discount rate d = i/(1+i), because the annuity is paid in advance and the assurance at the end of the year of death. This links the two standard functions through the equivalence of present values.

Question 30Actuarial Mathematics

A life aged 40 has a-double-dot_40 = 18.5 at an effective interest rate of 4% per year. Using the identity a-double-dot_x = (1 - A_x)/d, calculate A_40.

  1. 0.26000
  2. 0.31154
  3. 0.74000
  4. 0.28846 Correct answer.

Explanation

First compute d = 0.04/1.04 = 0.038462. Then A_40 = 1 - d * a-double-dot_40 = 1 - 0.038462 * 18.5 = 0.28846. The assurance factor is a probability-weighted discount and must lie between 0 and 1.

Question 31Actuarial Mathematics

A life aged 50 takes out a whole life assurance with sum assured 100,000 payable at the end of the year of death. Given A_50 = 0.30, calculate the single net premium.

  1. 70,000
  2. 3,000
  3. 30,000 Correct answer.
  4. 33,333

Explanation

The single net premium under the equivalence principle equals the expected present value of the benefit, which is 100,000 * A_50 = 100,000 * 0.30 = 30,000. The factor A_50 already incorporates both discounting and the probability of payment.

Question 32Actuarial Mathematics

A life aged 60 has q_60 = 0.01 and q_61 = 0.012. Assuming independent annual survival, calculate 2p_60, the probability of surviving two years.

  1. 0.97812 Correct answer.
  2. 0.02200
  3. 0.97800
  4. 0.99000

Explanation

Two-year survival is the product of annual survival probabilities: 2p_60 = p_60 * p_61 = (1 - 0.01)(1 - 0.012) = 0.99 * 0.988 = 0.97812. Multiplying rather than adding reflects that both years must be survived.

Question 33Actuarial Mathematics

A term assurance on (x) pays 1 at the end of the year of death if death occurs within n years. Which symbol denotes its expected present value?

  1. n|A_x
  2. A_{x:n}
  3. A^1_{x:n} Correct answer.
  4. A_x

Explanation

The symbol A^1_{x:n} denotes a term assurance where the benefit is payable only if death occurs within n years. The superscript 1 signals that only the death benefit within the term is included, unlike the endowment assurance.

Question 34Actuarial Mathematics

A 20-year endowment assurance on (x) has sum assured 50,000 payable at the end of the year of death or on survival to age x+20. Which expected present value formula applies?

  1. 50,000 * 20p_x
  2. 50,000 * A_{x:20} Correct answer.
  3. 50,000 * A_x
  4. 50,000 * A^1_{x:20}

Explanation

An endowment assurance pays on death within the term or on survival to the term's end, so its expected present value uses A_{x:n}, which combines the term assurance and pure endowment. Here n = 20 and the sum assured is 50,000.

Question 35Actuarial Mathematics

An insurer sets net premiums using the equivalence principle with an interest rate of 5%, then discovers the actual interest earned is only 3%. All other assumptions hold. Which effect on the net premium reserve is expected?

  1. Reserves will be zero because the policy becomes void when the interest assumption fails.
  2. Reserves will be higher than expected because lower interest increases the present value of benefits.
  3. Reserves will be lower than expected because investment income is insufficient to fund the assumed accumulation. Correct answer.
  4. Reserves will be unchanged because the equivalence principle guarantees adequacy at any interest rate.

Explanation

Lower-than-assumed interest means premiums accumulate more slowly than planned, so the reserve built up will fall short of the level the original basis required. The equivalence principle holds only under the pricing basis and does not guarantee adequacy if actual experience differs.

Question 36Actuarial Mathematics

A deferred whole life annuity-due of 1 per year is payable to (x) starting at age x+10. Which expression gives its expected present value?

  1. 10|a-double-dot_x = a-double-dot_x - a-double-dot_{x:10} Correct answer.
  2. a-double-dot_{x:10} - a-double-dot_x
  3. a-double-dot_x + a-double-dot_{x:10}
  4. 10 * a-double-dot_x

Explanation

A deferred annuity is the difference between the whole life annuity and the temporary annuity covering the deferment period: 10|a-double-dot_x = a-double-dot_x - a-double-dot_{x:10}. This subtracts the payments received during the deferment, which are zero.

Question 37Actuarial Mathematics

A life aged 45 has a-double-dot_45 = 20.0 and a-double-dot_{45:15} = 12.0. Calculate the expected present value of a whole life annuity-due of 1 per year deferred 15 years.

  1. 8.0 Correct answer.
  2. 20.0
  3. 32.0
  4. 12.0

Explanation

The deferred annuity equals the whole life annuity less the temporary annuity over the deferment period: 20.0 - 12.0 = 8.0. This isolates the payments beginning after the 15-year deferment.

Question 38Actuarial Mathematics

Which statement best explains why the net premium reserve for a whole life policy rises over time under a level premium?

  1. The level premium exceeds the cost of insurance in early years, and the excess accumulates to fund rising mortality costs later. Correct answer.
  2. The sum assured increases each year, requiring a larger reserve to be held.
  3. Mortality rates fall with age, so the cost of insurance decreases and the reserve must rise to offset it.
  4. Interest rates are assumed to rise with duration, increasing the reserve required.

Explanation

Under a level premium, early premiums exceed the current cost of insurance, and the surplus is accumulated with interest as the reserve. As mortality rises with age, the reserve funds the increasing cost of insurance in later years.

Question 39Actuarial Mathematics

A life aged 65 has q_65 = 0.02. Calculate the expected number of deaths in a group of 5,000 independent lives aged exactly 65 over the coming year.

  1. 4,900
  2. 5,000
  3. 98
  4. 100 Correct answer.

Explanation

The expected number of deaths is the group size times the death probability: 5,000 * 0.02 = 100. This is the mean of a binomial count with n = 5,000 and p = 0.02.

Question 40Actuarial Mathematics

Which expression equals the expected number of years of life remaining at age x under the curtate future lifetime random variable?

  1. e_x = l_x / l_{x+1}
  2. e_x = sum over k>=1 of k q_x
  3. e_x = 1 / q_x
  4. e_x = sum over k>=1 of k p_x Correct answer.

Explanation

The curtate expected future lifetime sums the survival probabilities over future years: e_x = sum of k p_x for k = 1, 2, and so on. This follows from the tail-sum formula for the expectation of a nonnegative integer-valued random variable.

Question 41Actuarial Mathematics

A life aged 55 pays a single premium for a whole life annuity-due of 10,000 per year. Given a-double-dot_55 = 15.0, calculate the single net premium.

  1. 150,000 Correct answer.
  2. 10,000
  3. 15,000
  4. 66,667

Explanation

The single net premium equals the annual payment times the annuity factor: 10,000 * 15.0 = 150,000. The annuity-due factor already accounts for survival and discounting.

Question 42Actuarial Mathematics

Under a select mortality table, which feature distinguishes the mortality of recently underwritten lives from that of the general population?

  1. Recently underwritten lives have identical mortality to the general population at all durations.
  2. Recently underwritten lives have mortality that is independent of the underwriting process.
  3. Recently underwritten lives have higher mortality that gradually converges to ultimate rates.
  4. Recently underwritten lives have lower mortality that gradually converges to ultimate rates after a select period. Correct answer.

Explanation

Select mortality reflects favorable underwriting, so recently selected lives experience lower mortality that grades into ultimate rates after the select period. Select tables use a double subscript such as q_[x]+t to show the duration since underwriting.

Question 43Actuarial Mathematics

A 10-year term assurance on (x) has sum assured 100,000 payable at the end of the year of death, with level annual net premiums payable in advance. Which equation determines the annual net premium under the equivalence principle?

  1. P * a-double-dot_{x:10} = 100,000 * A^1_{x:10} Correct answer.
  2. P * A^1_{x:10} = 100,000 * a-double-dot_{x:10}
  3. P * a-double-dot_x = 100,000 * A^1_{x:10}
  4. P = 100,000 * A^1_{x:10}

Explanation

The equivalence principle equates the expected present value of premiums, P times the temporary annuity-due, to the expected present value of the death benefit, 100,000 times the term assurance factor. Solving gives the level annual net premium.

Question 44Actuarial Mathematics

A life aged 40 has mu = 0.01 and delta = 0.05, both constant. Calculate the expected present value of a whole life assurance of 1 payable at the moment of death.

  1. 0.50000
  2. 0.16667 Correct answer.
  3. 0.20000
  4. 0.83333

Explanation

Under constant forces, the assurance value is mu/(mu+delta) = 0.01/(0.01+0.05) = 0.16667. The value is bounded between 0 and 1 as an expected discounted benefit.

Question 45Actuarial Mathematics

For two independent lives aged 45 and 50 with independent future lifetimes, which expression gives the probability that both survive one year?

  1. p_45 + p_50
  2. 1 - q_45 * q_50
  3. p_45 * p_50 Correct answer.
  4. q_45 * q_50

Explanation

Independence allows the joint survival probability to be the product of the individual survival probabilities: p_45 * p_50. This is the basis for joint-life functions in multiple-life problems.

Question 46Actuarial Mathematics

Which expression gives the variance of the present value of a whole life assurance of 1 payable at the end of the year of death, in terms of the assurance factor evaluated at a modified interest rate?

  1. Variance = A_x - (A_x)^2
  2. Variance = A_x * (1 - A_x) / (1 + i)
  3. Variance = (2A_x evaluated at rate i') - (A_x)^2, where 1 + i' = (1 + i)^2 Correct answer.
  4. Variance = (A_x)^2 - 2A_x

Explanation

The second moment of the discounted benefit is found by evaluating the assurance factor at the rate i' satisfying 1 + i' = (1 + i)^2, which squares the discount function. The variance is that second moment minus the square of the mean.

Question 47Actuarial Mathematics

An insurer prices a policy assuming q = 0.005 but actual mortality proves to be 0.006. Which consequence follows for a term assurance priced on the original assumption?

  1. The insurer incurs losses because the expected cost of benefits exceeds the premium collected. Correct answer.
  2. The insurer makes gains because higher mortality reduces the number of claims.
  3. The reserve falls to zero because the mortality assumption changed.
  4. The premium remains adequate because the equivalence principle is self-correcting.

Explanation

Higher actual mortality raises the expected cost of benefits above the level funded by the premium, producing losses on the block. The equivalence principle does not adjust premiums automatically when experience departs from the basis.

Question 48Actuarial Mathematics

A life aged 30 has l_30 = 98,000 and l_35 = 97,000. Calculate 5p_30, the probability of surviving from age 30 to age 35.

  1. 0.98980 Correct answer.
  2. 0.99000
  3. 0.01020
  4. 0.95000

Explanation

Five-year survival is l_35 / l_30 = 97,000 / 98,000 = 0.98980. The complement, 0.01020, is the probability of dying within the five years.

Question 49Actuarial Mathematics

Under a life table, which quantity is defined as the expected number of complete future years lived by a life now aged x?

  1. The curtate expectation of life, e_x Correct answer.
  2. The complete expectation of life, e_x
  3. The force of mortality at age x, mu_x
  4. The annual premium for a whole life assurance on (x)

Explanation

The curtate expectation of life, written e_x, is the expected value of the curtate future lifetime K_x, which counts only complete years lived. It equals the sum over k of k+1 p_x from k=0 onwards. The complete expectation, by contrast, uses the full continuous future lifetime.

Question 50Actuarial Mathematics

A life table gives l_20 = 95,000 and l_21 = 94,050. Calculate q_20, the probability that a life aged exactly 20 dies before age 21.

  1. 0.0100 Correct answer.
  2. 0.0090
  3. 0.0110
  4. 0.0125

Explanation

The one-year death probability is q_20 = (l_20 - l_21)/l_20. Here that is (95,000 - 94,050)/95,000 = 950/95,000 = 0.01. The complement p_20 = 94,050/95,000 = 0.99 confirms the result.