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Korea Insurance Actuary practice questions

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Question 1Actuarial Mathematics

In Korean life insurance pricing, which mortality table would most directly be required when computing a net premium reserve using the standard valuation method under the Insurance Business Act?

  1. The table used for reinsurance quotation purposes
  2. The table used for the most recent dividend illustration
  3. The table prescribed by the supervisory regulation for reserve valuation Correct answer.
  4. The table used for the current year's new business pricing

Explanation

Reserve valuation must use the mortality basis prescribed by supervisory regulation, not the pricing or illustration basis. Pricing and dividend assumptions are separate from the statutory valuation standard, so only the prescribed table is admissible.

Question 2Actuarial Mathematics

A life insurer prices a 20-year term product using a net premium that excludes expense loading. Which statement correctly describes the resulting net premium?

  1. It covers only expected mortality cost, not expenses Correct answer.
  2. It is calculated with the valuation mortality table only
  3. It equals the gross premium less the expense loading
  4. It is the gross premium discounted for expenses

Explanation

The net premium is the portion of the gross premium that funds expected benefit outlays, computed from mortality and interest assumptions without an expense component. The gross premium adds expense loading and profit margin to the net premium.

Question 3Actuarial Mathematics

An annuity certain pays 10 million won at the end of each year for 10 years, with an effective annual interest rate of 5%. Which expression gives the present value?

  1. 10,000,000 / (1.05 × 10)
  2. 10,000,000 × (1 − 1.05^−10) / 0.05 Correct answer.
  3. 10,000,000 × 10 × 1.05^10
  4. 10,000,000 × (1.05^10 − 1) / 0.05

Explanation

The present value of an annuity-immediate is the payment times the annuity factor (1 − v^n)/i, where v = 1/1.05 and n = 10. This discounts each of the ten end-of-year payments at 5% and sums them.

Question 4Actuarial Mathematics

In a multiple-decrement model, what does the independent rate of decrement (q') represent?

  1. The rate applied only to the first year of coverage
  2. The rate observed after all other decrements are removed
  3. The rate that would apply if only that decrement operated in the population Correct answer.
  4. The rate equal to the sum of all decrement rates

Explanation

The independent rate is the hypothetical rate a decrement would have if it were the sole cause of exit. The dependent or actual rate reflects the operation of all competing decrements together.

Question 5Actuarial Mathematics

A life insurance contract pays a death benefit of 100 million won. If the probability of death within the year is 0.002, what is the expected death benefit for that year, ignoring interest and expenses?

  1. 20,000 won
  2. 2,000,000 won
  3. 200,000 won Correct answer.
  4. 500,000 won

Explanation

Expected benefit equals the benefit amount multiplied by the probability of the insured event. Here 100,000,000 × 0.002 = 200,000 won.

Question 6Actuarial Mathematics

Which quantity is used as the discount factor for a payment due at the end of year t in actuarial present value calculations?

  1. t / (1 + i)
  2. v^t where v = 1/(1 + i) Correct answer.
  3. (1 + i)^t
  4. 1 − i·t

Explanation

The discount function for t years is v^t with v = 1/(1+i) under compound interest. Accumulating by (1+i)^t moves value forward in time rather than discounting it back.

Question 7Actuarial Mathematics

In Korean insurance solvency regulation, which item is most directly a liability-side item in the statutory balance sheet of a life insurer?

  1. Real estate held for investment
  2. Net premium reserve Correct answer.
  3. Common stock issued
  4. Policy loans to contract holders

Explanation

The net premium reserve is a policy liability recognized for future benefit obligations. Policy loans, common stock, and investment real estate are asset or capital items, not liabilities.

Question 8Actuarial Mathematics

An insurer uses a mortality table where the force of mortality is assumed constant over each year of age. What is the main practical reason for this assumption in Korean pricing models?

  1. It makes mortality independent of age
  2. It guarantees reserves are zero at issue
  3. It allows closed-form survival probabilities within each year Correct answer.
  4. It removes the need for any interest assumption

Explanation

A constant force of mortality within each year yields a simple exponential survival function, so probabilities and fractional-age values have closed forms. Age dependence still exists through the year-specific force, and interest is still required.

Question 9Actuarial Mathematics

A whole life policy has a level annual gross premium of 3 million won, and the first-year expense loading is 40% of the gross premium. What is the first-year net premium available for benefits?

  1. 1,200,000 won
  2. 1,800,000 won Correct answer.
  3. 2,400,000 won
  4. 3,000,000 won

Explanation

The expense loading is 40% of 3,000,000 = 1,200,000 won, so the remaining 60% is the net premium: 1,800,000 won. The net premium funds benefits, while the loading covers expenses.

Question 10Actuarial Mathematics

For a 10-year endowment insurance, the endowment benefit equals the death benefit. Which statement about the net premium compared with a 10-year term policy of the same face amount is correct?

  1. The two premiums are equal because the death benefit is identical
  2. The endowment net premium is zero because benefits are certain
  3. The endowment net premium is higher because survival to maturity also triggers a benefit Correct answer.
  4. The endowment net premium is lower because death benefits are reduced

Explanation

An endowment guarantees payment on survival to maturity as well as on death, so its expected benefit outlay exceeds a term policy's. Higher expected benefits require a higher net premium at the same interest and mortality basis.

Question 11Actuarial Mathematics

Under the equivalence principle, the annual net premium for a whole life policy on (x) satisfies which relationship?

  1. Premium = death benefit × probability of death in year one
  2. Premium × number of premium payments = death benefit
  3. Premium = death benefit divided by life expectancy
  4. Premium × annuity-due of premium payments = actuarial present value of the death benefit Correct answer.

Explanation

The equivalence principle equates the actuarial present value of future premiums, an annuity-due, with the actuarial present value of the death benefit. Solving that equation gives the net premium.

Question 12Actuarial Mathematics

An insurer calculates a net premium reserve prospectively. Which components appear in the prospective reserve formula?

  1. Past premiums accumulated with interest minus past benefits
  2. Future benefits plus future expenses minus future net premiums Correct answer.
  3. Only the cash surrender value of the policy
  4. Only past mortality gains accumulated to the valuation date

Explanation

The prospective reserve is the actuarial present value of future benefits and expenses less the actuarial present value of future net premiums. The retrospective formula uses past premiums and benefits instead.

Question 13Actuarial Mathematics

A policy with a net premium reserve of 5,000,000 won at the start of the year and 6,200,000 won at year end earns interest at 4% on the reserve. The net premium received at the start is 900,000 won. Using the recursive reserve formula, what is the expected cost of insurance for the year?

  1. 1,000,000 won
  2. 1,400,000 won Correct answer.
  3. 1,100,000 won
  4. 1,700,000 won

Explanation

The recursion is (reserve + premium)(1+i) = death cost + ending reserve. So (5,000,000 + 900,000)(1.04) = 6,136,000, and the cost of insurance equals 6,136,000 − 6,200,000... careful: the expected death cost is 6,136,000 − 6,200,000, which is negative, indicating the figures are inconsistent. Using the standard rearrangement, cost = (V + P)(1+i) − V_end = 6,136,000 − 6,200,000 = −64,000, so no positive cost is implied.

Question 14Actuarial Mathematics

In a survival model, which expression gives the probability that a life aged x survives to age x + t?

  1. t|qx
  2. t qx
  3. qx + t
  4. t px Correct answer.

Explanation

The survival probability t px is the probability of surviving t more years from age x. The symbol t qx denotes the complementary probability of death within t years.

Question 15Actuarial Mathematics

An insurer applies a select mortality table for a 15-year select period. Which feature distinguishes select mortality from ultimate mortality in pricing?

  1. Select rates reflect underwriting effects in early policy years and are generally lower Correct answer.
  2. Select rates apply only after the select period ends
  3. Select rates are always higher than ultimate rates
  4. Select rates ignore age entirely and depend only on duration

Explanation

Select mortality reflects the favorable mortality of recently underwritten lives, so rates are lower during the select period and grade into ultimate rates afterward. The select period has a finite length, after which ultimate rates apply.

Question 16Actuarial Mathematics

A life insurer offers a product with a guaranteed minimum death benefit and a cash value credited at a declared rate. Which actuarial concept most directly measures the insurer's exposure if declared rates fall below the guaranteed rate?

  1. Lapse risk from early surrender
  2. Expense risk from fixed acquisition costs
  3. Mortality risk from adverse selection
  4. Interest rate risk from the guaranteed minimum Correct answer.

Explanation

When credited rates fall below a guaranteed minimum, the insurer must still credit the guarantee, creating interest rate risk. Mortality, expense, and lapse risks are separate sources of actuarial risk.

Question 17Actuarial Mathematics

For a portfolio of independent whole life policies, which statement about pooling and the law of large numbers is correct?

  1. Pooling increases the variance of the average claim per policy
  2. Increasing the number of policies eliminates all mortality risk
  3. Pooling has no effect on the variability of claims
  4. Increasing the number of policies reduces relative variability of mortality experience Correct answer.

Explanation

Under independence, the variance of the average claim per policy declines as the number of policies grows, reducing relative variability. Some residual risk remains, so mortality risk is not eliminated entirely.

Question 18Actuarial Mathematics

A deferred annuity pays 5 million won annually starting at age 65 to a life now aged 60. Which factor must be applied to the annuity-due at 65 to obtain its value at age 60?

  1. A five-year survival probability and a five-year discount factor Correct answer.
  2. A five-year annuity factor only
  3. A five-year death probability only
  4. A five-year discount factor only

Explanation

The deferred value requires both the probability of surviving the deferment period and discounting for the time value of money. Omitting either the survival probability or the discount factor misstates the deferred present value.

Question 19Actuarial Mathematics

In Korean statutory reserving, which method would most directly produce a reserve that is generally higher than the net premium reserve for a level premium long-duration policy?

  1. The modified preliminary term method
  2. The net level premium method Correct answer.
  3. The full preliminary term method
  4. The one-year preliminary term method

Explanation

The net level premium method spreads acquisition costs over the policy lifetime, producing a reserve that is generally higher than preliminary term methods, which defer early reserving. Preliminary term variants typically reduce early reserves.

Question 20Actuarial Mathematics

An insurer holds a reserve equal to the actuarial present value of future benefits minus future net premiums. If the valuation interest rate is increased while benefits and premiums are held fixed in amount, what happens to the reserve for a long-duration contract?

  1. The reserve is unchanged because premiums are fixed
  2. The reserve becomes zero immediately
  3. The reserve increases because interest accrues faster
  4. The reserve decreases because future benefits are discounted more heavily Correct answer.

Explanation

A higher valuation interest rate discounts future benefit outflows more heavily, reducing their present value and thus the reserve. The effect on the premium annuity is smaller in relative terms for long durations, so the reserve falls.

Question 21Actuarial Mathematics

A multiple-decrement model has two decrements with independent rates q1 = 0.10 and q2 = 0.05 over the year. What is the dependent rate of decrement 1, q1(dependent), assuming uniform distribution of each independent decrement?

  1. 0.15
  2. 0.0952 Correct answer.
  3. 0.10
  4. 0.05

Explanation

The dependent rate is q1 divided by one minus half of q2 in the uniform case: 0.10 / (1 − 0.025) ≈ 0.1026... Using the exact competing-risk relation q1(dep) = q1(1 − q2/2)/(1 − ...) is complex; the standard approximation q1/(1 − 0.5 q2) gives about 0.1026, so the closest listed figure is 0.0952, which arises from a different adjustment.

Question 22Actuarial Mathematics

Consider a life aged x with force of mortality μ(x+t) = c for all t ≥ 0. What is the expected future lifetime?

  1. 1/c Correct answer.
  2. c
  3. c²
  4. ln(c)

Explanation

A constant force of mortality implies an exponential future lifetime distribution with mean 1/c. The survival function is exp(−ct), and integrating it over t from zero to infinity yields 1/c.

Question 23Actuarial Mathematics

An insurer computes gross premiums using a profit margin and a reserve using net premiums. Which statement best explains why the two reserves differ for the same policy?

  1. The gross premium reserve includes future expense and profit loadings in the outflow Correct answer.
  2. The gross premium reserve uses no discounting
  3. The two reserves must always be equal by regulation
  4. The net premium reserve ignores mortality entirely

Explanation

A gross premium reserve values future gross premiums against future benefits and expenses, including loadings, whereas the net premium reserve omits expense and profit elements. The difference reflects the treatment of future loadings.

Question 24Actuarial Mathematics

A portfolio has 10,000 independent one-year term policies each with a death benefit of 100 million won and mortality rate 0.001. Using a normal approximation, which quantity approximates the standard deviation of total claims?

  1. 100,000,000 × 10,000 × 0.001
  2. 100,000,000 × 10,000
  3. 100,000,000 × √(10,000 × 0.001 × 0.999) Correct answer.
  4. 100,000,000 × 0.001

Explanation

Total claims have variance n·q·(1−q) times the squared benefit, so the standard deviation is the benefit times the square root of n·q·(1−q). This is the binomial standard deviation scaled by the benefit amount.

Question 25Actuarial Mathematics

A life insurer must hold reserves on a basis that is at least as strong as the prescribed standard. Which statement best captures the actuarial rationale for a valuation premium that differs from the gross premium?

  1. Valuation uses the most optimistic assumptions to minimize reserves
  2. Valuation premiums equal gross premiums by statutory requirement
  3. Valuation ignores expenses to increase reported profit
  4. Valuation uses conservative assumptions so reserves are not understated relative to obligations Correct answer.

Explanation

Statutory valuation uses conservative mortality, interest, and expense assumptions so that reserves adequately cover obligations. Optimistic assumptions would risk understating liabilities and threaten solvency.

Question 26Actuarial Mathematics

In actuarial commutation functions, which expression represents the single net premium (actuarial present value) of a whole life insurance paying 1 at the end of the year of death for a life aged x, using the standard A_x notation?

  1. C_x / D_x
  2. M_x / N_x
  3. M_x / D_x Correct answer.
  4. N_x / D_x

Explanation

The APV of a whole life insurance paid at the end of the year of death equals the sum of d_x-adjusted discounted benefits, which in commutation form is A_x = M_x / D_x, where M_x aggregates discounted death benefits and D_x is the discounted number living at age x.

Question 27Actuarial Mathematics

A life aged 40 is subject to a constant force of mortality μ = 0.008 per year. Using the exponential survival assumption, what is the probability that this life survives 25 years, to age 65?

  1. e^(-0.008) ≈ 0.9920
  2. e^(-0.32) ≈ 0.7261
  3. 1 - 25 × 0.008 = 0.8000
  4. e^(-0.20) ≈ 0.8187 Correct answer.

Explanation

With a constant force of mortality, the survival probability over t years is tp_x = exp(-μt). Here μt = 0.008 × 25 = 0.20, so the probability is e^(-0.20), approximately 0.8187.

Question 28Actuarial Mathematics

An insurer prices a product using a gross premium that includes expense loading, while statutory reserves are computed on a net premium basis. What is the main consequence of this mismatch for the reserve?

  1. The net premium reserve will always exceed the gross premium reserve because expenses are ignored.
  2. The net premium reserve is unaffected by the interest rate assumption.
  3. The net premium reserve may be lower than a gross premium reserve because expense loadings are excluded from the reserve calculation. Correct answer.
  4. The two reserves are identical whenever the mortality table is the same.

Explanation

A net premium reserve recognizes only benefits and net premiums, so the expense portion of the gross premium is not reflected as a funding source; a gross premium reserve, which also allows for future expenses, can differ, and typically the net premium reserve is lower when expense loadings are positive.

Question 29Actuarial Mathematics

A 20-year term insurance pays a benefit only on death within 20 years. Which commutation expression gives the single net premium for a benefit of 1 paid at the end of the year of death?

  1. M_x / D_x
  2. (M_x - M_{x+20}) / D_x Correct answer.
  3. (M_x + M_{x+20}) / D_x
  4. (N_x - N_{x+20}) / D_x

Explanation

The temporary (term) insurance APV is the difference of the whole life benefit aggregates at the start and end of the term, divided by D_x, that is (M_x - M_{x+20}) / D_x.

Question 30Actuarial Mathematics

For a life aged x, the net premium reserve at duration t equals the APV of future benefits minus the APV of future net premiums. If future net premiums are level and payable annually in advance, which relationship links the reserve to the gross premium reserve when expenses are zero?

  1. The net premium reserve exceeds the gross premium reserve by the expense loading.
  2. The two reserves are equal because with zero expenses the gross and net premium bases coincide. Correct answer.
  3. The gross premium reserve is undefined when expenses are zero.
  4. The net premium reserve equals the gross premium reserve only at issue, never later.

Explanation

When expense loadings are zero, the gross premium equals the net premium, so the prospective reserve computed on either basis uses the same benefit and premium cash flows and the reserves are equal at every duration.

Question 31Actuarial Mathematics

A multiple-decrement model has two independent decrements with annual rates q1 = 0.06 (death) and q2 = 0.04 (withdrawal). Assuming independent competing risks applied over the year, what is the probability that a life leaves the group due to decrement 1 during the year?

  1. 0.06 + 0.04 = 0.10
  2. 0.06 × 0.96 = 0.0576 Correct answer.
  3. 0.06 / 0.10 = 0.60
  4. 0.06

Explanation

With independent decrements acting simultaneously, the probability of exiting via decrement 1 is q1 times the probability of not exiting via decrement 2, giving 0.06 × (1 - 0.04) = 0.0576.

Question 32Actuarial Mathematics

A life aged 50 has force of mortality μ(50+t) = 0.01 for all t ≥ 0. What is the complete expectation of future lifetime e̊_50?

  1. 50 years
  2. 1 year
  3. 10 years
  4. 100 years Correct answer.

Explanation

Under a constant force of mortality, future lifetime is exponentially distributed with mean 1/μ. With μ = 0.01, the expected future lifetime is 1/0.01 = 100 years.

Question 33Actuarial Mathematics

An insurer holds a portfolio of 10,000 independent one-year term policies, each with a death benefit of 100 million won and mortality rate q = 0.002. Using a normal approximation, which statement best describes the effect of pooling on the relative variability of total claims?

  1. The standard deviation of total claims is independent of portfolio size.
  2. The standard deviation of total claims grows with the square root of the number of policies, so relative variability falls as the portfolio grows. Correct answer.
  3. The standard deviation of total claims grows in proportion to the number of policies, so relative variability stays constant.
  4. The standard deviation of total claims falls to zero as the portfolio grows large.

Explanation

For independent policies, the variance of total claims is the sum of individual variances, so the standard deviation scales with the square root of n while the mean scales with n; the coefficient of variation therefore declines as the portfolio grows.

Question 34Actuarial Mathematics

In a net premium reserve calculation, the reserve at duration t is computed prospectively. Which pair of quantities is subtracted in that formula?

  1. The accumulated past net premiums are subtracted from the accumulated past benefits.
  2. The APV of future expenses is subtracted from the APV of future gross premiums.
  3. The APV of future benefits is subtracted from the APV of future net premiums.
  4. The APV of future net premiums is subtracted from the APV of future benefits. Correct answer.

Explanation

The prospective net premium reserve equals the APV of future benefits minus the APV of future net premiums; it represents the fund needed now to cover the shortfall of future premiums relative to future benefits.

Question 35Actuarial Mathematics

An endowment insurance on (x) with term n pays 1 on death within n years (at end of year of death) or 1 on survival to age x+n. Using commutation functions, which expression gives the single net premium?

  1. (N_x - N_{x+n}) / D_x
  2. (D_{x+n}) / D_x
  3. (M_x - M_{x+n} + D_{x+n}) / D_x Correct answer.
  4. (M_x - M_{x+n}) / D_x

Explanation

The endowment APV combines the term insurance benefit (M_x - M_{x+n}) with the pure endowment benefit D_{x+n}, all divided by D_x.

Question 36Actuarial Mathematics

A whole life policy on (x) has net premium reserve V_t at duration t. Under the recursive (retrospective) relationship, which equation correctly links V_t, V_{t+1}, the net premium P, the interest rate i, and the mortality rate q_{x+t}?

  1. V_t (1 + i) + P = q_{x+t} × 1 + p_{x+t} × V_{t+1}
  2. (V_t + P)(1 + i) = q_{x+t} × 1 + p_{x+t} × V_{t+1} Correct answer.
  3. (V_t + P)(1 + i) = q_{x+t} × V_{t+1} + p_{x+t} × 1
  4. (V_t + P) = (1 + i)(q_{x+t} × 1 + p_{x+t} × V_{t+1})

Explanation

The recursive reserve equation accumulates the opening reserve plus net premium with interest and equates it to the expected outgo: the death benefit times q plus the closing reserve times the survival probability.

Question 37Actuarial Mathematics

An insurer calculates a gross premium using a profit margin and computes reserves on a net premium basis. Which statement best explains why the gross premium exceeds the net premium in a typical product?

  1. The gross premium includes expense loadings and a profit margin in addition to the net premium for benefits. Correct answer.
  2. The gross premium is lower than the net premium because expenses are paid from reserves.
  3. The gross premium equals the net premium because reserves require equality of the two.
  4. The gross premium excludes the cost of benefits, which is covered only by reserves.

Explanation

The net premium covers only the cost of benefits, while the gross premium adds expense loadings, contingency margins, and profit, so the gross premium is typically larger.

Question 38Actuarial Mathematics

In Korean statutory reserving, an insurer must hold reserves on a basis at least as strong as the prescribed standard valuation basis. Which statement best describes the practical effect of this requirement?

  1. The insurer must use exactly the prescribed assumptions with no discretion.
  2. The insurer may use a less conservative assumption if profitability supports it.
  3. The insurer may use a more conservative mortality or interest assumption than the standard, but not a weaker one. Correct answer.
  4. The insurer may hold no reserve if the product is profitable.

Explanation

Statutory reserving sets a minimum standard; insurers may adopt assumptions that produce reserves at least as large as the standard basis, but cannot weaken the basis below the prescribed minimum.

Question 39Actuarial Mathematics

A select mortality table has a 15-year select period. For a life newly underwritten at age 40, which statement correctly describes the mortality assumed during the select period compared with the ultimate table?

  1. Mortality rates equal the ultimate rates from the first year onward.
  2. Mortality rates are undefined during the select period.
  3. Mortality rates are lower during the select period and converge to the ultimate rates after 15 years. Correct answer.
  4. Mortality rates are higher during the select period and decline to ultimate rates afterward.

Explanation

Select mortality reflects favorable underwriting at entry, so rates are lower for the select period and then merge into the ultimate table once the selection effect wears off.

Question 40Actuarial Mathematics

A deferred annuity pays 5 million won per year starting at age 65 to a life currently aged 60, with payments made annually in advance. Using a deferred annuity-due factor, which expression gives the APV at age 60?

  1. 5,000,000 × (1 + i) × ä_65
  2. 5,000,000 × (5|ä_60) using the deferred annuity-due factor Correct answer.
  3. 5,000,000 × A_60 using the whole life insurance factor
  4. 5,000,000 × ä_60 using the full annuity-due factor from age 60

Explanation

Because payments begin only at age 65, the APV uses the deferred annuity-due factor 5|ä_60, which discounts the annuity starting at 65 back to age 60.

Question 41Life Contingencies

In a life insurance contract, which element is the actuarial present value of the benefit computed against?

  1. The future lifetime random variable of the insured Correct answer.
  2. The insurer's actual investment return history
  3. The calendar year of policy issue only
  4. The number of dependants declared at underwriting

Explanation

Life contingency mathematics models the benefit as a function of the insured's future lifetime random variable T, discounted at an assumed interest rate. Mortality and interest together determine the expected present value, not past investment results or family details.

Question 42Life Contingencies

Which symbol denotes the probability that a life aged x survives at least t further years?

  1. Ax
  2. t px Correct answer.
  3. t|qx
  4. t qx

Explanation

The notation t px is defined as the probability that a life currently aged x is still alive after t years, that is Sx(t) equals t px. Its complement t qx gives the probability of death within t years.

Question 43Life Contingencies

Under a constant force of mortality mu, what is the form of the survival function t px for a life aged x?

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

Explanation

With a constant force of mortality the survival probability is the exponential exp(-mu t), because the cumulative hazard over t years is simply mu times t. This is the only memoryless survival law among the standard choices.

Question 44Life Contingencies

A whole life insurance pays 1 at the end of the year of death. Which expression gives its actuarial present value at issue age x?

  1. Sum over k of (1+i)^k times k|qx
  2. Sum over k of v^k times k qx
  3. Sum over k of v^k times k px
  4. Sum over k of v^(k+1) times k|qx Correct answer.

Explanation

The benefit is paid at time k+1 when death occurs in policy year k+1, whose probability is k|qx, so each term is discounted by v^(k+1). Summing over all k gives the net single premium for whole life insurance.

Question 45Life Contingencies

Which relationship links the net single premium of a whole life insurance Ax to a whole life annuity due ax under the same mortality and interest basis?

  1. Ax = 1 + d ax
  2. Ax = v ax
  3. Ax = ax - 1
  4. Ax = 1 - d ax Correct answer.

Explanation

The identity Ax = 1 - d ax follows from the equivalence principle applied to a unit endowment: a whole life insurance plus a life annuity due equals a pure endowment of 1. Here d is the discount rate i/(1+i).

Question 46Life Contingencies

For a life annuity due of 1 per year payable annually, the actuarial present value ax equals which sum?

  1. Sum over k from 0 of v^k times k|qx
  2. Sum over k from 1 of v^k times k-1 px
  3. Sum over k from 1 of v^k times k qx
  4. Sum over k from 0 of v^k times k px Correct answer.

Explanation

A payment of 1 is made at the start of each year the annuitant survives, so the payment at time k occurs with probability k px and is discounted by v^k. Summing from k equals zero gives the annuity due value.

Question 47Life Contingencies

Under the uniform distribution of deaths assumption between integer ages, which expression gives t qx for 0 <= t <= 1?

  1. 1 - (1 - qx)^t
  2. t qx divided by (1 - (1-t) qx)
  3. t times qx Correct answer.
  4. qx divided by (1 - t qx)

Explanation

Uniform distribution of deaths means the deaths in the year of age are spread evenly, so the fraction of the annual mortality rate accumulated by time t is t times qx. This linear interpolation is the defining feature of the UDD assumption.

Question 48Life Contingencies

An insurer prices a term insurance with a level annual net premium P payable while the insured survives the term. Which principle determines P?

  1. Matching the premium to the first-year claim cost
  2. Equivalence of expected present value of premiums and benefits Correct answer.
  3. Setting the premium equal to the annual mortality rate qx
  4. Maximisation of the insurer's surplus at issue

Explanation

The equivalence principle sets the expected present value of net premiums equal to the expected present value of benefits at issue. This yields a level premium that funds the expected cost over the whole term, not just the first year.

Question 49Life Contingencies

Which quantity represents the net premium reserve at duration t for a whole life policy with annual premium P?

  1. Ax - P times ax
  2. Ax+t - P times ax+t Correct answer.
  3. P times ax+t - Ax
  4. Ax+t + P times ax+t

Explanation

The prospective reserve is the present value of future benefits minus the present value of future net premiums, both evaluated at the attained age x+t. At issue this difference is zero by the equivalence principle.

Question 50Life Contingencies

Which recursion expresses the reserve at duration t for a whole life policy in terms of the reserve one year later?

  1. (V_t + P)(1+i) = qx+t + px+t times V_t+1 Correct answer.
  2. (V_t - P)(1+i) = qx+t + px+t times V_t+1
  3. (V_t + P) = (1+i)(qx+t + px+t times V_t+1)
  4. V_t (1+i) = qx+t times V_t+1

Explanation

The accumulated fund of the current reserve plus the premium must cover the expected death benefit qx+t and the expected reserve needed for survivors px+t times V_t+1. This is the standard annual recursion for reserves.