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principles of risk management

Questions and Answers of Principles Of Risk Management

Given a^iy + ay\x = 4, la^y + 2^x1;^. + a^i^ = 24, and 2(7;,^ -\- a^\y + 2ay\x = 22, find the present value of a 10,000 lastsurvivor annuity to two individuals aged x and y.
Given 4= 120 — x, find the probability that a reversionary annuity bought by Alphonse, aged 60, for his girlfriend Brigitte, aged 19, will be in payment status after 10 years.
Marcella, aged 35, and Maria, aged 40, purchase a life insurance policy which will pay annual benefits to the survivor upon the death of one of the women, first payment at the end of the year of
On January 1, 1998, Martha, aged 40, purchases a reversionary annuity to benefit her child, aged 15. Any payment made on or before December 31, 2008, will be 10,000, and subsequent payments will be
In Question 27, what is the present value of the benefits payable to Bertha's husband after her death in each of the two cases?
Bertha, aged 40, wishes to purchase an annuity. She is offered two equivalent plans:(a) 500 per month for life, with 300 continuing to her husband after her death.(b) 600 per month for life, with 150
Show that ax\y — ay\x = ay — a^.
Can you calculate the present value of a reversionary annuity payable to Paul upon his wife's death, from the data given in Question 24? If not, what additional information is required?
Paul, aged 65, purchases a reversionary annuity of 1000 per year for his wife, aged 60. Find the net single premium for this annuity given the values A^6i = 1350, D(,o = 300, iV6i:66 = 620 andZ)60 65
Smith is age 50 and Brown is age 51. A life annuity of 100 per year is payable to Smith. The net single premium for this annuity is equal to the net single premium for an annuity of K per year
George and Sarah purchase 100,000 of whole life insurance.Assuming ^^ — 05 for George, iiy — .04 for Sarah, and 6 = .08, find the net single premium for this insurance in each of the following
George and Sarah purchase a continuous annuity of 20,000 per year payable as long as at least one of them survives. Assuming lix = .05 for George, //^ — .04 for Sarah, and 6 — .08, find the net
If two independent lives (x) and (y) are both subject to the force of mortality yix+t = M>;+/ = 03, < / < 10, find the probability that the last survivor status (3cy) will survive for 6 years.
Rank the following in increasing order of magnitude: ax, a^y,^yx-> ^^^xy•
An «-year temporary annuity pays 1 while both lives survive, 4 if the first life only survives, and 4 if the second life only survives. If the ages are x and y, find the present value in terms of
Express the answer to Question 15 in terms of single-life probabilities.
Do Example 1 1.8 if the annuity pays 1000 as long as exactly one of Julio or Harold is alive.
George and Sarah purchase a continuous annuity of 20,000 per year, payable as long as both of them survive. Assuming 11^ — .05 for George, iiy = .04 for Sarah, and 6 — .08, find the present value
Repeat Question 1 1 if the 40-year-old is subject to ^^ = -OOOSx and the 50-year-old is subject to ^y — M\y.
Given two independent lives, each subject to a force of mortality/i;^ = ^' determine the probability that the joint life status(40, 50) fails during the next 5 years.
The curtate-future-lifetime of a given situation is said to be n if the situation survives for n years but does not survive for n+ 1 years. If ^;c = 05, ^;,+ i = .07, qy = M and ^^^i^.H, determine
Repeat Question 8 if the first 3 annuity payments are guaranteed.Assume A^63 68 = 1280 and / == .09.
Repeat Question 7 if the first payment occurs immediately.
Francis, aged 65, purchases a life annuity paying 1000 at the end of each year provided she and her husband, aged 60, are both alive. Find the net single premium for this annuity if iV60:65 = 1500
If 6 students all graduate at age 22, find expressions for each of the following:(a) The probability that not more than 3 of the students will be alive at age 50.(b) The probability that not less
If two lives are both governed by the formula 4 = 120 — x, for< X < 120, find /i45+/5o+/.
Show that the probability that two lives, one aged 30 and the other aged 40, will die at the same age last birthday is equal to 10/?30(l+^40:40) " 2 •1 l/'SoCl +^40:41 ) + 04o)(l l/'SoX 1+^4 1:41
Show that ^?g^ = (px)(n-iPx+i,)-
Mary is 15 years old and Helen and Harry are 25-year-old twins.Find expressions for each of the following:(a) The probability that all three live 40 years.(b) The probability that at least one does
In Questions 31(a) and 32(a), find the minimum number of policies of the type described that an insurer must sell in order to be 99% certain that the total loss will be negative.
Let L be the loss random variable for a fully continuous whole life insurance of 1 purchased by a life age x. Assume that the net annual premium is determined by the equivalence principle, but that
(a) In a fully discrete setting, prove that Q = Px if and only if E[L] = 0.(b) In a fully continuous setting, prove that Q = P(Ax) if and only if E[L] = 0.
Let L be the loss random variable for a fully continuous whole life insurance of 1 purchased by a life age 40. Find E[L] and Var(L) in each of the following cases:(a) 6 = .06, fix = .03 for all x,
Let L be the loss random variable for a fully discrete (i.e., annual premiums and benefit payable at the end of the year of death)whole life insurance of 1 purchased by a life age 25. Find £[1]and
Redo Question 28 if the insurance is payable at the moment of death.
Redo Question 28 if there are 1000 lives instead of 100. Explain the relative magnitude of your answers to the last two exercises.
100 independent lives, each age 20, contribute equal amounts to establish a fund which will pay 10,000 at the end of the year of death of each individual. Given ^/?20 = (-98/ and / = .06, find the
Redo Question 25 if 10 policies with death benefit 100 each are sold.
Redo Question 25 if a single policy with death benefit 1000 is sold.
In Question 1, assume 1000 policies of the type described are sold to different individuals and let Z denote the present value of aggregate future benefits for all these policies. Find £'[Z] and
Redo Question 21 if the death benefit of Question 1 is payable at the moment of death.
Using the result of Question 22, redo Questions 19 and 21 if the death benefit is 1000 instead of 1. (This could also be done using Question 2 but the suggested method is preferable.)
Show that the standard deviation of rZ, where r is any positive number, is equal to r times the standard deviation of Z.
Assuming a normal distribution in Question 1, find the constant c such that (a) Pr{Z
What value of k is needed in Chebyshev's Rule in order to guarantee that Pr{\Z - E[Z\\ < ka) > .95?
In Question 1, use Chebyshev's Rule to find an interval for Z such that the probability that a specified value lies in the interval ois at least (a) .75; (b) g. Are these intervals useful?
Given 4 = 1000(^1 - y^] and / = .07, find the probability that aj^> 10, where T represents the future lifetime of a life aged 50.
Find the median of 7(40) in each of the cases in Question 16.
Find the mean and variance of 7(40) in each of the following cases:(a) £x=\00-x,0
Given that /x^ and 6 are both constant, find expressions for each of the following:(a) The mean, variance and median of the present value random variable for a whole life insurance of 1 payable at
Given that p.^ is constant, 6 — .09, and -^A^ = .10, calculate ax and -^a^. (Recall that the superscript to the left denotes a doubling of the force of interest.)
A three-year temporary life annuity will pay 2 at the end of the first year, 5 at the end of the second year, and 4 at the end of the third year. Given that pso = 98, ps\ = .978, ps2 = 975 and/ =
Do Question 1 1 if the annuity is continuous.
Determine the mean and variance of the present value random variable for a life annuity of 1 per year, first payment in one year, given / = .09 and tPx — {-915y for all t.
Let Z denote the present value random variable for a whole life policy of 1 payable at the moment of death. Find E\Z\ and Var{Z) if fi = fix and S are both constant and if6 = 2//.
Do Question 7 assuming (5 = .1 1 and 4 = 100 - jc, < x < 100.
Do Question 7 if the policy is for 40-year term insurance.
Let Z denote the present value random variable, at policy issue, for a whole life policy with a death benefit of 1 payable at the moment of death purchased by a 30-year-old, assuming ^^ — .03 for
Do Question 1 if the death benefit is payable at the moment of death.
Do Question 1 if the insurance is deferred for 30 years.
Do Question 1 if the policy is for 30-year endowment insurance.
Do Question 1 if the policy is for 30-year term insurance.
Do Question 1 if the death benefit is 1000 instead of 1.
Let Z denote the present value random variable, at policy issue, for a whole life policy with a death benefit of 1 payable at the end of the year of death purchased by a 30-year-old. Find E[Z]and
Under a multiple decrement table, at each age from 50 to 70 the absolute rate of death is 2% and the absolute rate of termination for all causes other than death is 4%. Find a good approximation to
A company is affected by two preretirement decrements, mortality(d) and disability (/). Assume 4o = l^'^OO and 4V = 9300. If the absolute rate of disability at age 50 is .06, find ct^q .
In the first year following training for soldiers, the central death and withdrawal rates are rrix — a and mx =b, respectively.What is the probability that a soldier just finishing training will
Consider a population in which three decrements are acting, mmQ-\y{a\{b\{c).(i) Write an expression for the probability that a 40-year-old will leave the population because of decrement {b) at age 47
In a small country with a stationary population, a special system is used for supporting the elderly. On January 1, each person whose age is between 20 and 65 contributes 100 to a pool. On the same
Find an expression for the expected age at death of a person who survives to age 40 and either dies before age 50 or dies after age 75.
An army of mercenaries has a constant size of 1000. Each year all new entrants are at exact age 25, and any soldier reaching age 55 must retire. No one can leave the army except by death or
In a stationary population of 120,000 lives, the number of deaths is 2000 annually. The complete expectation of life for a 40-year-old is 30 years. 60% of the population is under age 40.(a) What is
We are given the values £50 = 8200, ^50 = 20, £70 = 3000, and eio = 10.(a) Find the average age at death for those surviving to age 50.(b) Find the average age at death for those surviving to age
You are given the following values for a stationary population:X 4 55 70 10,000 8,250 5,380 69.0 19.5 10.3 Find the average age at death for those who survive to age 55 but die before age 70.
If 4 = lOOOf 1 — TTiri), calculate ^90 and ego (both exactly and approximately).
Before 1980, a stationary population of 500,000 was maintained by 10,000 annual births. 40% of the population was under age 15.Beginning in 1980, annual births increased to 12,000. Assuming mortality
Verify that W;, = /i;c+i/2 for the function 4 = lOOOf^l - ^V
Prove that rux > q^ for all ages x. When does equality occur?
Estimate m^ if 4 = 3825 and 4+i = 5713.
A service club has a constant membership of 1,000. X new entrants are added each year at exact age 35. Withdrawals are either by death or by retirement. 40% of those who reach age 50 retire at that
During the first 12 months of life, infants in a developing country are subject to a force of mortality given by fix = -^ \_ , where jc is measured in months. Calculate the probability that a newborn
Show that if uniform distribution of deaths over year of age jc is assumed, then tPxI^x+t = ^^ for all < / < 1.
If //;c = .0017 for 20 < jc < 30, find each of the following:(a) P20 (b) sPio (c) qi2> (d) 5^23 (e) 41^23 (0 4|3^23
If 4 = 250(64 - .80jc)'/\ find each of the following:(a) lopo(b) /X70(c) The terminal age of the population.
If 4 = 100,000 f^^] and £35 = 44,000, find each of the following:(a) The value of c.(b) The terminal age in the life table.(c) The probability of surviving from birth to age 50.(d) The probability
Show that, under de Moivre's formula, n\qx is independent of «.(See Question 7-10 for the definition of this notation.)
(a) Starting with Gompertz' formula 4 = ^S^\ verify that the force of mortality is yi^ = Bc^ for suitable B.(b) Starting with Makeham's formula 4 = ks^g^\ verify that/i;r = ^ + Bc^ for suitable A and
Show that Makeham's formula for 4 implies px = sg^^^~^\
Show that Gompertz' formula for £x implies /7;f = gc'(c-i)
Obtain an expression for /jLx if 4 = ks^w^ g^. (This is called Makeham's second formula.)
Prove that in the general case of de Moivre's formula, given by(7.7), we have ^ix = u^-
For the 4 given in Question 13, calculate general formulae for px, qx and ^x- Then sketch graphs of all four functions.
Given 4 = lOOOf 1 — jJtv ), determine each of the following:(a) £o (b) ^120 (c)
The probability that a person age 10 will survive to age 30 is .80.Sixty per cent of the deaths between ages 10 and 40 occur after age 30. The probability that three lives aged 30, 50 and 70 will all
Let nlm^x denote the probability that a person aged x will die between ages x -\- n and x -'r n + m. (When w = 1 it is omitted in this notation.)(a) Show that ,U^, = ^-^n-Un^m(b) Show that
Derive each of the following approximations, where < / < 1.(a) 4+/ = £x - i- dx(b) 4+/ = (1-0 -4 +^-4+](c) ip^= \ -tq^AppendixLO1
Find an expression for the probability that a 30-year-old will die in the second half of the year following her 35th birthday.AppendixLO1
If/7;c = -95 for all jc, find each of the following:(a) /?20 (b) 2^30(c) The probability that a 20-year-old dies at age 50 last birthday.(d) The probability that a 20-year-old dies between age 50 and
Explain, both mathematically and verbally, why the following are true.(a) 4 = ^x 4-
Four persons are all aged 30.(a) Find an expression for the probability that any three of them will survive to age 60, with the other dying between age 50 and age 55.(b) Find an expression for the

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