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principles of risk management

Questions and Answers of Principles Of Risk Management

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
80% of people age 25 survive to age 60. 40% of people who die between age 25 and age 60 do so before age 45. Find the probability that a 45-year-old will die before reaching age 60.AppendixLO1
Write expressions for each of the following:(a) The probability that a 20-year-old lives 25 years.(b) The probability that a 20-year-old reaches age 25.(c) The probability that a 20-year-old dies
For a certain type of insect, we find that ^q = -70, ^i = .30, qi = .40 and ^3 = 1.0. Starting with £0 = 1000, construct a life table.AppendixLO1
Assume in Question 17 that, in addition to the information given, there is also a 5000 deposit to the fund on July 1, 1997.(a) Find the dollar-weighted annual rate of investment return for the fund,
Emily's trust fund has a value of 100,000 on January 1, 1997. On April 1, 1997, 10,000 is withdrawn from the fund, and immediately after this withdrawal the fund has a value of 95,000. On January 1,
The present value of a series of payments of 1 at the end of every 3 1 2S years forever is equal to -^. Find the effective rate of interest per year.AppendixLO1
A trust company pays 7% effective on deposits at the end of each year. At the end of every four years, a 5% bonus is paid on the balance at that time. Find the effective rate of interest earned by an
John buys a TV for 600 from Jean. John agrees to pay for the TV by making a cash down payment of 50, then paying 100 every four months for one year (i.e. three payments of 100), and finally making a
Bemie borrows 5000 on January 1, 1995, and another 5000 on January 1, 1998. He repays 3000 on January 1, 1997, and then finishes repaying his loans by paying 10,000 on January 1, 2000.What effective
Find the effective rate of interest if payments of 300 at the present, 200 at the end of one year, and 100 at the end of two years accumulate to 800 at the end of three years.AppendixLO1
A consumer purchasing a refrigerator is offered two payment plans:Plan A: 150 down, 200 after 1 year, 250 after 2 years Plan B: 87 down, 425 after 1 year, 50 after 2 years Determine the range of
John pays Henry 500 every March 15 from 1996 to 2000 inclusive.He also pays Henry 300 every June 15 from 1998 to 2001 inclusive. Assuming f^"^ =17, find the value of these payments on (a) March 15,
Fund A accumulates at 9% effective and Fund B at 8% effective.At the end of 10 years, the total of the two funds is 52,000. At the end of 8 years, the amount in Fund B is three times that in Fund
How long should 1000 be left to accumulate at /= .12 in order that it may amount to twice the accumulated value of another 1 000 deposited at the same time at 8% effective?AppendixLO1
In return for payments of 400 at the end of 3 years and 700 at the end of 8 years, a woman agrees to pay X at the end of 4 years and IX at the end of 6 years. Find X'xii — .14.AppendixLO1
(a) The present value of 2 payments of 1000 each, to be made at the end oi n years and n -\- A years, is 1250. If / = .08, find n.(b) Repeat part (a) if the payments are made at the end of n years
A vendor has two offers for a house: (i) 40,000 now and 40,000 two years hence, or (ii) 28,750 now, 23,750 in one year, and 27,500 two years hence. He makes the remark that one offer is"just as good"
Payments of 800, 500 and 700 are made at the ends of years 2, 3 and 6 respectively. Assuming i = .13, find the point at which a single payment of 2100 would be equivalent.AppendixLO1
Boswell wishes to borrow a sum of money. In return, he is prepared to pay as follows: 200 after 1 year, 500 after 2 years, 500 after 3 years and 700 after 4 years. If / = .12, how much can he
Eileen borrows 2000 on January 1, 1997. On January 1, 1998, she borrows an additional 3000. On January 1, 2001, she repays 4000.Assuming / = .13, how much does she owe on January 1, 2005?AppendixLO1
Brenda deposits 7000 in a bank account. Three years later, she withdraws 5000. Two years after that, she withdraws an additional 3000. One year after that, she deposits an additional 4000. Assuming /
If 30% of each gross premium is required for loading (when paying for a deferred annuity), what is the ratio of the gross premium to the net premium?
Repeat Question 53 if 1000 out of the first premium is required for issue expenses, and 20% of each subsequent premium of X is required for administrative upkeep.
Again consider Eric's purchase of the deferred life annuity in Question 53. This time, if Eric dies before reaching age 60, net premiums paid prior to death are refunded with interest. Using the same
Eric, aged 40, purchases a deferred life annuity which will provide monthly payments of 300 at the beginning of each month commencing at age 60. Eric pays a premium at the beginning of each year for
Repeat Example 8.18 if Arabella is paying for the annuity with semiannual instead of annual premiums.
Let {Ia)x denote the present value of a continuous life annuity (to a person age jc) which pays at the rate of 1 per year during the first year, at the rate of 2 per year during the second year, and
Find formulae for n\(J^)x and (Da)^-, in terms of commutation symbols.
Express in commutation symbols the present value at age .x of a life annuity which commences with a payment of 10 at age x, increases annually by 1 for 5 years to a maximum of 15, and then decreases
Repeat Question 41 if, instead of increasing, the payments decrease by 400 per year until reaching zero. Assume 5*42 = 185,000 and7V42 = 15,000.
Repeat Question 41 if payments increase to a maximum of 5600 and then remain constant thereafter. Make the same assumptions as in Question 42.
Repeat Question 41 if a maximum of 10 payments is to be made.Assume 5*47 = 160,000 and N47 = 9,500.
Pauline purchases a life annuity which will pay 2000 in one year's time, with annual payments that will increase by 400 per year thereafter. Find the present value of this annuity if Pauline is aged
A man is offered the choice of a continuous life annuity paying 20,000 per year or a continuous 5-year annuity with certain payments at X per year, followed by a continuous life annuity paying X per
(a) Find the present value of a continuous life annuity of 1000 per year for a 40-year-old if we assume that 6 = .06 and/jLx = 04 for all x.(b) Redo part (a) if the annuity is temporary, lasting for
Express a^;^ and nt^^ in terms of commutation functions.
Edgar was entitled to a monthly life annuity of 400 at age 65, but died on the day before the first payment was due. A death benefit, equal in value to half the value of the annuity, was payable to
Repeat Questions 33 and 34 if death is possible before retirement, and we assume that 20A5 = • ^4.
Repeat Question 33 if the yearly pension payment is to be divided into twelve equal monthly payments, the first occurring one month after retirement.
Marilyn, aged 45, works for the same company as Jeannette(Question 31) and will have the same retirement benefit, beginning at age 65. Marilyn's current salary is 25,000 per year and her salary will
Repeat Question 31 if the yearly payment is to be divided into twelve equal monthly payments, the first occurring in one month.
Repeat Question 29 if no payments are guaranteed. Assume, in addition, that A^50 = 83,500.8-3 1 . Jeannette, aged 65, is about to retire. Her salary is 70,000 per year and, because of her long
Marvin, aged 50, purchases an annuity of k per month, the first payment to be made immediately. For the first 60 months, payments will be guaranteed (i.e., will be made independent of Marvin's
Exp^ress cr 4 in terms of commutation functions. Give two anx:n\swers, one involving A^ terms, and one using regular Ny terms.
Explain verbally why the formula a - = ^ + ^^~\ is incorrect.
(a) How much money must be invested to provide John, aged 50, with monthly payments of 400 for life if ^50 = 16.5?The first payment will occur in exactly one month.(b) Repeat part (a) if the payment
Assume / = .08 and that we are dealing with a four-year select period. We know that ^po] = •40,/>[30]+i = .80, ^[30]+2 =10 and^[30^_^3 = . 10. Also D34 = 1000 and ^35 = 920. Find the probability
A select-and-ultimate disabled life table has a select period of two years. Select probabilities are related to ultimate probabilities by the rules py^^ ^ \ • Px and p^^^^x = !•/?;,+ ,. Given dis
Harold, aged 60, purchases a life annuity which will provide annual payments of 1000 commencing at age 61. For the year beginning at age 60 only, Harold is subject to a higher risk of death, namely
Given A^;, = 2000, 7V;,+ , = 1900, A^^+2 = 1820 and / = .ll, find qx.
State verbally the meaning of each of the following.(,) fc^ + 10 -40^30(b)A^x+l + -g^+2+---
Brenda, aged 38, purchases a 20,000 life annuity with the first payment in 10 years.(a) Find Brenda's net single premium in terms of commutation functions.(b) Find Brenda's net single premium in
Andrea, aged 25, purchases a life annuity of 2000 per year. From tables, we find Nis = 2450, 7^27 = 2290 and Die = 75.(a) Find the net single premium Andrea should pay for this annuity if the first
Assume in Question 13 that Noreen's mortality is also governed by /A = .3(4-0.(a) Find the value at r = of a life annuity paying 1000 at the end of each year as long as at least two of Julio, Harold
Repeat Question 13 if the annuity is paid as long as either Julio or Harold is still alive.
Julio's mortality for 1 < / < 4 is governed by tp^ — 3(4 — /), and Harold's mortality for 1 < r < 5 is governed by tPx — -25(5 — f).If / — .07, fmd the value at time of an annuity which
Give verbal explanations for each of the identities in Question 1 1.
For a given population, 4 = 120 — jc. Given that / = .07, find the net single premium at age 60 for a deferred life annuity with annual payments of 1000 commencing at age 70 if at most twenty
Do Question 7 ifp, = .96 for all x.
Find an expression for the present value of a life annuity which will pay 500 at the end of every two years if / = .13 and the annuity is sold to a person aged 45.
Do Question 4 if the maximum number of payments will be 40.
Do Question 4 if the first payment is deferred until age 40.
Elaine, aged 30, purchases a contract which provides for three payments of 2000 each at ages 40, 50 and 55, if she is alive.Given i^ = 1 10 — x and / = .09, find the net single premium for this
Henri, 1 1 years old, wins first prize in the Parisian Lottery. He can have 1,000,000 francs if alive at age 21, or X francs today.FindXif/ = .13andio;?ii -= -975.
Charles buys a 1000 face value 20-year bond redeemable at par with semiannual coupons at 12% convertible semiannually. He determines his purchase price to yield 14% convertible semiannually, and to
Agatha pays 770 for a 1000 face value bond paying interest at 1 1%)convertible semiannually, and redeemable at par in 20 years. If her desired yield was 12% convertible semiannually, what rate of
The Happy Finance Company experiences a 10% default rate on one-year loans. The Super Finance Company experiences a 7%default rate on one-year loans. If the Super Finance Company charges 16%) on
.A 20-year 100 face value bond, redeemable at par, is offered for sale. The coupon rate is 12% convertible semiannually. Find the purchase price to yield 8% convertible semiannually, if the
Do Question 17 for a perpetuity of 3000 if the probability of surviving t years is estimated to be ^
An insurance company sells an annuity to a person whose probability of surviving 1 year is .65, of surviving 2 years is .45, and whose probability of surviving 3 years is negligible. If the annuity
The All-Mighty Bank wishes to lend 100,000,000 to a South American country and would like to earn \2% on its investment.Repayment of the loan will be by two equal annual payments, the first due in
How much would you lend a person today if he promised to repay 2000 at the end of each year for the next 1 years? Assume there is a 3% chance of default in any year and you wish to earn 1 \% on your
Mrs. Trudeau is interested in ensuring that her newborn son will have sufficient funds for higher education. A certain plan will award a 5000 scholarship if her son survives to age 18 and enters a
Friendly Finance Company wishes a yield rate of 15%, but charges 22% on loans repayable with a single payment at the end of one year. What default rate is being assumed?
Mrs. Kelly wants to borrow some money. She wishes to repay the loan with a single payment of 3000 in two years' time. It is fek that there is a 5% chance she will not repay the loan. How much will
Mr. Hill wishes to borrow 5000. He will repay the loan with a single payment at the end of one year. The lending agency has a"risk-free" rate of interest of 13%, but estimates there is an 8%chance
The Trustworthy Trust Company would like to obtain a yield of 16% on their loans. Past experience indicates that 5% of all loans are not paid. What rate of interest should Trustworthy Trust charge?
A perishable product is purchased by a retailer for 5 and sold for 9. Based on past experience, it is estimated that 4 of the time 10 items can be sold, k of the time 1 1 items are sold, and A of the
A current quiz program gives the contestants 5 true/false statements and awards 5 for each correct answer. If all 5 are answered correctly, the contestant gets 1 ,000 extra. Find the expected value
For a 1 -dollar ticket, a lottery offers the following prizes:1 prize of 25,000;20 prizes of 1,000 each;50 prizes of 100 each;100 prizes of 25 each;1500 prizes of 5 free tickets.If 100,000 tickets
A box contains 3 lO-doUar bills, 6 5-dollar bills, and 4 1 -dollar bills. You are allowed to pull two bills (without replacement)from the box. If both bills are of the same denomination you can keep
A single die is thrown. If a 1, 2 or 3 turns up, player A wins that amount of money (1, 2 or 3). If a 4, 5 or 6 turns up, player B wins the amount of money showing. Find the expected value for each
The probability of a 45-year-old surviving to age 80 is 4. The probability of a 45-year-old dying between 60 and 80 is #. Find the probability of a 45-year-old surviving to age 60.

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