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Introduction, This collection of exercises in life insurance mathematics replaces the collection of. Steen Pedersen and all other exercises and problems in any text or article in the. FM0L curriculum, The following abbreviations are being used for the contributors of exercises. AM Bowers et al Actuarial Mathematics, Society of Actuaries Itasca Il 1986. BM Bjarne Mess, BS Bo S ndergaard, FW Flemming Windfeld. JC Jakob Christensen, JH Jan Hoem Element r rentel re.

Universitetsforlaget Oslo 1971, MSC Michael Schou Christensen. MS Mogens Steffensen, RN Ragnar Norberg, SH Svend Haastrup. SK Stephen G Kellison The theory of Interest, Richard D Erwing Inc Homewood Il 1970. SP Steen Pedersen Opgaver i livsforsikringsmatematik. SW Schwartz Numerical Analysis Wiley 1989, Copenhagen August 2 1997. Bjarne Mess, Jakob Christensen, 4 Exercises in Life Insurance Mathematics.

1 Interest, FM0 S91 1 FM0 S92 1 FM0 S93 1, FM0 S93 2 FM0 S94 1. Exercise 1 1 Show that an an a n when i 0, Exercise 1 2 Show that an n decreases as n increases and i 0. Exercise 1 3 Show that an i1 an i2 decreases as n increases if i1 i2. Exercise 1 4 A man needs approximately 2 500 and raises in that connection a. loan in a bank The principal which is to be fully repaid after 6 months is 2 600. From this amount the bank deducts the future interest 84 50 and other fees of 5 70. so that the bank pays out the man 2 509 80 cash The interest rate of the bank is. a What is the effective interest rate p a for the bank. b What is the effective interest rate p a for the borrower. JH 4 rev 1971, Exercise 1 5 One day a company receives an american loan offer Principal of. 5 000 000 rate of course 99 and nominal interest rate 6 5 The loan is free of. installments for 5 years and is after that to be amortized over 15 years Interests. and installments are due annually Assume that the dollar rate of exchange decreases. exponentially from DKK 6 75 at the initial time to DKK 4 75 at the end of the loan. How can one determine the effective interest rate of the loan. Interest 5, Exercise 1 6 By calculation of the interest rate for a fraction of a year a bank will. usually calculate with linear payment of interest instead of exponential payment of. interest If the interest rate is i p a and we have to calculate interest for a period. of time 0 1 the bank will calculate interest as i per kr 1 in capital. instead of calculating the interest as, per kr 1 in capital Is this for the benefit of the borrower.

Exercise 1 7 A man is going to buy new furniture on an installment plan In the. hire purchase agreement he finds the following account. Cash payment for the furniture 6 306 00, In advance to the salesman 2 217 00. Net balance 4 089 00, Installment fee for 18 mths 501 00. Net balance for installment 4 590 00, The monthly installments are 255 00. What effective interest rate p a, JH 10 1971, Exercise 1 8 A man has been promised some money He can choose from two. alternatives for the payment, Under alternative i A5 4 495 and A9 5 548 are paid out after 5 and 9 years.

respectively, Under alternative ii B7 10 000 is paid out after 10 years. Denote the market interest rate by i, For which value values of i is i just as good as ii and when is i more profitable. for the man What if A9 5 562 i e 14 more, JH 11 1971. 6 Exercises in Life Insurance Mathematics, Exercise 1 9 A says to B I would like to borrow 208 in one year from today. In return for your kindness I will pay 100 cash now and 108 15 in two years from. today by the end of the loan, What is the effective interest rate p a for B if he accepts this.

JH 12 1971, Exercise 1 10 Consider a usual annuity loan with principal H interest rate i and. n installments Show that the installment which falls due in period t is. and find an expression for the remaining debt immediately after this period. Exercise 1 11 Consider a linearly increasing annuity At time t 1 2 the. amount t is being paid The present value of this cash flow is denoted by Ian. a Show that, Ian t an t, and interpret this equation intuitively. b Give an explicit expression for Ian, c What does the symbol Ia n mean Give expressions corresponding to the ones. from a and b, Exercise 1 12 A person has a table of annual annutities with different interest rates. and durations to his disposal However he needs some present values of half year. annuities These annuities do all have the same interest rate and the corresponding. whole year annuities can be found in the table, What is the easiest way to find the desired annuities.

Interest 7, Exercise 1 13 A debtor is going to pay an amount of 1 some time in the future. He does not know this point of time in advance he only knows that it is a stochas. tic variable T with a known distribution Now consider the expected present value. denoted by, a Show that the variance of the present value is given by. Var e T A2 2, Now assume that creditor has to pay a continuous T year annuity The present value. is a T Let the expected present value be denoted by. b Show that, and interpret the equation, c Give an expression for the variance Var a T corresponding to the one from a. d Show that, and find a similar inequality for a T Hint Use Jensen s inequality.

e Find at least two situations where these considerations are relevant. Exercise 1 14 Consider a loan principal H nominal interest rate i 1 rate of course. k and installment Ft in period t where t 1 N Show that the effective interest. rate ie satisfies, Exercise 1 15 A loan with principal H and fixed interest rate i 1 has to be amortized. annually over a period of N years The borrower can each year deduct half the interest. 8 Exercises in Life Insurance Mathematics, expenses on his tax declaration Construct the installment plan in a way so that the. amount of amortisation minus deductible actual net payment will be the same in. all periods assume tax is payable by the end of each year Find the annual net. installment, RN Opgaver til FM0 rentel re 18 05 93. Exercise 1 16 Consider a general loan Show that for t 1 we have. At 1 i1 Rt 1 Rt, v1t Rt H 1 2, with the use of standard notation Formula 1 1 is called the prospective formula for. the remaining debt Formula 1 2 is called the retrospective formula for the remaining. JH 20 1971, Exercise 1 17 A loan is being repaid by 15 annual payments The first five install.

ments are 400 each the next five 300 each and the final five are 200 each Find. expressions for the remaining debt immediately after the second 300 installment. a prospectively, b retrospectively, SK 1 p 122 1970. Exercise 1 18 A loan of 1 000 is being repaid with annual installments for 20 years. at effective interest of 5 Show that the amount of interest in the 11th installment. SK 10 p 123 1970, Exercise 1 19 A borrower has mortgage which calls for level annual payments of 1 at. the end of each year for 20 years At the time of the seventh regular payment he also. Interest 9, makes an additional payment equal to the amount of principal that according to the. original amortisation schedule would have been repaid by the eighth regular payment. If payments of 1 continue to be made at the end of the eighth and succeding years. until the mortgage is fully repaid show that the amount saved in interest payments. over the full term of the mortgage is, SK 16 p 124 1970. Exercise 1 20 A man has some money invested at an effective interest rate i At. the end of the first year he withdraws 162 5 of the interest earned at the end of. the second year he withdraws 325 of the interest earned and so forth with the. withdrawal factor increasing in arithmetic progression At the end of 16 years the. fund exhausted Find i, SK 40 p 127 1970, Exercise 1 21 A loan of a25 is being repaid with continuous payments at the annual.

rate of 1 p a for 25 years If the interest rate i is 0 05 find the total amount of interest. paid during the 6th through the 10th years inclusive. SK 42 p 127 1970, Exercise 1 22 After having made six payments of 100 each on a 1 000 loan at. 4 effective the borrower decides to repay the balance of the loan over the next five. years by equal annual principal payments in addition to the annual interest due on. the unpaid balance If the lender insists on a yield rate of 5 over this five year. period find the total payment principal plus interest for the ninth year. SK 45 p 127 1970, Exercise 1 23 A student has heard of a bank that offers a study loan of L 10 000. kr The rate of interest is 3 p a and the student applicates for the loan on the. following conditions, i The first m 5 years he will only pay an interest of 300 kr per year. ii After that period of time he will pay interests and installments of 900 kr per year. until the loan is fully amortized the last installment may be reduced. a For how long N does he have to pay installments and how big is the last amount. of amortisation, 10 Exercises in Life Insurance Mathematics. b Which amount t has to be paid at time t if the loan with interest earned is to. be fully paid back at time t t 1 2 N, Aktuarembetseksamen Oslo 1960.

Aggregate Mortality 11, 2 Aggregate Mortality, Exercise 2 1 Let T be a stochastic variable with distribution function F Assume F. is concentrated on the interval a b and that F is continuous with continuous density. f Assume F t 1 for t a b Define, We say that is the intensity of F. a Show that a t dt, b Can we conclude that t for t b. Let T be the life length of a newly born Let a 0 and b where is the maximum. life length, c Show that is the force of mortality. Exercise 2 2 Use the decrement tables of G82M to find the following probabilities. a The probability that a 1 year old person dies after his 50th year but before his. b The probability that a 30 year old dies within the next 37 years. c The probability that two persons now 26 and 31 years old and whose remaining. life times are assumed to be stochastically independent both are alive in 12 years. Exercise 2 3 Explain why each of the following functions cannot serve in the role. indicated by the symbol, x 1 x 3 x 0, 22x 11x2 7x3.

F x 1 0 x 3, 12 Exercises in Life Insurance Mathematics. f x xn 1 e x 2 x 0 n 1, AM 3 4 p 77 1986, Exercise 2 4 Consider a population where the distribution functions for a man s. and a woman s total life lengths are x q0M and x q0K respectively Assume that these. probabilities are continuous so that the forces of mortality M K. x and x are defined, Let s0 denote the probability that a newly born is a female Assume moreover that s 0. and the forces of mortality are not being altered during the period of time considered. in this exercise, a Find the distribution function x q0 for the total life time for a person of unknown. sex and find t qx Find the force of mortality x for a person of unknown sex. b What is the probability sx that a person aged x is a woman. Using decrement series M K, x and x for men and women respectively work out a decre.

ment serie x for the total population, c How should one appropriately choose M K. d Express ax in terms of aM K, Exercise 2 5 Consider a random survivorship group consisting of two subgroups. 1 The survivors of 1 600 births, 2 The survivors of 540 persons joining 10 years later at age 10. An excerpt from the appropriate mortality table for both subgroups follows. If 1 and 2 are the numbers of survivors under the age of 70 out of subgroups 1. and 2 respectively estimate a number c such that P 1 2 c 0 05 Assume. the lives are independent, Aggregate Mortality 13, AM 3 13 p 78 1986. Exercise 2 6 When considering aggregate mortality the probability that an x year. old person is going to die between x s and x s t is denoted by the symbol s t qx. a Express this probability by the distribution function of the person s remaining life. b Is there a connection between s t qx and t qx, c Show that.

s t qx u px x u du, and interpret this expression, When t 1 we write s qx s 1 qx. d Show that for integer x and n we have, e Show that s t qx can be expressed similarly use the function instead of d. f Prove the following identities, n m qx n px n m px. n qx n px qx n, n m px n px m px n, Exercise 2 7 Let e x n denote the expected future lifetime of x between the ages. of x and x n Show that, e x n tt px x t dt nn px, This is called the partial life expectancy.

14 Exercises in Life Insurance Mathematics, AM 3 14 p 78 1986. Exercise 2 8 The force of mortality x is assumed to be. Three persons are x y and z years old respectively What is the probability of dying. in the order x y z, Tentamen i fo rsikringsmatematik Stockholms Ho gskola 1954. Exercise 2 9 If F x 1 x 100 0 x 100 find x F x f x and P 10. AM 3 5 p 77 1986, Exercise 2 10 If x 0 0001 for 20 x 25 evaluate 2 2 q20. AM 3 7 p 77 1986, Exercise 2 11 Assume that the force of mortality x is Gompertz Makeham i e. x cx For at certain cause of death the force of mortality is given by 1 1 cx. Show that the probability of dying from the above disease for an x year old is. Tentamen i fo rsikringsmatematik Stockholms Ho gskola 1954. Exercise 2 12 Show that constants a and b can be determined so that. x a log 1 qx b log 1 qx 1, when x can be put as a linear function for x t x 2.

Tentamen i fo rsikringsmatematik Stockholms Ho gskola 1954. Introduction This collection of exercises in life insurance mathematics replaces the collection of Steen Pedersen and all other exercises and problems in any text or