An Introduction To Actuarial Mathematics 1st Edition Arjun K. Gupta ,Tamas Varga - Solutions

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3.2. A 25 year term insurance of $8000 issued to a life aged 50 is purchased by annual premiums. Find the Zillmer maximum for the insurance. Compute the modified net premiums and find the expressions for the modified net premium reserves at duration t = 1,2, ... ,25 if an initial expense of $50 is
3.1. A 20 year endowment of $6000 issued to a life aged 40 is purchased by annual premiums. Obtain the Zillmer maximum for the insurance. Find the modified net premiums if an initial expense of $100 is paid from the first premium. Furthermore, derive the expressions for the modified net premium
2.6. An insurance company issues 15 year annuity-immediates of $1500 per annum with a guaranteed payment period of 10 years to lives aged 50 on January I, 1990 when the single premiums are payable. Determine the mortality profit or loss for the first year of the insurance if from the 830 insured
2.5. An insurance company issues 15 year term certain insurances of $7000 to lives aged 40 on January I, 1980. The premiums are payable annually. From the 800 policyholders alive at the beginning of 1990, 796 survive to the end of the year. Find the mortality profit or loss for the year 1990 based
2.4. An insurance company issues 20 year annuities-due to lives aged 55 on January I, 1982 when the single premiums are payable. The amount of the total annuity payments if $1,460,000 in 1990 dropping to $1,430,000 in 1991. Using a 6% annual rate of interest, obtain the mortality profit or loss for
2.3. An insurance company issues 10 year endowment insurances to lives aged 45 on January I, 1990. The premiums are payable annually. On January I, 1992, the total sum at risk is $2,400,000. During the year of 1992, the total death benefit payment is $8000. Based on a 6% annual interest rate,
2.2. An insurance company issues 15 year annuities-due of $2000 per annum deferred 4 years to lives aged 40 on January I, 1980. The premium is payable yearly during the 4 years of the deferment period. If the number of policies in force is 950 on January I, 1982, 945 on January I, 1983, 942 on
2.1. An insurance company issues 25 year term insurances of $12,000 to lives aged 30 on January I, 1985. The premiums for the insurances are payable annually. On January I, 1990, the number of policyholders still alive is 540 from whom 2 die during the year. Based on a 6% annual rate of interest,
1.18. Using the recursive formula, evaluate the reserves at the end of each policy year for the insurance given in Problem 1.6.
1.17. Use the recursive formula to obtain the reserve at the end of years 7 and 8 for the pure endowment insurances defined in Problem 1.9.
1.16. A 25 year endowment insurance of $5000 is purchased for a life aged 40 by 10 annual premiums. Using the recursive formula, obtain the reserves at the end of each policy year. Base the computations on a 6% annual interest rate.
1.15. Assume that the premiums for the family income benefit, given in Problem 1.5, are payable yearly in advance. Determine the longest premium paying period under which the reserves are not negative.
1.14. The death benefit of a 20 year term insurance issued to a life aged 40, is $8000 in the first year and decreases linearly to $400 in year 20.a) Evaluate the reserves at the end of each policy year if the premium is payable annually over the whole term of the insurance.b) Find the longest
1.13. A 10 year term certain insurance of $8000 is issued to a life aged 30. Based on a 6% annual rate of interest, find the reserve at the end of year 3, ifa) the premium is payable annually for the whole term of the insurance.b) the premium is payable annually for a term of 5 years.
1.12. A 20 year annuity-due of $8000 per annum, deferred 5 years is purchased for a life aged 60 by monthly premiums payable in the 5 years of the deferment period. Based on a 6% annual rate of interest, find the reserves at the end of years 3, 5, and 10.
1.11. Repeat Problem 1.3 if the premium is payable continuously throughout the term of the insurance.
1.10. Repeat Problem 1.2 ifa) an annual premium is payable for the whole term of the insurance.b) an annual premium is payable for a term of 5 years.
1.9. Repeat Problem 1.1 if 271a) an annual premium is payable for the whole term of the insurance.b) an annual premium is payable for a term of 10 years.
1.8. Based on a 6% annual rate of interest, obtain 1a) 3V3S:101b) 20V40c) 7V4S:1S1d) S V (A30:201) ---e) lOV ( A SO) (4) 1 £) 14 V 20:3s1 S 1 g) 2V 60:201 h) 10V(12) S 30·
1.7.a) Show that tV x = 1 - (1 - 1 V x) (1 - 1 V X+ 1) .,. (1 - IV X+t-I) andb) tVx:nl = 1- (1-1 Vx:nV (1- 1 Vx+l:n-IV ... (1-1 Vx+t-1:n-t+IV'
1.6. A 20 year annuity-immediate of $1000 per annum, with the first 15 payments guaranteed, is purchased for a life aged 40 by a single premium. Find the expressions for the prospective and retrospective reserves at the end of each policy year. Based on a 6% annual rate of interest, evaluate the
1.5. A 10 year family income benefit of $6000 per annum is purchased for a life aged 50 by a single premium. Derive the expressions for the prospective and retrospective reserves at the end of each policy year and calculate them numerically at the end of year 5. Use a 6% annual rate of interest.
1.4. A 10 year annuity of $300 per annum payable yearly in advance is purchased for a life aged 45 by a single premium. Determine the expressions for the prospective and retrospective reserves at the end of each policy year. Based on a 6% annual rate of interest, evaluate the reserves numerically
1.3. The death benefit of a 15 year endowment insurance of $4000, issued to a life aged 25 is payable at the moment of death. The premium is payable at the commencement of the insurance. Derive the expressions for the prospective and retrospective reserves at the end of each policy year. Based on a
1.2. The premium for a whole life insurance of $8000, issued to a life aged 40 is payable at the commencement of the insurance. Obtain the expressions for the prospective and retrospective reserves at the end of each policy year. Calculate the reserves numerically at the end of year 1 and year 5.
1.1. A 20 year pure endowment insurance of $2000, on a life aged 30, is purchased by a single premium. Find the expressions for the prospective and retrospective reserves at the end of each policy year. Evaluate the reserves numerically at the end of year 6 and year 18. Use a 6% annual rate of
2.5. The premiums for a 25 year endowment insurance of $6000 on a life aged 40 are payable yearly in advance. Expenses to be allowed for are initial expenses of $200, renewal expenses of 0.3% of the sum insured, and 5% of the annual premium. The renewal expenses are incurred every year. Based on a
2.4. A 20 year pure endowment insurance of $5000 is issued to a life aged 30. Find the gross annual premium if there are initial expenses of 0.2% of the sum insured and renewal expenses of $10, plus 7% of the annual premium. The renewal expenses are incurred every year. Use a 6% annual rate of
2.3. The premiums for a life annuity-immediate of $2000 per quarter, deferred 10 years on a life aged 50, are payable yearly in advance for a term of 5 years. Expenses to be allowed for are initial expenses of $100 and renewal expenses of 3% of each premium including the first. Find the gross
2.2. The premiums for a 15 year endowment insurance of $4000 issued to a life aged 35 are payable quarterly in advance. There is an initial expense of 2% of the sum insured. At each premium payment time, including the first, there are renewal expenses of 0.3% of the sum insured and 5% of the
2.1. A 20 year term insurance of $5000 on a life aged 40 is purchased by a single premium. There are initial expenses of $100, renewal expenses of 0.1% of the sum insured, incurred at the beginning of each year including the first, and claims expenses of $30. Based on a 6% annual interest rate,
1.16. Assume the premium for the endowment insurance, given in Problem 4.4 of Section 3.4, is payable continuously. Determine the constant premium payment rate per annum so that the insurance company makes a profit with 95% probability on a group ofa) 50 peopleb) 120 peoplec) 1000 peopled) 9000
1.15. Find the net annual premium for the pure endowment insurance, given in Problem 3.3 of Section 3.3, so that the insurance company makes a profit with 95% probability on a group ofa) 70 peopleb) 300 peoplec) 5000 peopled) 10,000 people
1.14. Determine the net single premium for the temporary insurance, given in Problem 3.10 of Section 3.3, so that the insurance company makes a profit with 95% probability on a group ofa) 60 peopleb) 150 peoplec) 2000 peopled) 8000 people
1.13. A whole life insurance on a life aged 40 pays a death benefit of $4000 at the end of the year of death. Based on a 6% annual rate of interest find the single premium so that the policy is profitable to the company with 95% probability.
1.12. A life annuity of $6000 per annum, payable monthly in advance with a deferment period of 10 years, is purchased for a life aged 50 by a single premium. Based on a 6% annual rate of interest,a) obtain the single premiumb) obtain the single premium under the condition it is returned without
1.11. A 15 year pure endowment insurance of $5000 is issued to a life aged 30. If the insured dies before the age of 45, the sum of the annual premiums received by the insurance company are returned without interest at the end of the year of death. Determine the annual premium based on a 6% annual
1.10. A term certain insurance of $5000 is taken out at the age of 50 for a term of 10 years. Based on a 6% annual rate of interest, find the annual premium if it is payable a ) over the full term of the insuranceb) for the first 5 years.
1.9. The monthly premium for a 20 year term insurance of $9000 is increased by 20% after the first 10 years. Find the monthly premium, if the insurance is issued to a life aged 50. Use a 6% annual rate of interest.
1.8. A continuous life annuity of $2000 per annum deferred 15 years is purchased for a life aged 50 by monthly installments payable for 5 years. Determine the monthly premium using a 6% annual interest rate.
1.7. A 15 year term insurance with a $3000 death benefit payable at the end of the year of death is issued to a life aged 50. The premium is payable continuously for a term of 5 years. Find the annual rate of the premium payment based on a 6% annual interest rate.
1.6. A 30 year endowment insurance of $5000, whose death benefit is payable at the end of the year of death, is issued to a life aged 20. Determine the monthly premium based on a 6% annual rate of interest.
1.5. A whole life insurance is issued to a life aged 40, with a death benefit of $3000, payable at the moment of death. Based on a 6% annual rate of interest, determine the annual premium if it is paida) yearly in advanceb) continuously for the whole term of the insurance.
1.4. Based on a 6% annual rate of interest, obtain 1a) P40:25J 1b) P30:101c) P50:151d) P60 (12) 1e) P 25:lOJf) P~~:5J g) lOP20 h) 20P25:30J i) P( A 50:20J)j) k) -1 p(4)( A 40:101) p (A20) 1) P ( A 50:201) m) lOP(lO I ii30) n) p(4)«(IA)40:101).
1.3. An insurance issued to a life aged 40 pays a death benefit of $5000 at the moment of death, if death occurs within 20 years and $7000 on survival to age 60. Determine the single premium based on a 6% annual interest rate.
1.2. The annual payment of a 15 year annuity immediate is $800 in the first 5 years and $1400 for the rest of the term. Find the single premium of the annuity at the age of 30 using a 6% annual rate of interest.
1.1. A whole life insurance is issued to a life aged 50. The death benefit payable at the end of the year of death is $5000, if death occurs in the first 10 years and $2000 afterwards. Obtain the single premium at the age of 50, based on a 6% annual rate of interest.
5.42. Find the present value of a 15 year complete annuity-due of $400 per quarter purchased for a life aged 30, based on a 6% annual rate of interest.
5.41. A 10 year complete annuity-due of $1500 per month is issued to a life aged 50. Determine the present value of the annuity using a 6% annual rate of interest.
5.40. A 15 year family income benefit is payable continuously at a rate of $8000 per annum. Find the present value of the insurance at the age of 45 based on a 6% annual interest rate.
5.39. A 25 year family income benefit, payable monthly, is purchased by a person aged 30 for $3000. Determine the amount of the monthly payments based on a 6% annual rate of interest.
5.38. The payments of a 20 year annuity-due of $800 per month are guaranteed in the first 10 years. Find the present value of the annuity at the age of 40, based on a 6% annual rate of interest.
5.37. A 25 year annuity-immediate of $1000 per annum with guaranteed payments in the first 10 years is issued to a life aged 45. Find the present value of the annuity based on a 6% annual interest rate.
5.36. The rate of payment for a 15 year continuous life annuity is $1200 per annum in the first year and increases by $200 each year. Determine the present value of the annuity at the age of 35, based on a 6% annual interest rate.
5.35. A varying continuous life annuity is issued to a life aged 40. The rate of payment is $3500 per annum in the first year and increases by $400 each year. Find the present value of the annuity at a 6% annual rate of interest.
5.34. Based on a 6% annual rate of interest, obtaina) S 65b) (Iahoc) (I a 55:10l
5.33. A continuous life annuity of $2000 per annum issued to a life aged 50 is deferred 10 years. Determine the present value of the annuity, based on a 6% annual interest rate.
5.32. A 30 year life annuity payable continuously at a rate of $4000 per annum is purchased for a life aged 20. Determine the present value of the annuity at a 6% annual rate of interest.
5.31. A continuous life annuity of $3600 per annum is issued to a life aged 40. Find the present value of the annuity based on a 6% annual rate of interest. Also find the standard deviation of the annuity.
5.30. Based on a 6% annual rate of interest, evaluatea) N40b) a 20c) V(a 20)d) a 30:2sle) V(a 30:2sl)f) 10 I Ii 40 g) 151 Ii 2s:30l
5.29. Prove (76) by general reasoning.
5.28. Prove the following identities:a) b)c) mlli l=1i -Ii l' x:n x:m+n l x:m m Iii l = mEx' Ii x+m - m+nEx' Ii x+m+n, x:n m I Ii l = mEx . Ii l'
5.27. A 20 year life annuity makes quarterly payments in arrears. In the first year, the quarterly payment is $600 and it increases by $50 each year. Find the present value of the annuity at the age of 40, using a 6% annual interest rate.
5.26. Based on a 6% annual rate of interest, obtaina) (Ia) ~4Jb) (2) (Ia) 40:10l
5.25. A life annuity of $12000 per annum payable quarterly in arrears between the ages of 50 and 65, is issued to a life aged 45. Obtain the present value of the annuity at a 6% annual rate of interest.
5.24. Find the present value of a life annuity paying $300 monthly in arrears purchased for a life aged 40. Use a 6% annual rate of interest.
5.23. Based on a 6% annual rate of interest, evaluatea) (2) a50b) (4) a 30:251c) 10 I a~\2)d) (12) 251 a40:101
5.22. Prove the following identities:a) (p) (p) (p) m I ax:n 1 == ax:m+n 1- ax:m 1b) (p) - (p) (p) m I ax:n 1- mEx . ax+m - m+nEx . ax+m+wc) (p) (p) m I ax:n 1 == mEx . ax+m:n rd) I (p) I"(P) E E m ax:n 1 == m ax:n 1- m x + m+n X·
5.21. A life annuity payable monthly in advance is purchased for a life aged 20. The annual payment is $3000 in the first year and increases by $600 each year. Find the present value of the annuity, based on a 6% annual rate of interest.
5.20. Based on a 6% annual rate of interest, determinea) b) (Iii) W (12) (Iii) 55:101' 191
5.19. A 10 year life annuity pays $800 monthly in advance. Determine the present value of the annuity at the age of 55 based on a 6% annual interest rate.
5.18. Find the present value of a life annuity of $4000 per annum payable quarterly in advance issued to a life aged 35. Use a 6% annual rate of interest.
5.17. Based on a 6% annual interest rate, obtaina) .. (12) a30b) .,(4) a 4O:20lc) 1.,(2) 10 a2sld) I .,(4) 10 a3S:1Sl
5.16. Prove the following identities: ) I .,(p) Jp) Jp) a m ax:n l = ax:m+n l- ax:m Vb) I .,(p) E .. (p) E .. (p) m ax:n l = m x' ax+m - m+n x' ax+m+w I .,(p) E .. (p)c) m ax:n l = m x' ax+m:n l
5.15. A 25 year life annuity purchased for a life aged 40 makes a payment of $2000 at the end of the first year and increases by $150 each year. Find the present value of the annuity, based on a 6% annual interest rate.
5.14. Based on a 6% annual rate of interest, obtaina) (Iaho,b) (Ia)60:1Sl
5.13. A temporary life annuity of $2000 per annum is payable yearly in arrears between the ages of 40 and 55. Find the present value of the annuity at the age of 35 at a 6% annual rate of interest.
5.12. A life annuity-immediate of $1500 per annum is issued to a life aged 45. Find the present value and the standard deviation of the annuity based on a 6% annual interest rate.
5.11. Based on a 6% annual rate of interest, finda) a3Sb) V(a3S)c) a 2S:lOld) V(a 2S:lOl)e) 20 I a40f) 10 I a45:20l
5.10. If we look at (17), we might think a similar identity holds for annuities-immediate. Show that this is not true; that is, 1 ax '* -:- (I-Ax)· l
5.9. Prove the following identities:a) b)c) d) la =a -a m x:n l x:m+n l x:m r m I a l = mEx . ax+m - m+nEx . ax+m+n, x:n la = mEx' a l' m x:n l x+m:n m I a x:n l = m I ax:n+ 1 - mEx·
5.8. A 20 year life annuity-due issued to a life aged 50 pays $1500 in the first year and increases by $100 each year. Find the present value of the annuity based on a 6% annual rate of interest.
5.7. Based on a 6% annual rate of interest, finda) 540b) (Iabsc) (Ia) 40:1Sl
5.6. A temporary life annuity of $3000 per annum payable yearly in advance, between the ages of 55 and 70, is purchased for a life aged 50. Find the present value of the annuity, based on a 6% annual rate of interest.
5.5. A life annuity-due of $1500 per annum is payable after the age of 60. Find the present value of the annuity at the age of 40, based on a 6% annual rate of interest.
5.4. A temporary life annuity of $2500 per annum is payable yearly in advance between the ages of 35 and 50. Find the present value of the annuity at the age 35, based on a 6% annual rate of interest. Also find the standard deviation of the annuity.
5.3. A life annuity-due of $2000 per annum is issued to a life aged 30. Find the present value of the annuity, based on a 6% annual rate of interest. Also find the standard deviation of the annuity.
5.2. Based on a 6% annual interest rate, evaluatea) N35b) ii25c) V(ii25)d) ii 45:151e) V (ii 45:151)f) 25 I ii30 g) 51 ii25:101 CHAPTER 3
5.1. Prove the following identities:a) ml ii l=ii -ii x:n x:m+n 1 x:m 1b) I·· E" m a x:n 1 = m x' a x+m:n 1
4.4. The death benefit of a 15 year endowment insurance of $4500 is payable at the moment of death. Based on a 6% annual rate of interest, find the present value and the standard deviation of the insurance at the age of 45.
4.3. Based on a 6% annual rate of interest, obtaina) A 3S:2Slb) V(A 3S:2Sl)c) 10 I A2S:30l
4.2. The death benefit of a 20 year endowment insurance of $3000 is payable at the end of the year of death. Based on a 6% annual rate of interest, obtain the present value and the standard deviation of the insurance at the age of 30.
4.1. Based on a 6% annual rate of interest, obtaina) A30:10lb) V(A30:10l)c) lsi A40:20l
3.14. The death benefit of an insurance issued to a life aged 40 is payable at the moment of death. The death benefit is $3500 in the first year and increases by $100 each year. Find the present value of the insurance at a 6% annual interest rate.
3.13. Based on a 6% annual rate of interest, finda) R 60b) (IA )40 - 1c) (I A )30:151
3.12. A death benefit of an insurance is $3000 payable at the moment of death if death occurs between the ages of 55 and 65. Find the present value of the insurance at the age of 40, based on a 6% annual rate of interest.
3.11. An insurance on a life aged 50 pays $7000 at the moment of death if death occurs after the age of 55. Based on a 6% annual rate of interest, find the present value and the standard deviation of the insurance.
3.10. Find the present value of a 10 year temporary insurance issued to a life aged 40 with a death benefit of $2000 payable at the moment of death. Also, obtain the standard deviation of the insurance. Use a 6% annual interest rate.
3.9. The death benefit of a whole life insurance on a life aged 40 is payable at the moment of death. Determine the sum insured if the purchase price is $560 and a 6% annual interest rate is used.
3.8. Based on a 6% annual rate of interest, obtain the approximate values of the following expressions.a) C 50b) M 35c) A 45d) V(A 45) -1e) A 30:201 -1f) V(A 30:201) g) 351 A 20 h) V(35 1 A 20) -1 i) 10 1 A 50:151

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An Introduction To Actuarial Mathematics 1st Edition Arjun K. Gupta ,Tamas Varga - Solutions

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