99package - ACT348F - TERM TEST 1 - OCTOBER 20, 1999 Print...

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ACT348F - TERM TEST 1 - OCTOBER 20, 1999 Print your name and student number clearly on each page Do all work on this question paper, with work for question 1 on this page (use both sides if necessary), work for question 2 on the next page, etc. All calculations should be done to . at least 6 digit accuracy Answers are expected to have at least 4 digit accuracy. 1. A two-year fully discrete insurance with face amount $1000 is issued at age , with term B level annual premium of payable at the start of each year. The mortality probabilities are U ; &±#² ; &±$ B B³" , and the annual effective interest rate is 10%. (a) Write out the point-by-point expression for the issue-date loss random variable (3-point P random variable). (b) Suppose that . Find . U&#´ ! IµP¶ (c) The annual benefit premium (equivalence principle) is Find if this is the ##!±" ! Z +<µP¶ premium. Solution: (a) if P& "!!!@· U O&! ²; &±# "!!! @ ·U—"³@ ¸ O&" ² ; &—±)¸—±$¸&±#% ·U—"³@ ¸ O¹# ²: &—±)¸—±(¸&±´’ & B # "l B # B & *!*±!*· U O&!² ±# )#’±%%·"±*!*" U O&"² ±#% ·"±*!*" U O¹#² ±´’ & (b) IµP¶&—*!*±!*·#´!¸—±#¸³—)#’±%%·—"±*!*"¸—#´!¸¸—±#%¸·"±*!*"—#´!¸—±´’¸ &· ´"±’’ (c) Z+<µP¶&IµP ¶·—IµP¶ ¸ &IµP¶ # ## (since from the equivalence principle) IµP¶&! IµP ¶&—*!*±!*·##!±" ¸ —±#¸³—)#’±%%·—"±*!*"¸—##!±"¸ ¸ —±#%¸ # ³—"±*!*"—#´!¸ ¸ —±´’¸&#’#²""$ # . 2. . T &+² T &,² T &- Bº8l " Bº8l B³8 Find an expression for in terms of and the discount rate . + +²,² -. ±± B Solution: . +& + ³I »+ ± ± ± ± B 8 B B³8 Bº8l + & & ² + & ± ± Bº8l " " "" T , T .³- B³8 Bº8l B³8 T · T & T & pI &—,·+¸» Bº8l Bº8l Bº8l I + .³, 8B " Bº8l Then, + ³I» ³—,·+¸» ¸ —¸ ± ± ± ± B 8 B B³8 Bº8l " , , .³-
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.²+³,³- —.³,´—.³-´
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3. Find an expression in terms of life annuity and life insurance values (and constants) for the annual benefit premium (equivalence premium) for each of the following fully discrete policies T issued at age . Also, find expressions in terms of standard , etc. symbols) B T&T B B±8l " (a) Whole life insurance - death benefit is 1000 up to age , and death benefit is 500 after B²8 age . Level annual premiums of are payablefor years. 8 T8 (b) Whole life insurance - death benefit is 1000 for life. Level annual premiums of are T payablefor the first years, followed by after years. 8 " # Solution: ³!!— E ²E ´µT + pT µ µ³!!— T ² T´ ¶¶ B±8 l B±8l "" B B 8B ³!!— E ²E´ + B±8l "B B±8l (b) "!!! Eµ T— + ²+´µT + ²T+ B BB ## B±8 l B±8l 8l pT µ µµ #!!! E E #!! !T + ² + # + ²+ #²T — +´ B B B±8 l B±8l 8 lB 8l . 4. An insurer issues 1000 independent policies each of face amount 1. The policies are identical (except that they are issued to 1000 different individuals). Suppose that each policy is a fully discrete whole life insurance with face amount 1 issued at age , with annual benefit premiums based on the equivalence principle. You are given that B 3µ¶" &E µ¶ $ E µ¶"³ # and .
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This note was uploaded on 09/27/2009 for the course ACT 348 taught by Professor Broverman during the Fall '04 term at University of Toronto- Toronto.

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99package - ACT348F - TERM TEST 1 - OCTOBER 20, 1999 Print...

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