99package - ACT348F TERM TEST 1 Print your name and student...

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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°#&! I´Pµ (c) The annual benefit premium (equivalence principle) is . Find if this is the ##!±"! Z +<´Pµ premium. Solution: (a) if if if P ° "[email protected] ¶ U O ° ! ² ; °±# "[email protected] ¶U—"³@· O ° " ² ; °—±)·—±$·°±#% ¶U—"³@· O ¸ # ² : °—±)·—±(·°±&’ ° B # "l B # B if if if ° *!*±!* ¶ U O ° ! ² ±# )#’±%%¶"±*!*"U O ° " ² ±#% ¶"±*!*"U O ¸ # ² ±&’ ° (b) I´Pµ°—*!*±!*¶#&!·—±#·³—)#’±%%¶—"±*!*"·—#&!··—±#%·¶"±*!*"—#&!·—±&’· ° ¶ &"±’’ (c) Z +<´Pµ°I´P µ¶—I´Pµ· °I´P µ # # # (since from the equivalence principle) I´Pµ°! I´P µ°—*!*±!*¶##!±"· —±#·³—)#’±%%¶—"±*!*"·—##!±"·· —±#%· # # # ³—"±*!*"—#&!·· —±&’·°#’#²""$ # . 2. . T ° + ² T ° , ² T ° - B¹8l " B¹8l B³8 Find an expression for in terms of and the discount rate . + + ² , ² - . ±± B Solution: . + ° + ³ I º + ±± ±± ±± B 8 B B³8 B¹8l + ° ° ² + ° ° ² ±± ±± B¹8l " " " " .³T .³, .³T .³- B³8 B¹8l B³8 T ¶ T ° T ° p I ° — , ¶ + · º B¹8l B¹8l " " B¹8l I + .³, ±± 8 B " 8 B B¹8l Then, + ° + ³ I º + ° ³ — , ¶ + · º — ·— · ±± ±± ±± B 8 B B³8 B¹8l " " " .³, .³, .³-
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° ± .¶+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 ² T B B¹8l " (a) Whole life insurance - death benefit is 1000 up to age , and death benefit is 500 after B ³ 8 age . Level annual premiums of are payablefor years. B ³ 8 T 8 (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 T 8 " # Solution: (a) &!!—E ³ E · ° T + p T ° °&!!—T ³ T · ±± B¹8l B¹8l " " B B 8 B &!!—E ³E · + ±± B¹8l " B B¹8l (b) "!!!E ° T—+ ³ + · ° T + ³ T + ±± ±± ±± ±± B B B " " # # B¹8l B¹8l 8l p T ° ° ° #!!!E #!!!E #!!! T + ³+ #+ ³ + ±± ±± ±± ±± #³T— + · ±± B B 8 B B¹8l B¹8l B B 8l B 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).
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