# W13Tute 1001 - 03 with probability 0 . 4 with probability 0...

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Examples 1. Jade is 18 years old and has current salary of \$20000. The continuous compounding salary growth rates for each year are independent and identically distributed with distribution δ i = ( 0 . 05 with probability 0 . 4 0 . 02 with probability 0 . 6 . 1.1 Calculate E [ S 19 ] 1.2 Explain why the growth rate can be expressed as δ i = 0 . 02 + 0 . 03 X , where X Binomial(1 , 0 . 4) 1.3 Hence, or otherwise, show that E [ S 18+ t ] = 20000 e 0 . 02 t (0 . 4 e 0 . 03 + 0 . 6) t You may use the fact that if X and Y are independent random variables, then E [ f ( X ) g ( Y )] = E [ f ( X )] E [ g ( Y )] for arbitrary functions f and g . 1.4 Contributions of 1% of Jade’s salary are made into the fund each year. The eﬀective rate of interest is 10% p.a. and t ( ap ) 18 = e - 0 . 02 t . Calculate the expected present value of the next 10 contributions, if the ﬁrst contribution is made in one year’s time. UNSW Week 13 ACTL1001 Tutorial

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Examples 1.1, 1.2 E [ S 19 ] = 20000 e 0 . 05 × 0 . 4 + 20000 e 0 . 02 × 0 . 6 = 20652 . 58485 Since X is Binomial(1, 0.4) it can be expressed as X = ( 1 with probability 0 . 4 0 with probability 0 . 6 Hence 0 . 03 X = ( 0 .

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Unformatted text preview: 03 with probability 0 . 4 with probability 0 . 6 and . 02 + 0 . 03 X = ( . 05 with probability 0 . 4 . 02 with probability 0 . 6 UNSW Week 13 ACTL1001 Tutorial Examples 1.3 E [ S 18+ t ] = E [ S 18 e 1 e 2 . . . e t ] = 20000 E [ e 1 ] E [ e 2 ] . . . E [ e t ] due to independence = 20000( E [ e 1 ]) t since the deltas are identically distributed = 20000(0 . 4 e . 05 + 0 . 6 e . 02 ) t = 20000 e . 02 t (0 . 4 e . 03 + 0 . 6) t UNSW Week 13 ACTL1001 Tutorial Examples 1.4 We want . 01 10 X i =1 E [ S 18+ t ] t ( ap ) 18 1 + i t = 0 . 01 10 X i =1 20000 e . 02 t (0 . 4 e . 03 + 0 . 6) t e-. 02 t (1 + i ) t = 200 10 X i =1 . 4 e . 03 + 0 . 6 1 . 1 t = 200 10-1 -1 = 1302 . 0336 where = . 4 e . 03 + 0 . 6 1 . 1 UNSW Week 13 ACTL1001 Tutorial...
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## This note was uploaded on 06/12/2011 for the course ASB 1001,2522, taught by Professor Nicole during the One '09 term at University of New South Wales.

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W13Tute 1001 - 03 with probability 0 . 4 with probability 0...

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