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ct010s07 - UNSW ACTL1001 Actuarial Studies and Commerce...

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UNSW ACTL1001 Actuarial Studies and Commerce Sample Solutions 7 Exercise 1 1. For Investment A we have E [ A ] = 21 100 × 6250 + 74 100 × 12500 + 5 100 × 18750 = 11500 Variance = 21 100 × (6250 - 11500) 2 + 74 100 × (12500 - 11500) 2 + 5 100 × (18750 - 11500) 2 = 9 , 156 , 250 standard deviation = p 9 , 156 , 250 = 3025 . 9 For Investment B we have E [ B ] = 21 100 × 18750 + 74 100 × 12500 + 5 100 × 6250 = 13500 Variance = 21 100 × (18750 - 13500) 2 + 74 100 × (12500 - 13500) 2 + 5 100 × (6250 - 13500) 2 = 9 , 156 , 250 standard deviation = p 9 , 156 , 250 = 3025 . 9 1
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2. Expected utilities are E [ U ( A )] = 21 100 × 6250 - 0 . 000005 × 6250 2 + 74 100 × 12500 - 0 . 000005 × 12500 2 + 5 100 × 18750 - 0 . 000005 × 18750 2 = 10793 E [ U ( B )] = 21 100 × 18750 - 0 . 000005 × 18750 2 + 74 100 × 12500 - 0 . 000005 × 12500 2 + 5 100 × 6250 - 0 . 000005 × 6250 2 = 12543 3. Investment B is preferred. If we look at the expected returns and stan- dard deviations we see that both investments have the same standard deviation, hence both have the same variability, but Investment B has a higher expected return. Also Investment B has a higher expected utility. In passing note that Investment A pays off more in the bad state of the world. If this was a state when we had low total wealth from other investments it might be possible that we would prefer Investment A.
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