2010_PASS_wk10 - PASS for Actuarial Studies and Commerce...

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Unformatted text preview: PASS for Actuarial Studies and Commerce (ACTL1001) 2010 Session 2 – Week 10 Risk Management 1. Suppose we have two iid random variables these two random variables as ̅ ( ̅ ). (a) Find ( ̅ ) and (b) By comparing the above results with 2. Suppose that and are independent. ( )? (a) What is ()() (b) Show that ( ) ( ) () (c) Show that and and that have mean and variance . Denote the average of , explain why risk pooling is helpful in insurance. () 3. * Suppose that we have insurance policies. Each policy has a probability of claim of 0.08. If a claim occurs, the size of that claim is exponential with mean 5. Claims between policies are independent. Find the mean and variance of the average amount of claims for and . 4. Suppose that two random variables and are both normally distributed with correlation between and is . (a) Find given that ( ) . (b) What does the negative sign indicate? and . The 5. An unbiased eight-sided dice is tossed, and represents the numerical value of the outcome. If the outcome is odd, is set equal to 0; otherwise it is set equal to 1. Calculate the correlation coefficient of and . 6. For random variables and : () (a) Show that ( ) (b) Show that ( ) (c) Show that ( ), where is a constant. ( ), where and are constants. () ( ) ( ). 7. * There are two stocks, and , which offer random returns of and . The expected returns are ( ) and ( ) , the standard deviations are and , and the correlation between the returns is . (a) Yuna wants to invest her money in these two stocks. She chooses to invest a proportion of in stock and ( ) of the return on her portfolio. in stock . Find the expected return ( ) and variance ( )) () ( ). Find the optimal value (b) Suppose that she has a utility function given by ( ( ) of . You may find the results in Q6 useful. 8. An NCD system involves three levels of discount: 0%, 10% and 30%. In the event of a claim-free year, the policyholder moves to the immediately higher discount level. A claim causes the policyholder to move back one level. The probability of having no claims in a year is 0.8. (a) Write down the transition probability matrix. (b) Given that he begins in the 0% discount level in the first year, find the probability that he is still in the 0% level in two years. (c) Determine the long run proportion of policyholders in each level. Solutions to Exercises – (2009 Session 2 – Week 10) Question 1 ( ̅) ( ) ( ) ( ̅) ( ( ) ) [( )] [( )] ,( ) ( ( ( ) )( ( ( ) ) ( ( )- )) ) Pooling helps to reduce risk since the average variance is lower than the individual variance. Question 2 (a) ( ) ( ) (b) ( ( ( ( ( ( ))( ( ))] () () ( ) ( )) ( )) ( )) ( ( ( ( ) ( )) ()() ()() ()() ()() [( ) ) ) ) ()() (c) ( ,( )- , ( ( ) () () * ( ) , ( )- + () () ) ( ) () () ),() ( )( ) , ( )( ) ( ) , ( )*() ( ) ( )+ * ( ) , ( )- + ( ) Question 3 ̅ ∑ ∑ ( Where () () ) () () () () ( ( ̅) ( () ()) ( ( )) ) () ( ̅) () { ( ( ) ) Question 4 ( ) ( ) ()() The negative sign indicates that and Question 5 ( ) ( ) () ( ( ) ) ()() () ( () ( ) ) ( ) ( () () ) √ √ tend to move in opposite directions. ( ) Question 6 (a) ( ) ( [( ))] ( ))] [( ( ( )) ] [( [ ( )) ] () (b) ( ) [( ( [( ( ))( ( ))( ) [( ( ))( ( ))] ( ))] ( ))] (c) See answers for Q2(c) Question 7 ( () () ) ( () ( ) ( ) ) ( ( () () ( ) ( ) ( ) ) ( ) ( ) ) () , - This is a quadratic with negative leading coefficient, and is maximised at ( ) Here the interpretation is that we borrow an amount of stock B worth 624.5% of our initial wealth and sell it, then spend 724.5% of our initial wealth on stock A. So if we had $100 to invest, we would borrow $624.5 of stock B from someone and sell it, then use that money to purchase $724.5 of stock A. We still owe that person $624.5 of stock B (and this amount will change as the value of stock B changes). Question 8 (a) ( ) (b) ( ) ( ) ( ) ( Alternatively, we can begin with the state vector ( )( ( ( ) )( ) (c) Long run proportions are given by: ∑ ( ) ), since we begin in the 0% state. ) ( ) ...
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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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