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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 eightsided 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 claimfree 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.
 One '09
 Nicole

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