Course 3, #36 (Fall 2001)

[MathGuy]

[p]36. WidgetsRUs owns two factories. It buys insurance to protect itself against major repair costs. Profit equals revenues, less the sum of insurance premiums, retained major repair costs, and all other expenses. WidgetsRUs will pay a dividend equal to the profit, if it is positive.

You are given:

(i) Combined revenue for the two factories is 3.

(ii) Major repair costs at the factories are independent.

(iii) The distribution of major repair costs for each factory is:

<table align=center>
<tr><td>k <td>Prob(k)
<tr><td>0 <td>0.4
<tr><td>1 <td>0.3
<tr><td>2 <td>0.2
<tr><td>3 <td>0.1
</table>

(iv) At each factory, the insurance policy pays the major repair costs in excess of that
factory’s ordinary deductible of 1. The insurance premium is 110% of the expected
claims.

(v) All other expenses are 15% of revenues.

Calculate the expected dividend.

(A) 0.43
(B) 0.47
(C) 0.51
(D) 0.55
(E) 0.59

<font size=-1>[ This Message was edited by: MathGuy on 2001-11-08 11:33 ]</font>

[MathGuy]

Here are my calculations:

Expected Claim Size (after deductible) = .4(0)+.3(0)+.2(1)+.1(2)=.4
Insurance Premium = 1.1(.4) = .44
Retained Claims Costs = .4(0)+.3(1)+.2(1)+.1(1)=.6

Premium for 2 factories = 2*.44 = .88
Retained Costs for 2 factories = 2*.6 = 1.2
Expenses = .15*3 = .45
Revenue = 3

Dividend = Profit = Revenue - Premium - Retained Costs - Expenses
Dividend = 3 - .88 - 1.2 - .45 = .47 (B)

The answer key says (D) 0.55. I note that .55 - .47 = .08. It appears that the SOA solution does not factor in the fact that "insurance premium is 110% of expected claims".

[MathGuy]

Please feel free to correct me.

[kelly]

i got B also

[Anonymous]

I solved it the same way as well!

[Anonymous]

The answer was actually E.

The probability that the dividend is 1.67 is .4x.4=.16

The probability that it is .67 is 2*.4*.6=.48

and the probability that it is 0 is .6*.6=.36

The expected divided is therefore 1.67*.16 + .67*.48 + 0*.36 = .59 E

[Caramel]

Hello MathGuy,

The following is my solution to #36:

Premium charged by insurance company is
1.1 * E[(x-1)+]
= 1.1 * { E[x] - [ 1 - Fx(0) ] }
= 1.1 * ( 1 - 0.6 )
= 0.44

Retained major loss is
E[X ^ 1] = 1 * 0.3 + 1 * [ 1 - Fx(1) ] = 0.6

Other expenses = 0.15* Revenue = 0.15 * 3
= 0.45

E[dividend] = E[profit]
= 3 - 2*0.44 - 2*0.6 - 0.45 = 0.47 which is
(B)

My solution may not be right but go through it and tell me if you agree?

Thanks

[Caramel]

Hello Course3,

Could you please show more details on how did you find dividend of 1.67, 0.67 ?

Thanks.

[MathGuy]

Here is what Course 3 is getting at. Revenue, premium and expense are all constant. Revenue - Premium - Expense = 3 - .88 - .45 = 1.67. If the retained losses are equal to 0, then the dividend is 1.67. If the retained losses are 1,then the dividend is 0.67. Otherwise, the dividend is 0.

P(Retained loss = 0) = Prob(Both factroies have no losses) = .4*.4 = .16.

P(Retained Loss = 1) = Prob(Exactly one factory has a loss) = 2*.4*.6 = .48

Thus, the expected dividend is 1.67*.16 + .67*.48 = .5888. Most intriguing.

[Anonymous]

Sure-

Revenue=3

Premium= Expected losses excess of ded. x 2 x 110% = .88

Expenses= .15 x 3 = .45

The probability that the combined factories pay 0 in losses is the prob that they both have 0 losses = .4 x .4 = .16
so there is a .16 probability that the dividends are = 3-.88-.45-0 = 1.67

The prob that the factories pay 1 is the probability that one of them has no losses and the other has atleast $1 in losses which is = 2 x .4 x .6 = .48 so there is a .48 prob that the div = 3-.88-.45-1 = .67

The prob that the factories pay 2 is the prob that they both have atleast $1 in losses which is .6 x .6 = .36 so there is a .36 prob that the div = 3-.88-.45-2 = -.33
The div cannot be negative so there is a .36 chance that the div = 0

[MathGuy]

So where did we go wrong?

In calculating the expected retained loss as .6, and then doubling, we included the possibility that we would have a retained loss of 2, which would make the profit -.33. In effect, we included the -.33 in our calculation:

Dividend = .16(1.67) + 2(.4)(.6)(.67) + .36(-.33) = .47.

Well, that was fun. :smile: