5. I buy an insurance policy for my company. Frequency follows a Discrete Unifor

Question

5. I buy an insurance policy for my company. Frequency follows a Discrete Uniform distribution (0,1,2,3. Each claim has a total loss (before applying deductibles and limits) of $200,000. Before applying and per-claim deductibles and limits, what is the Expected Value, Variance, and Co-efficient of variation of the Claim losses? (A per-claim deductible means a deductible applied to each claim. An aggregate limits means a benefit limit applied to the total claims from a policy). My policy has a Per-Claim deductible of $5,000, and an aggregate limit of $450,000. What is the Expected Value, Variance, and Co-efficient of variation of the Claim losses? Are the Variance and Co-efficient of variation higher or lower than in (a.)? Why is this? The next year, my frequency distribution remains the same, but due to inflation my severity distribution increases 5%. The deductibles/ limits stay at the same R amount. What is the Expected Value, Variance, and Co-efficient of variation of the Claim losses? a. b. c.

Explanation / Answer

Since frequency flow follows discrete uniform distribution, thus, probability of occuring of each event is equal

i.e. P(X=0) = P(X=1) = P(X=2) = P(X=3) = 0.25

Mean = .5(n+1) =0.5*5 =0.25

Variance = ((n-1)(n+1))/12 =(3*5)/12 =1.25

Standard Deviation = 1.118

1) Each claim has a total loss of $200,000

Total Loss = Loss for each claim * Number of claims

Loss = 0 .........when claim=0

200,000 .........when claim=1

400,000 .........when claim=2

600,000 .........when claim=3

Total Loss = Loss for each claim * Number of claims

= 0*P(claim=0) +200,000*P(claim=1) +400,000*P(claim=2) +600,000*P(claims=3)

Thus, E( Total Loss) = 200,000*(0+1+2+3)*0.25

= 200,000*6*0.25

= 600,000

And, Variation( Total Loss) = 200,000**2 *(0+1**2+2**2+3**2)*Var(number of claim)

= 4*10**14*1.25

= 70 * 10**6

Standard Deviation = 8366.6

Coefficient of variation = standard Deviation / Mean =71.7

2) Each claim has a total limit of $450,000 and per claim deductible is $5,000, thus, loss for each claim is $445,000

Thus, Loss for claim count will be as follows,

Loss = 0 .........when claim=0

445,000 .........when claim=1

440,000 .........when claim=2

435,000 .........when claim=3

Total Loss = Loss for each claim * Number of claims

= 0*P(claim=0) +445,000*P(claim=1) +440,000*P(claim=2) +435,000*P(claims=3)

Thus, E( Total Loss) = 0.25(0+445000+440000+435000)=330,000

And, Variation( Total Loss) = 1.25(0+445000**2+440000**2+435000**2)

= 1.25*580850*10**6

=726062 *10**6

Standard Deviation = 852000

Coefficient of variation = standard Deviation / Mean =852000/330,000= 2.58

Here Variance is high because of increase in loss amount . However, coefficient of variation is high but it cant be compared with the previous coefficient of variation.  

3) The increase in inflation will lead to increase in the claim loss. However, the insurance company will pay a fixed ampunt only. Thus, there will be no impact

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