Calculate the Probability of Ruin for the Life Insurance Fund Within 3 Years

Calculate the Probability of Ruin for the Life Insurance Fund Within 3 Years

(7 points) Bauer Blasting Company (BBC) self insures its employee life insurance. At the end of 2006, the life insurance fund has a surplus of 5,000. At the beginning of each calendar year, BBC contributes 20,000 into the life insurance fund. The fund earns interest at 4%.

If a BBC employee dies on the job, BBC provides a death benefit of $10,000 per death at the end of the calendar year. The distribution of deaths each year is:

Deaths / Probability
1 / 0.2
2 / 0.5
3 / 0.3

Calculate the probability of ruin for the life insurance fund within 3 years.

(7 points) You are given the following sample of claims:

X: 8, 9, 12, 12, 13, 13, 13, 15, 17, 21

The sum of X is 133and the sum of X2 is 1895.

H0 is that μx= 10.5 and H1 is that μx ≠ 10.5.

Calculate the z statistic, the critical value(s) assuming a significance level of 1%, and the p value. State your conclusion with regard to the Hypothesis Testing.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Failures / Drops to Failure / Failures
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

(3 points) Calculate the unbiased estimator of the mean of the true Drops to Failure distribution.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Failures / Drops to Failure / Failures
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

(4 points) Calculate Ĥ(4) using the Nelson-Åalen estimator.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Failures / Drops to Failure / Failures
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

(5 points) Calculate 1000[Ŝ(4) - S100(4)], where Ŝ(x) is the survival function from the Nelson Åalen estimate and S100(x) is the empirical survival function.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Failures / Drops to Failure / Failures
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

(5 points) Calculate the Var[Ĥ(4)] where Ĥ(4) is calculated using the Nelson-Åalen estimator.

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Failures / Drops to Failure / Failures
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

(5 points) Using the linear confidence interval, calculate the 92% confidence interval for H(4) using the Nelson-Åalen estimator.

Ledbetter Life Insurance Company is completing a mortality study on a 3 year term insurance policy. The following data is available:

Life / Date of Entry / Date of Exit / Reason for Exit
1 / 0 / 0.2 / Lapse
2 / 0 / 0.3 / Death
3 / 0 / 0.4 / Lapse
4 / 0 / 0.5 / Death
5 / 0 / 0.5 / Death
6 / 0 / 0.5 / Lapse
7 / 0 / 1.0 / Death
8 / 0 / 3.0 / Expiry of Policy
9 / 0 / 3.0 / Expiry of Policy
10 / 0 / 3.0 / Expiry of Policy
11 / 0 / 3.0 / Expiry of Policy
12 / 0 / 3.0 / Expiry of Policy
13 / 0 / 3.0 / Expiry of Policy
14 / 0 / 3.0 / Expiry of Policy
15 / 0 / 3.0 / Expiry of Policy
16 / 0.5 / 2.0 / Lapse
17 / 0.5 / 1.0 / Death
18 / 1.0 / 3.0 / Expiry of Policy
19 / 1.0 / 3.0 / Expiry of Policy
20 / 2.0 / 2.5 / Death

(6 points) Calculate S20(1.0) using the Kaplan-Meier product-limit estimator where death is the decrement of interest.

9.Ledbetter Life Insurance Company is completing a mortality study on a 3 year term insurance policy. The following data is available:

Life / Date of Entry / Date of Exit / Reason for Exit
1 / 0 / 0.2 / Lapse
2 / 0 / 0.3 / Death
3 / 0 / 0.4 / Lapse
4 / 0 / 0.5 / Death
5 / 0 / 0.5 / Death
6 / 0 / 0.5 / Lapse
7 / 0 / 1.0 / Death
8 / 0 / 3.0 / Expiry of Policy
9 / 0 / 3.0 / Expiry of Policy
10 / 0 / 3.0 / Expiry of Policy
11 / 0 / 3.0 / Expiry of Policy
12 / 0 / 3.0 / Expiry of Policy
13 / 0 / 3.0 / Expiry of Policy
14 / 0 / 3.0 / Expiry of Policy
15 / 0 / 3.0 / Expiry of Policy
16 / 0.5 / 2.0 / Lapse
17 / 0.5 / 1.0 / Death
18 / 1.0 / 3.0 / Expiry of Policy
19 / 1.0 / 3.0 / Expiry of Policy
20 / 2.0 / 2.5 / Death

(6 points) Calculate the Var[S20(1.0)] using the Greenwood approximation where death is the decrement of interest.

10.Schneider Trucking Company had the following losses during 2006:

Amount
of Claim / Number
of Payments / Total Amount
of Losses
0 – 10 / 8 / 60
10-20 / 5 / 70
20-30 / 4 / 110
30 + / 3 / 200
Total / 20 / 440

(7 points) Calculate F20(x) and f20(x)for 20 ≤ x ≤ 30 from the ogive and histogram.

11.Schneider Trucking Company had the following losses during 2006:

Amount
of Claim / Number
of Payments / Total Amount
of Losses
0 – 10 / 8 / 60
10-20 / 5 / 70
20-30 / 4 / 110
30 + / 3 / 200
Total / 20 / 440

(5 points) Calculate E(XΛ30) minus E[(X-20)+].

Wang Warranty Corporation is testing iPods. Wang starts with 100 iPods and tests them by dropping them on the ground. Wang records the number of drops before each iPod will no longer play. The following data is collected from this test:

Drops to Failure / Number / Drops to Failure / Number
1 / 6 / 9 / 7
2 / 2 / 10 / 6
4 / 3 / 11 / 7
6 / 4 / 12 / 7
7 / 4 / 13 / 48
8 / 5 / 22 / 1

Ledbetter Life Insurance Company is completing a mortality study on a 3 year term insurance policy. The following data is available:

Life / Date of Entry / Date of Exit / Reason for Exit
1 / 0 / 0.2 / Lapse
2 / 0 / 0.3 / Death
3 / 0 / 0.4 / Lapse
4 / 0 / 0.5 / Death
5 / 0 / 0.5 / Death
6 / 0 / 0.5 / Lapse
7 / 0 / 1.0 / Death
8 / 0 / 3.0 / Expiry of Policy
9 / 0 / 3.0 / Expiry of Policy
10 / 0 / 3.0 / Expiry of Policy
11 / 0 / 3.0 / Expiry of Policy
12 / 0 / 3.0 / Expiry of Policy
13 / 0 / 3.0 / Expiry of Policy
14 / 0 / 3.0 / Expiry of Policy
15 / 0 / 3.0 / Expiry of Policy
16 / 0.5 / 2.0 / Lapse
17 / 0.5 / 1.0 / Death
18 / 1.0 / 3.0 / Expiry of Policy
19 / 1.0 / 3.0 / Expiry of Policy
20 / 2.0 / 2.5 / Death

Schneider Trucking Company had the following losses during 2006:

Amount
of Claim / Number
of Payments / Total Amount
of Losses
0 – 10 / 8 / 60
10-20 / 5 / 70
20-30 / 4 / 110
30 + / 3 / 200
Total / 20 / 440