Optimizing Auto Insurance Deductible: A Simulation Study

Suppose you own an expensive car and purchase auto insurance. This insurance has a $1000 deductible, so that if you have an accident and the damage is less than $1000, you pay for it out of your pocket. However, if the damage is greater than $1000, you pay the first $1000 and the insurance pays the rest. In the current year there is probability 0.025 that you will have an accident. If you have an accident, the damage amount is normally distributed with mean $3000 and standard deviation $750.

 

1.Use Excel to simulate the amount you have to pay for damages to your car. This should be a one-line simulation, so run 5000 iterations by copying it down. Then find the average amount you pay, the standard deviation of the amounts you pay, and a 95% confidence interval for the average amount you pay. (Note that many of the amounts you pay will be 0 because you have no accidents).

 

2.Continue, the simulation by creating a two-way data table, where the row input is the deductible amount, varied from $500 to $2000 in multiples of $500. Now find the average amount you pay, the standard deviation of the amounts you pay, and a 95% confidence internal for the average amount you pay for each deductible amount.

 

3.Do you think it is reasonable to assume that damage amounts are normally distributed? What would you criticize about this assumption? What might you suggest instead?

 

4.Suppose instead that the damage amount is triangularly distributed with parameters 500,1500, and 7000. That is, the damage in an accident can be as low as $500, and there is definite skewness to the right. (It turns out, as you an verify in @RISK, that the mean of this distribution is $3000). Use @RISK to simulate the amount you pay for damage. Run 5000 iterations, then answer the following questions. In each case, explain how the indicated event would occur.

 

a.What is the probability that you pay a positive amount but less than $750?
b.What is the probability that you pay more than $600?
c.What is the probability that you pay exactly $1000(the deductible)?

According to the given information, The Pulsometer Pump Company has three alternative options and can anticipate three different market situations. For each of these combinations, they have already identified expected profit. Hence, the payoff matrix criteria has been applied to decide which strategy needs to be applied by them.

The payoff calculations have been done as mentioned below:

Maximax Criteria
	Market Factors
Strategy	15%	Stable	-10%	Max
S1	240	130	0	240
S2	210	150	70	210
S3	170	150	70	170
Maximax	240
Decision	S1 strategy needs to be applied

 

Maximin Criteria
	Market Factors
Strategy	15%	Stable	-10%	Min
S1	240	130	0	0
S2	210	150	70	70
S3	170	150	70	70
Maximin	70
Decision	Either S1 or S2 needs to be applied

 

Minimax Criteria
	Market Factors
Strategy	15%	Stable	-10%	Max
S1	0	20	70	70
S2	30	0	0	30
S3	70	0	0	70
Minimax	30
Decision	S2 needs to be applied

								15%
								0.6	£ 2,40,000.00
				S1 Strategy				Stable
				EMV	£ 1,83,000.00			0.3	£ 1,30,000.00
								-10%
								0.1	0
								15%
								0.6	£ 2,10,000.00
Pumping Equipment				S2 Strategy				Stable
Production				EMV	£ 1,78,000.00			0.3	£ 1,50,000.00
								-10%
								0.1	£    70,000.00
								15%
								0.6	£ 1,70,000.00
				S3 Strategy				Stable
				EMV	£ 1,54,000.00			0.3	£ 1,50,000.00
								-10%
								0.1	£    70,000.00

As per this decision tree and EMV calculations, it can be said that strategy 1 will be the feasible option to attain maximum profit. This method has given confidence to the management, however, they wants further investigation and thus a third party consultation service option has been assessed though below calculations. 

EMV as per consultation Report
Strategy	EMV	Consultancy Fees	Net Value
S1	£ 1,08,600.00	£               5,000.00	£ 1,03,600.00
S2	£    53,600.00	£               5,000.00	£    48,600.00
S3	£    24,100.00	£               5,000.00	£    19,100.00

The above table has shown that even after considering third party consultation service, the option will remain the same, that is, they should go with strategy 1. Hence, it is not necessary to go for consultation service.

Finally, they wants to apply bays theorem option to ensure the decision making process. Accordingly posterior probability of each of the strategy has been calculated and basis these probabilities, EMV has been calculated:

EMV as per bays theorem
Strategy	EMV	Consultancy Fees	Net Value
S1	£          2,54,464.29	£               5,000.00	£ 2,49,464.29
S2	£          1,32,194.55	£               5,000.00	£ 1,27,194.55
S3	£             38,633.86	£               5,000.00	£    33,633.86

This also confirms the above decision and thus it can be concluded that strategy 1 should be chosen as final decision.

According to the given information, the insurance will be applicable if the car experienced and accident and the damages crossed $1000 amount. In such case beyond the $1000, everything will be covered by the insurance authority. Now, there is a probability of 0.025 that accident will happen.

Here, first of all excel simulation has been performed considering 5000 iterations and the average amount you pay, the standard deviation of the amounts you pay, and a 95% confidence interval for the average amount you pay are calculated as mentioned below:

	Details
Average Amount Pay	$   48.28
Standard Deviation of Pay	$329.64
Confidence Interval	0.95
Standard Error	$     0.07
Lower Limit	$   39.15
Upper Limit	$   57.42

 The above table has shown that the average amount pay has to incur is $48.28 and it will remain in between $39.15 and $57.42

Now, as per requirement, further study has been done considering changes in deductible amount, varied from $500 to $2000 in multiples of $500. A two way table has been shown the implication of such changes on the average amount you pay, the standard deviation of the amounts needs to be paid, and a 95% confidence internal for the average amount needs to be paid for each deductible amount. It is shown as mentioned below:

	Average Amount Pay	Standard Deviation of Pay	Lower Limit	Upper Limit
	$48.28	$329.64	$39.15	$57.42
$500.00	$63.69	$416.13	$52.15	$75.22
$1,000.00	$50.38	$331.49	$41.19	$59.56
$1,500.00	$40.18	$268.98	$32.73	$47.64
$2,000.00	$28.38	$207.98	$22.62	$34.15

Considering damages amount is normally distributed is not always holds true. The reason is that there is a significant amount of probabilities involved behind the extent to which the accident can take place.

Now, the information has shown that instead of normal distribution, if damages due accident follows triangular distribution with value, $500, $1500 and $7000, then using @RISK software, following result can be found:

Insurance Deductible Amount	$1,000.00
Probability of Accident	0.025
Average damages	$3,000.00
Stdev of damages	$    750.00
Damages [Triangular Distribution]	$2,908.23
Amount payable	$1,908.23
P(probability<$750)	0.200
P(probability>$600)	0.852
P(probability=$1000)	0.699

From the above payoff calculations, it has found that three different options are highlighted applying three different payoff matrix techniques. Hence, the organisation further applies probability aspects associated with each three strategies and performed EMV calculations (Schwalbe, 2015). Further these EMVs have been shown using decision tree as mentioned bellow.

As per this table, it can be said that the probability of damages less than $750 is 0.200. Similarly, the probability that the damages will be more than $600 is 0.852 and the probability that damages amount will be exactly $1000 is 0.699.

1. Calculation of Expected Time and Mean Time

Activity No	Task Name	Optimistic Duration	Most Likely Duration	Pessimistic Duration	Immediate Predecessor	Expected Time	Variance
1	Team meeting	0.5	1	1.5		1	0.027778
2	Hire Contractors	6	7	8	1	7	0.111111
3	Network Design	12	14	16	1	14	0.444444
4	Order Ventilation system	18	21	30	1	22	4
5	Install Ventilation system	5	7	9	4	7	0.444444
6	Order new racks	13	14	21	1	15	1.777778
7	Install racks	17	21	25	6	21	1.777778
8	Order power supplies and cables	6	7	8	1	7	0.111111
9	Install power supplies	5	5	11	8,12	6	1
10	Install cables	6	8	10	8,12	8	0.444444
11	Renovation of data centre	19	20	27	2,3	21	1.777778
12	City inspection	1	2	3	2,5,7	2	0.111111
13	Facilities	7	8	9	10	8	0.111111
14	Operations/System	5	7	9	10	7	0.444444
15	Operations/Telecommunications	6	7	8	10	7	0.111111
16	System & applications	7	7	13	10	8	1
17	Customer service	5	6	13	10	7	1.777778
18	Power check	0.5	1	1.5	9,10,11	1	0.027778
19	Install test servers	5	7	9	1,2,13,14,15,16	7	0.444444
20	Management safety check	1	2	3	5,18,19	2	0.111111
21	Primary systems check	1.5	2	2.5	20	2	0.027778
22	Set date for move	1	1	1	21	1	0
23	Complete move	1	2	3	22	2	0.111111

Network Diagram

Calculation of Variance and SD

Activity No	Optimistic Duration	Most Likely Duration	Pessimistic Duration	Var	SD
	(a)	(b)	(c)	V=((c-a)/6)^2	Sqrt(V)
1	0.5	1	1.5	0.028	0.167
2	6	7	8	0.111	0.333
3	12	14	16	0.444	0.667
4	18	21	30	4.000	2.000
5	5	7	9	0.444	0.667
6	13	14	21	1.778	1.333
7	17	21	25	1.778	1.333
8	6	7	8	0.111	0.333
9	5	5	11	1.000	1.000
10	6	8	10	0.444	0.667
11	19	20	27	1.778	1.333
12	1	2	3	0.111	0.333
13	7	8	9	0.111	0.333
14	5	7	9	0.444	0.667
15	6	7	8	0.111	0.333
16	7	7	13	1.000	1.000
17	5	6	13	1.778	1.333
18	0.5	1	1.5	0.028	0.167
19	5	7	9	0.444	0.667
20	1	2	3	0.111	0.333
21	1.5	2	2.5	0.028	0.167
22	1	1	1	0.000	0.000
23	1	2	3	0.111	0.333

Probability Calculation

Expected time for project completion is T (mean) = 69 (from critical path analysis)

Variance (V) = 5.833

SD = 2.415

Z= -1/2.415

Since, Z= -0.4140

Therefore, applying normal distribution function for Z,

Probability= 33.94%

4. Measurement of Project Risk Exposure

Risk assessment is a crucial method of identification and analysis of the factors of risk in a project (Kerzner and Kerzner, 2017). The risk monitoring tools have been helpful for forming the expectation of the risk factors in project.

Project Risk Exposure can be mitigated by the use of the high level of risk mitigation and planning. The risk management and analysis is done for taking care of the impact of risk in projects for the deployment of the effective mitigation strategies in project (Harrison and Lock, 2017).

1. Risk Identification can be implied for identifying the factors of risk favouring the alignment of successful risk mitigation (Nicholas and Steyn, 2017)
2. Risk Impact Assessment is helpful for analysing the consequences of cost, schedule and technical performance of project activities
3. Risk prioritization is done for analysing the impact of risk in the project for developing effective implication of the operations