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Actuarial Application of
Monte Carlo Simulation
A stochastic Approach to Pricing a Life Insurance Policy
By: Adam Conrad
Table Of Contents
• Introduction……………………………………………………………
3
• Problem
Statement………………….........................................
Introduction
• Actuaries are mathematicians that put a price on risk.
• Actuaries use statistical data to develop mathemat...
Problem Statement
• A husband and wife are looking to purchase the
following life insurance policy:
– Term: 20 Years
or
In...
Problem Statement
• If 1000 similar policies are to be sold in
the next year, determine the premium
price that will consis...
Age of Husband at Purchase (HA)
6
Age of Husband at Purchase (HA)
7
Code Simulation Run
Calculations
*All future standard deviations will be
calculated in t...
Age of Wife at Purchase (WA)
• Below is a table of the data collected on the age
difference between a husband and a wife i...
Age of Wife at Purchase (WA)
• Assumptions:
– 32.4% of wives are the same age as their husband.
– 55.4% of wives are young...
Age of Wife at Purchase (WA)
10
Code Simulation Run
Age of Husband and Wife at Purchase
11
Simulation Run
Age of Husband at Death (HD)
• Assumptions:
– The average age of death of a male in America is 76 years.
– Ages 65 to 100 ...
Age of Husband at Death (HD)
13
Simulation RunCode
Calculations
Age of Wife at Death (WD)
• Assumptions:
– The average age of death of a female in America is 81 years.
– Ages 65 to 100 m...
Age of Wife at Death (WD)
15
Simulation RunCode
Premiums Collected Over Claims Paid
(POC)
16
• Lifespans of Husband and Wife after the
policy purchase.
Code
Simulation Run
Premiums Collected Over Claims Paid
(POC)
17
Events after 20 years Claim Profit
Only Husband Survives $250,000 (Premium)*(...
Monte Carlo Simulation Results
18
Selected Premium Expected Profit Standard Deviation Worst Case Profit
50.00$ 1,430,807.0...
Monte Carlo Simulation Results
19
Selected Premium Expected Profit Standard Deviation Worst Case Profit
50.00$ 1,430,807.0...
Monte Carlo Simulation (Code for Copying and Pasting into MATLAB)
20
Bibliography
• http://www.cdc.gov/nchs/data/series/sr_10/sr10_260.pdf
• http://www.cincinnati-oh.gov/health/environmental-...
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Actuarial Application of Monte Carlo Simulation

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Actuarial Application of Monte Carlo Simulation

  1. 1. Actuarial Application of Monte Carlo Simulation A stochastic Approach to Pricing a Life Insurance Policy By: Adam Conrad
  2. 2. Table Of Contents • Introduction…………………………………………………………… 3 • Problem Statement…………………..............................................4 • Random Variable Generation………………………………………. – Age of Husband at Purchase……………………………………………………..6 – Age of Wife at Purchase..………………………………………………………...8 – Age of Husband at Death………………………………………………………..12 – Age of Wife at Death…………………………………………………………......14 – Premiums Collected Over Claims Paid………………………………………...16 • Monte Carlo Simulation Results.……………………………….…18 • Simulation Coding……..………………………………..................20 • Bibliography…………………………………………………………21 2
  3. 3. Introduction • Actuaries are mathematicians that put a price on risk. • Actuaries use statistical data to develop mathematical models. • Most actuaries work for insurance companies to set policy rates. • Assumptions made for this problem were made though research 3
  4. 4. Problem Statement • A husband and wife are looking to purchase the following life insurance policy: – Term: 20 Years or In the event a death – Benefit: $250,000 – Premium: Joint Monthly Payment 4
  5. 5. Problem Statement • If 1000 similar policies are to be sold in the next year, determine the premium price that will consistently result in total premiums being more than total claims. 5
  6. 6. Age of Husband at Purchase (HA) 6
  7. 7. Age of Husband at Purchase (HA) 7 Code Simulation Run Calculations *All future standard deviations will be calculated in the same fashion
  8. 8. Age of Wife at Purchase (WA) • Below is a table of the data collected on the age difference between a husband and a wife in America. 8
  9. 9. Age of Wife at Purchase (WA) • Assumptions: – 32.4% of wives are the same age as their husband. – 55.4% of wives are younger than their husbands. – 12.3% of wives are older than their husbands. – The average age difference is 2.3 years with the husband being older than the wife. – If the husband is older, 1 to 17 year makes up 99.7% of a normal distribution (3 standard deviations) – If the wife is older, 1 to 7 years makes up 99.7% of the a normal distribution (3 standard deviations) 9
  10. 10. Age of Wife at Purchase (WA) 10 Code Simulation Run
  11. 11. Age of Husband and Wife at Purchase 11 Simulation Run
  12. 12. Age of Husband at Death (HD) • Assumptions: – The average age of death of a male in America is 76 years. – Ages 65 to 100 make up 99.7% of the male deaths – The pdf is normal until it reaches the mean, and then it is exponential 12
  13. 13. Age of Husband at Death (HD) 13 Simulation RunCode Calculations
  14. 14. Age of Wife at Death (WD) • Assumptions: – The average age of death of a female in America is 81 years. – Ages 65 to 100 make up 99.7% of the female deaths – The pdf is normal until it reaches the mean, and then it is exponential 14
  15. 15. Age of Wife at Death (WD) 15 Simulation RunCode
  16. 16. Premiums Collected Over Claims Paid (POC) 16 • Lifespans of Husband and Wife after the policy purchase. Code Simulation Run
  17. 17. Premiums Collected Over Claims Paid (POC) 17 Events after 20 years Claim Profit Only Husband Survives $250,000 (Premium)*(Months Wife Lived) - 250000 Only Wife Survives $250,000 (Premium)*(Months Husband Lived) - 250000 Both Survive $0 (Premium)* (240 Months) The table below shows a list of possible outcomes under the policy The coding on the right determines whether or not the nth couple filed a claim before the term expired and how much premium was collected throughout the policy. It then runs the simulation 10 times so that the variance can be determined.
  18. 18. Monte Carlo Simulation Results 18 Selected Premium Expected Profit Standard Deviation Worst Case Profit 50.00$ 1,430,807.00$ 1,816,279.35$ (4,018,031.05)$ 60.00$ 3,267,329.01$ 1,570,115.48$ (1,443,017.43)$ 70.00$ 6,024,908.71$ 1,890,279.74$ 354,069.49$ 80.00$ 8,497,356.64$ 1,631,740.58$ 3,602,134.90$ 90.00$ 11,472,155.66$ 1,002,633.00$ 8,464,256.66$ 100.00$ 13,489,612.32$ 1,482,734.74$ 9,041,408.10$ Monte Carlo Simulation The simulation was run with premiums set at $50/month up to $100/month and the expected profits and variances were outputted. Using the variances of each run, the profit earned given the worst case scenario was calculated by subtracting 3 standard deviations from the expected value.
  19. 19. Monte Carlo Simulation Results 19 Selected Premium Expected Profit Standard Deviation Worst Case Profit 50.00$ 1,430,807.00$ 1,816,279.35$ (4,018,031.05)$ 60.00$ 3,267,329.01$ 1,570,115.48$ (1,443,017.43)$ 70.00$ 6,024,908.71$ 1,890,279.74$ 354,069.49$ 80.00$ 8,497,356.64$ 1,631,740.58$ 3,602,134.90$ 90.00$ 11,472,155.66$ 1,002,633.00$ 8,464,256.66$ 100.00$ 13,489,612.32$ 1,482,734.74$ 9,041,408.10$ Monte Carlo Simulation Risk was mitigated when the monthly premium was 70$/month ($35 per person) Which means that that is the minimum price that this policy can be set at so that there is no risk. The decision about how much to raise the price is dependent on the needs of the business department.
  20. 20. Monte Carlo Simulation (Code for Copying and Pasting into MATLAB) 20
  21. 21. Bibliography • http://www.cdc.gov/nchs/data/series/sr_10/sr10_260.pdf • http://www.cincinnati-oh.gov/health/environmental-health/cincinnati-neighborhood-specific- life-expectancy-data/ • http://fivethirtyeight.com/datalab/whats-the-average-age-difference-in-a-couple/ • http://www.allcountries.org/uscensus/56_married_couples_by_differences_in_ages.html • https://www.jrcinsurancegroup.com/best-age-life-insurance-for-male/ • http://www.nejm.org/doi/full/10.1056/NEJMsa1211128 • http://www3.imperial.ac.uk/statistics/msc/optionalcourses • http://www.worldlifeexpectancy.com/usa/life-expectancy-male 21

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