Insurance vs. No Insurance: A Comparative Analysis

Scenario 1: No Insurance

• Expected Loss per Individual

  • Each person has a $1\%$ chance of losing $1,000$.

  • Calculated as: $0.01 \times \$1,000 = \$10$ per person.

• Expected Loss for the Society

  • With $2,000$ people, the total expected loss is: $2,000 \text{ people} \times \$10 = \$20,000$.

• Capital Needed per Person for 99.9% Confidence

  • To ensure each person can cover a $1,000$ loss with only a $0.1\%$ chance of default, each person must set aside the full $1,000$ dollars.

• Capital Needed for the Whole Society

  • If every person needs $1,000$ dollars, the total capital needed for $2,000$ people is: $2,000 \text{ people} \times \$1,000 = \$2,000,000$.

• Guarantee of Loss Payments to Individuals

  • Yes, losses are guaranteed if each person individually sets aside the full $1,000$. However, this method is inefficient because most people will not experience a loss.

Scenario 2: With Insurance

• Premium per Person

  • Expected Loss: $10$.

  • Adding $20\%$ Expense Margin: $10 \times 1.20 = \$12$.

  • Adding $5\%$ Profit Margin: $12 \times 1.05 = \$12.60$ per person.

• Total Premium Collected

  • From $2,000$ people: $2,000 \text{ people} \times \$12.60 = \$25,200$.

• Total Expense Margin

  • $20\%$ of expected loss per person: $10 \times 0.20 = \$2$.

  • Total for society: $2,000 \times \$2 = \$4,000$.

• Total Profit Margin

  • $5\%$ of premium after expenses per person: $12 \times 0.05 = \$0.60$.

  • Total for society: $2,000 \times \$0.60 = \$1,200$.

• Capital Insurance Company Needs for 99.9% Confidence

  • Methodology: Uses a binomial distribution with parameters $n = 2,000$ (number of people) and $p = 0.01$ (probability of loss).

  • Expected Number of Losses: $n \times p = 2,000 \times 0.01 = 20$ people.

  • Expected Monetary Losses: $20 \text{ people} \times \$1,000 = \$20,000$.

  • Standard Deviation (of number of losses): $\sqrt{n \times p \times (1-p)} = \sqrt{2,000 \times 0.01 \times (1-0.01)} = \sqrt{19.8} \approx 4.4497$ losses.

  • Standard Deviation (monetary): $\sqrt{19.8} \times \$1,000 \approx \$1,407$.

  • Capital for 99.9% Confidence: For this confidence level, the z-score is approximately $3.09$. The capital needed is calculated as Expectation + ($z$-score $\times$ Standard Deviation):

    • $\$20,000 + 3.09 \times \$1,407 \approx \$24,352$.

• Additional Capital Needed per Person

  • None. The insurance company efficiently pools the risk and holds the necessary capital on behalf of all policyholders.

• Guarantee of Loss Payments to Individuals

  • Yes, individuals are guaranteed their losses will be paid, provided the insurance company maintains adequate capital and solvency.

General Questions

• Which is the Better Option?

  • Insurance is demonstrably the better option.

• Why is Insurance Better?

  • Reduced Individual Burden: Individuals only pay a premium of $12.60$ dollars, rather than setting aside a full $1,000$ dollars for a potential loss.

  • Efficient Risk Sharing: Risk is spread among a large pool of people, meaning fewer individuals need to hold large emergency funds, leading to greater societal financial efficiency.

• Capital Freed Up by Insurance

  • Without Insurance (Societal Capital): Approximately $2,000,000$.

  • With Insurance (Insurer's Capital): Approximately $24,352$.

  • Capital Freed: Approximatley $\$2,000,000 - \$24,352 = \$1,975,648$.

• Three Reasons Insurers Should Maintain Adequate Capital

  • To be able to pay all claims, even in the event of extreme, high-loss scenarios.

  • To maintain trust with policyholders and comply with regulatory requirements.

  • To prevent bankruptcy and protect the financial interests of policyholders.

• Two Reasons Insurers Shouldn't Hold Excess Capital

  • Excess capital ties up resources that could otherwise be invested to generate returns or be distributed back to shareholders.

  • Holding too much capital can lead to inefficient allocation of funds and potentially lower financial profitability for the insurer.

• Two Fundamental Principles of Insurance (Key Takeaways)

  • Risk Pooling: This principle involves sharing financial risks among a large number of participants. By doing so, the collective impact of losses is distributed, significantly reducing the individual financial burden on any single person.

  • Confidence Level Capitalization: This refers to the practice of maintaining a sufficient amount of capital or reserves to cover expected and unexpected losses with a very high degree of certainty. This ensures the financial stability of the insurer and builds trust among policyholders, guaranteeing that claims will be paid.