Demand for Insurance v1Oct2023

1. Specficy Risk Preferences Utility Function: Log or Power Fn Choose 1 for LOG and 2 for Power 1 Natural LOG LN(w) 2. Specify the Risk Class Low Risk Proportion 0.5 High Risk Proportion 0.5 Probability of Loss : LOW Risk 0.25 Probability of Loss : HIGH Risk 0.5 3. Initial Wealth W(0) 120 Potential Loss L 80 4. Loss Exposure Loss State 1 (No Loss) 0 State 2 (Loss) p 80 Proportion of Low and High ρ Risk-Pooling Premium Rate θ 0.375 5. Budget Constraint: SLOPE -1.67 6. Aggregate Premium Rate θ 0.375 Loading λ Premium Rate with Loading 0.375 Coverage Q 5.00 (1 - p ) (1+ λ )p Notes: 1. This spreadsheet analyzes the demand for insurance with heterogene 3. Two risk-averse utility functions are included: LN(W) and 𝑊^(−𝑎) . 3. The demand for insurance under Expected Utility Analysis is found by 4. The model setup in this analysis is based on the lecture on "DEMAND

7. Risk Class to Investigate 1 for LOW or HOMOGEN; 2 for HIGH 1 or 2 1 8. Wealth W Probability State 1 (No Loss) W(1) 0.75 State 2 (Loss) W(2) 0.25 9. Expected Utility Analysis Expected Utility w/o Insurance E[U(w)] 4.513 Expected Utility with Insurance E[U(w)] 4.520 Loss Exposure L 80 Demand for Insurance 5.00 10. Results Can Insurance improve utility ? YES Insurance Coverage PARTIAL PERCENT Coverage 6.3% Insurance Premium 1.88 Risk Exposure Retained 75.00 Certainty Equivalent 91.18 Implied Max. Insurance premium 28.82 Q ≤ L

W(No Insurance) W(Insurance) 120 118.13 #VALUE! 40 43.13 #VALUE! < -- Change Q to maximize E[U(w)]; USE SOLVER

− ? ? ) ? 1 − ( 1 + ? ) ???? ????? 1 ( ?? ???? ) ? + ? − ( 1 + ? ) ???? ????? 2 ( ???? ) (𝑁𝑜 𝐿𝑜𝑠𝑠)@𝑊_2=𝑊_0−𝐿+𝑄 −(1 ) +𝜆 𝑝𝑄 𝑖𝑛 𝑠𝑡𝑎𝑡𝑒 2 (𝐿𝑜𝑠𝑠))┤ 1−𝜋)?(𝑊_1 ) ) )+(1 _0−(1 ) ) 𝜆 𝑝𝑄 −𝜋)?(𝑊 +𝜆 𝑝𝑄 e petitive insurance market. eneous. s and it's the risk-pooling rate.

24 200.00 24 26 196.67 26 28 193.33 28 30 190.00 30 32 186.67 32 34 183.33 34 36 180.00 36 38 176.67 38 40 173.33 40 42 170.00 42 44 166.67 44 46 163.33 46 48 160.00 48 50 156.67 50 52 153.33 52 54 150.00 54 56 146.67 56 58 143.33 58 60 140.00 60 62 136.67 62 64 133.33 64 66 130.00 66 68 126.67 68 70 123.33 70 72 120.00 72 74 116.67 74 76 113.33 76 78 110.00 78 80 106.67 80 82 103.33 82 84 100.00 84 86 96.67 86 88 93.33 88 90 90.00 90 92 86.67 92 94 83.33 94 96 80.00 96 98 76.67 98 100 73.33 100 102 70.00 102 104 66.67 104 106 63.33 106 108 60.00 108 110 56.67 110 112 53.33 112 114 50.00 114

116 46.67 116 118 43.33 118 120 40.00 120 122 36.67 122 124 33.33 124 126 30.00 126 128 26.67 128 130 23.33 130 132 20.00 132 134 16.67 134 136 13.33 136 138 10.00 138 140 6.67 140 142 3.33 142 144 0.00 144 146 -3.33 146

Driver Population Potential Loss L 1,000,000 Proportion of High-Risk Driver R(H) 50% 3.0% 1.0% Fair Prmium (Low Risk) 10,000 Fair Premium (High Risk) 30,000 Pooled Premium P* P* 20,000 Determination of Deductibles LB for Difference in Deductibles 666,667 UB forDifference in Deductibles 2,000,000 Deductible for High-Risk 0.00 Margin on Low-Risk Deductible 1000.00 Deductible for Low-Risk 667,667 Determination of Base Premium Base Premium (Low Risk) B(L) 10,000 Incentive Compatibility IC 9,970 Expected Insurance Cost to Driver Low Risk High Risk Policy L 16,677 30,030 Policy H 30,000 30,000 Insurance Profit Expected Profit Policy L 6,677 Policy H - #VALUE! Prob of Accident p (H) p (H) Prob of Accident p (L) p (L) p (H) x L p (L) x L δ (L) - δ(H) δ (H) δ (L)

Constraints 1. Incentive Compatibility B(L) >= IC 2. High Risk Driver Chooses Policy H ? 3. Low Risk Driver Chooses Policy L ? 4. Insurance Profit: Policy L ? 5. Insurance Profit: Policy H ? #VALUE! < -- Change B(L) until all constraints are YES

Driver Population Potential Loss L Proportion of High-Risk Driver R(H) Fair Prmium (Low Risk) Fair Premium (High Risk) Pooled Premium P* P* Determination of Deductibles LB for Difference in COINSURANCE UB forDifference in COINSURANCE COINSURANCE for High-Risk Margin on Low-Risk COINSURANCE (%) COINSURANCE for Low-Risk Determination of Base Premium Base Premium (Low Risk) B(L) Incentive Compatibility IC Expected Insurance Cost to Driver Low Risk Policy L 16,667 Policy H 30,000 Insurance Profit Expected Profit Policy L 6,667 Policy H - Prob of Accident p (H) p (H) Prob of Accident p (L) p (L) p (H) x L p (L) x L α (L) - α(H) α (H) α (L)

1,000,000 50% 3.0% 1.0% 10,000 Constraints 30,000 1. Incentive Compatibility: B(L) >= IC 20,000 2. High Risk Driver Chooses Policy H ? 3. Low Risk Driver Chooses Policy L ? 4. Insurance Profit: Policy L ? 0.6667 5. Insurance Profit: Policy H ? 2.0000 6. Coinsurance for Low Risk: 1 > α(L) > 0 ? 66.67% #VALUE! 10,000 < -- Change B(L) until all constraints are YES 10,000 High Risk 30,000 30,000 #VALUE!

Payoff Prob Utility State 1 150 0.5 5.01 State 2 50 0.5 3.91 Expected W 100 Exp Utility 4.46 Payoff Prob Utility State 1 110 0.5 4.70 State 2 68.2 0.5 4.22 Expected W 89.1 Exp Utility 4.46 Diff Exp U(w) 0.00