6.9.Kreps

• Swap Risk-Free Investment for Risky Investment
• Purchase Put Options

• Safety Constraint: funds available at year end are at least equal to the specified loss safety level (IRR ≥ r_f)
• Investment Variance Constraint: variance of reins + inv strategy should be no more volatile than a direct investment in risky assets

Writing reinsurance can never really result in dist of capital that are really the same as investment dist because investment returns are strictly limited to having worst-case returns of -100%

• (1 + IRR)A = (1 + r_f)(P + A) - L
• R = (y - r_f)/(1 + r_f) * A
• Safety Constraint
• A = (s - μ_L) / (1 + y)
• s = e^{μ+ N_-1(pct)σ}
• σ = √ln(1 + CV²)
• μ = ln(mean) - σ²/2
• Variance Constraint
• A = σ_L/σ_y

• F = (P + A) / (1 + r)
• R = A[(1 + r)(1 + y) - (1 + i)/(1 + i)] + μ_L[(1 + r)/(1 + i) - 1/(1 + r_f)]
• Safety Constraint
• A = 1 / (1 + y) * [(1 + i)/(1 + r_f) * s - μ_L]
• Variance Constraint
• A = [b + √(b² + ac)] / a
• a = σ_y²(1 + i)² - σ_i²(1 + y)²
• b = μ_L(1 + y)σ_i²
• c = μ_L²σ_i² + σ_L²(1 + i)²