continous EPD with variable losses,

assuming normal dist

d_L = kφ(-c/k) - cΦ(-c/k)

• k = CV of losses
• c = capital/loss ratio
• Φ() = cum standard normal dist
• φ() = standard normal density function

continous EPD ratios, variable assets

assuming normal distribution

d_A = [1/(1-c_A)] *[k_aφ(-c_A/k_A) - c_AΦ(-c_A/k_A)

• k_A = CV of assets
• c_A = capital/asset ratio
• Φ() = cum standard normal dist
• φ() = standard normal density function

EPD ratio variable losses

assuming lognormal loss

d_L = Φ(a) - (1+c)Φ(a-k)

• k = CV of losses
• a = k/2 - ln(1+c)/k
• Φ() = cum standard normal dist
• φ() = standard normal density function

EPD ratio, continous case with variable assets

assuming lognormal dist 

d_A = Φ(b) - Φ(b-k_A)/(1-c_A)

• k_A = CV of assets
• b = k_A/2 - ln(1-c_A)/k_A
• Φ() = cum standard normal dist
• φ() = standard normal density function

simplifaction for correlations

C = [Σ C_i² + ΣΣp_ijC_iC_j]^0.5

Be careful, because correlated items on opposite sides of the balance sheet should be multiplied by -1. I.E., if loss and assets are 100% correlated, subtract them.

inherently, accounting figures biased since paper value doesn't necessarily equal realizable value (market)

For RBC, market value accounting is best, since it relys on current market value = price in event of insurer failure

1. It should be the same for all classes of insured, all types of insurers
2. It should be objectively measured
3. It should discriminate between quantifiable sources of risk

Density Function of Standard Normal Distribution

φ(x)

e^{^}(-x²/2)

/√(2∏)

note this is the probability of the option being exercised

= 1-N(d₂)

d₂ = [ln(S₀/K) + (r-c²/2) * T ]/

(σ*√T)

K is loss to be paid

S₀ = Beginning Assets

r =risk free interest rate

σ_v = volatility of assets

S₀ * N(d₁) - K*e^{^-rt}*N(d₂)

where d₂ =  

d₁ - (σ*√T)