Mildenhall All Formulas Flashcards
(14 cards)
Wtd Avg Cost of Capital
Reinsurance Cost (%), WACC
Reinsurance Cost = (Ceded Prem * (1+r) - E[Recoveries]) / Capital Benefit
* Capital Benefit ~ reduction in regulatory capital
WACC = wtd average (capital, cost)
* Equity = assets - liabilities
* Reinsurance Capital: capital benefit received from reinsurance
* Debt Capital: Bonds/debt issued by company
Cost of equity know as target ROE
PCPs
Net Premium
Expected Value / Constant Loading
Max Loss
P = E[X]
P = E[X] + π * E[X]
P = E[X] + π * max(X)
PCPs
VaR / Percentile
TVaR
Exponential
Semideviation / Dutch
Higher-order semideviation / Fisher
P = VaR_π_[X]
P = TVaR_π_[X]
P = ln(E[exp(πX)])/π
P = E[X] + πE[(X - E[X])+]
P = E[X] + πE[(X - E[X])+^p]^(1/p)
PCPs
Mean
Variance
Standard Deviation
Semivariance
P = E[X] + π * E[X]
P = E[X] + π * Var[X]
P = E[X] + π * SD[X]
P = E[X] + πVar+[X] where Var+[x] = integral E[X] to inf (x - E[X])^2 f(x)dx
only look at the variance of losses above the expected loss
Ferrari
Total Returns (TR)
TR = Income / Equity = (U+I)/Q
= U/P * P/Q + I/a * a/Q
= UW Margin * UW Leverage + Investment Return * Investment Leverage
Plug in a = Q+R, a/Q = 1 + R/Q:
= U/Q + I/a * (1 + R/Q)
= I/a + U/Q + I/a * R/Q
= I/a + R/Q(U/R + I/a)
R/a = policyholder-funded asset leverage
Leverage is beneficial if (I/a + U/R) > 0
R = loss reserves and UEPR
U = UW income | P = Prem | I = Invest Income | Q = Equity | a = Assets
Calculate Premium using DCF
Assume 1 year policy
* L / (1+RL) - discount the expected loss at risk adjusted rate
* t * Rf (P + Q) / (1+Rf) - discount the taxes on (risk-free) investment income at the risk free rate
* t * P / (1+Rf) - discount the taxes on the premium at the risk free rate
* - t * L / (1+RL) - discount the taxes on the losses at the risk adjusted rate
P = sum of everything above
P = L /(1+RL) + t * Rf * Q / [ (1+Rf)(1-t) ]
Q = capital invested
Year-end Value of Firm, V1
Again, assume 1 year
* P * (1+Rf) - premium invested at the risk free rate
* Q * (1+Rf) - capital invested at the risk free rate
* - L - minus losses
* - t * (P - L) - minus taxes on the (premium - loss)
* - t * Rf * (P+Q) - minus taxes on the investment income of (P + Q)
V1 = value at end of year 1 = sum of above
Q = capital invested
Portfolio Constant Cost of Capital (CCoC) Pricing
With default, w/o default, with friction
- based on simplified DCF model with no taxes
- Premium = Loss Cost + Cost of Capital
- Cost of Capital = target return on capital * amount of capital
P(a) = (E[X ^ a(X)] + i * a) / (1+i) - w/ default
P(a) = (E[X] + i * a) / (1+i) - w/o default
1 + i * = (1+i)(1+t) - adjusted cost of capital for frictional cost
where t = frictional cost (%)
i = cost of capital | a = assets
P(a), a, S(a), M(a), F(a), i(a), δ(a), v(a), Markup
Prem = P(a) = S(a) + M(a) = Loss + Margin
Assets = a = P(a) + Q(a) = Prem + Capital
Residual Value = F(a) = a - S(a) = M(a) + Q(a) = Marginal + Capital = E[(a-X)+]
Cost of Capital = Risk Return = i(a) = M(a) / Q(a)
Risk Discount Rate = δ(a) = M(a) / F(a) = P(a) - S(a) / a - S(a)
Markup = 1 / LR = P(a) / S(a)
So we know a - S(a) = M(a) + Q(a)
* δ(a) * (a - S(a)) = M(a)
* v(a) * (a - S(a)) = Q(a)
* δ(a) = 1 - v(a)
a=1 for bernoulli layer
Natural Allocation, Bodoff Percent Layer of Capital (PLC)
E[X(a)], Premium ($), α, β, M, Q, i, Capital ($)
Shared Liability in Layer (a’1)
E[X(a)] = sum[p * X(a)]
Prem = sum[Δg(s) * X(a)]
α = sum(p * X1/X) / sum(p) - for all rows below it
E[X(a)] = sum[α * S * ΔX(a)]
β = sum(Δg(s) * X1/X) / sum(Δg(s)) - for all rows below it
Prem = sum[β * g(s) * ΔX(a)]
M = Prem - Loss = g(s) - S
M1 = β * g(s) - α * S
Q = Assets - Premium = 1 - g(s)
i = M/Q
Q1 = M1/i
Shared Liability in Layer
a’1 = α1 * [1-S] * ΔX(a)
Q1 = α1 * [1-g(s)] * ΔX(a)
Distortion Functions
Constant Cost of Capital (CCoC) and TVaR
Constant Cost of Capital
* g(s) = (s + i) / (1 +i) when s>0, otherwise 0
TVaR
* g(s) = min[ 1, s / (1-p) ]
Distortion Functions
Porportional Hazard (PH)
Proportional Hazard (PH) Transform
* g(s) = s^a, where a < 1 (so that this is concave)
* Smaller a values –> greater risk adversion
* hazard rate is proportionately increased by a factor of a
Distortion Function
Dual Moment
Dual Moment Distortion / MAXVAR Distortion
* g(s) = 1 - (1 - s)^m, where m ≥ 1
* Larger m values –> greater risk aversion
Distortion Function
Wang Transform
Wang Transform
* g(s) = ϕ(ϕinv(s) + λ), where λ > 0
* ϕ is CDF of standard normal distribution
* Larger λ –> greater risk adversion
