Mildenhall Chapter 14 + 15 Flashcards
(13 cards)
Linear Natural Allocation for Discrete Distribution
Steps
- Add a X0 = 0 with probability = 0
- Sort events Xj in ascending order
- Group by Xj and take p-weighted averages for X1 and X2
- Calculate P(X>x) = Sj = S(Xj) for each Xj
- Distort the survival function, compute g(Sj)
- Risk adjusted probability = g(Sj-1) - g(Sj)
- Sum-Product risk adjusted p * X1 or X2, etc to get the allocation
Xj = X1 + X2 + etc
Lifted Natural Allocation for Discrete Distribution
Steps
- Add a X0 = 0 with probability = 0
- Sort events Xj in ascending order
- Group by Xj and take p-weighted averages for X1 and X2
- Calculate X ^ a
- X1(a) = X1 if <a or X1 * a/Xj
- Distort the survival function, compute g(Sj)
- Risk adjusted probability = g(Sj-1) - g(Sj)
- Sum-Product risk adjusted p * X1(a) or X2(a), etc to get the allocation
Xj = X1 + X2 + etc
Kernel Estimate
Steps
- Add a X0 = 0 with probability = 0
- Sort events Xj in ascending order
- Calculate K(u)
- Calculate p * K (used in demon)
- Calculate X1 * p * K (used in numerator)
sum[ X1 * p * K(u) ] / sum[ p * K(u) ]
where u = [ (X1 - X) / h ]
Xj = X1 + X2 + etc
Discrete Example
Expected Proportion of Recoveries for Unit i: αi
Steps, j(a), Expected Value w/ and w/o accounting for default, Si(a) bar
- Add a X0 = 0 with probability = 0
- Sort events Xj in ascending order
- Calculate ΔX = X(j+1) - X(j), p, S(Xj), S * ΔX = total loss for the book
- Calculate X1/X and X2/X
- Calculate α1 and α2
- Calculate α1 * S * ΔX and α2 * S * ΔX and sum
α1 = sumprod(p, X1/X) / sum(p) - for only the rows BELOW the current row
The sum of step 6 column is the expected losses of unit 1 and 2 losses WITHOUT accounting for default
j(a) = find when X first > a, then take the previous row #
Si(a) bar = E[Loss] accounting for default = sum [ step 6 for j<j(a) ] + [ a - Xj(a) ] * α(Xj(a)) * Sj(a)
Xj = X1 + X2 + etc
Continuous Example
Expected Proportion of Recoveries for Unit i: αi
αi(a) * S(a) = E[ Xi/X * 1(X>a) ] = E[ Xi/X * 1(X>a) ]
Si(a) = αi(a) * S(a) = unit i loss density in asset layer a
Si(a) bar = E[Xi(a)] = integral 0 to a [ αi(x) S(x) dx ]
Continuous Example
Risk-Adjusted Expected Proportion of Recoveries for Unit i: βi
βi * g(S(a)) = E[ Xi/X * Z(X) * 1(X>a) ]
where Z(x) = [ g(S(x-)) - g(S(x)) ] / [ S(x-) - S(x) ]
Pi(a) = βi(a) * g(S(a)) = unit i premium density in asset layer a
Pi(a) bar = E [ Xi(a) ] = integral 0 to a [ βi(x) * g(S(x)) dx ]
Discrete Example
Risk-Adjusted Expected Proportion of Recoveries for Unit i: βi
Steps
- Add a X0 = 0 with probability = 0
- Sort events Xj in ascending order
- Calculate ΔX = X(j+1) - X(j), p, S(Xj), g(S) , g(S) * ΔX = total prem for the book
- Calculate X1/X and X2/X
- Calculate β1 and β2
- Calculate β1 * g(S) * ΔX and β2 * g(S) * ΔX
β1 = sumprod(Δg(S), X1/X) / sum(p) - for only the rows below than the current row
The sum of step 6 column is the allocated premiums of unit 1 and 2 losses WITHOUT accounting for default
j(a) = find when X first > a, then take the previous row #
Pi(a) bar = E[Prem] accounting for default = sum [ step 6 for j<j(a) ] + [ a - Xj(a) ] * β(Xj(a)) * g(S(j(a))
now it’s premiums instead of loss because we are using risk adjusted prob
Margin
Margin, Margin by layer, Steps
Margin = Prem - Loss
* Mi(a) bar = Pi(a) bar - Si(a) bar
Margin by Layer
* Mi(a) = Pi(a) - Si(a) = βi(a) * g(S(a)) - αi(a) * S(a)
Steps:
1. Add a X0 = 0 with probability = 0
2. Sort events Xj in ascending order
3. Calculate ΔX = X(j+1) - X(j), p, S(Xj), g(S)
4. Calculate M = g(S) - S, M * ΔX = Total margin for the book
5. Calculate M1 and M2 = βi(a) * g(S(a)) - αi(a) * S(a)
6. Calculate M1 * ΔX and M2 * ΔX
Sum(step 6) = total margin for unit 1 or 2 without default = P bar - S bar
j(a) = find when X first > a, then take the previous row #
Mi(a) bar = E[Margin] accounting for default = sum [ step 6 for j<j(a) ] + [ a - Xj(a) ] * M(j(a))
Discrete Example
Capital (Q) and Cost of Capital (i) by Unit
Formula and Steps, Average cost of capital
Cost of Capital = i(a) = M(a) / Q(a)
=[ P(a) - S(a) ] / [1 - P(a) ]
=[ g(S(a)) - S(a) ] / [1 - g(S(a)) ]
To ensure our pricing is law invariant, the CoC for the book is also M(a) / Q(a)
Natural allocation of capital to unit i = Qi(a) = Mi(a) / i(a) = Mi(a) / M(a) * Q(a)
where Mi(a) = Mi(a) = Pi(a) - Si(a) = [ βi(a) * g(S(a)) - αi(a) * S(a) ] = margin by layer
* The total capital is allocated to each unit by [ margin unit i / total margin ]
- Calculate X, ΔX, M, Q = 1 - g(S), i = M / Q
- Calculate M1 * ΔX and M2 * ΔX
- Calculate Q1 and Q2 = M1 * ΔX / i
- Calculate Q1 * ΔX and Q2 * ΔX
sum(step 4) is the natural allocation of capital in unit 1 and 2 without accounting for default
j(a) = find when X first > a, then take the previous row #
Qi(a) bar = Capital accounting for default = sum [ step 4 for j<j(a) ] + [ a - Xj(a) ] * Q(j(a))
Average CoC for unit 1 = M1(a) / Q1(a)
Continuous Example
Capital and Cost of Capital by Unit
Qi(a) bar= integral 0 to a [ Qi(x) dx ]
Average CoC for unit i at asset level a =
ii(a) bar = Mi(a) bar / Qi(a) bar
Discrete Example
Percentile Layer of Capital (PLC)
Approach
MSteps:
1. Add 0 if needed
2. Calculate X = X1 + X2, sorted ascending, calculate p and S(x)
2. Calculate X ^ a and Δ(X^100) = “j+1 - j”
3. Calculate X1(a) = X1 if <a or X1 * a/Xj (Unit limited expected loss)
4. Calculate X1/X, X2/X, α1, and α2
5. a’1 = α1 * (1-S) * Δ(X^100)
α1 = sumprod(p, X1/X) / sum(p) - for only the rows below than the current row
Total shared liability = sum(step 5) = capital + margin = assets - E[Loss]
δ = i / (1+i) where i = cost of capital (CoC)
v = 1- δ
Portfolio Premium, Portfolio Margin, Portfolio Capital
* M bar = Portfolio Margin = Total Shared Liability * δ
* Q bar = Portfolio Capital = Total Shared Liability * v (that’s why Margin + Capital = Total Shared Liability)
* Portfolio Limited Expected Loss = sumprod(p, X ^ a)
* P bar = Portfolio Premium = Portfolio Limited Expected Loss (step 2) + Portfolio Margin
* Check: Portfolio Margin / Capital = i = CoC
Unit Premium, Unit Margin, Unit Capital
* M1 bar = a’1 / (a’1 + a’2) * M bar
* Q1 bar = a’1 / (a’1 + a’2) * Q bar
* P1 bar = Unit Premium = Unit Limited Expected Loss (step 3) + M1 bar
* Check: P1 bar / Q1 bar = i = CoC
Percentile Layer of Capital (PLC)
Assets in the layer 1(X>x) are allocated to unit i in proportion to its share of portfolio losses, E(Xi/X | X > x) = αi(x)
* Total assets allocated across all layers to unit i = integral 0 to a [ αi(x) dx ]
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 = α * g(s) - β * S
Q = Assets - Premium = 1 - g(s)
i = M/Q
Q1 = M/i
Shared Liability in Layer
a’1 = α1 * [1-S] * ΔX(a)
Q1 = α1 * [1-g(s)] * ΔX(a)
