Mildenhall Chapter 12: Allocations

Question 1

Desirale Properties of Allocations

Answer 1

1. Should work with any level of granularity, even down to the policy
2. Should be decomposable: the allocation to a sum of random variables = the sum of their allocations
3. Should be computed using a simple, consistent formula

Question 2

Expected Value Allocation

Formula + Definition + Exogenous or Endogenous

Answer 2

ai = a(X) * E[Xi] / E[X]
The total amount a(X) is allocation to each unit i in proportion to expected loss

Exogenous allocation b/c the allocation is based on expected value

a(X) is any risk measure

Question 3

Proportional Allocation / Haircut Allocation

Formula + Definition + Exogenous or Endogenous

Answer 3

ai = a(X) * a(Xi) / sum[ a(Xi) ] - endogenous
ai = a(X) * p(Xi) / sum[ p(Xi) ] - exogenous

exogeneous ~ excluding ~ you are not using the risk measure itself in the allocaton proportion

endogenous ~ including ~ you are using the risk measure itself in the allocation proportion

Question 4

Equal Risk Allocations

Exogenous or Endogenous

Answer 4

Endogenous solves for sum[ a(Xi, p * )] = a(X) for p * and sets ai = a(Xi, p * )
Exogenous solves for sum[ p(Xi, p * )] = a(X) for p * and sets ai = p(Xi, p * )

Capitalize each unit to the same probability of default of (1- p * )
In other allocation methods, the probability of default differs by unit i

This allocation often used in Lloyd’s/London market

Question 5

Marginal BU Allocation / Merton-Perold Method

Answer 5

Difference in risk capital is the “allocated capital” for that unit
NOT additive

Examples:
* p(X1 + X2) = 80 and p(X1) = 60 THEN p(X2) = 20
* p(X1 + X2) = 80 and p(X2) = 40 THEN p(X1) = 40

Sum of individual unit allocations is < the total capital
p(X1) + p(X2) < p(X1 + X2)

Question 6

Marginal Business Euler Gradient Allocation

Total Required Capital, Standalone Capital Charges

Answer 6

Given capital formula p(P, R, A) - Prem, reserves, assets
* Total Required Capital = p(P, R, A)
* Standalone Capital Charge = factor * (P, R, A)
* Allocation: derivative * (P, R, A) - take partial derivatives and plug in (P,R,A)

Question 7

2 and 3 Unit Shapley Allocation

Answer 7

2 Unit:
ai = 50% * a(Xi) + 50% * [ a(X) - a(X excluding i) ]
* Check both units (a1, a2) add up to total a(X)

3 Unit: Make a table:
1. S - subsets without player i
2. |S| how many are in each subset
3. Increment = c(S U {i}) - c(S)

Shapley weight/value = ci = sum[ |S|! (n-|S|-1)! * (c(S U {i}) - c(S)) / n! ]
where n = # of subsets without player i (step 1)

c1 + c2 = c(1, 2) - each player sum up to the total portfolio cost

Question 8

Relationship Between Merton-Perold (Marginal), Shapley and Stand-Alone

Answer 8

Average(stand-alone, marginal) = shapley

Question 9

Shapley Allocation Advantages/Disadvantages

Answer 9

Advantages
* It is additive
* It is symmetric
* It is linear in game theory
* It allocates no capital to a constant risk
* It is homogeneous if c is homogeneous
* If c is sub-additive, then Shapley value satisfies the no-undercut property

Disadvantages
* To allocate to n units, we must compute 2^n marginal impacts, which is impractical
* If a unit is sub-divided further into 2 new units, then allocations assigned to the other unit changes

Question 10

Allocate Loss Payments in Default

Answer 10

Aggregate: X = (X ^ a) + (X-a)+
Allocate (equal priority) Xi = Xi/X * (X ^ a) + Xi/X * (X-a)+
=payments to unit i + default borne by unit i
* each unit shares in the default proportionately

Final payments to unit i
Xi = Xi when X ≤ a
Xi = Xi * a/X when X > a

X-a > 0