Mildenhall Ch 12: Classical Price Allocation Theory

Question 1

List the 4 sets of variables in Portfolio CCoC Allocation method

Answer 1

1. Premium (P)
2. Expected Loss (l)
3. CoC Target Return (i)
4. Assets (a)

Question 2

List the 5 relationships between the 4 variables in Portfolio CCoC Allocation method

Answer 2

1. P = l + d*(a-l)
2. Pi = li + d*(ai-li)
3. P = sum of Pi
4. l = sum of li
5. a = sum of ai

Question 3

Briefly describe the portfolio CCoC Allocation method

Answer 3

Allocated CCoC pricing fixes target return and allocated assets to determine premium.

Question 4

Contrast Portfolio CCoC allocation method and SRMs

Answer 4

Portfolio CCoC fixes target return and allocated assets to determine premium.

SRMs fix target return by layer and allocate premium to determine assets and average return by unit.

Question 5

Describe 2 approaches to risk-adjust target return and fix amount of capital.

Answer 5

1. RAROC (risk-adjusted return on capital): return varies with risk
2. RORAC (return on risk-adjusted capital): return is constant when capital reflects risk. Combines a constant CoC with capital allocation that normalizes for risk and is the industry practice.

Question 6

Does Allocated CCoC Pricing method uses RAROC or RORAC?

Answer 6

RORAC

Question 7

True or False?
Weighted CoC across layers is the same for all units with allocated CCoC pricing.

Answer 7

False.

Even if all units have a constant CoC within a layer, the weighted CoC across layers will differ since each unit has different mix of capital.

Question 8

Briefly explain how to determine the CCoC to be used in RORAC

Answer 8

WACC:

To determine the CCoC to be used, we rely on an estimate of the insurer’s weighted average CoC (between debt, reinsurance and equity).

We can easily quantify debt and reinsurance costs.

This leaves equity cost as most important unknown input.

Question 9

Briefly explain how to determine the risk-adjusted capital

Answer 9

To determine the risk-adjusted capital (aka economic capital or risk capital), there is no widely accepted practice.

Question 10

Briefly explain why allocated capital is artificial

Answer 10

Because the entire capital base of the insurer is available for each individual unit.

We still need it since it influences the decisions insurers make by unit.

Question 11

Describe the goal of the allocation of non-additive functions

Answer 11

The goal is to define a function that applies a risk measure function to loss amounts and uses those to allocate some sort of total risk measure to each unit.

Question 12

List 3 desirable properties of an allocation

Answer 12

1. It should be work at any level of granularity
2. It should be decomposable which means the alloc to a sum of res equals sum of their allocations
3. It should be computed using a single consistent formula.

Question 13

Contrast endogenous and exogenous allocations

Answer 13

When the same risk measure is used to determine total AND to allocate it, we call it endogenous allocation.

When same risk measure is not used, we call it exogenous allocation.

Question 14

Is amount of capital of a firm endogenous or exogenous?

Answer 14

Tot amount of capital of a firm is largely exogenous.

Regulator, rating agency or market consensus establishes amount of capital, but insurer determines its allocation.

Question 15

Describe the Expected Value Allocation method. Is it exogenous or endogenous?

Answer 15

ai = a(X)*E(Xi)/E(X)

Tot amount a(X) is allocated to each unit I in proportion to expected loss.

This is exogenous allocation since it is based on an auxiliary measure.

When applied to premium, it results in equal LR across all units.

Question 16

Describe the Proportional Allocation method. Is it exogenous or endogenous?

Answer 16

ai = a(X)*a(Xi) / sum of a(Xi)

Proportional allocation is endogenous because same risk measure used to calculate a(X) and individual a(Xi).

Each unit is evaluated on a stand-alone basis (a(Xi)) and total is allocated based on these stand-alone measures.

Since it uses stand-alone measures, it is not influenced by dependence between Xi.

Question 17

Describe the Haircut Allocation method. Is it exogenous or endogenous?

Answer 17

ai = a(X)*rho(Xi) / sum of rho(Xi)

Haircut allocation is exogenous because stand-alone measures are based on general rho.

Like proportional allocation, it is not influenced by dependence between Xi.

When applied to premium, premium is higher for riskier LOBs (ex: home vs auto), thus riskier LOB has lower implied LR and higher margin to reflect riskiness.

Question 18

Describe the Equal Risk Allocation method. Is it exogenous or endogenous?

Answer 18

Can be written endogenously or exogenously:
(Endogenous) sum of a(Xi, p) = a(X)
(Exogenous) sum of rho(Xi, p) = a(X)

The idea is to capitalize each unit to the same probability of default 1-p* (hence equal risk).

Under the other allocation methods, prob of default differs by unit.

This allocation is often used in Lloyd’s/London Market

Each unit, stand-alone, has an expected excess loss beyond its allocated capital.

Objective is to allocate capital to minimize sum of these expected excesses.

Question 19

Describe the Marginal Business Unit Allocation method (aka Merton-Perold method).

Answer 19

ai = a(X) - a(X excluding i)

Attributes to each unit the reduction in capital from dropping it from portfolio.

We compute the risk capital for the total portfolio with and without business unit. Dif in risk capital is the allocated capital for that unit.

This method is not additive (there is an unallocated portion).

It mimics the actions of a corporate manager who optimizes atomically (keeping or dropping whole units).

Question 20

Describe the Marginal Business Euler Gradient Allocation method. Is it exogenous or endogenous?

Answer 20

a(X) = sum of Xi * da/dXi

Unlike Merton-Perold, we do not calculate the change in capital from removing an entire business unit.

Instead, we calculate the marginal change in capital given a marginal change in the amount of unit i written.

This is equivalent to taking the derivative of capital with respect to unit i. Then, we apply this derivative to the unit total to obtain the allocation.

This method is endogenous.

Question 21

Which allocation method is the only one suitable for performance measurement? Explain.

Answer 21

A prior study showed that Euler allocation is the only one suitable.

When Euler allocation is used, growing (or shrinking) lines with a higher (or lower) RORAC always improve the average return.

This is not the case with the other allocation methods.

Question 22

Describe the Aumann-Shapley Allocation method.

Answer 22

Based on game theory.

ci = sum over S of |S|! (n-|S|-1)! (c(S U i) - c(S)) / n!

|S| denotes the number of elements in S.

Unit A will accept any premium less than its stand-alone premium to join B and form AB.

min(premA) = premAB - premB

When P is SA, players are best served by creating a pooled portfolio.

Question 23

Describe game theory.

Answer 23

There are n units and a cost function c on subsets of (1,2,…,n).

Suppose S is subset of (1,2,…,n), then c(S) is the cost of operating the units in S together.

In a game, business units can join together or stand alone.

The central premise is that we want an allocation such that no unit is allocated more than it would incur going in alone.

This is because there is no incentive in joining others if there is no benefit from doing so.

Question 24

Briefly describe the no-undercut property.

Answer 24

Shows how to allocate tot cost to players s.t. every is better off joining together rather than going alone or joining sub-coalitions.

An allocation where no unit is allocated more than its stand-alone cost satisfies the no-undercut property.

The set of allocations that satisfies the no-undercut property is called the CORE of the game.

Question 25

Contrast atomic and fractional game.

Answer 25

The game is called atomic when each unit is completely in or completely out.

The game is called fractional when units can form fractional coalitions.

Question 26

Identify 6 desirable qualities of Shapley allocation

Answer 26

1. It is additive (efficiency property: no portion unallocated)
2. It is symmetric (if 2 units I and j increase the cost of every S that contains neither i nor j by the same amount, then ci = cj)
3. It is linear in game theory (decomposable in the sense of Kreps)
4. It is homogenous if c is.
5. If c is SA, then Shapley value satisfies no-undercut property.
6. It allocates no capital to a constant risk (null player or riskless property)

Question 27

List 2 drawbacks from Shapley allocation

Answer 27

1. To allocate n units, we must compute 2^n marginal impacts, which is impractical.
2. If a unit is sub-divided further in 2 new units, then allocations assigned to the other units also change.

Question 28

Briefly explain a nice feature of Shapley allocation under two-player game

Answer 28

The Shapley allocation for each unit is the average of the stand-alone capital for the unit and the Merton-Perold marginal capital of removing the unit from total.

ai = 50%a(Xi) + 50%(a(X) - a(X excl. i))

ci = 50%c(i) + 50%(c(i,j) - c(j))

Question 29

Describe the co-measures allocations

Answer 29

Each unit’s allocation is based on the allocation condition for the whole firm.

If a risk measure rho can be written in form rho(X) = E(h(X)L(X)) where h is an additive function, then the co-measure E(h(Xi)L(X)) can be used as an allocation to unit i.

Question 30

Give 2 examples of co-measures.

Answer 30

1. Covariance
   Let rho = kVar(X), we can write rho as E(h(X)L(X)) where h(X) = (X-E(X)) and L(X) = k(X-E(X)), thus E(h(Xi)L(X)) = kcov(Xi,X) is called the covariance allocation.
   All things equal, line with lowest LR is the riskiest.
   Cov between LOB and total higher means largest losses occur when insurer total losses are the highest.
2. coTVaR
   Let rho = TVaRp(X), we can write rho as E(h(X)L(X)) where h(X) = X and L(X) = 1/(1-p) * 1_x>=xp(X), thus E(h(Xi)L(X)) = 1/(1-p)E(Xi1_x>=xp(X)) is called the coTVaR allocation.
   It is the avg of Xi over the cases where total X greater or equal to Xp.

Question 31

Contrast co-measures and standalone risk measures.

Answer 31

Co-measures are always additive (sum of co-measures is 100% of the aggregate risk measure).

Standalone risk measures do not sum up to agg risk measure which is why proportional method needs to be used to allocate.

Question 32

Describe 1 situation in which exogenous allocations are more appropriate.

Answer 32

Exogenous allocations are helpful when total risk measure is difficult to allocate.

For ex: rating agency capital models combining factor-based charges for non-cat lines with VaR-based charges for cats are difficult to allocate by same risk model. Instead, we can use an auxiliary measure rho to allocate total capital charge from the model.

Question 33

Explain 2 methods for deriving premiums implied by capital allocation.

Answer 33

1. Allocated CCoC Pricing
   Allocate assets (or capital) to each unit and then use those figures to calculate premium based on CCoC formulas.
2. Direct allocation of PCPs
   Allocate tot premium to each unit directly using the allocation methods.

Question 34

Briefly describe 5 things to consider when selecting allocation method.

Answer 34

1. The allocation should be fair, which means stakeholders consider the allocation reflective of reality and not overly influenced by factors not specific to the problem being solved by allocation.
2. The allocation is not overly tail-focused.
3. Professional actuarial standards should be followed.
4. The practitioner should strive for simplicity, transparency, fairness, objectivity and best practice.
5. Stakeholder concerns should be addressed.

Question 35

Explain how to calculate losses to unit i in presence of default.

Answer 35

In the agg tot portfolio, losses are split between insurer payments and insurer default as follows:
X = X lim to a + (X-a)+

Our goal is to allocate actual agg payments (X lim to a) to each unit. We assume equal priority, which means each unit shares in default proportionately: Xi = Xi(X lim to a)/X + Xi(X-a)+/X

Final payments are:
Xi = Xi if X smaller or equal to a
Xi = Xi*a/X if X greater than a