Mildenhall CH12-15

Question 1

Two approaches to risk-adjust the target return and fix the amount of capital

Answer 1

1. Risk-adjusted return on capital (RAROC) methods that say return varies with risk
2. Return on risk-adjusted capital (RORAC) methods that say return is constant when capital reflects risk

Question 2

Describe RAROC (risk-adjusted return on capital)

Answer 2

When a single constant CoC is combined with a capital allocation that normalizes for risk.
We call this the allocated CCoC pricing method.

Question 3

What’s the problem with constant CoC?

Answer 3

Since insurers are funded by different types of capital with varying cost characteristics, each unit will consume a unit mix of capital. So even all units have a constant CoC within a layer, the weighted cost of capital across layers will differ since each unit has a different mix of capital.

Question 4

How to determine the constant CoC to be used

Answer 4

We rely on the estimate of the insurer’s weighted average cost of capital (between debt, reinsurance, and equity)
We can quantify debt and reinsurance costs, so leaving the equity cost of capital as the most unknown input.

Question 5

3 desirable properties of an allocation.

Answer 5

1. It should work at any level of granularity, even down to the policy
2. It should be decomposable, meaning that the allocation to a sum of random variable equals the sum of their allocations
3. it should be computed using a single, consistent formula

Question 6

What is endogenous allocation

Answer 6

When the same risk measure is used to determine the total and allocate it.

Question 7

What is exogenous allocation

Answer 7

When the same risk measure is not used to determine the total risk measure and allocate it. When the amount is stipulated by an external risk measure and allocated using a auxiliary measure.
Occurs when a regulator or rating agency determines the amount of capital, but an insurer determines its allocation.

Question 8

What allocations is the solution to optimization problem

Answer 8

Equal risk VaR.
Each unit, stand-alone, has an expected loss beyond its allocated capital. The objective is to allocate capital to minimize the sum of these expected excesses.

Question 9

Describe marginal business unit or Merton-Perold method

Answer 9

it attributes to each unit the reduction in capital from dropping it from the portfolio.
This method is not additive: generally the sum of allocation is less than total capital.

Question 10

Describe marginal business Euler gradient allocation

Answer 10

It’s based on the marginal change in capital given a marginal change in the amount of unit i written.
This is endogenous allocation.

Question 11

Why is the Euler gradient allocation is only one suitable for performance measurement

Answer 11

The growing lines with higher return to marginal capital always improve the average return.
This assumes that the average return is sufficient for deciding whether a portfolio’s pricing has improved, it does not allow for the possibility that a change in portfolio composition causes a change in the cost of capital.

Question 12

What is no-undercut property

Answer 12

an allocation where no unit is allocated more than its stand-alone costs

Question 13

When is a game called atmoic

Answer 13

when each unit is in or out

Question 14

When is a game called fractional or fuzzy

Answer 14

when units can form tractional coalitions (quota share)

Question 15

Desirable properties of Shapley allocation

Answer 15

1. It’s an allocation, so it’s additive. In game theory, this is called the efficiency property
2. It’s symmetric. if two units i and j increases the cost of every S that contains neither i nor j by the same amount, then cj = cj
3. It’s decomposable. It’s called linear in game theory
4. It’s homogeneous if c is
5. if c is subadditive then Shapley value satisfies the no-undercut property
6. It allocates no capital to a constant risk, called the null player or riskless allocation property.
   Shapley value is the only allocation that is additive, symmetric, linear and allocates no capital to constant risks

Question 16

2 drawbacks of Shapley allocation

Answer 16

1. To allocate to n units involves computing 2^n marginal impacts, which quickly becomes impractical.
   2 If a unit is subdivided further into two new units then allocations assigned to the other units typically change.

Question 17

Characteristics of Aumann-Shapley allocation

Answer 17

it satisfies linearity, decomposable, and either continuity or non-negativity when a is non decreasing

Question 18

When is Exogenous allocation helpful

Answer 18

When a is determined by a risk measure that is difficult to allocate.

Question 19

Two ways to derive premiums using capital allocation methods

Answer 19

1. Allocated CCoC pricing (allocate assets or capital to each unit and then use those to calculate premium based on the CCoC formulas)
2. Direct Allocation of PCPs (allocate total premium to each unit directly using the allocation methods)

Question 20

Considerations when choosing an allocation method

Answer 20

1. All allocation must be considered fair. It means the stakeholders consider the allocation reflective of reality and not overly influenced by factors not specific to the problem the allocation is supposed to solve.
2. The allocation is not overly tail-focused. Aumann-Shapley allocation can produce an overly tail-focused answer. The body of the risk distribution also matters.
3. Professional actuarial standards should be followed
4. Recognizing the scope of the larger task within which allocation is a part
5. The value of simplicity, transparency, fairness, objectivity and precedent.
6. Sensitivity to stakeholder concerns.
7. Communication and persuasion.

Question 21

Describe natural allocation of coherent risk measure

Answer 21

It’s an explicit allocation formular that applies to coherent risk measures.
The allocation is natural because it entails no additional choices (no new probability threshold for example).
It follows the finance philosophy to adjust the probabilities and act risk neutral.

Question 22

When is marginal allocation method useful?

Answer 22

It’s useful for performance management, meaning they give the right grow or shrink signal for portfolio optimization in a constant CoC world.

Question 23

Why co-measure allocation is useful

Answer 23

It’s intuitive, practical, and easy to apply.
They can be interpreted as risk-adjusted probabilities, and this interpretation provides a philosophical alignment with finance. and they are closely related to the natural allocation.

Question 24

Requirements for the risk measure for linear natural allocation

Answer 24

The risk measure must be law invariant.
The second constraint restricts attention to risk measures where it does hold.
This can be achieved by assuming the risk measure is comonotonic additive, so meaning the risk measure is spectral risk measure.

Question 25

Two observations about unit margins by layer

Answer 25

1. Units where alphai(x) increase with x always have a positive margin 2. A thin-tailed unit aggregated with a thick-tailed unit can have a negative margin for lower asset layers

Question 26

What does law invariant mean?

Answer 26

for a law invariant pricing approach the layer CoC must be the same for all units. Law invariance implies the risk measure is only concerned with the attachment probability of the layer and not with the cause of loss within the layer.

Question 27

Observations for strictly concave distribution with decreasing CoC

Answer 27

1. Lower layers of assets, around the expected losses, have a high CoC, but offsetting this, they are mostly funded by premium and require little supporting investor capital. 2. Higher layers of assets have a lower CoC but higher capital content; they are mostly funded by capital. 3. Lowe volatility units tend to have losses close to their expected value, regardless of the value of total losses, and so consume relatively more of the expensive lower layer capital and a smaller proportion of cheaper, higher asset layers. Offsetting this there is more capital in higher layers than lower ones. 4. High volatility units tend to be a larger proportion of total losses when the total is large, and so they consume a greater proportion cheaper, higher layer capital

Question 28

The Percentile Layer of capital (PLC) approach to capital allocation has what features

Answer 28

1. It allocates capital to all losses, rather than allocating capital only to extreme losses 2. It does not allocate capital in proportion to average loss. Instead, it allocates a disproportionate amount of capital to severe losses. 3. It does not rely on specialized parameters to allocate capital. it's very practical approach.