06_Mildenhall_pt3

Question 1

Formula for Premium Under the Allocated CCoC Method

Answer 1

The total premium is calculated as follows:
𝑃 = 𝑙 + 𝛿(𝑎 − 𝑙)
The allocated premium to each unit 𝑖 is calculated as follows:
𝑃_i = 𝑙_i + 𝛿_i(𝑎_i − 𝑙_i)

Question 2

RORAC Method

Answer 2

The RORAC method combines a constant cost of capital with a capital allocation that normalizes for risk. We call this the allocated CCoC pricing method.

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

There is no widely accepted practice for determining the risk-adjusted capital.
There are a number of allocation methods that could be used to determine the risk-adjusted capital for each unit.

Question 3

Three Desirable Properties of an Allocation

Answer 3

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

Question 4

Endogenous vs. Exogenous Allocation

Answer 4

When the same risk measure is used to determine the total ρ AND allocate it, we call it an endogenous allocation.
When a different risk measure is used, we call it an exogenous allocation.

Question 5

Marginal Business Euler Gradient Allocation

Answer 5

Unlike the Merton-Perold method, we do not calculate the change in capital from removing an entire business unit with this method. 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 𝑖 total to obtain the allocation. This method is endogenous.

Question 6

Explain why Euler Allocation is the only allocation suitable for performance measurement.

Answer 6

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

Question 7

Definition of a Game

Answer 7

• There are 𝑛 units
• There is a cost function 𝑐 on subsets of {1, 2,…, n}. Suppose 𝑆 is a subset of {1, 2,…, 𝑛}. Then, 𝑐(𝑆) is the cost of operating the units in 𝑆 together
• 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 (like a quota-share)
• The set of allocations that satisfies the no-undercut property is called the core of the game

Question 8

Six Desirable Qualities of the Shapley Allocation

Answer 8

1. It is additive
2. It is symmetric. If two units 𝑖 and 𝑗 increase the cost of every 𝑆 that contains neither 𝑖
   nor 𝑗 by the same amount (i.e., 𝑐(S ∪ {i}) = c(S ∪ {j})), then c_i = c_j
3. It is linear in game theory
4. It is homogenous if 𝑐 is
5. If 𝑐 is sub-additive, then the Shapley value satisfies the no-undercut property
6. It allocates no capital to a constant risk

Question 9

Two Drawbacks of the Shapley Allocation

Answer 9

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

Question 10

Two Methods for Deriving Premiums Implied by a Capital Allocation

Answer 10

1. Allocated CCoC Pricing: Allocate assets (or capital) to each unit and then use those figures to calculate premium based on the 𝑃_i = 𝑙_i + δ_i(𝑎_i − 𝑙_i) formula
2. Direct Allocation of PCPs: Allocate total premium to each unit directly using the allocation methods

Question 11

Briefly describe five things that should be considered when selecting an allocation method.

Answer 11

1. The allocation should be fair, which means that stakeholders consider the allocation reflective of reality and not overly influenced by factors not specific to the problem being solved by the 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 12

Briefly describe a major criticism of allocated CCoC pricing.

Answer 12

It is criticized for being heavily weighted towards tail risk (i.e., it tends to allocate higher capital amounts to businesses that are heavily skewed).
The heavy weighting to tail risk is caused by the choice of capital risk measure and applying the same cost of capital to all layers. It is good from a regulatory standpoint but is less useful for managing the interest of the firm’s stakeholders.

Question 13

Two Observations Regarding Unit Margins by Layer in the Context of α_i(𝑥)

Answer 13

1. Units where α_i(𝑥) increases with 𝑥 always have a positive margin by layer
2. A thin-tailed unit aggregated with a thick-tailed unit can have a negative margin for lower asset layers. This is because a thin-tailed unit aggregated with thick-tailed units will have a decreasing α_i(𝑥) with asset layer 𝑥

Question 14

Four Observations for a Strictly Concave Distortion Function with a Decreasing Cost of Capital by Layer

Answer 14

1. Lower layers of assets (around expected losses) have a high CoC. However, they are mostly funded by premium and require little investor capital
2. Higher layers of assets have a low CoC but higher capital content (i.e., they are mostly funded by investor capital)
3. Low volatility units tend to have losses close to their expected values. Thus, they consume more of the expensive, lower layer capital and less of the cheaper, higher layer capital
4. High volatility units tend to be a larger proportion of total losses when the total loss is large. Thus, they consume less of the expensive, lower layer capital and more of the cheaper, higher layer capital

Question 15

Three Features of the Percentile Layer of Capital (PLC) Approach

Answer 15

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. In other words, it’s a very practical approach

Question 16

Can the unit premiums be calculated directly under the PLC approach?

Answer 16

No! A big difference between the PLC approach and the lifted natural allocation is that we can calculate the premium directly using the lifted natural allocation.
This is NOT possible under the PLC approach. We have to allocate the portfolio margin first before we calculate unit premiums.

Question 17

When allocating capital, moving from one approach to another, allocated capital increases.
What implications would be to the LOB?

Answer 17

More capital assigned to it
Increasing cost of capital
So need to include more margin in the premium in order to break even