Bodoff - capital allocation by layer

Question 1

amount of capital that an insurer must hold as well as required rate of return on this capital are often determined by external forces in environment

Answer 1

1. regulators and rating agencies influence amount of capital
2. investors influence the amount of capital and required return on capital

Question 2

insurer incurs an “overhead” cost related to the required return on capital and management will need to

Answer 2

allocate this cost to the different parts of the business

Question 3

this allocation of cost of capital will have significant impact on many factors

Answer 3

Measured profitability of segment

Pricing levels

Volume of business that firm can write

Question 4

cost of capital should be allocated to

Answer 4

business units/products/perils and policies that contribute to loss scenarios that “use” capital

Question 5

proposed an alternate interpretation of 99th percentile capital requirement

Answer 5

firm holds enough capital even for 99th percentile loss

key difference between this and prior definition is that this also considers losses at lower percentiles

Question 6

similar interpretation can be applied to TVaR

Answer 6

firm can hold capital even for average loss beyond 99th percentile

Question 7

incremental capital amount between 99th and 98th percentile layer can be attributed to losses that

Answer 7

exceed 98th percentile

Question 8

Bodoff assumes that the capital required at loss percentile =

Answer 8

that loss level

Question 9

each layer of capital is used by

Answer 9

loss events that pierce the lower bound of the layer, but not by losses that are under the lower bound

• each layer of capital should only be allocated to events that penetrate the layer
• some of the losses that penetrate the layer are more likely to do so than others

Question 10

each event should receive an allocation based on

Answer 10

its conditional exceedance probability

Question 11

capital allocation is essentially proportional

Answer 11

to the peril’s average loss

-this is not always appropriate ie CAT perils should be allocated a greater portion of capital even if the average loss from peril is relatively low

Question 12

following procedure would be used to allocate a percentile layer of capital across loss events

Answer 12

1. Layer of Capital = [x(α+j)-x(α)]
2. allocation to loss event x(i) = [x(α+j)-x(α)]*prob(x=x(i))/prob(x>x(α))
3. sum across all loss events that have losses greater than α

Question 13

Two Alternative Views of Capital Allocation by Percentile Layer

Answer 13

horizontal procedure

vertical procedure

Question 14

Horizontal procedure:

Answer 14

allocationg each layer of capital to loss events that penetrate the layer

-need to perform this for all layers of capital

Question 15

Vertical procedure

Answer 15

allocates capital to each loss event based upon layers that it penetrates

-need to perform this allocation across all loss events

Question 16

this procedure of capital allocation by layer dictates that any loss event’s allocated capital depends on:

Answer 16

1. probability that loss occurs f(x)
2. severity of loss or extent to which loss penetrates the layer of capital: integration goes until x
3. number of events with which capital is shared 1/(1-F(y)); as this number reduces, capital allocated to each event increases

Question 17

2 factors simultaneously affect the allocated capital

Answer 17

1. allocated capital will increase because the loss will receive allocation from an additional layer of capital
2. allocated capital will change to extent that loss is less likely to occur and therefore will receive a lower allocation of capital

Question 18

we can derive the total cost of the event, given that it occurs, by

Answer 18

adding the loss amount x to cost of capital

Question 19

capital allocation by percentile layer recognizes

Answer 19

that even though capital held by firm is targeted for a CAT scenario, some of the capital would also benefit more likely, moderate events

Question 20

there are other methods that allocate capital to a broader range of events that consume capital

Answer 20

they do this by allocating based on conditional probability

-issue is that because methods are driven by relative probability, they allocate insufficient capital to the severe yet unlikely events

Question 21

capital allocation by percentile layer will allocate more capital cost

Answer 21

to these unlikely events

Question 22

capital allocation by percentile layer described can be based on TVaR

Answer 22

• in order to implement this, capital allocation up to 99th will be the same as allocation for VaR & there is additional layer of capital that needs to be allocated to losses that exceed VaR: TVaR-VaR
• this additional layer would need to be allocated in proportion to each event’s average loss in excess of TVaR

Question 23

when deriving the cost of capital, we should recognize the benefit

Answer 23

from premium

-premium less expenses would be contributed to capital and so we need to subtract the premium net of expenses from capital before applying the cost of capital rate

Question 24

risk load increases with respect to

Answer 24

loss amount at an increasing rate

Question 25

even if x is very small, risk load is positive

Answer 25

-this implies that even if the event is small and less than the mean, it should be allocated some capital and have a positive risk load

-tail based methods would allocate the greatest amount of capital to severe but unlikely events but capital allocation by percentile would not

Question 26

advantages of capital allocation by percentile layer

Answer 26

1. allocates capital to entire range of loss events rather than just most extreme
2. tends to allocate more capital to events that are more likely
3. tends to allocate significantly more capital to events that are more severe
4. eliminates the need to select an arbitrary percentile level to use as a basis for allocating capital as it is based on all relevant percentile threshold
5. always allocates 100% of capital
6. provides a method to allocate capital by layer

Question 27

disutility function

Answer 27

given a loss event x after accounting for its cost of capital

Premium = disutility function * f(x)

P=f(x)( x + additive risk load)

P = E[L] + r/(1+r)(allocated capital - E[L])