Bodoff - capital allocation by layer Flashcards
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
- regulators and rating agencies influence amount of capital
- investors influence the amount of capital and required return on capital
insurer incurs an “overhead” cost related to the required return on capital and management will need to
allocate this cost to the different parts of the business
this allocation of cost of capital will have significant impact on many factors
Measured profitability of segment
Pricing levels
Volume of business that firm can write
cost of capital should be allocated to
business units/products/perils and policies that contribute to loss scenarios that “use” capital
proposed an alternate interpretation of 99th percentile capital requirement
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
similar interpretation can be applied to TVaR
firm can hold capital even for average loss beyond 99th percentile
incremental capital amount between 99th and 98th percentile layer can be attributed to losses that
exceed 98th percentile
Bodoff assumes that the capital required at loss percentile =
that loss level
each layer of capital is used by
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
each event should receive an allocation based on
its conditional exceedance probability
capital allocation is essentially proportional
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
following procedure would be used to allocate a percentile layer of capital across loss events
- Layer of Capital = [x(α+j)-x(α)]
- allocation to loss event x(i) = [x(α+j)-x(α)]*prob(x=x(i))/prob(x>x(α))
- sum across all loss events that have losses greater than α
Two Alternative Views of Capital Allocation by Percentile Layer
horizontal procedure
vertical procedure
Horizontal procedure:
allocationg each layer of capital to loss events that penetrate the layer
-need to perform this for all layers of capital
Vertical procedure
allocates capital to each loss event based upon layers that it penetrates
-need to perform this allocation across all loss events
this procedure of capital allocation by layer dictates that any loss event’s allocated capital depends on:
- probability that loss occurs f(x)
- severity of loss or extent to which loss penetrates the layer of capital: integration goes until x
- number of events with which capital is shared 1/(1-F(y)); as this number reduces, capital allocated to each event increases
2 factors simultaneously affect the allocated capital
- allocated capital will increase because the loss will receive allocation from an additional layer of capital
- allocated capital will change to extent that loss is less likely to occur and therefore will receive a lower allocation of capital
we can derive the total cost of the event, given that it occurs, by
adding the loss amount x to cost of capital
capital allocation by percentile layer recognizes
that even though capital held by firm is targeted for a CAT scenario, some of the capital would also benefit more likely, moderate events
there are other methods that allocate capital to a broader range of events that consume capital
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
capital allocation by percentile layer will allocate more capital cost
to these unlikely events
capital allocation by percentile layer described can be based on TVaR
- 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
when deriving the cost of capital, we should recognize the benefit
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
risk load increases with respect to
loss amount at an increasing rate
even if x is very small, risk load is positive
-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
advantages of capital allocation by percentile layer
- allocates capital to entire range of loss events rather than just most extreme
- tends to allocate more capital to events that are more likely
- tends to allocate significantly more capital to events that are more severe
- 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
- always allocates 100% of capital
- provides a method to allocate capital by layer
disutility function
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])
