Kreps - Riskiness Leverage Models Flashcards
each insurer has number of different liabilities and assets which are all supported by
a single pool of shared capital
typically the mean is supported
by the reserves
variability is supported
by surplus
capital often has to be allocated but
really it is the allocation of cost of capital which is relevant
allocation of cost of capital which is relevant can help
derive an appropriate risk load
Several desirable qualities for an allocatable risk load formula
- it can be able to be allocated down to any level
- risk load of any sum of random variables should = sum of risk loads allocated individually
- same additive formula can be used to calculate risk load for any subgroup or groups of groups
if desirbale qualities apply
senior management would be able to allocate capital to regions and then have regional management allocate their share of capital to the LOBs
-these allocations @ LOB level should add to original capital
Riskiness leverage ratio
arbitrary selection by management, allowing their views towards risk be incorporated
capital to support X (sum of liabilities)
C = u + R
u = mean of X
R = risk load for X
we can think of this in balance sheet terms: capital would be = total assets, mean = booked liabilities, risk load = surplus
Kreps states that riskiness leverage models have the following form
-should use Monte Carlo simulation to calculate the above integrals
riskiness leverage L depends only
on sum of individual variables
riskiness leverage models equations indicate
that some variables may have negative risk loads if they are below their mean when riskiness leverage is large
desirable feature of variable that have negative risk loads
these variables can be considered to be hedges -> should not require a risk load and in fact should reduce overall required risk load
Several different ways to express the equation for R
- expresses the risk load as the probability weighted average of risk loads over outcomes of the total loss
- this riskiness leverage factor would reflect that all dollars are not equally risky
- those that trigger analyst or regulatory tests would be considered to be more risky - expresses risk load as integral over risk load density
- advantages of this formula is that it shows which outcomes contribute most to risk load
Properties of Risk Load
- risk load for a constant c, is 0; R(c)=0
- risk load with scale with a currency change; R(λX)=λR(X)
it is possible to make L a function of x/S where S is total surplus
- S should be readily available, liquefiable capital
- makes sense that S should be incorporated
- for example, assume that loss is normally distributed with mean of 100 and SD of 5; is insurer subject to risk of ruin? We could conclude that it was if S=105 but on other hand if S=200, it is not
Examples of Leverage Models
Risk neutral
Variance
VaR
TVaR
Semi-variance
mean downside deviation
proportional excess
Risk neutral
Risk load = 0
L(x) is constant. Excess is small relative to capital.
-according to Kreps, this is appropriate if range of x where f(x) is significant is small compared to available capital or alternatively if capital levels are infinite
Variance
whole distribution is relevant
TVaR
-riskiness leverage ratio is 0 up to a point and then constant
measure only attributes risk to the portion of the distribution which exceeds the defined threshold -> coincides with the rating agency downgrade
because the impact of how much the expectation was not met also matters. TVaR starts at the loss that would fail to meet expectation and then gets larger as the loss increases.
VaR
-only the selected percentile is relevant -> shape of loss distribution doesn’t matter
riskiness leverage measure is 0 except at one point (the selected percentile). Since we are only concerned about one point that would cause a rating agency downgrade, VaR is appropriate if the percentile selected corresponds to the level at which downgrade would occur
the insurer is most likely concerned about a particular level of losses. Losses below this level would not threaten surplus. And losses higher would not be as relevant (once the losses exceed surplus, the insurer is already in trouble)
semi-variance
risk load is only positive for losses that exceed the mean
-thought here is that risk is only relevant for bad results
don’t care about favorable deviations – semi-variance only considers deviations above the mean
mean downside deviation
-Kreps believes that this is most natural naïve measure, as it essentially assigns capital for bad outcomes in proportion to how bad they are
Allocates riskiness based on how bad the outcome is
Interested in bad deviation from the mean (reserve)
proportional excess
-allocation for any outcome is pro-rata to its contribution to excess over the mean
Risk load is proportional to the amount the claim exceeds the reserve
