Kreps - Riskiness Leverage Models

Question 1

each insurer has number of different liabilities and assets which are all supported by

Answer 1

a single pool of shared capital

Question 2

typically the mean is supported

Answer 2

by the reserves

Question 3

variability is supported

Answer 3

by surplus

Question 4

capital often has to be allocated but

Answer 4

really it is the allocation of cost of capital which is relevant

Question 5

allocation of cost of capital which is relevant can help

Answer 5

derive an appropriate risk load

Question 6

Several desirable qualities for an allocatable risk load formula

Answer 6

1. it can be able to be allocated down to any level
2. risk load of any sum of random variables should = sum of risk loads allocated individually
3. same additive formula can be used to calculate risk load for any subgroup or groups of groups

Question 7

if desirbale qualities apply

Answer 7

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

Question 8

Riskiness leverage ratio

Answer 8

arbitrary selection by management, allowing their views towards risk be incorporated

Question 9

capital to support X (sum of liabilities)

Answer 9

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

Question 10

Kreps states that riskiness leverage models have the following form

Answer 10

-should use Monte Carlo simulation to calculate the above integrals

Question 11

riskiness leverage L depends only

Answer 11

on sum of individual variables

Question 12

riskiness leverage models equations indicate

Answer 12

that some variables may have negative risk loads if they are below their mean when riskiness leverage is large

Question 13

desirable feature of variable that have negative risk loads

Answer 13

these variables can be considered to be hedges -> should not require a risk load and in fact should reduce overall required risk load

Question 14

Several different ways to express the equation for R

Answer 14

1. 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
2. expresses risk load as integral over risk load density
   - advantages of this formula is that it shows which outcomes contribute most to risk load

Question 15

Properties of Risk Load

Answer 15

• risk load for a constant c, is 0; R(c)=0
• risk load with scale with a currency change; R(λX)=λR(X)

Question 16

it is possible to make L a function of x/S where S is total surplus

Answer 16

• 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

Question 17

Examples of Leverage Models

Answer 17

Risk neutral

Variance

VaR

TVaR

Semi-variance

mean downside deviation

proportional excess

Question 18

Risk neutral

Answer 18

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

Question 19

Variance

Answer 19

whole distribution is relevant

Question 20

TVaR

Answer 20

-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.

Question 21

VaR

Answer 21

-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)

Question 22

semi-variance

Answer 22

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

Question 23

mean downside deviation

Answer 23

-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)

Question 24

proportional excess

Answer 24

-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

Question 25

question of appropriate measure of risk to insurer that generates a surplus need

Sources of risk

Answer 25

Risk of not making plan

Risk of serious deviation from plan

Risk of not meeting investor analysts’ expectations

Risk of downgrade from rating agencies

Risk of triggering a regulatory notice

Risk of going into receivership

Risk of not getting a bonus

Question 26

management’s desired properties of riskiness leverage ratio/measure may be that it

Answer 26

Be a down side measure

Be roughly constant for excess that is small compared to capital

Become much larger for excess that significantly impacts capital

Reduces to 0 or at least doesn’t increase for excess that significantly exceeds capital (once you are buried, it doesn’t matter how much dirt is on top)

It should be allocated to any desired level of definition

The risk load allocated for any sum of random variables should be sum of the risk load amounts allocated individually

Same additive formula is used to calculate the risk load for any subgroup or group of groups

Question 27

regulator’s preferences for riskiness leverage

Answer 27

Be 0 until capital is seriously impacted

Not decrease for excess that significantly exceeds capital because of risk to state guaranty fund (regulator would be concerned about these higher level of losses, because these would need to be paid by the guaranty fund or alternatively will be borne by the insureds)

Question 28

when determining whether to purchase reinsurance

Answer 28

insurer should compare reduction in cost of capital to cost of reinsurance