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1

Total assets and what is supported by each component (2)

(Kreps)

total assets = reserves + surplus

reserves support mean of assets & liabilities

surplus supports variability of assets & liabilities

2

Total capital (C)

(Kreps)

total capital = mean outcome + risk load

3

Desirable qualities for an allocatable risk load (3)

(Kreps)

1. ability to be allocated to any level
2. allocated risk load for a sum of random variables should = sum of individually allocated risk load amounts
3. same additive formula is used to calculate risk loads for any sub-group or grouping

4

General form of riskiness leverage models

(Kreps)

R = integral of f(x) * (x - mu) * L(x) dx
f(x)dx can be called dF-bar for joint probability distributions

where f(x) = joint probability distribution
and L(x) = riskiness leverage function for total losses

5

Risk load (R) and capital (C) across multiple LOB

(Kreps)

R = sum of R(k)'s
C = sum of C(k)'s

where k = individual LOB

6

Advantage of co-measures

(Kreps)

they are automatically additive

7

Disadvantage of co-measures

(Kreps)

can be challenging to find appropriate forms of the riskiness leverage function L(x)

8

Conditions for negative risk loads (2) and when it is desirable

(Kreps)

1. x(k) < mean
2. large L(x)

desirable for hedges, occurs when there is a low correlation with total losses

9

Properties of riskiness leverage models (4)

(Kreps)

1. desirable qualities for allocatable risk loads are satisfied
2. no risk load for constant variables - R(c) = 0
3. risk load will scale with change in currency - R(lambda * x) = lambda * R(x)
4. may not produce a coherent risk measure

10

Coherent risk measures

(Kreps)

satisfy sub-additivity requirement

R(x + y) <= R(x) + R(y)

11

Super-additivity

(Kreps)

R(x + y) > R(x) + R(y)

(not coherent)

12

Types of riskiness leverage functions, L(x) (7)

(Kreps)

1. risk-neutral
2. variance
3. VaR
4. TVaR
5. semi-variance, SVaR
6. mean downside deviation
7. proportional excess

13

Risk-neutral form of the riskiness leverage function, L(x)

(Kreps)

L(x) = c

14

Situations when a risk-neutral form of L(x) might be appropriate (2)

(Kreps)

1. risk of ruin if risk of ruin is very small compared to capital OR capital is infinite
2. risk of not meeting plan if indifferent about making plan

15

Variance form of the riskiness leverage function, L(x)

(Kreps)

L(x) = (beta / surplus) * (x - mu)

16

Relevant part of the distribution when using the variance form of L(x)

(Kreps)

entire distribution (just as much risk associated with good & bad outcomes)

17

Surplus (S) when using the variance form of L(x)

(Kreps)

S = sqrt(beta * var(x))

18

Forms of the riskiness leverage function, L(x) where the risk load increases quadratically (2)

(Kreps)

1. variance
2. semi-variance, SVaR

19

TVaR form of the riskiness leverage function, L(x)

(Kreps)

L(x) = theta(x - x(q)) / (1 - q)

where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0

x(q) = value of x so F(x(q)) = q

20

Relevant part of the distribution when using the TVaR form of L(x)

(Kreps)

only the high end of the distribution is relevant

21

VaR form of the riskiness leverage function, L(x)

(Kreps)

L(x) = delta(x - x(q)) / f(x(q))

where delta(x) = 0 everywhere except 0 and integrates to 1

22

Coherent riskiness leverage function, L(x)

(Kreps)

TVaR

23

Capital (C) when using the VaR form of L(x)

(Kreps)

C = x(q) = VaR

x(q) = value of x so F(x(q)) = q

24

Relevant part of the distribution when using the VaR form of L(x)

(Kreps)

only the single VaR point is relevant

25

Semi-variance, SVaR, form of the riskiness leverage function, L(x)

(Kreps)

L(x) = (beta / surplus) * (x - mu) * theta(x - mu)

where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0

26

Risk load when using the risk-neutral form of L(x)

(Kreps)

risk load = 0

27

Risk load when using the semi-variance, SVaR, form of L(x)

(Kreps)

risk-load = semi-variance (only non-zero for results worse than the mean)

28

Relevant part of the distribution when using the semi-variance, SVaR, form of L(x)

(Kreps)

only bad results are relevant

29

Mean downside deviation form of the riskiness leverage function, L(x)

(Kreps)

L(x) = beta * theta(x - mu) / (1 - F(mu))

where theta is a step function with:
theta(x) = 0 for x <= 0 and
theta(x) = 1 for x > 0

30

Risk load when using the mean downside deviation form of L(x)

(Kreps)

risk load = multiple of mean downside deviation

31

Condition when the mean downside deviation = TVaR form of the riskiness leverage function, L(x)

(Kreps)

x(q) = mu

32

Riskiness leverage ratio for the mean downside deviation form of L(x)

(Kreps)

riskiness leverage ratio = 0 below the mean and constant above the mean

33

Capital allocation using the mean downside deviation form of L(x)

(Kreps)

assigns capital for bad outcomes in proportion to how bad they are

34

Proportional excess form of the riskiness leverage function, L(x)

(Kreps)

L(x) = h(x) * theta(x - (mu + delta)) / (x - mu)

35

Capital allocation using the proportional excess form of L(x)

(Kreps)

individual allocation for any given outcome is pro rata to its contribution to the excess over the mean

36

Sources of risk that could impact riskiness leverage ratio selection (7)

(Kreps)

1. not making plan
2. serious deviation from plan
3. not meeting investor expectations
4. ratings downgrade
5. triggering regulatory notice
6. going into receivership
7. not getting a bonus

37

Management properties of a riskiness leverage ratio (4)

(Kreps)

must:
1. be a downside measure
2. be approximately constant for excess losses that are small compared to capital
3. become much larger for excess losses significantly impacting capital
4. go to 0 for excess losses significantly exceeding capital

38

Regulator properties of a riskiness leverage ratio (2)

(Kreps)

must:
1. be 0 until capital is seriously impacted
2. not decrease (due to risk to state guaranty fund)

39

Reinsurance premium

(Kreps)

reinsurance premium = E[ceded loss] + load % * std. dev(ceded losses)

40

Reinsurance net cost

(Kreps)

reinsurance net cost = reinsurance premium - E[ceded loss]

41

Impact of reinsurance on total return and capital

(Kreps)

positive net cost of reinsurance will reduce total return, but allow firm to release capital

42

Total return

(Kreps)

total return = income / surplus

43

Surplus released from purchasing reinsurance

(Kreps)

surplus release = current surplus - target surplus

44

Reduction in cost of capital from purchasing reinsurance

(Kreps)

reduction in cost of capital = cost of capital * surplus release

45

Determining whether reinsurance treaties are worth pursuing

(Kreps)

if reduction in cost of capital > net avg cost of reinsurance, then reinsurance is worth pursuing

46

Potential actions to take if return on surplus < target return on surplus (3)

(Kreps)

1. write less business (shift volume to more profitable LOB)
2. purchase reinsurance to reduce required surplus
3. raise premium by increasing the profit load

47

Potential problems with changing LOB volume when return on surplus < target (3)

(Kreps)

1. LOBs may be two parts of an indivisible policy
2. regulatory requirements
3. may not be cost effective to undergo major UW effort

48

Risk load, R(k), for a line of business using riskiness leverage models

(Kreps)

R(k) = average of all L(x) * (x(k) - mu(k)) for individual loss amounts

**L(x) is always for total losses, NOT individual LOB