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CAS Exam 9 > Mildenhall Ch 5: Properties of Risk Measures > Flashcards

Mildenhall Ch 5: Properties of Risk Measures Flashcards

1
Q

Define Insurance Event and provide an example

A

Set of circumstances likely to result in insurance losses.

Ex:
1. Occurrence of a category 4 hurricane
2. Cluster of bad traffic

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2
Q

Define Realistic Disaster Scenario (RDS) and provide an example

A

Specific type of insurance event that is potentially disastrous but plausible.

Ex:
1. category 3 hurricane similar to one that has occured before

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3
Q

Define Probability Event and provide an example

A

Possible state of the world to which probability is assigned

Ex:
1. 2 category 4 hurricanes and an earthquake in same year
2. No natural catastrophe occur in a year

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4
Q

Define Conditional Probability Scenario. How is it calculated?

A

Best estimate probability in states of the world where insurance Ek occurs.

We are assuming Ek occurs, thus the conditional probability scenario.

Qk(A) = P(A&Bk)/P(Bk)

Bk is the set of all states of the world where Ek occurs.
P is the objective probability of the prob event.

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5
Q

Briefly explain how probability scenarios can be used to set capital

A

When a disaster occurs, we want to ensure we have enough money on hand to pay out our claims.

To do this, we can define a set of r RDSs and set our risk measure as:
rho_c(X) = max(E_Q1(X),…,E_Qr(X)) where each E_Qk is a conditional expectation of X given that the specific RDS has occured.

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6
Q

Identify 2 types of uncertainty about probability function P

A
  1. Statistical Uncertainty
  2. Information Uncertainty
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7
Q

Define Statistical Uncertainty

A

P is an estimate subject to the usual problems of estimating cost determining best estimate expected losses.

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8
Q

Complete the sentence:
Statistical Uncertainty concerns…

A

Estimates of objective probabilities.

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9
Q

Define Information Uncertainty

A

P is based on a limited & filtered subset of ambiguous information.

Reflects information asymmetry between insured & insurer and between insurer & investor.

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10
Q

True or False?
Both Statistical and Information uncertainties are diversifiable.

A

False

Only Statistical Uncertainty diversifies across large portfolio and is managed by law of large numbers.

Information Uncertainty is more unavoidable.

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11
Q

Complete the sentence:
Information Uncertainty concerns…

A

Risk aversion & estimates of subjective probabilities.

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12
Q

Briefly explain why uncertainty around P is not a big issue for capital management but is for pricing

A

Any movements in P from best estimate tend to offset each other in tail of X. Hence, measures like TVaR do not differ much with & without P uncertainty.

However, pricing risk measures are focused on risk in estimate of the mean E(X) rather than risk of an RDS outcome (tail event). In this case, uncertainty in P matters.

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13
Q

Describe Generalized Probability Scenarios and provide an example

A

Since information uncertainty is the main issue (cannot be diversified), we can create generalized probability scenarios that reflect it.

They incorporate additional information & not necessarily conditional probability related to P.

Ex:
1. Insureds being systematically misclassified
2. Adverse selection

These additional events are included as part of state of the world & probability associated with them are more subjective.

By incorporating them into pricing risk measure, we have more confidence that premium will cover expected losses.

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14
Q

Identify 3 qualities of coherent risk measures

A
  1. Intuitive and easy to communicate
  2. Can be used for capital & pricing
  3. Has properties alignes with rational risk preferences
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15
Q

Define Translation Invariant (TI) risk measure

A

rho(X+c) = rho(X) + c

Increasing a loss by a constant c increases risk by c.

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16
Q

Provide 2 examples of TI risk measures

A
  1. Mean
  2. VaR
  3. TVaR
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17
Q

Provide an example of non-TI risk measure and explain

A
  1. Variance ( V(X+c) = V(X))
  2. Standard deviation
  3. Factor-based measures such as RBC (since factor is applied to a constant)
  4. All higher central limit theorems
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18
Q

True or False?
All coherent risk measures are TI

A

True since E_Q(X+c) = E_Q(X) + c

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19
Q

Define Normalized (NORM) risk measure

A

rho(0) = 0

The risk of an outcome with no gain or loss equals zero.

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20
Q

Define acceptable risks

A

Assuming rho is normalized, risk is preferred to doing nothing if rho(X) negative.

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21
Q

Define acceptance set of risks

A

Set of risks preferred to doing nothing.

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22
Q

Provide 2 examples of NORM risk measures.

A
  1. TVaR
  2. VaR
  3. Mean
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23
Q

True or False?
All coherent risk measures are NORM

A

True since E_Q(0) = 0

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24
Q

Define Monotone (MON) risk measure

A

If X smaller than Y in all outcomes, than X is preferred over Y.

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25
Q

Provide 2 examples of MON risk measures

A
  1. TVaR
  2. VaR
  3. Mean
  4. Scenario losses
  5. Higher central moments
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26
Q

Provide an example of non-MON risk measure

A

Standard deviation

If X is uniform(0,1) and Y=1, X is smaller than Y for all outcomes, but rho(X) = sigma greater than rho(Y) = 0

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27
Q

True or False?
All coherent risk measures are MON

A

True since E_Q(X) smaller than E_Q(Y) when X smaller than Y

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28
Q

Define the no rip-off property

A

If X smaller than c, rho(X) smaller than c

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29
Q

True or False?
All MON risk measures have no rip-off property

A

True

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30
Q

Define Positive Loading

A

rho(X) greater or equal to E(X)

Reinsurance if a part of insurance portfolio with negative loading.

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31
Q

True or False?
All coherent risk measures have positive loading.

A

False

Coherent risk measures may or may not have positive loading property.

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31
Q

Define Monetary Risk Measure (MRM)

A

Risk measure has a monetary unit.

32
Q

Provide an example of MRMs

A
  1. Mean
  2. TVaR
  3. VaR
33
Q

True or False?
All coherent risk measures are MRM

A

True, by definition coherent risk measures are MRMs

34
Q

Provide an example on non-MRM risk measure

A

Variance since squared units

35
Q

MRM implies automatically which 2 mathematical properties

36
Q

Define Positive Homogeneous (PH)

A

rho(lambdaX) = lambdarho(X) for all positive lambdas

Implies that rho is scale invariant.

37
Q

Provide 2 examples of PH risk measures.

A
  1. VaR
  2. Standard deviation
  3. Scenario Losses
38
Q

PH implies automatically which other mathematical property?

A

NORM since rho(0) = rho(0X) = 0rho(X) = 0

39
Q

Provide an example of non-PH risk measure

A

Variance since V(lambda*X) = lambda^2 * V(X)

40
Q

True or False?
All coherent risk measures are PH

A

True since E_Q(lambdaX) = lambdaE_Q(X)

41
Q

Briefly explain why PH is a controversial axiom.

A

Some argue that risk varies with scale (i.e. not scale invariant)

For example a portfolio that is 10 times larger may have risk that is more than 10 times greater because it is more difficult to liquidate large investment portfolios.

42
Q

Define Lipschitz continuous risk measure

A

absolute value of rho(X)-rho(Y) smaller or equal to max of absolute value of X(w)-Y(w) over all states of the world w.

Diff in risk between 2 random variables is at most the max of the absolute value of the difference of their outcomes.

43
Q

Lipschitz continuity also implies which mathematical property

A

Continuity since LC is a stronger condition than continuous

44
Q

Define subadditive (SA) risk measure

A

rho(X+Y) =< rho(X) + rho(Y)

The risk of the pool is at most the sum of the risk of the parts.

45
Q

True or False?
Mergers increase risk.

A

Not without controversy since regulators can find too much diversification benefit.

46
Q

Provide 2 examples of SA risk measures.

A
  1. TVaR
  2. Standard deviation
47
Q

Provide an example of non-SA risk measure

A
  1. VaR
  2. Variance (V(X+Y) = V(X) + V(Y) + 2 Cov(X,Y) which is higher than V(X) + V(Y) for positive correlations)
48
Q

True or False?
All coherent risk measures are sub additive.

A

True, max(E_Q(X+Y)) = max(E_Q(X) + E_Q(Y)) smaller or equal to max(E_Q(X)) + max(E_Q(Y))

49
Q

Define sublinear risk measure

A

Means that PH and SA both hold.

50
Q

Complete the sentence:
Sublinear risk measures have ___ bis-ask spread

A

Positive.

Bid-ask spread = rho(X) - rho(-X)

51
Q

Define Comonotonic Additive (COMON) risk measure

A

Variables are comonotonic and rho(X+Y) = rho(X) + rho(Y) (additive)
Comonotonic variables provide no hedge against one another (no diversification)

52
Q

Complete the sentence:
2 random variables X and Y are comonotonic if…

A

X = g(z) and Y=h(z) for increasing functions g and h and common variable z.

Said differently, (X(w1)-X(w2))*(Y(w1)-Y(w2)) positive so the differences have the same sign.

Ex: if X & Y are different XS layers of same risk Z, then they are comonotonic since indemnity functions are increasing.

53
Q

Provide 2 examples of COMON risk measures

54
Q

Provide an example of non-COMON risk measure

55
Q

True or False?
All coherent risk measures are COMON

A

False, coherent risk measure may or may not be COMON

56
Q

Define Independent Additive risk measure

A

rho(X+Y) = rho(X) + rho(Y) if X and Y are independent.

57
Q

Provide an example of independent additive risk measure

A

Variance since when indenepent random variables, Cov = 0

58
Q

Provide an example of non-independent additive risk measure

A

Standard deviation

59
Q

True or False?
All coherent risk measures are independent additive

A

False, in general, coherent risk measures are not independent additive.

60
Q

Define Law Invariant (LI) risk measure

A

Means that rho(X) is a function of F(X)

If X and Y have same distribution function, than rho(X) = rho(Y)

61
Q

True or False?
LI risk measure can only assess risk given explicit representation.

A

False, LI risk measures can assess risk given implicit or dual implicit representations, does not need to be explicit.

62
Q

True or False?
Cause is relevant in Law Invariance

A

False, LI is motivated by the fact that entities risk of insolvency depends only on its distribution of future changes in surplus, cause is irrelevant.

63
Q

Why is LI desirable for regulatory capital risk measures

A

Coupled with continuity, LI enables risk to be estimated statistically which is appropriate for regulatory capital risk measures.

64
Q

Briefly explain why LI risk measures may not be appropriate for pricing (or CAPM)

A

Since in pricing and CAPM, underlying scenario (cause) matters

65
Q

Provide 2 examples of LI risk measures

A
  1. VaR
  2. TVaR
  3. Standard deviation
66
Q

Law Invariant is also known as…

A

Objectivity

67
Q

True or False?
All coherent risk measures are LI

A

False, coherent risk measures may or may not be LI

68
Q

Define Coherent (COH) risk measure

A

Coherent if MON, TI, PH and SA

69
Q

Provide 2 examples of coherent risk measures

A
  1. TVaR
  2. Average of TVaR at different thresholds
70
Q

Provide an example of non-COH risk measure

A
  1. VaR (fails SA property)
  2. Variance (fails MON, PH and SA properties)
71
Q

Define Spectral Risk Measure (SRM)

A

Means that risk measure is COH, LI and COMON

72
Q

Provide 1 example of SRM risk measure

73
Q

Provide 1 example of non-SRM risk measure

74
Q

Define Compound Risk Measure

A

rho_a(X) = rho(X limited to a(X))

75
Q

For which use are compound risk measures interesting

A

In pricing, we might combine a pricing risk measure with a capital risk mesure to produce a compound risk measure

76
Q

Which 4 mathematical properties are implied by rho_a if both rho and a have them.

A
  1. PH
  2. NORM
  3. TI
  4. MON
77
Q

Describe 5 problems with Utility Theory as a model of firm decision making

A
  1. Assumes a diminishing marginal utility of wealth but does not apply in reality.
  2. Assumes firm preferences are relative to wealth, while they are absolute
  3. Combines attitudes to wealth and risk where they should be separate.
  4. Utility functions are not linear, thus expected utility is not MRM
  5. Based on combination through mixing, with no pooling, which does not align with insurance operations.
78
Q

Describe how dual utility theory addresses each of the 5 issues.

A
  1. Utility is linear with wealth, thus no marginal diminishing utility of wealth
  2. Reflects absolute firm preferences regardless of wealth
  3. Allows firms to maximize profits (wealth) while being risk averse
  4. Linear in outcomes based on distorted probabilities which means consuming 2 goods is equal to the sum of the utilities of consuming each individual good.
  5. Based on combination through mixing and pooling which alignes with insurance purposes.

CAS Exam 9 (24 decks)

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