04_Mildenhall_pt1

Question 1

Risk Definitions

Answer 1

• Risk – the effect of uncertainty on objectives, where an effect is a deviation from what is expected
• Pure or Insurance Risk – a risk that has a potential bad outcome but no good outcomes (ex. insurance policy)
• Speculative or Asset Risk – a risk that has both good and bad outcomes (ex. gambling)
• Financial Risk – a risk whose outcomes are denominated in monetary units

Question 2

Describe diversifiable risk. Provide an example in the context of insurance.

Answer 2

Diversifiable risk is a risk that can be reduced by diversifying a portfolio. This diversification benefit relies on independence. When independent units are added to a portfolio, the overall risk is increased by a much smaller figure than what the stand-alone risks represent.

Insurance example: By selling a large number of policies, an insurer’s “bad outcomes” from some policyholders are offset by “no outcomes” from other policyholders.

Question 3

Describe systematic risk. Provide an example in the context of insurance.

Answer 3

Systematic risk (i.e., non-diversifiable risk) is caused by a common cause across all individual units (i.e., not independent) or by a single unit that heavily influences the total loss. Systematic risk cannot be reduced through diversification.

Insurance example: A catastrophe that affects multiple policyholders at the same time.

Question 4

Describe systemic risk. Provide an example in the context of insurance.

Answer 4

Systemic risk occurs when an event causes a chain reaction of consequences that impacts an entire financial system due to the structure of the system.

Insurance example: A widely used reinsurer could cause systemic risk if it fails and causes several insurers to fail as a result.

Question 5

Describe non-systemic risk. Provide an example in the context of insurance.

Answer 5

Non-systemic risk is any risk that does not meet the definition of systemic risk.

Insurance example: A catastrophe is non-systemic because it is not caused by the operation of the insurance system.

Question 6

Define objective and subjective probabilities.

Answer 6

Objective probabilities are probabilities that can be precisely determined based on repeated observations.

Subjective probabilities represent a degree of belief and are not based on repeated observations.

Question 7

Process Risk vs. Parameter Risk vs. Uncertainty

Answer 7

Process risk is due to the inherent variability of the process being studied and is based on an objective probability model.

Parameter risk is due to the estimation of unknown parameters in a known probability model.

Uncertainty is a general term that applies when there is no objective probability model or defined set of outcomes.

Question 8

Briefly describe three ways to represent risk outcomes.

Answer 8

1. Explicit representation – identifies the risk outcome using detailed facts and circumstances
2. Implicit representation – identifies the risk outcome with its value
3. Dual implicit representation – identifies the risk outcome 𝑋 = 𝑥 with its non-exceedance probability, 𝐹(𝑥) = Pr (𝑋 < 𝑥)

Question 9

Provide one advantage and one disadvantage of dual implicit representation.

Answer 9

Advantage – it makes comparisons easy since 𝐹(𝑥) lies between 0 and 1 for all outcomes

Disadvantage – it is hard to aggregate

Question 10

Provide an insurance example for each of the three risk outcome
representations.

Answer 10

1. Explicit representation – a commercial multi-peril insurer exposed to hurricanes and earthquakes might label simulated catastrophes using a catastrophe identifier number, hurricane/earthquake flag, hurricane windspeed, earthquake latitude and longitude, multi-peril portfolio gross loss, etc.
2. Implicit representation – the same insurer from above writes homeowners insurance. The actuaries handling the homeowners line label simulated catastrophes using a hurricane/earthquake flag and the homeowners portfolio loss
3. Dual implicit representation – rating agency models charge for catastrophe risk using events defined by exceedance probability

Question 11

Risk Measure vs. Risk Preference

Answer 11

A risk measure is a real-valued function on a set of random variables that quantifies a risk preference.

A risk preference models the way we compare risks and how we decide between them.

Question 12

Three Properties of a Risk Preference

Answer 12

1. Complete (COM) – we must be able to compare any two prospects 𝑋 and 𝑌, where a prospect is an uncertain outcome that involves a choice
2. Transitive (TR) – if 𝑋 is preferred over 𝑌 and 𝑌 is preferred over 𝑍, then 𝑋 is preferred over 𝑍
3. Monotonic (MONO) – If 𝑋 ≤ 𝑌 in all outcomes, then 𝑋 is preferred over 𝑌

Question 13

Describe three components of a risk random variable quantified by a risk measure.

Answer 13

1. Volume – A smaller risk is preferred. The expected value of a risk is volume-based measure
2. Volatility – A less variable risk is preferred. Variance and standard deviation are volatility-based measures
3. Tail – A risk with a thinner tail (i.e., smaller probability of an extreme outcome) is preferred. VaR and TVaR are tail-based measures

Question 14

Capital Risk Measure vs. Pricing Risk Measure

Answer 14

Capital Risk Measure
* Used to determine the required assets (and hence, the required capital) needed to support a portfolio at a given level of confidence

Pricing Risk Measure
* Determines the expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolio’s risk

Question 15

Two Issues when Defining Quantiles

Answer 15

1. The equation 𝐹(𝑥) = 𝑝 might not have a unique solution if 𝐹 is not strictly increasing (ex. 𝐹 might have a flat spot)
2. The equation 𝐹(𝑥) = 𝑝 may have no solution if 𝐹 is not continuous (ex. 𝐹 jumps from below 𝑝 to above 𝑝; this can happen with discrete distributions)

Question 16

Provide four advantages of using 𝑉𝑎𝑅(𝑋) as a risk measure.

Provide one disadvantage of using 𝑉𝑎𝑅(𝑋) as a risk measure.

Answer 16

Advantages
1. Simple to explain
2. Can be estimated robustly
3. Always finite
4. Widely used by regulators, rating agencies, and companies

A disadvantage is that it does not always recognize diversification (i.e., not always sub-additive).

Question 17

Three Different Versions of Probable Maximum Loss (PML)

Answer 17

1. Severity 𝑉𝑎𝑅 (PML) – the lower 𝑝-quantile of the severity distribution. It is the smallest loss that Pr (𝑋 ≤ 𝑉𝑎𝑅p(𝑋)) ≥ 𝑝
2. Occurrence PML – the smallest loss such that the probability of one or more losses of 𝑋 ≥ 𝑉𝑎𝑅!(𝑋) in a year is less than or equal to 1 − 𝑝 =1/𝑛. It reflects both the likelihood and size of a loss. It is the lower 𝑝-quantile of the severity distribution at an adjusted probability
3. Aggregate PML – the smallest aggregate loss A such that Pr (𝐴 ≤ 𝑉𝑎𝑅p(𝐴)) ≥ 𝑝. In other words, is the aggregate value at risk

Question 18

Formula for the Approximate 𝑉𝑎𝑅!(𝑋) with Independent Thin-Tailed (𝑋0) and Thick-Tailed (𝑋1) Random Variables

Answer 18

𝑉𝑎𝑅!(𝑋) ≈ 𝐸(𝑋0) + 𝑉𝑎𝑅p(𝑋1)
Where 𝑋 is the sum of independent random variables 𝑋0 and 𝑋1, 𝑋0 is thin tailed and 𝑋1 is thick-tailed. This approximation improves as 𝑝 increases.

Question 19

Thin-Tailed vs. Thick-Tailed

Answer 19

• Thin-Tailed: If a distribution is bounded OR has a log concave density, then the distribution is thin-tailed
• Thick-Tailed: If a distribution is sub-exponential (i.e., log convex density), then the distribution is thick-tailed

Question 20

Three Ways in Which 𝑉𝑎𝑅 can Fail to be Sub-Additive

Answer 20

1. The dependence structure is of a particular, highly asymmetric form

[cross pairing]

2. The marginals have a very skewed distribution

[for thin tailed right skewed distribution, the quantile can behave inconsistently when combining assets, leading to a situation where the risk of a portfolio can be higher than the sum of individual risks]

3. The marginals are heavy-tailed

Question 21

Co-Monotonic Pairing vs. Crossed Pairing

Answer 21

Under co-monotonic pairing, we pair the largest value of 𝑋1 with the largest value of 𝑋2, the second largest value of 𝑋1 with the second largest value of 𝑋2, and so on. This pairing produces the greatest variance and the worst TVaR for the total 𝑋 = 𝑋1 + 𝑋2. This pairing never fails sub-additivity.

Under crossed pairing, we start by identifying the top [(1-𝑝)𝑛 + 1] values for both 𝑋1 and 𝑋2. Then, we pair the largest value of 𝑋1 with the smallest value of 𝑋2, the second largest value of 𝑋1 with the second smallest value of 𝑋2, and so on. This pairing produces the largest 𝑽𝒂𝑹𝒑 for the total 𝑋 = 𝑋1 + 𝑋2.

Question 22

Explain when 𝑉𝑎𝑅 is sub-additive.

Answer 22

The value at risk is sub-additive for log-concave (i.e., sufficiently thin-tailed) distributions.

Question 23

Two Alternative Risk Measures to 𝑇𝑉𝑎𝑅

Answer 23

1. Conditional Tail Expectation (𝐶𝑇𝐸p)
   o Lower CTE = 𝐸(𝑋|𝑋 ≥ 𝑉𝑎𝑅p(𝑋))
   o Upper CTE = 𝐸(𝑋|𝑋 ≥ 𝑞+(𝑝))
2. Worst Conditional Expectation (𝑊𝐶𝐸p) = maximum 𝐸(𝑋|𝐴) given that Pr(𝐴) > 1 − 𝑝. In other words, if we take the conditional expected value of all sets 𝐴 that have probability of at least 1 − 𝑝, the maximum of those expected values is the 𝑊𝐶𝐸p

Question 24

Relationship Between 𝑉𝑎𝑅_p, 𝑇𝑉𝑎𝑅_p, 𝐶𝑇𝐸_p, and 𝑊𝐶𝐸_p

Answer 24

In general, 𝑉𝑎𝑅_p ≤ 𝐶𝑇𝐸_p ≤ 𝑊𝐶𝐸_p ≤ 𝑇𝑉𝑎𝑅_p
For continuous distributions, 𝑇𝑉𝑎𝑅_p = 𝐶𝑇𝐸_p = 𝑊𝐶𝐸_p

Question 25

𝑇𝑉𝑎𝑅 Features: Sub-Additivity & Optimization

Answer 25

1. 𝑇𝑉𝑎𝑅 is sub-additive. The total firm’s 𝑇𝑉𝑎𝑅 is less than or equal to the sum of the individual unit 𝑇𝑉𝑎𝑅s 2. 𝑇𝑉𝑎𝑅 optimally balances the cost providing capital, 𝑥, against the cost of a shortfall, 𝐸[(𝑋 − 𝑥)+]

Question 26

Define insurance event, realistic disaster scenario (RDS), and probability event. Provide an example of each.

Answer 26

* Insurance event – a set of circumstances likely to result in insurance losses. For example, the occurrence of a category 4 hurricane or a cluster of bad traffic are both insurance events * RDS – a specific type of insurance event that is potentially disastrous but plausible. For example, a category 3 hurricane that is similar to one that has occurred before * Probability event – a possible “state of the world” to which a probability is assigned. For example, one possible “state of the world” is that two category 4 hurricanes and an earthquake occur in the same year

Question 27

Explain how conditional probability scenarios can be used to set capital.

Answer 27

* 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 𝜌!(𝑋) = max (𝐸_Q1(𝑋),…, 𝐸_Qr(𝑋)., where each 𝐸_𝑄k(𝑋) is a conditional expectation of 𝑋 given that the specific RDS has occurred * Any risk measure 𝜌_c of this form is called a coherent risk measure

Question 28

Two Types of Uncertainty About the Probability Function 𝑃

Answer 28

* Statistical Uncertainty – 𝑃 is an estimate subject to the usual problems of estimation risk. The type of uncertainty diversifies across large portfolios * Information Uncertainty – 𝑃 is based on a limited and filtered subset of ambiguous information. This type of uncertainty reflects information asymmetry between the insured and the insurer and between the insurer and investor. It is more unavoidable than statistical uncertainty

Question 29

Generalized Probability Scenario

Answer 29

The probability function 𝑃 provides a best estimate probability. This means it is an expected value and has uncertainty. [Pricing risk measure are focused on the risk in the estimate of the mean E(X) rather than the risk of an RDS outcome (tail), we can create generalized probability scenario to reflect that] A generalized probability scenario reflects information uncertainty in 𝑃 (since information uncertainty is the main issue). It incorporates subjectiveness into the probabilities and is used for pricing risk measures focused on the mean.

Question 30

Advantages of a Coherent Risk Measure

Answer 30

* Intuitive and easy to communicate * Can be used for capital and pricing * Has properties that are aligned with rational risk preferences

Question 31

Translation Invariant (TI), Normalized (NORM), and Monotone (MON)

Answer 31

* TI o Means that 𝜌(𝑋 + 𝑐) = 𝜌(𝑋) + 𝑐. Increasing a loss by a constant 𝑐 increases the risk by c o Examples of TI risk measures: mean, 𝑉𝑎𝑅, 𝑇𝑉𝑎𝑅 o Examples of non-TI risk measures: variance, standard deviation * NORM o Means that 𝜌(0) = 0. The risk of an outcome with no gain or loss equals zero o Example of a NORM risk measure: 𝑇𝑉𝑎𝑅 * MON o Means that if 𝑋 ≤ 𝑌 in all outcomes, then 𝑋 is preferred to 𝑌 o Example of a MON risk measure: 𝑇𝑉𝑎𝑅 o Example of a non-MON risk measure: standard deviation

Question 32

Positive Loading, Monetary Risk Measure (MRM), and Positive Homogenous (PH)

Answer 32

* Positive Loading o Means that 𝜌(𝑋) ≥ 𝐸(𝑋) o Reinsurance is part of an insurance portfolio with a negative loading * MRM o Means that the risk measure has a monetary unit o Example of an MRM: 𝑇𝑉𝑎𝑅 o Example of a non-MRM: variance * PH o Means that if 𝜌(𝜆𝑋) = 𝜆𝜌(𝑋) for 𝜆 ≥ 0 o Example of a PH risk measure: 𝑇𝑉𝑎𝑅 o Example of a non-PH risk measure: variance

Question 33

Lipschitz Continuous, Sub-additive (SA), and Sublinear

Answer 33

* Lipschitz Continuous o Means that |𝜌(𝑋) − 𝜌(𝑌)| ≤ max |𝑋(𝜔) − 𝑌(𝜔)| over all states of the world 𝜔. The difference in risk between two random variables is at most the maximum of the absolute value of the different in their outcomes * SA o Means that 𝜌(𝑋 + 𝑌) ≤ 𝜌(𝑋) + 𝜌(𝑌). The risk of the pool is at most the sum of the risk of the parts o Example of a SA risk measure: 𝑇𝑉𝑎𝑅 o Example of a non-SA risk measure: 𝑉𝑎𝑅 * Sublinear o Means that PH and SA both hold

Question 34

Comonotonic Additive (COMON), Independent Additive, and Law Invariant (LI)

Answer 34

* COMON o Comonotonic variables provide no hedge against one another (i.e., no diversification). If 𝑋 and 𝑌 are comonotonic and 𝜌(𝑋 + 𝑌) = 𝜌(𝑋) + 𝜌(𝑌), then ρ is comonotonic additive (COMON) o Example of a COMON risk measure: 𝑇𝑉𝑎𝑅 o Example of a non-COMON risk measure: variance * Independent Additive o Means that 𝜌(𝑋 + 𝑌) = 𝜌(𝑋) + 𝜌(𝑌) if 𝑋 and 𝑌 are independent o Example of an independent additive risk measure: variance o Example of a non-independent additive risk measure: standard deviation * LI o Means that if 𝑋 and 𝑌 have the same distribution function 𝐹, then 𝜌(𝑋) = 𝜌(𝑌) o Example of an LI risk measure: 𝑇𝑉𝑎𝑅

Question 35

Coherent (COH) and Spectral (SRM) Risk Measures

Answer 35

* COH o Means that the risk measure is MON, TI, PH, and SA o Example of a COH risk measure: 𝑇𝑉𝑎𝑅 o Examples of non-COH risk measures: 𝑉𝑎𝑅, variance * SRM o Means that the risk measure is COH, LI, and COMON o Example of an SRM: 𝑇𝑉𝑎𝑅 o Examples of a non-SRM: 𝑉𝑎𝑅

Question 36

Compound Risk Measure

Answer 36

In pricing, we might a combine a pricing risk measure 𝜌 with a capital risk measure 𝑎 to produce a compound pricing risk measure given by 𝜌_a(𝑋) =𝜌(𝑋 ⋀ 𝑎(𝑋)).

Question 37

Acceptable Risk

Answer 37

Suppose a risk is NORM, which means that 𝜌(0) = 0. Then, a risk is preferred to doing nothing if 𝜌(𝑋) ≤ 0. A risk preferred to doing nothing is called an acceptable risk.

Question 38

No Rip-Off Property

Answer 38

The no rip-off property means that if 𝑋 ≤ 𝑐, then 𝜌(𝑋) ≤ 𝑐. All MON risk measures have the no rip-off property.

Question 39

Positive Homogeneous (PH) risk measures assume that risk scales proportionally with the business. Provide a counterargument to this.

Answer 39

Some argue that risk varies with scale. As a finance example, support it's more difficult to liquidate large investment portfolios. This lack of liquidity suggests that a portfolio that is ten times larger may have a risk that is more than ten times greater.

Question 40

Five Problems with Utility Theory (i.e., Expected Utility Representation) as a Model of Firm Decision Making

Answer 40

1. Utility theory assumes a diminishing marginal utility of wealth. In reality, firms do NOT have a diminishing marginal utility of wealth. As wealth increases, shareholders continue to crave more wealth at the same level 2. Utility theory assumes that firm preferences are relative to a wealth level 3. Utility theory combines attitudes to wealth and to risk, whereas firm decision-making should separate wealth from risk 4. Utility functions are not linear. Thus, the expected utility is not a monetary risk measure. 5. Utility theory is based on combination through mixing, with no pooling, which does not align with insurance operations

Question 41

Dual Utility Theory’s Answers to the Five Utility Theory Problems

Answer 41

1. Under dual utility theory, utility is linear in wealth. Thus, there is no marginal diminishing utility of wealth 2. Dual utility theory reflects absolute firm preferences, regardless of wealth 3. Dual utility theory allows firms to maximize profits (i.e., wealth) while being risk averse. Under utility theory, risk aversion leads to a diminishing marginal utility of wealth (i.e., combined attitude to wealth and risk) 4. Dual utility theory is linear in outcomes based on distorted probabilities 5. Dual utility theory is based on combination through mixing, with pooling, which aligns with insurance operations Note that spectral risk measures correspond to dual utility theory.

Question 42

Briefly describe three ways to select a risk measure.

Answer 42

1. Ad Hoc Method – start with a reasonable risk measure and rationalize it by establishing it has the desired properties (ex. percentage loading) 2. Economic Method – utility-based methods that solve an optimization problem (hardest method to apply) 3. Characterization Method – start with a list of desired properties and then determine which risk measures have those properties (more scientific than ad hoc method and easier to apply than economic method)

Question 43

Briefly describe five important properties of a risk measure from a practical point of view.

Answer 43

1. Diversification – the risk measure must reflect diversification 2. Allocation – when applied in the aggregate, the risk measure must allow a practical allocation to its individual parts 3. Theoretical Soundness – the risk measure must align well with theory 4. Explainable – the risk measure must be easily explained to users 5. Backtesting – the measured risk as defined by the risk measure should be consistent with observations

Question 44

Five Desirable Characteristics of Risk Margins

Answer 44

1. The less that is known about the current estimate and its trend, the higher the risk margin should be 2. Low frequency, high severity risks should have higher risk margins than high frequency, low severity risks 3. For similar risks, longer-duration contracts should have higher risk margins than short-duration contracts 4. Risks with a wide probability distribution should have higher risk margins than risks with a narrower distribution 5. To the extent that emerging experience reduces uncertainty, risk margins should decrease (and vice versa)

Question 45

Eight Degrees of Tail-Thickness

Answer 45

From thickest to thinnest: 1. No mean 2. Mean, but no variance 3. Mean and variance, but only finitely many moments 4. All moments exist, but sub-exponential 5. Exponential tail (this lives between thick- and thin-tailed) 6. Super-exponential 7. Log-concave density 8. Bounded

Question 46

Briefly describe five intended users of risk measures. And what components of risk each of the users may focus on

Answer 46

1. Pricing actuary – price adequacy, price competition, fair allocation of cost and capital 2. Insurer management – portfolio optimization, reinsurance strategy, capital structure 3. Insured – value of insurance, solvency of insurer 4. Regulator – setting minimum capital standards 5. Rating agency – evaluating capital standards Focus on: Pricing actuary - prospective amount and timing uncertainty Management- UW income, net income and cash flow at group and legal entity level Regulation - default risk encompassing all sources of risk at a legal entity level

Question 47

Briefly describe five desirable properties of premium calculation principles (PCPs).

Answer 47

1. Explainable – PCPs should be reasonable, transparent, and explainable 2. Estimable – PCP parameters should be easy to estimate from market prices 3. Computable – PCPs should be easy and efficient to compute 4. Robust – PCP values should be robust to the ambiguity in the underlying risk distributions 5. Allocation – Aggregate PCPs should have a logical allocation methodology

Question 48

Briefly describe five desirable properties of capital risk measures.

Answer 48

1. Robustness to regulatory arbitrage – capital risk measures should balance complexity against the data available 2. Simplicity & explainability – capital risk measures must be simple to help regulators communicate objectives to stakeholders 3. Standardization – a standard approach, along with reliance on public data (the public data piece applies to regulators and rating agencies), should be used for capital risk measures to ensure comparison across entities 4. Backtesting – it should be possible to determine if a portfolio was managed to a risk measure tolerance 5. Portfolio optimization – portfolio optimization against a regulatory standard is important for insurers

Question 49

Explain why regulators may prefer EPD this risk measure over others such as VaR

Answer 49

VaR ignores any losses greater than VaR. EPD accounts for the degree of default relative to the promised payments. VaR calculates the loss associated with a given confidence level EPD calculates the expected value of the difference between the obligation owed to the claimant and the amount actually paid