Mildenhall - CH1-6 Flashcards
1
Q
Differences between pure risk and speculative risk
A
- Pure risk or insurance risk has a potential bad outcome but no good outcomes.
- Speculative risk or asset risk has both good and bad outcomes.
The loss on an insurance policy is a pure risk, but the net position (premium minus loss and expenses) is a speculative risk
2
Q
Describe a prospect
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It’s an uncertain outcome that involves a choice.
3
Q
Describe a financial risk
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It’s a prospect with outcomes denominated in a monetary unit.
Examples of financial risk: insurance loss, the future value of a stock or bond, the present value of future lifetime earnings.
It can have timing uncertainty and amount uncertainty
4
Q
Risk is time separable if
A
if a measure of the magnitude of the risk of an amount at a future time can be expressed as the product of (1) the magnitude of the risk of the amount if immediately due, times (2) a discount factor.
5
Q
Differences between Diversifiable (idiosyncratic) risk and systematic risk
A
- Diversifiable risks is where the risk of the sum is less than the sum of the risks. A diversification benefit occurs when adding independent units to a portfolio increases its risk by much less than what the standalone risks represent.
- Systematic risk means that there is a common underlying cause or other source of dependence risk to multiple unit losses. such as catastrophe
6
Q
Ways to identify dependence risk in a simulation context
A
Where are you sharing variables?
Any variable whose value is shared between units introduces dependence and systematic risk.
Examples of shared variables: weather and loss trend assumption
7
Q
Difference between systemic risk and non-systemic risk
A
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. Systemic risk is non-diversifiable.
Non-systemic risk is any risk that does not meet the definition of systemic risk. (Although a catastrophe is a systematic risk, it’s non-systemic because it is not caused by the operation of the insurance system)
8
Q
Difference between objective probabilities and subjective probabilities
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Objective probabilities can be precisely determined based on repeated observations (such as coin toss, dice roll). Insurance is largely based on objective probabilities from repeated observations (loss data).
Subjective probabilities provides a way of representing a degree of belief and are not based on repeated observations (such as election, horse race, or economic outcome)
9
Q
Describe process risk
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Process risk is also called aleatoric uncertainty, is due to the inherent variability of the process being studied and is based on an objective probability model.
10
Q
Describe unceratinty
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Uncertainty is sometimes called Knightian uncertainty, is a general term that applies when there is no objective probability model or defined set of outcomes.
11
Q
Describe Epistemic uncertainty
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It’s caused by a knowledge gap, possibly one that could in principle be filled.
12
Q
Describe parameter risk
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It’s intermediate between process risk and uncertainty. It’s due to the estimation of unknown parameters in a known probability model.
13
Q
Describe explicit representation of risk outcome
A
- Identifies the risk outcome using detailed facts and circumstances.
- Mathematically, we present the set of all possible variable value combinations.
- It’s the most detailed representation and allows for easy aggregation, critical in reinsurance and risk management.
- It can distinguish between different events even if they cause the same loss outcome.
14
Q
Describe implicit representation of risk outcome
A
- It identifies an outcome with its value, creating an implicit event.
- It’s easy to understand but hard to aggregate because there is no way to link outcomes.
- There is no easy way to specify dependence.
- it’s impossible to distinguish between implicit events that cause the same loss outcome.
15
Q
Describe Dual Implicit Representation of risk outcome
A
- It identifies the outcome with its non-exceedance probability.
- Identifying an outcome with its probability is scale invariant and straightforward. It’s easy to make comparisons since F(x) lies between 0 and 1.
- It’s hard to aggregate
16
Q
Define risk measure
A
Risk measure is a real-valued functional on a set of random variables that quantifies a risk preference - the way an individual or group of individuals decides risk questions
17
Q
3 characteristics of a risk random variable quantified by risk measures
A
- Volume (smaller risk is preferred. Example of risk measure: expected value)
- Volatility (less variable risk is preferred. Example of risk measure: variance, standard deviation)
- Tail (risk with a lower likelihood of extreme outcomes (thinner tail) is preferred. Example of risk measure: VaR, TVaR)
18
Q
Why a risk measure must reflect volume
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Because we want it to mirro a risk preference satisfying the MONO property (smaller size risks are preferred even if the small risk is more volatile)
19
Q
Two distinct risk measure categories for insurance company operations
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- Capital risk measure (setting the needed amount of capital)
- Pricing risk measure (determining its cost)
20
Q
Describe capital risk measure
A
- It determines the assets necessary to back an existing or hypothetical portfolio at a given level of confidence.
Example: VaR or TVaR at some high confidence level - Capital risk measures must be sensitive to tail risk to ensure solvency.
21
Q
How capital risk measure are being used by different parties?
A
- Management use it to determine the economic capital
- Regulator use it to determine a minimum capital requirement
- Rating agency use it to opine on the adequacy of held capital.
22
Q
Describe pricing risk measure
A
- It determines the expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolio’s risk
They are also called premium calculation principles (PCP) - Given a level of conservatism, they determine a premium or capital requirement.
Given price or capital requirement, they evaluate its implied level of conservatism.
23
Q
Describe law invariant
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Risk measures that depend only on the distribution X rather than its value on each specific cause
24
Q
two issues when defining quantiles
A
- The equation F(x) = p may fail to have unique solution when F is not strictly increasing. (F might have a flat spot)
- When F is not continuous, the equation F(x) may have no solution (F jumps from below p to above p)
25
Advantage of VaR
- simple to explain
- exists for all random variables (even those without a mean)
- can be estimated robustly
- widely used by regulators, rating agencies, and companies in their internal risk management
26
Disadvantage of VaR
it does not always recognize diversification (it's not always sub-additive)
27
How do we know if a random variable X is thick-tailed or thin-tailed?
- If a distribution is bounded OR has a log concave density (the log of the density is concave) ,then the distribution is thin-tailed
- If a distribution is sub-exponential (log convex density), then the distribution is thick-tailed
28
3 ways that VaR can fail to be subadditive
1. When the dependence structure is of a particular, highly asymmetric form
2. When the marginals have a very skewed distribution
3. When the marginals are very heavy-tailed.
29
2 dependence structure
1. Co-monotonic paring
2. Crossed Paring
30
Describe Co-monotonic
- We pair the largest value of X1 with the largest value of X2
- Positive dependence between variables increases the risk of their sum
- It has the greatest variance and worst TVaR characteristics
- This paring NEVER fails sub-additivity because it assumes 100% dependence (no diversification benefit)
- The p percentile of the sum is simply the sum of the p percentiles of X1 and X2.
31
Describe crossed paring
- Pair the largest value of X1 with the smallest value of X2.
- Failure of subadditivity because each term in the crossed paring is greater than the sum of the individual VaRs.
- This dependence structure does not have extreme right tail dependence because it does not combine the worst value of X1 with the worst value of X2. it just gives the highest p-VaR.
- This paring is tailored to a specific value of p and does not work for other ps. It produces relatively thinner tail for higher values of p than either the co-monotonic colula or independence.
- It works for any non-trivial marginal distributions X1 and X2.
- It's hard to generalize to three or more marginal distributions.
32
Describe a situation when VaR sub-additivity fails driven by heavy-tailed marginals
Sub-additivity fails for a range p when we have two skewed, thin-tailed iid marginal distributions.
Sub-additivity fails for all p above a threshold when we have two thick-tailed iid marginal distributions.
33
When is VaR subadditive?
VaR subadditivity holds for long-concave (thin-tailed) distributions.
- iid variables with tails like x^-a for a>1 are sub additive for p sufficiently close to 1
34
Describe TVaR
it's the conditional average of the worst 1-p proportion of outcomes
- TVaR is a well-behaved function of p:
It's continuous, differentiable almost everywhere, and equal to the integral of its derivative.
35
What is PELVE
it's a constant c so that TVaR at 1-Ce is the same as VaR at 1-e
36
Why does regulators like EPD risk measure
It accounts for the degree of default relative to promised payments.
A smaller EDP ratio means a stricter standard.
37
Define insurance event
It's a set of circumstances likely to result in insurance losses
38
Define realistic disaster scenario
it's an insurance event that is potentially disastrous but plausible.
39
Define probability event
it's a possible "state of world" to which a probability is assigned.
40
How conditional probability scenarios are useful for setting capital
- When a disaster occurs, we want to ensure we have enough money on hand to pay out our claims.
- We can define a set of RDS (Realistic disaster scenarios) and set our risk measure where each event is a conditional expected of X given that specific RDS has occurred.
- Any risk measure of this form is called a coherent risk measure (TVaR is a coherent risk measure
41
Two uncertainty categories about P
1. Statistical uncertainty: P is an estimate subject to the usual problems of estimation risk. It concerns estimates of objective probabilities
2. Information uncertainty: P is based on a limited and filtered subset of ambiguous information. It concerns risk aversion and estimates of subjective probabilities
42
Define generalized probability scenarios
Probability scenarios that reflect information uncertainty, and don't need to be conditional expectation of P.
It reflects events such as insureds being systematically miss-classified, adverse selection.
43
Characteristics of coherent risk measure
1. it's intuitive and easy to communicate
2. Can be used for capital and pricing
3. has properties as a risk measure that are aligned with rational risk preferences
4. any measure with those properties is a pc for some set of probability scenarios
44
Describe expected utility representation
Utility theory describes how individuals ran order choices. If we assign a utility to each choice, along with an associated probability, then we can calculate the expected utility in the same way we would calculate the expected value
45
What undesirable properties does the expected utility representation have when applied to firm?
1. Utility theory assumes a diminishing marginal utility of wealth, which means as an individual's wealth increases, less additional satisfaction is acquired from consuming another unit of good. 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. Firm preferences are not relative to a wealth level (which is what utility theory assumes). Instead, firm preferences are absolute
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, so the expected utility is not a monetary risk measure.
5. Utility theory is based on combination through mixing, with no pooling, counter to the operation of insurance.
46
How Dual utility theory address the shortcomings of expected utility?
1. Under dual utility theory, there is no marginal diminishing utility of wealth.
2. Dual utility theory separates a firm's attitude toward wealth from risk.
3. Dual utility theory allows firm to maximize profit (wealth) while being risk adverse
4. Dual utility theory is linear in outcomes based on distorted probabilities
5. Dual utility theory is based on combination through mixing, with pooling, with aligns with insurance operations.
Since spectral risk measures correspond to dual utility theory, they are well suited for insurance purposes.
47
3 ways to select risk measures
1. Ad hoc method: start with a reasonable risk measure and rationalize it by establishing it has the properties desired. (such as percentage loading (constant underwriting margin))
2. Economic method: use a rigorous economic theory to select a risk measure. (include utility-based approaches that set up and solve an optimization problem). This is the most rigorous selection process but also the hardest to apply in practice.
3. Characterization method: start wit ha list of desirable properties and then determine which risk measures have those characteristics. This method is more scientific than the ad hoc method and easier to apply than the economic method.
48
What are the most important/desirable properties of risk measure from a practical point of view
1. Monotone and translative invariance are essential for the measure to reproduce intuitive concepts of risk. both are theoretically and practically sound.
2. Diversification must be reflected. Both coherent and convex risk measures reflect diversification (in different ways)
3. Allocation: a risk measure applied in aggregate must allow a practical allocation to its parts. Coherent risk measures perform well in this.
4. Theoretic soundness and consistency with a general theory is desirable. Coherent risk measures perform this well.
5. Explainable: Coherent risk measures perform this well.
6. Elicitability: the risk measure can be estimated by regression-like techniques.
7. Robustness or continuity: a small change in inputs should result in a small change in measured risk.
8. Backtesting: measured risk is consistent with observations.
49
Is VaR a good fit for risk measure?
The VaR's failure to be subadditive may not be a severe issue in practical applications. The fact that VaR ignores the tail is more serious but makes it more robust.
If we consider VaR for a range of thresholds, then we somewhat capture the tail.
TVaR reflects diversification (subadditive) but it's also hard to elicit and difficult to backtest.
50
5 desirable characteristics of risk margins
1. The less that is known about the current estimate and its trend, the higher the risk margins should be.
2. Risks with low frequency and high severity have higher risk margins than risks with high frequency and low severity
3. For similar risks, contracts that persist over a longer time-frame have higher risk margins than those of shorter duration
4. Risks with a wide probability distribution have higher risk margins than risks with a narrower distribution
5. To the extent that emerging experience reduces uncertainty, risk margins decrease, and vice versa.
51
5 Different approaches to estimate risk margins
1. Outcome methods
2. Cost of capital methods
3. Discount-related methods
4. Explicit assumptions
5. Conservative assumptions
52
8 degrees of tail-thickness (from thicket to thinnest)
1. No mean (thick-tailed. VaR is not subadditive for distribution with no mean. Law of large numbers does not apply for these risks. These risks are impossible to insure)
2. Mean, but no variance (still thick-tailed. law of large numbers applies but the central limit theorem does not.)
3. Mean and variance, but only finitely many moments (thick-tailed, law of large numbers and central limit theorem both apply. Higher moments such as skewness and kurtosis may exist.)
4. All moments exist, but subexponential (thick-tailed. A distribution is subexponential if its survival function decays slower than an exponential.)
5. Exponential tail (lives between thick and thin tailed)
6. Super-exponential (faster than exponential) tail (thin-tailed. They have all moments and the central limit theorem and law of large numbers apply.)
7. Log-concave density: (thin-tailed. These distributions are very well behaved, with sample averages tightly clustered around sample mean. Risk in a portfolio of log-concave densities is entirely driven by dependency risk)
8. Bounded (no tail risk in a bounded distribution)
53
Our choice of risk measure should reflect its intended purpose and users. Describe intended purpose and users
- Intended purpose is the goal or question, whether generalized or specific, addressed by the model within the context of the assignment.
- Intended user is any person whom the actuary identifies as able to rely on the model output.
54
3 components of a model
1. information input component, which delivers data and assumptions to the model
2. A processing component, which transforms input into output
3. A result component, which translate the output into useful business information.
55
Intended purpose of individual risk pricing risk measures
quoting and evaluation of market pricing
56
intended purposes of classification rate mating risk measures
setting profit margins and allocation of cost of capital
57
intended purposes of portfolio management risk measures
reinsurance purchasing, portfolio optimization, ORSA, insurance vs. asset risk split, strategic planning
58
intended purposes of capital risk measures
determining risk capital or evaluating held capital
59
What risk measure do underwriting or pricing actuary concern about?
price adequacy, price competition, fair allocation of cost of capital
60
what risk measure do insurer management and ERM function concern about?
relative performance, portfolio optimization, retention and reinsurance strategy, capital structure, signaling competence to external users
61
What risk measures do insured concern about
value of insurance, total cost of risk, solvency of insurer
62
what risk measures do regulator concern about
setting binding objective minimum capital standards
63
what risk measure do rating agency concern about
evaluating actual capital
64
what risk measure do reinsurer concern about
pricing inwards business, often with substantial catastrophe risk and potential aggregations
65
What risk measure do investor concern about
balance and compare risks and returns, solvency, franchise value, dilution
66
What should be considered in a premium calculation principles (PCP), AKA pricing risk measures?
1. Explainable (PCP should have a reasonable, transparent, and explainable basis and are consistent with theory)
2. Estimable (PCP parameters should be easy to estimate from market prices)
3. Computable (PCP should be easy and efficient to compute)
4. Robust: (PCP values should be robust to the ambiguity in the underlying risk distribution)
5. Allocation (aggregate PCPs should have a natural and logical allocation methodology)
6. Optimal (PCPs should behave well when used in optimization algorithms) Convex risk measures are particularly helpful.
7.Diversification (PCPs should respect insurance risk diversification)
8. Law invariant (PCPs should be law invariant) can be violated through the use of a reference portfolio, such as in catastrophe risk portfolios
9. Theoretically sound (PCPs should be consistent with economic, financial theory, and behavioral considerations)
67
How are capital risk measures being used
- used by management to estimate economic capital
- used by regulators to determine minimum capital standards (like RBC, Solvency I)
- used by rating agencies to set target capital standards and to assess the adequacy of held or adjusted capital. (like parallel capital adequacy ratio models)
68
Considerations when using capital risk measures
1. Robustness to regulatory arbitrage (Capital risk measures should balance complexity against the data available)
2. Simplicity and explainability (to help the regulator communicate their objectives to different stakeholders)
3. Standardization and reliance on public data enables comparison across entities.
4. Backtesting (it should be possible to determine if a portfolio was managed to a risk measure tolerance)
5. Portfolio optimization against a regulatory standard is very important for insurers. Convex risk measures have good optimization behavior.
