Mildenhall Ch 3: Risk & Risk Measures

Question 1

Define Risk (ISO)

Answer 1

Effect of uncertainty on objectives.

Where an effect is a deviation from what is expected.

Risk is caused by events and have consequences.

Question 2

Define uncertainty (ISO)

Answer 2

The state, even partial, of deficiency of information related to, understanding or knowledge of, an event, its consequence, or likelihood.

Question 3

Contrast speculative risk and pure risk

Answer 3

A pure risk (insurance risk) has a potential bad outcome but no good outcomes.
It is a possible loss with no chance of gains.
Ex: insurance policies are designed to put the insured, at best, in the same position they would have been without a loss.

A speculative risk (asset risk) has both good and bad outcomes.
It can be a loss or a gain.

Question 4

Is it possible to convert a pure risk into speculative risk?

Answer 4

Yes, using Reframing.

The loss on an insurance policy is a pure risk. But the net position, premium less loss and expenses, is a speculative risk.

Question 5

Define prospect

Answer 5

Incertain outcome that involves a choice.

A prospect is relative to a reference point. The uncertainty in your bonus is relative to what you expect, not to zero. Business is evaluated relative to plan, not insolvency.

Question 6

Complete the sentence:
The existence of different reference points can lead to…

Answer 6

Framing bias problems.

Question 7

Define financial risk

Answer 7

Prospect with outcome denominated in a monetary unit.

Question 8

Identify 3 examples of financial risk

Answer 8

1. Insurance loss
2. Future value of a stock or a bond
3. Present value of future lifetime earnings

Question 9

Identify and describe the 3 types of uncertainty a financial risk can have.

Answer 9

1. Timing:
   Payment of a known amount at a random time.
   Ex: Benefit payment on a whole or term life insurance policy.
2. Amount
   Payment of a random amount at a known time.
   Ex:
   Payment on a pure endowment policy (pays if insured survives a certain age).
   Payment of a YE employee bonus if employer profit target is met.
3. Both
   Payment of a random amount at a random future time.
   Ex: Loss payment on a typical property-casualty insurance policy.

Question 10

Briefly explain how can uncertainty be reduced in financial risk.

Answer 10

By specifying payment dates or applying limits and deductibles to loss amounts.

Accounting rules often require that reinsurance contract transfers both timing and amount risk.

Question 11

Describe the conditions for a risk to be time separable.

Answer 11

Risk is time separable 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.
(2) a discount factor.

Under this book, we will assume risk is time separable.

Question 12

Complete the sentence:
Under time separability, a risk measure becomes….

Answer 12

A measure of amount of risk.

Question 13

The 3 risks relevant to pricing are…

Answer 13

1. Pure UW
2. Reserving
3. Catastrophe

The others tend to be background risks that are hard to distinguish between units and therefore not relevant to pricing.

Question 14

Identify the 3 dimensions to categorize risks.

Answer 14

1. Diversifiable (idiosyncratic) vs systematic
2. Systemic vs Nonsystemic
3. Objective vs Subjective probability and uncertainty

Question 15

Define the diversification basis of insurance.

Answer 15

The risk of the sum is less than the sum of the risks.

Question 16

Briefly explain how do we determine if a risk is diversifiable.

Answer 16

Risks diversify when each unit is small relative to the total and their losses exhibit a material degree of independence from one another.

Question 17

Briefly explain when does a diversification benefit occurs.

Answer 17

A diversification benefit occurs when adding independent units to a portfolio increases its risk by much less than what the standalone risks represent.

Question 18

Complete the sentence:
The central limit theorem ensures that….

Answer 18

pooling is an effective mechanism to manage diversifiable risk.

Question 19

Briefly explain systematic risk (non-diversifiable)

Answer 19

The failure to diversify means that there is a common underlying cause or other source of dependence risk to multiple unit losses, or there is a single unit heavily influencing the total loss.

Ex: catastrophes affecting multiple units simultaneously.

Question 20

Fill the blank?
The presence of systematic risk means there is ______ diversification benefit than in its absence.

Answer 20

Less

Question 21

Briefly explain how dependence risk can be identified.

Answer 21

It is easy to identify in a simulation context where are we sharing variables.

Variables resimulated in each iteration for each unit diversify, at least to some extent.

Any variable whose value is shared between units introduces dependence and systematic risk.

Ex: weather and loss trend assumptions.

Question 22

Briefly explain why only systematic risk matters in investment.

Answer 22

Because the well-diversified investor can make idiosyncratic risk go away by adding many stocks to a portfolio.

Note: does not apply to the problem of pricing insurance risk.

Question 23

True or False?
The discounting impact of timing remains even when amount risk diversifies.

Briefly explain.

Answer 23

True.

Timing risk tends to be quite tame since payout patterns follow a predictable claim settlement process, regulated by the cadences of medicine and law.

The historical development of insurance pricing reflects this distinction: in many cases amount risk is largely irrelevant but estimating the appropriate discount rates and investor/insured CFs remains paramount.

Question 24

Briefly describe systemic risk.

Answer 24

Systemic risk affects financial system consisting of many interacting agents or firms and is created by that system’s operation or structure.

It occurs when an event causes a chain reaction of consequences.

It needs to be caused or exacerbated by the operation of system.

Question 25

Are systemic risk diversifiable or non-diversifiable?

Answer 25

Systemic risks are by nature nondiversifiable.

Question 26

Identify two triggers of systemic risk.

Answer 26

1. Oil crisis of 1970s
2. Failure of Long-Term Capital Management
3. October 1987 Black Monday crash
4. Global Financial Crisis (GFC) of 2008

Question 27

True or False?
P&C insurers are systematically important financial institutions.

Answer 27

False. Insurers themselves are generally not regarded as systematically risky because they have liquid assets and illiquid uncallable liabilities.

Although some large life insurers and AIG were designated as SIFIs (Systematically Important Financial Institutions)

Question 28

How could a reinsurer generate systemic risk?

Answer 28

A large highly connected reinsurer could generate systemic risk if, for example, its failure would cause knock-on insolvencies.

Question 29

Describe Objective probabilities.

Answer 29

Objective probabilities are amenable to precise determination.

It applies the law of large numbers, the central limit theorem, Bayesian statistics and credibility theory to make precise predictions about samples.

Ex: Insurance is largely based on objective probabilities from repeated observations (loss data).

Question 30

Describe Subjective probabilities.

Answer 30

Subjective probabilities provide a way of representing a degree of belief.

Subjective probabilities are applied to non repeatable events.

Ex: election, horse race, economic outcome, game theory.

Insurance largely avoids pricing based on subjective probabilities.

Question 31

Contrast process risk and uncertainty.

Answer 31

Process risk is defined in an objective probability model while uncertainty occurs when there is not probability model or even no clearly defined set of possible outcomes.

Question 32

Define parameter risk.

Answer 32

Parameter risk is intermediate between process risk and uncertainty. It refers to a known model with unknown parameters.

Actuaries often use Bayesian models to introduce parameter risk.

Question 33

Briefly explain unknown unknowns.

Answer 33

Represents an extreme form of uncertainty.

Question 34

Identify the 3 representations of outcomes.

Answer 34

1. Explicit
2. Implicit
3. Dual implicit

Question 34

Briefly describe explicit representation of outcome.

Answer 34

A risky outcome can be labeled explicitly by describing the facts and circumstances causing it.

With sufficient detail of identifying variables, we have an explicit representation of risk outcomes.

Question 35

Identify 2 pros of explicit event representation.

Answer 35

1. Enables outcomes to be linked across a book of business, thus dependence risk can be modelled without making assumptions.
2. Useful when events are not too numerous and affect significant portions of portfolio.
3. Most detailed representation & allows for easy aggregation, which is critical in reinsurance and risk management.
4. Can distinguish between different events even if they cause the same loss outcome.

Question 36

Identify 1 con of explicit representation.

Answer 36

1. When events are very numerous and affect only small portions of portfolio, explicit event rep does not provide enough benefit to justify its greater complexity.
2. Suffers from being arbitrary, especially regarding detail communicated by sample points, and the complexity of defining events, especially for high volume lines where it is unrealistic to tie an event to each individual policyholder.

Question 37

Describe the implicit representation of outcome.

Answer 37

Identifies an outcome with its value, creating an implicit event.

Relabels the sample space by identifying the event with the sample point x, creating a new r.v. on the sample space of outcome values.

Question 38

Identify 2 pros of implicit representation.

Answer 38

1. It is easy to understand.
2. When risk is solely a function of loss outcome, rather than the cause of loss, it can be appropriate to work with implicit events.

Question 39

Identify 2 cons of implicit representation.

Answer 39

1. It is hard to aggregate because there is no way to link outcomes.
2. No easy way to specify dependence.
3. It is impossible to distinguish between implicit events that cause the same loss outcome.

Question 40

Briefly explain dual implicit representation of outcome.

Answer 40

Identifies the outcome with the probability of observing no larger value.

If we do not care about the monetary outcome but care only about the rank of outcomes, we can use dual implicit representation and identify an outcome with its nonexceedance probability.

Question 41

Identify 2 pros of dual implicit representation.

Answer 41

1. Scale invariant
2. Straightforward
3. Easy to make comparisons, no matter the range of X, F(X) lies between 0 and 1

Question 42

Identify 2 cons of dual implicit representation.

Answer 42

1. Hard to aggregate
2. Suffers from being relative to an often unspecified reference portfolio.

Question 43

Define risk measure

Answer 43

Real-valued functional on a set of random variables that quantifies risk preference.

Numerical representation of risk preferences.

Question 44

Define risk preference.

Answer 44

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

It captures our intuitive notions of riskiness and converts them into a form we can use to predict future preferences.

Question 45

A risk preference for insurance loss outcomes need to have which 3 properties.

Answer 45

1. Complete
   Any pair of prospects can be compared.
2. Transitive
   If X preferred to Y and Y preferred to Z, then X preferred to Z
   Ensures risk preference is logically consistent.
3. Monotonic
   Reflects the reality that large positive outcomes for losses are less desirable than small ones.
   Also ensures the risk preference takes into account the volume/size even when there is no variability.

Question 46

Identify the 3 characteristics quantified by risk measures.

Answer 46

1. Volume
   Smaller risk is preferred.
   Mean measures volume.
2. Volatility
   A risk exhibiting less volatility is preferred.
   Variance and SD measure volatility.
   Volatility is two-sided.
3. Tail
   A risk with lower likelihood of extreme outcomes is preferred.
   Tail risk is one-sided.

Question 47

Which one is preferred?
Small volatile risk
Large certain risk

Answer 47

Small, always

Question 48

Define deviation

Answer 48

Measure of variability or tail risk that ignores volume.

Question 49

Identify 2 types (different uses) of risk measures.

Answer 49

1. Risk-based capital formulas.
   Many of them are volume based.
   They compute target capital by applying factors to premium, reserve or asset balances.
   Ex: NAIC RBC, and most rating agency capital adequacy models.
2. Classification rating plans.
   They compute a premium as a function of risk characteristics, such as building value and location, construction, occupancy, protection, and use for property insurance.

Question 50

Insurance company operations are governed by the interaction of which two risk measures.

Answer 50

1. A capital risk measure setting the needed amount of capital.
   Determines the assets necessary to back an existing or hypothetical portfolio at a given level of confidence.
2. A pricing risk measure determining its cost.
   Determines the expected profit insureds need to pay in total to make wi worthwhile for investors to bear the portfolio’s risk.

Question 51

Contrast the sensitivity trigger of capital versus pricing risk measures.

Answer 51

1. Capital risk measures are sensitive to tail risk to ensure solvency.
2. Pricing risk measures are sensitive to volatility to ensure solvency and consider earnings risk.

Question 52

Identify 3 ways to generalize the mean.

Answer 52

1. Adjust outcomes by a factor depending on the outcome value and explicit sample point omega.
2. Adjust the probabilities to create a new measure. The new measure can make previously impossible events possible and vice versa.
3. Adjust with a function of loss and not omega. SD has this form, h(x) = x^0.5 and g(x) = (x-u)^2.
4. Adjust outcomes independently of omega and leave probability unchanged. If the function y is increasing and convex then this form is called an expected utility risk measure.