Mildenhall Flashcards
What is risk?
Effect of uncertainty on objectives
effect is a deviation from what is expected
caused by events which have consequences
What does it mean for a risk to be 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 if immediately due (2) a discount factor
What are some types of risk?
Asset
catastrophe
underwriting
reserve
operational
strategic
reputational
compliance
credit
market
What is systemic risk?
affects a financial system consisting of many interacting agents or firms
occurs when event causes a chain reaction
SIFIs generate systemic risk (too big to fail companies)
P&C insurers not usually SIFIs, but AIG (life) was in ‘08
interaction of rating plans is a source of systemic risk for insurers (combo of adverse selection and the winners curse)
cats are SYSTEMATIC, not systemic.
What’s the difference between objective and subjective probabilities?
Objective: amenable to precise determination (repeated obs), applies LLN, CLT, bayesian stats to make precise predictions about samples
Subjective: provide a way of representing a degree of belief (applied to nonrepeatable events: election, horse race, economic outcome)
What is another term for process risk?
Aleatoric uncertainty
Epistemic refers moreso to model risk (knowledge gap)
What is the explicit representation of risk outcomes?
represent outcomes with a unique identifier such as pol num, VIN, date and time of loss, GPS location of accident
an element x of sample space
strength = enables outcomes to be linked across a book of business (model dependence risk)
What is the implicit representation of risk outcomes?
identifies an outcome with its value
easy to understand, hard to aggregate
cant distinguish between implicit events with the same loss outcome
What is the dual implicit representation of risk outcomes?
if we only care about the rank of the outcomes
identify outcome X = x with its nonexceedence probability F(x)
or we can identify with exceedence probability S(x)
disadv: relative to an often unspecified reference portfolio
e.g. bond default probability groups
What is a risk measure?
a real-valued functional on a set of random variables that quantifies a risk preference
risk measure conducts a “taste-test” so to speak
given two, which is preferred? (lower risk)
examples: NAICs RBC or a rating plan
numerical representation of risk preferences
What is a risk preference?
Something that models the way we compare risks and decide between them
defined on a set of loss RVs S
X >= Y if X is preferred to Y if X>=Y and Y>= X we are neutral
What three properties should a risk preference for insurance outcomes have?
Complete => for any pair of prospects either X>=Y, Y>=X or both
Transitive if X>= Y and Y>=Z then X>=Z
Monotonic if X <= Y in all outcomes then X >= Y
What three characteristics of a random variable do risk measures capture?
(1) Volume => smaller risks are preferred
(2) Volatility => less volatility is preferred
(3) Tail => lower likelihood of extreme outcomes is preferred (one-sided)
What is a capital risk measure?
determines assets needed to back an existing or hypothetical portfolio at a given level of confidence
used by company to determine economic vapital
regulator to determine MCR
What is a pricing risk measure?
determines expected profit insureds need to pay in total to make it worthwhile for investors to bear the portfolios risk
aka premium calculation principles (pcps)
What form of a risk measure is known as dual utility theory?
Adjusting probabilities by the rank of their loss but don’t change the loss
leads to spectral risk measures g: [0,1] => [0,1] must satisfy g(0) = 0 and g(1) = 1
What does the law invariance property refer to?
The risk measure relies solely on the distribution function F(x).
What are the advantages of VaR?
Simple to explain
estimated robustly
always finite
widely used by regulators, rating agencies, and companies in internal risk assessment
How is PML different from MFL
PML refers to the largest loss in a building likely to suffer from a single fire if critical protection systems function as expected
MFL is if the protection systems all fail
adjusted probability formula for occurrence PMLs
1 + ln(p) / lambda
Aggregate VaR adjusted probability
1 - (1-p) / lambda
Formula to approximate sum of VaRs
when Xo is thin-tailed
when X1 is an independent thick-tailed
Var(Xo + X1) ≈ E[Xo] + Var(X1)
better as p gets higher.
When is a distribution thin-tailed?
When it’s bounded or has a log concave density
aka superexponential
What are three ways VaR can fail to be subadditive?
With a highly asymmetric dependence structure
when the marginals are heavily skewed
when the marginals are very thick-tailed
What’s the worst pairing of X1 and X2 in terms of subadditivity?
not the comonotonic pairing
given 10,000 pairs, we want the 100th largest to be maximized for 0.99 VaR
crossed pairing is the solution: 1st with 100th, 2nd with 99th… etc
works for any 2 nontrivial marginals
values below p var DONT matter
creates risk ONLY at p.
How can VaR subadditivity fail for skewed marginals?
Suppose 2 iid exponentials with mean 1. X1 + X2 ~ Gamma(alpha = 2)
Var xi = 1.204
varsum = 2.439
subadditive for p < 0.7.
Whats the formula for a normal distributions TVaR?
mu + [sigma * phi(zp)] / (1 - p)
phi = [e^-(z^2/2)] / sqrt(2pi)
Whats the formula for a lognormal distributions TVaR?
[E[X] * dist_func(sigma - zp)] / (1-p)
TVaR for variables with density c(a) x^a g(x)
g(x) doesn’t involve powers of x alone or a
= c(a) / c(a+1) * [1 - F(q(p): a+ 1)] / [1 - F(q(p): a]
examples: gamma, generalized beta
What is PELVE?
Probability equivalent level of VaR and TVaR
constant c such that TVaR(1 - c*epsilon) = VaR(1 - epsilon) for small epsilon
c usually > 2.5
What are the formulas for the upper and lower CTE?
LOWER: 𝖢𝖳𝖤𝑝(𝑋) ∶= 𝖤[𝑋 ∣ 𝑋 ≥ 𝖵𝖺𝖱𝑝(𝑋)]
UPPER: 𝖤[𝑋 ∣ 𝑋 ≥ 𝑞+(𝑝)]
What is the formulas for the WCE?
Worst conditional expectation, aka highest E[X] over a subset of the sample space with at least probability 1- p
𝖶𝖢𝖤𝑝(𝑋) ∶= sup {𝖤[𝑋 ∣ 𝐴] ∣ 𝖯𝗋(𝐴) > 1−𝑝}
When will WCE, CTE, and TVaR not all be the same?
For discrete distributions when p coincides with a mass point
Whats the equality for WCE, CTE, TVaR, and VaR?
𝖵𝖺𝖱𝑝(𝑋)≤𝖢𝖳𝖤𝑝(𝑋)≤𝖶𝖢𝖤𝑝(𝑋)≤𝖳𝖵𝖺𝖱𝑝(𝑋)
What optimization problem do TVaR and VaR solve?
𝖳𝖵𝖺𝖱𝑝(𝑋) = min𝑥 {𝑥 +(1−𝑝)−1𝖤[(𝑋 −𝑥)+]}
𝖵𝖺𝖱𝑝(𝑋) = argmin𝑥 {𝑥 +(1−𝑝)−1𝖤[(𝑋 −𝑥)+]}.
TVaR balances the cost of providing capital against the cost of a shortfall
What is the formula relating expected policyholder deficit and TVaR?
𝖤[(𝑋−𝑎)+]=(1−𝐹(𝑎))(𝖳𝖵𝖺𝖱𝐹(𝑎)(𝑋)−𝑎).
𝖳𝖵𝖺𝖱𝑝(𝑋) = 𝖵𝖺𝖱𝑝(𝑋)+ 𝖤[(𝑋 −𝖵𝖺𝖱𝑝(𝑋))+] / (1- p)
What is the Lloyd’s definition of a realistic disaster scenario (RDS)?
An insurance event that is potentially disastrous but plausible
Events: E1…. EN
What is the formula for a conditional probability scenario?
𝖰𝑘(𝐴) ∶= 𝖯(𝐴∩𝐵𝑘) / 𝖯(𝐵𝑘)
𝐵𝑘 ∶= 𝐵(𝐸𝑘) to be the set of all sample points where the insurance event 𝐸𝑘 occurs, i.e., those with the value 1 in the 𝑘th place.
Coherent risk measure form:
𝜌𝑐(𝑋) ∶= max{𝐸𝖰𝟣[𝑋],…,𝐸𝖰𝗋[𝑋]}
What are the two types of uncertainty associated with P?
- Statistical uncertainty: 𝖯 is an estimate subject to the usual problems of estimation risk
- Information uncertainty: 𝖯 is based on a limited and filtered subset of ambiguous information
What are generalized (non-conditional) probability scenarios?
Scenarios such as insureds being systematically misclassified, adverse selection, parameter error, take the expected losses of all and compute the maximum E[X]
What are some advantages of the risk measure pc (allows generalized probability scenarios)?
● is intuitive and easy to communicate,
● can be used for capital and pricing,
● has properties as a risk measure that are aligned with rational risk preferences, and
● any measure with those properties is a 𝜌𝑐 for some set of probability scenarios.
Translation invariance
p(X+c) = p(X) + c
requires p to be in monetary units
VaR, TVaR and a scenario loss at TI
sd, variance, and higher order central moments aren’t TI
Normalized property of a risk measure
p(0) = 0
risk of an outcome with no gain or loss is zero
a risk is preferred to doing nothing if p(X) <= 0 (aka “acceptable”)
Monotone property
If X<=Y for all outcomes then X >= Y (preferred over Y)
all pc risk measures are monotone
sd is not monotone.
What is the no ripoff property?
if X <= c then p(X) <= c
Positive loading property
p(X) >= E[X]
not all pc have this property, generally larger Q means more likely it does
Monetary risk measures
satisify MON and TI
Positive homogeneous
p(lambda X) = lambda * p(X)
scale invariance
PH implies NORM since p(0) = p(0 x X) = 0 * p(X)
Lipschitz Continuous property
the difference in risk between random variables X and Y is at most the maximum of their outcome differences
|p(x) - p(y)| <= MAX(|X - Y|)
Subadditive property
𝜌(𝑋+𝑌) ≤ 𝜌(𝑋)+𝜌(𝑌)
the risk of the pool is less than or equal to the sum of its parts
mergers do not increase risk
implies p(0) >= 0
Sublinear property
PH and SA hold.
sublinear pricing risk measures have a positive bid-ask spread
ask price = p(X)
bid price = -p(-X)
Comonotonic property
X and Y are COMON if X = g(Z) and Y = h(Z) for increasing functions g and h and a common variable Z
different excess layers of a risk Z are comonotonic
Comonotonic additive property
p(X+Y) = p(X) + p(Y) if X and Y are comonotonic
Independent additive property
p(X+Y) = p(X) + p(Y) if X and Y are independent
Variance is an example
Where is law invariance not appropriate?
for some pricing applications, certain risks may have the same distribution but one is definitely more risky (florida hurricane risk vs auto liability)
CAPM is not LI. same return distribution but correlation to the market is what matters.
Coherence requirements
monotone
translation invariant
positive homogeneous
subadditive
Spectral risk measure requirements
coherent
law invariant
comonotonic additive
What is a compound risk measure?
a combination of a pricing risk measure (p) and a capital risk measure (a)
p(x) = p(X ^ a(x))
What are some issues with the expected utility representation when applied to firms?
firms dont have diminishing marginal utility of wealth
firm preferences aren’t relative to a wealth level
utility theory combines attitudes to wealth and risk
not linear and thus expected utility isnt a monetary risk measure
based on combination through mixing, not pooling, counter to insurance
What is an event in insurance?
a set of circumstances likely to result in insurance losses
occurrence of a hurricane
traffic jam
What is the ad hoc method for selecting a risk measure?
start with a seemingly reasonable risk measure and rationalize it by establishing it has properties desired (or argue against it)
e.g. constant underwriting margin
What is the economic method for selecting a risk measure?
use a rigorous economic theory to select a risk measure
set up and solve an optimization problem (utility-based approaches)
What is the characterization method for selecting a risk measure?
start with a list of desirable properties and then determine which risk measures have those characteristics
What are the most important properties of a risk measure?
Monotone and TI (intuitive)
ability to be allocated
diversification must be reflected
Backtesting (consistent with observations)
Explainable (sell to users)
Elicitability (estimated via regression-like techniques)
Robustness / continuity
theoretic soundness and consistency
MAD BEER (T)
What are 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.
What are the 8 gradations of tail-thickness?
(1) No mean. VaR is not subadditive. LLN doesn’t apply. Impossible to insure.
(2) Mean no variance: LLN applies by CLT doesnt
(3) mean and var finitely many moments: LLN and CLT apply
(4) all moments, sub exponential: decays slower than exponential e^kx * S(x) => inf for all k>0
(5) exponential tail (dividing line between thick and thin)
(6) super exponential : thin-tailed e^kx*S(x) => 0 for all k > 0
(7) log concave density: proportional to -x^2 (sample averages tightly clustered around the sample mean)
(8) bounded.
What is the intended purpose and the intended user of a model?
Intended purpose: The goal or question, whether generalized or specific, addressed by the model within the context of the assignment.
Intended User: Any person whom the actuary identifies as able to rely on the model output.
What are some intended purposes of risk measures?
Individual risk pricing: quoting and evaluation of market pricing
classification rate making: setting profit margins and allocating cost of capital
portfolio management: reinsurance purchase, ORSA,
Capital: determine risk capital or evaluating held capital
