Mildenhall Section 1

Question 1

VaR Pros and Cons

Answer 1

Pros
-Simple to explain
-Can be estimated robustly
-Always finite
-Widely used

Con:
-Not always sub-additive (aka doesn’t always reflect diversification)

Question 2

Probable Maximum Loss Types

Answer 2

Occurrence PML = VaR at adjusted p

Aggregate PML = VaR at 1 - (1-p)/lambda

note: VaR at just p is Severity

Question 3

Sub-Additivity

Answer 3

Total Measure <= Sum of Unit Measures
rho(x+y) <= rho(x) + rho(y)

Question 4

3 Ways VaR can fail Sub-Additivity

Answer 4

1. Highly asymmetric dependence structure
2. Very skewed marginals
3. Heavy-tailed marginals

Question 5

Translation Invariant

Answer 5

rho(x+c)=rho(x)+c

Question 6

Normalized

Answer 6

rho(0)=0

Question 7

Monotone

Answer 7

if X<=Y in all outcomes, we prefer X over Y

Question 8

Positive Loading

Answer 8

rho(x) >= E[X]

Question 9

Positive Homogeneous

Answer 9

rho(cx)=c*rho(x)

Question 10

Sublinear

Answer 10

If both positive homogeneous and sub-additive
rho(x) - rho(-x) > 0

Question 11

Comonotonic Additive

Answer 11

if
1. x=g(z)
2. y=h(z)
3. rho(x+y) = rho(x) + rho(y)

Question 12

Independent Additive

Answer 12

If X,Y independent then
rho(x+y) = rho(x) + rho(y)

Question 13

Law Invariant

Answer 13

if have same distribution function then
rho(x) = rho(y)

Question 14

Coherent Measure

Answer 14

Coherent if: TIMPHSA
-Translation Invariant
-Monotone
-Positive Homogeneous
-Sub-additive

Question 15

Spectral Measure

Answer 15

SRM if: (COHLICOMON)
-Coherent
-Law Invariant
-Comonotonic Additive

Question 16

Utility Theory vs Dual Utility Theory

Answer 16

Both assign every choice a utility and a probability

Dual Theory better represents firms (linear utility in wealth vs diminishing marginal utility for individuals)

Question 17

Ways to Select a Risk Measure

Answer 17

1. Ad Hoc (pick reasonable then rationalize)
2. Economic Method (rigorous theory to select)
3. Characterization Method (list desirable properties then pick one that fits, best in practice)

Question 18

Desirable Properties of Risk Margins

Answer 18

Higher if:
-less info known
-low freq, high sev
-longer duration contracts
-wide prob dist.

Should decrease when emerging info reduces uncertainty

Question 19

Desirable Properties of Risk Measures

Answer 19

-Coherent/Reflects diversification
-Practical for allocation
-Explainable

-Can be estimated with regression-like techniques
-Robust (small change in input=small change in measure)
-Consistent with past observations

Question 20

Uses of Risk Measures

Answer 20

-Individual risk pricing
-Class ratemaking
-Portfolio management
-Determining capital

Question 21

Users of Risk Measures

Answer 21

-Actuary (pricing)
-Management (reinsurance strategy)
-Regulator (Set min capital standards)
-Rating Agency (evaluating capital)
-Investor (compare risk/return)

Question 22

Pricing vs Capital Risk Measures

Answer 22

Pricing aka Premium Calculation Principles (PCPs)

Capital Measures should be:
-Standardized to compare companies
-Portfolio optimization
-Back-testing
-Simple to explain to stakeholders

Pricing should be:
-Computable
-Allocation
-Diversification

Both:
-Robust
-Optimizable
-Explainable

Question 23

Degrees of Tail Thickness

Answer 23

No mean (thickest tail)
No variance
Finite moments
Sub-exponential
Exponential (middle)
Super-exponential
Log-concave density
Bounded (aka no tail)