ERA 3 TIA

Question 1

When implementing an Internal Risk Model, firms often underestimate? (3)

Answer 1

• Resource Commitment - Staff, Systems, Software
• Timelines
• Organization impact

Question 2

When implementing an Internal Risk Model - Staff Considerations (5)

Answer 2

• Reporting Lines should be clear
• Leader should have a reputation for Fairness
• Functions Represented: U/W, Planning, Finance, Actuarial, Risk
• Full Time Staff vs. Part Time Staff (also have day to day job)
• Permanent Staff vs. Temporary Staff (for implementation)

Question 3

Implementing IRM - Scope considerations? (4)

Answer 3

• Underwriting Year
• Reserves
• Assets
• Low Detail on Company OR High Detail on pilot segment

Question 4

Parameter Estimation is difficult because (4):

Answer 4

• Low Data Quality
• Low Data Volume
• Unique Characteristics of Firm
• Differing Risk Attitudes

Question 5

Correlation Assessment in an IRM is difficult because (4):

Answer 5

• Lack of Data
• High Political Sensitivity
• spans Multiple Business Units
• Significant impact on Company Risk Profile and Capital Allocation

IRM team recommends correlation assumptions
Owned by CRO/CEO/CUO

Question 6

Why is Validation of an Internal Risk Model difficult?
How can we Validate?

Answer 6

• No current model to compare to
• Review a series of Complementary variables over an
  extended period

Question 7

How should a Pilot Test be done for the
implementation of an Internal Risk Model?

Answer 7

• Provide output in parallel to current decision metrics
  (allows user to get comfortable with new metrics)
• High Level of the Company OR Detail of a Pilot Segment
• Provide Education on New Metrics
• Each Quarter increase Weight that is given to new metrics

Question 8

Recommendations for Integration and Maintentance
of an Internal Risk Model (4)

Answer 8

• Integrate into the Corporate Calendar that already exists
• Major Updates - no more than twice a year
• Minor Updates - via scaling
• Input/Output - Ownership and Control must be very clear

Question 9

Formula for Coefficient of Variation (CV) of Losses

Answer 9

Question 10

What is Superimposed Inflation?

Answer 10

Severity Trend less General Inflation

[Claim Severity Trend] =
[General Inflation] + [Superimposed Inflation]

Question 11

For Projecting Annual Loss Trend, the author
recommends an AR(1) process with what parameters?

Answer 11

Question 12

What is the preferred method to estimate parameters
for Frequency and Severity Distributions?

Answer 12

Maximum Likelihood Estimator (MLE)

Among Unbiased estimators, it has the lowest Estimation
Error (for large data sets)

Question 13

When Estimating Parameters how we can estimate
correlations?

Answer 13

Question 14

How do the authors recommend we model parameter
estimates and their dependencies?

Answer 14

Model the parameter estimates as Joint LogNormal
with correlations from the Information Matrix

Question 15

Why do we prefer to use Joint LogNormal to model
estimates of parameters? (2)

Answer 15

• Removes Negative Values from possible
  simulated values
• Parameter Estimates have a heavy tail -
  LogNormal captures this

Question 16

What is Model Risk?

Answer 16

Risk that the selected model is not the correct one

Question 17

When estimating parameters - we can calculate the
Likelihood of the data given the selected parameters

What does the slope of the Negative Log Likelihood
tell us about our estimate of the parameters?

Answer 17

A steep slope tells us we are quite certain of our estimate of
the parameters

A shallow slope tells us we are not certain of our estimate of
the parameters

Question 18

A method to account for Model Risk

Answer 18

• When comparing different models use the HQIC to
  compare them
• Hannan-Quinn Information Criterion (HQIC) is a
  compromise on the # of parameters penalty
• Multiple models can be chosen for a pool of possible
  models
• For each simulation, draw one of these models, and then
  parameters from that model

Question 19

Formula for Copula Density

Answer 19

Question 20

How can we interpret the values of the copula density
c(u, v)?

Answer 20

If c(u, v) = 1, then the density is the same as if u & v were independent

c(u, v) > 1 Density is higher
c(u, v) < 1 Density is lower

Question 21

Gumbel τ

Answer 21

τ = 1 - 1/a

Question 22

HRT τ

Answer 22

τ = 1 / (2a + 1)

Question 23

Frank τ

Answer 23

(Don’t need to memorize)

not closed form

Question 24

Normal τ

Answer 24

τ = 2 * arcsin(a) / pi

Question 25

Formula for the Conditional Distribution of v|u

Answer 25

Question 26

Method to simulate Frank’s Copula

Answer 26

• Draw u, p ∼ Uniform[0, 1]
• set p = C1(u, v)
• solve for v
• (u, v) are drawn from Frank‘s copula

Question 27

Method to simulate Normal Copula

Answer 27

• Draw x, y ∼ Normal(0, 1) -> invert e.g. if x = 0, then u = 0.5
• z = ax +√ (1 − a^2) · y
• u = Φ^−1(x) & v = Φ^−1(z)
• (u, v) are drawn from the Normal Copula

Question 28

Method to simulate Gumbel Copula

Answer 28

*Draw p,r ∼ Uniform[0, 1]
* Solve for s:

Question 29

What is the Left Tail Concentration Function

Answer 29

L(z) = P(U < z|V < z) = C(z, z)/z

Probability that U is small, given that V is small

Question 30

What is the Right Tail Concentration Function

Answer 30

R(z) = P(U > z|V > z) = (1 − 2z + C(z, z))/(1 − z)

Probability that U is large, given that V is large

Question 31

Which of the 4 copulas has a Right Tail Concentration
Function that approaches zero in the limit?

Answer 31

Normal & Frank

Question 32

Which of the 4 copulas has the lightest left tail?

Answer 32

HRT

Question 33

Which Copula has a heavy right tail;
and a left tail density similar to Normal?

Answer 33

Gumbel

Question 34

What copulas are available to model dependencies
among multiple variables?

Answer 34

• t-copula
• Normal

Question 35

What advantages does the t-copula have over other
copulas? (4)

Answer 35

• Can be strongly correlated in the tails - this is controlled by the n parameter
• Can join multiple variables
• Has density where one variable is high and the other is low
• approaches the Normal copula for large n

Question 36

What is the limit of J(z)?

Answer 36

lim J(z) = τ
as z→1

Question 37

How do we use J(z) to fit a copula?

Answer 37

• Graph the empirical J(z)
• Compare to graphs of J(z) for possible copulas

Question 38

The limit of χ(z)

Answer 38

lim χ(z) = R
as z→1

Question 39

How do we use χ(z) to fit a copula?

Answer 39

• Graph the empirical χ(z)
• Compare to graphs of χ(z) for possible copulas