Risk Diversification

Question 1

Name some market risks a firm may face

Answer 1

Interest rate risk, stock market risk

Question 2

What are some risks life insurers may face?

Answer 2

Life firms: mortality risk, longevity risk, catastrophe risk.

Question 3

Under Solvency 1 how were stress test results combined together/aggregated?

Answer 3

We call stress test results ‘one-way’ tests, as they test one risk
at a time. Under Solvency 1 to combine them into an aggregate
capital requirement they generally were added together. This could be interpreted as all your risks are going to happen at once - are you prepared in terms of capital requirements.

Question 4

Why would it be naive to take the average of all stress tests results to find aggregate capital requirements?

Answer 4

As stresses have different effects and magnitudes of effect on a business.

Question 5

What is the reasoning behind not adding up all the stress tests results? - Solvency 2 approach

Answer 5

Might be sensible if all the risks were highly correlated with each other to add them all. However there is diversification meaning risks don’t all happen at the same time. Diversification is fundamental to have at work. Solvency 2 states that we don’t think you should have to add together all risks to get capital requirements. Insurers are allowed to take credit in the capital requirements for the fact that all risks won’t happen at once

Question 6

Give an example of two risks that are highly correlated that might make sense to add together. - why would you add them together?

Answer 6

Example of highly correlated uk and Irish stock market are quite linked
because of the amount of trade between the two countries. Hence adding capital requirements might make sense as two risks are likely to coincide as firms should be able to withstand both at once.

Question 7

Suppose two risks are mutually exclusive how should the capital requirements be determined

Answer 7

For example, interest rate rise and interest rate fall, then capital requirements should be the most onerous of rise and fall stress tests - based on most effected stress.

Question 8

Give two examples of risk interactions

Answer 8

Example ; writing annuities and the longevity assumption gets stronger. This changes the
duration of your cash flow and it makes you more exposed to interest rate risk as a
result.
If the USD strengthens relative to EUR, this can exacerbate a Florida
hurricane loss (in EUR).

Question 9

What does risk interactions mean?

Answer 9

Risks are said to interact when the occurrence of one risk changes
a firm’s exposure to another risk. Risk interactions are NOT the same as correlations

Question 10

Why can risk interactions sometimes be invisible in models

Answer 10

Risk interactions are only visible with combined stress scenarios
where more than one risk is tested simultaneously.
Therefore, no aggregation method for one-way capital requirements can accurately capture risk interactions.
Most internal models generate random multi-way stresses generated randomly
and would pick up the interactions, more accurate. Standard formula will ignore interactions as it conducts one way tests.

Question 11

Will all firms have capital requirements be les sunder Solvency 2?

Answer 11

Under solvency 1 you had to add all risks for capital requirements. Not the case now in solvency 2. However individual risk stress tests
are made more onerous in solvency 2 than they were previously. So even with additional diversification benefits firms suffer more onerous stress tests.

Question 12

What is the formula for capital aggregate requirements for two risks under Solvency 2

Answer 12

Cagg = sqrt(C1^2 + C2^2+2C1C2*Correlation)
This does give just addition of the capital requirements if the correlation is 1.

Question 13

What is the formula for capital aggregate requirements for two risks under Solvency 2 in the general case (multiple risks)

Answer 13

Can use matrices.
Cagg = sqrt(∑Ci^2 + 2+2 ∑i<j ρijCi
Cj)

In the first expression, the sum is taken over all pairs i,j for
the risks being aggregated, including where i=j. In the second
sum we count only the pairs where i<j.

Question 14

What is the diversification credit?

Answer 14

Cagg /Csum or
Aggregate capital requirement/Sum of all capital requirements with give the diversification factor.

Question 15

What does diversification mean in context of solvency 2 risk aggregation and what affect will it have.

Answer 15

Diversification means taking many different risks in the
expectation that not all will materialise at once.
Allowing for diversification usually reduces the capital
requirement (the sub-additivity criterion). The difference is the diversification credit.
Diversification credit is smaller if correlations are close to 1,
or if risk is concentrated in a small number of risk types.

Question 16

Define pre and post diversification capital requirement

Answer 16

The pre-diversification capital requirement is Csum = ∑ Ci
* The post-diversification capital requirement is Cagg

Question 17

What affect does the diversification credit have on insurers and customers?

Answer 17

Diversification credit gives insurers an incentive to diversify their risks which ultimately improves consumers’ chance of being paid in full.

Question 18

Why was there controversy over the new Solvency 2 regime bringing int he diversifying credit?

Answer 18

Large insurers have lobbied influentially for diversification credit because it gives them a competitive advantage relative to less diversified smaller companies. Big companies may have 50/60% diversification credit but if you are smaller firm and only exposed to one or two risks you don’t get much diversification credit. Plus solvency 2 one way stress tests became more onerous. So small firms capital requirements went up while big companies capital requirements went down.

Question 19

For every line of business in an insurance firm explain the two main categories of risk short term excluding catastrophe risk

Answer 19

For every line of business two categories of risks, premium and reserve.
Reserve risk means having to increase provisions for past accidents
Premium risks relates to future accidents on in force policies.
Ex; winter storm outs
lots of ice on road. Also accounts for premium inadequacy risk where
you’re not charging enough.

Question 20

How are the correlation matrices formed?

Answer 20

Correlations are written in Eu law and are all multiple of 1/4 depending on whether they are classified as high, moderate or low correlations. These figures were formed by opinion rather than data.

Question 21

What are the five different aggregation structures to consider for insurers depending on their business to get the overall solvency capital requirement?

Answer 21

Non life
Life risk
Health risk
Market risk
Counterparty default aggregation - treated differently

Question 22

Name stress tests conducted for a life firm

Answer 22

Mortality up
Mortality down
Sickness/Disability
Expense
Lapse
Mortality catastrophe

Question 23

Why is health risk aggregation structure a little bit more complicated?

Answer 23

Split into short term, long term and catastrophe. Can be more complex as policies can be written for 1 year time period or multi year depending on company and country policies. Therefore it takes elements form both life and non life aggregation structures.

Question 24

Corr(Pandemic, Mass Accident)

Answer 24

Assumed to be zero

Question 25

Corr(Premium Risk, Reserve Risk)

Answer 25

Assumed to be = 0.5 for all segments,

Question 26

What are the market risk stresses that need to be aggreagted?

Answer 26

Interest up, interest down, equity (aggregate of listed and unlisted), property, spread, currency

Question 27

What are the three approaches to counterparty default risk in the standard formula

Answer 27

Counterparty default risk is the most difficult risk to
aggregate in the Solvency II Standard Formula.
* The Gauss copula model
* The Basel ASRF Model
* The Solvency II Correlation Approach

Question 28

What does EAD, LGD and PD stand for?

Answer 28

Counterparty default models are usually based on the
following three concepts:
* EAD: Exposure at default (maximum amount of loss on
default if no residual value recovered).
* LGD: Loss given default ( = 1 – proportion recovered).
* PD: Probability of default.

Question 29

What three figures are we using measuring counter party default?

Answer 29

EAD, LGD, PD - we will have a large number of these parameters

Question 30

What is a name in a counter party default model?

Answer 30

Name is a person who owes you money. You can be owed
money in a lot of ways but you collect aggregate those into
one name, one firm

Question 31

Why can you not disregard correlations with counter party default?

Answer 31

Correlation is key to understanding counterparty default risk. You will not get any sort o sensible model if you ignore correlation in relation to credit risk.

Question 32

What is the process under the gauss copula model?

Answer 32

Under the Gauss Copula model (also called CreditMetrics) each name j is associated with a
latent random variable Zj ~ N(0,1)
Name defaults if CDF(Zj)<=PDj
This event evidently has probability PDj
The vector {Zj} is multivariate normal with correlation matrix {ρjk}.

Question 33

How can the gauss copula model be made more tractable?

Answer 33

The Gauss copula model is more tractable when the
correlations have a multiplicative structure:
* ρij = βi*βj for 0 ≤ β ≤ 1
* The basis for the ASRF (asymptotic single risk factor) model.

New method:
Define a single risk driver X~ N(0,1)
Set Zj = βj*X + sqrt(1- βj^2)Yj
Where the Yj ~ N(0,1) independent of X and of each other (so is specific to each name)
Again Name defaults if CDF(Zj)<=PDj where it can be verified Z is standard normal

Question 34

Explain the Basel accord ASRF Model

Answer 34

For large portfolios, the law of large numbers applies (conditional on the single risk driver X), cancelling out individual Yjs each name has, and
the portfolio defaults become a (decreasing) function of that risk driver.
The Basel loss stress formula is based on the expected losses given that X takes its 0.1%-ile value.
Also Individual loan stress
already allows for
diversification. so there is no further diversification benefit from aggregation.

Question 35

What does ASRF stand for

Answer 35

Asymptotic single risk factor

Question 36

Who primarily uses basel accord model

Answer 36

Banks - applied to banks as they have an immense amount of names to account for.

Question 37

How can the ASRF model be derived

Answer 37

Using Gauss copula with the number of names going to infinity

Question 38

How does the Basel loss stress formula work?

Answer 38

The Basel loss stress formula is based on the
expected losses given that X takes its 0.1%-ile
value. The contribution of exposure j is LGDjEADj expected defaults given X takes its 0.1%-ile value)

Question 39

How is diversification considered int he Basel formula ASRF? - What affect does this have

Answer 39

Diversification is implicit int he formula. It is assuming you have a diversified portfolio and therefore will be inadequate in terms of calculating the aggregate capital requirement is you are not diversified.

For insurers many may need the diversification assumption to be explicit as they may not have that many counterparties at play. And hence should not incorporate diversification into the model. Should use diversification benefit after capital aggregation.

Question 40

In contrast in bank capital regulation what is the basis for insurance capital for defaults risk?

Answer 40

In contrast to bank capital regulations, insurance
capital for default risk uses the correlation matrix.
The correlation between two default risks depends
on the two probabilities of default

Question 41

Compare ASRF to Solvency II. Kindve Banking vs Insurance

Answer 41

ASRF (Basel)
* Loss stress for a portfolio
is the sum of loss stresses
for each loan.
* Individual loan stress
already allows for
diversification.
* No further diversification
benefit from aggregation.

Solvency II Default Risk
* Individual loan stresses
reflect PD on that loan
(only)
* Diversification benefit
from holding multiple
exposures.
* Loss stresses (post
diversification) are not
additive.

Question 42

What is the overall theory of capital aggregation formula

Answer 42

The capital aggregation formula is a way of producing
capital numbers between the extremes of mutually
exclusive risks and totally correlated risks.
Most of the correlations come from what might be called ‘judgement’ which is why they are
all multiples of 25%.

Question 43

What is an elliptical distribution?

Answer 43

A random d-vector X has an elliptical distribution with
location vector m and scale matrix V if:
1. For all components Xi the standardized components have a common probability distribution
2. For an arbitrary vector a≠0, the standardized version of aTX, also shares that common distribution.

One familiar example is the multivariate normal
distribution, with mean m and variance-covariance
matrix (proportional to) V.
Elliptical distributions can also be constructed based on correlated
(some) other symmetrical distributions

Question 44

What is the percentile required in solvency 2 for percentile based stress tests?

Answer 44

Under Solvency II we might have
p=0.995 for the stress test to represent the pth
quantile of the loss for each component,

Question 45

Why can it be hard to validate a model dealing with rare events?

Answer 45

If you say smoothing is a rare event and it doesn’t happen in your working life time - that doesn’t necessarily prove your model is any good. Especially if event is a 1/200 year event say

Question 46

If we have vectors corresponding to losses of each risk type and we want to find the aggregate risk what is the formula to use under percentile based aggregation?

Answer 46

Denote the distribution function of the standardised one-dimensional distribution by F.

We calculate the p-quantile of the total losses:
* Cagg = 1^Tm + sqrt(1^TV*1)F-1(p)

Where m is the mean of all losses. Where R is correlation matrix corresponding to V
Can be written:
= 1^Tm + sqrt[(C-m)^TR*(C-m)]

Question 47

When does the percentile based aggregation look like the solvency II formula?

Answer 47

This looks like the Solvency II formula if m=0 (mean = 0 so you are as likely to have a loss as you are to have a profit)

Question 48

Expain the cascade form idea

Answer 48

How Solvency II formulas aggregate
nearly forty stress test results into an overall Solvency Capital Requirement.
We did not use one big matrix. Instead, we used a series of smaller matrices to aggregate to intermediate stages.
There is no 40×40 correlation matrix that would do the same job in a single step for every input vector.

Question 49

Could an internal model avoid cascade form?

Answer 49

Internal model may use a 1 matrix approach

Question 50

What are some limitations of Solvency II aggregations

Answer 50

The formula does not capture the whole distribution of the losses.
The formula works for zero-mean profits and losses; it does not include elements of expected profit.
* Firms might design contracts deliberately to arbitrage the formula - such as permitting very severe losses beyond the stated 99.5%-ile.
The calibration of the correlations is rough and reflects political haggling
Some of the capital elements are obtained by multiple aggregation
steps; it is not clear that the same correlation would be appropriate for top level aggregation.
It is a fallacy that the formula relies on normal distributions: it does not!

Question 51

Explain how Solvency II formulation for aggregation does not capture the whole distribution of the losses

Answer 51

It is not theoretically possible to calculate the 99.5%-ile of aggregate losses given only 99.5%-ile information on constituent risk drivers, except under restrictive assumptions.
The formula may not cope well with non-linear exposures such as some reinsurance or derivative contracts.