Module 15: Introduction to risk modelling

Question 1

5 Examples of quantifiable risks

Answer 1

• enterprise risk
• insurance / underwriting risk
• market / economic risks
• credit risk
• liquidity risk

Question 2

Type of model used for analysing:

Enterprise risk

Answer 2

Dynamic Financial Analysis
and
Financial Condition Reports

Question 3

Type of model used for analysing:

Insurance / underwriting risk

Answer 3

underwriting models or reviews

Question 4

Type of model used for analysing:

market / economic risks

Answer 4

• VaR,
• Tail VaR,
• interest rate models,
• scenario testing

Question 5

Type of model used for analysing:

credit risk

Answer 5

Credit risk models

Question 6

Type of model used for analysing:

liquidity risk

Answer 6

Asset-Liability Models

Question 7

4 Issues in risk quantification

Answer 7

• difficulties in assessing possible emerging risks and future extreme events (although EVT may be useful)
• imperfect data (eg limited volume, heterogeneity)
• difficulties in assessing the interdependence of risks (although multivariate distributions or copulas may be useful)
• how to deal with unquantifiable risks (although broad risk ranges or buckets may be used)

Question 8

Correlation

Answer 8

A measure of how different variables relate or associate to each other:

• if risks all have high positive correlation with a given risk factor then this is evidence of risk concentration
• negative correlation implied (partially) offsetting risks
• low correlation implies (some degree of) diversification

Question 9

Pearson’s rho

Answer 9

A measure of linear dependence between variables.

It is a function not only of the joint distribution of the variables but also of the marginal distributions.

It is only valid if the marginal distributions are jointly elliptical.

Question 10

Rank correlation coefficients

Answer 10

Based on the positions (or ranks) of items of observed data rather than the data values themselves:

• they are independent of the marginal distributions of the variables
• take a value of zero if the random variables are independent
• take the value of 1 if the ranks are perfectly aligned (comonotonic)
• take the value of -1 if the ranks are precisely reversed (countermonotonic)
• Examples are Spearman’s rho and Kendall’s tau

Question 11

Tail correlation

Answer 11

Focuses on the extreme values.

For example, it may be defined as the correlation based on the lowest and highest k% of observations in a sample.

Question 12

Deterministic model

Answer 12

Uses a set (or sets) of assumptions that are pre-determined.

Under each set, each variable takes a unique value.

Deterministic models can be back-tested by using historical data and comparing the results with what actually happened.

Question 13

Deterministic modelling approaches include: (3)

Answer 13

• sensitivity testing
• scenario analysis (with links to Business Continuity Management)
• stress testing

Question 14

Stochastic modelling

Answer 14

Treats one or more of the assumptions as random variables.

The model is run several times (simulations) each drawn randomly from the distribution(s).

The output is a distribution of outcomes.

Question 15

2 Main approaches to stochastic modelling

Answer 15

• Bootstrapping (generating simulations using historical data)
• Forward-looking (using Monte Carlo simulation)

Question 16

Forward-looking stochastic models can be: (2)

Answer 16

• factor-based (modelling causal links between drivers and key variables)
• data-based (modelling key variables directly, eg time-series approach)

Question 17

Dynamic Financial Analysis

Answer 17

Involves modelling the risks to which the enterprise as a whole is exposed and the relationships between these risks.

The outputs from these complex models is generally in the form of cashflow information used to produce projected balance sheets and profit and loss accounts.

The method of assessing a company’s capital is typically a form of dynamic financial analysis.

Question 18

Financial Condition Report

Answer 18

A report into the current solvency position of a company and its possible future development.

This requires the company to consider the risks to which it is exposed and, in particular, involves looking at projections of the expected level and profitability of new business, including any unusual features it may have.

Question 19

Define insurance risk and underwriting risk

Answer 19

Insurance risk relates to deviations in the timing, frequency and severity of insured events from those expected at underwriting.

Underwriting risk relates to the possible errors in the selection, approval or pricing of insurance risks.

Question 20

Describe ALM in the context of modelling liquidity risk

Answer 20

Asset-Liability Modelling is a method of projecting both the assets and liabilities of an institution within the same model, using consistent assumptions, in order to assess how well the assets match the liabilities, and to understand the probable evolution of future cashflows.

In the context of liquidity risk, we are interested in the level of cash held in each period to ensure that short-term liabilities can continue to be met with a desired level of confidence.

Question 21

Describe what is meant by the term “black swan event”

Answer 21

One-off events which are rare (beyond normal expectation), hard-to-predict and high-impact have come to be known as “black swan” events.

They are sometimes characterised as those events that are “predictable with hindsight”.

Question 22

Outline two processes that could help us respond appropriately to rare events when they happen

Answer 22

To help us respond to rare large-impact events, we could:

1. use previous experiences and incorporate any learning points from past events into our ERM strategy with an aim of becoming better able to react appropriately to ‘surprising’ events
2. develop an emerging risks register of potential future issues.

Question 23

Outline possible limitations of data upon which risk assessment is to be based

Answer 23

Data upon which a risk assessment is to be based might be limited in volume and/or be heterogeneous.

Alternative data sources (eg external published data) may help to reduce such problems but could introduce new issues (eg lower reliability).

Question 24

Describe how unquantifiable risks might be analysed using a risk map

Answer 24

The use of risk ranges or risk buckets is one possible approach to recognising the lack of granularity in a risk analysis.

The ranges might be quantitative (eg 0% to 20%, 20% to 40%, etc) but could be qualitative (eg low, medium, high).

The results of the analysis could be displayed on a risk map.

Question 25

Define correlation

Answer 25

Correlation is a measure of how different variables relate or associate to each other. In the context of ERM, we will be interested in how different risks respond to changes in a given risk factor. If the different risks respond similarly, then this is evidence of a concentration of risk, which may require careful monitoring and management.

Question 26

Define Pearson's rho (in words)

Answer 26

Pearson's rho, ρ, takes values in the range [-1, 1] and is a measure of linear dependence between the variables. I.e. if Y depends on X linearly (namely, Y = aX + b for some numbers a and b), then |ρ(X, Y)| = 1.

Question 27

State what it means when Pearson's rho takes each of its highest and lowest possible values

Answer 27

A value of +/- 1 indicates the variables have perfect linear correlation, ie that one variable is a linear function of the other. A value of +1 indicates perfect positive linear correlation. A value of -1 indicates perfect negative linear correlation.

Question 28

Outline the limitations of linear correlation

Answer 28

- The value of the linear correlation coefficient is not unchanged under the operation of a general (non-linear) strictly increasing transformation. - Pearson's rho is a valid measure of correlation only if the marginal distributions are jointly elliptical (eg multivariate normal) - Linear correlation is not defined where var(X) or var(Y) is indefinited. Hence, it cannot be used for some heavy-tailed distributions which may be of interest in ERM. - Independent variables are uncorrelated, ie ρ = 0, but not all uncorrelated variables are independent. (ie ρ = 0 does not imply that there is no relationship between the variables, only that there is no linear relationship). - Given the marginal distributions of a pair of random variables, X and Y, and a specified value of ρ, it is not necessarily the case that we will be able to put together a joint distribution to combine all this information. In effect, the value of ρ may be one that is unattainable, ie incompatible with the marginal distributions.

Question 29

State the key advantage of measures of rank correlation over linear correlation

Answer 29

The value of the linear correlation is dependent not only on the joint distribution (or copula), but also on the marginal distributions. The rank correlation of a bivariate distribution, however, is independent of the marginal distributions, which means that, as a measure of dependence, it has more attractive properties.

Question 30

Properties of both Kendall's tau and Spearman's rho

Answer 30

- They take values in the interval [-1, 1] - They are symmetric, ie p(X, Y) = p(Y, X) - They give a value of zero if the random variables are independent - They take the value 1 if the ranks of X and Y are perfectly aligned (comonotonic) and the value - 1 if the ranks of X and Y are precisely in reverse (countermonotonic). Note that this is not true for linear correlation, due to its limited attainable correlations.

Question 31

Define what is meant by a deterministic model

Answer 31

A deterministic model uses a set (or sets) of assumptions that are predetermined. Each set of assumptions uniquely determines the value to be taken by each variable in the model. For each single set of assumptions, the output from the model is fully determined - there is no random element. By contrast, the output from a truly stochastic model (ie not one based on pseudo-randomness) is unpredictable.

Question 32

State how prudence is allowed for in a deterministic model

Answer 32

Prudence is allowed for through the particular choice of assumptions, eg by adding margins to (best-estimate) assumptions.

Question 33

Sensitivity analysis

Answer 33

Involves varying each input assumption one at a time to quantify the effect each has independently on the model's output. It may be performed for every input, or just for those that are considered key to the model's operation.

Question 34

Outline 3 key reasons a company might use sensitivity analysis as part of risk assessment

Answer 34

1. to develop an understanding of the risks faced 2. to provide insight into the dependence of the output on subjective assumptions 3. to satisfy a supervisory authority's requirements

Question 35

Outline 1 key limitation of using sensitivity analysis as part of risk assessment

Answer 35

A key limitation of sensitivity analysis is that no probabilities are assigned to each of the options used; the options are merely viewed as possibilities of what might happen in certain circumstances.

Question 36

Scenario analysis

Answer 36

Similar to sensitivity analysis, except that instead of varying just one input at a time, we change multiple inputs simultaneously. Scenario analysis is therefore concerned with looking at the results from a model under various scenarios.

Question 37

Define precisely what is meant by a scenario

Answer 37

A scenario is a set of model inputs that represents a plausible and internally-consistent set of future conditions.

Question 38

Outline 4 steps a company should follow when conducting a scenario analysis within a risk-management framework

Answer 38

1. DECIDE ('TOP-DOWN') ON THE SCENARIOS TO BE MODELLED. could be based on: - historical events. - asking participants what the worst plausible event is that they can imagine. 2. ESTABLISH THE IMPACT ON RISK FACTORS, IE MODEL INPUTS. Run the risk model to give a feel for the overall effect of the scenario. 3. TAKE ACTION BASED ON RESULTS. - review the results from step 2 - decide what plans to put in place to minimise the effect of the scenario. - identify early warning indicators 4. REVIEW THE SCENARIOS TO ENSURE THEY REMAIN RELEVANT OVER TIME.

Question 39

Outline the advantages of scenario analysis

Answer 39

- It facilitates the evaluation of the potential impact of plausible future events on an organisation. - It is not restricted to consideration of what has actually happened in the past and can therefore include assessment of its vulnerabilities to high impact, low probability events. - it provides useful additional information to supplement traditional models based on statistical information - it can facilitate the production of action plans to deal with possible future catastrophes by assessing the possible impact both pre- and post impementation of a specified mitigation strategy.

Question 40

Outline the disadvantages of scenario analysis

Answer 40

- its potential complexity as a process - its reliance upon successfully generating hypothetical extreme but also plausible events - the uncertainty as to whether the full set of scenarios considered is representative or exhaustive - as with sensitivity analysis, the absence of any assigned probabilities to any of the scenarios

Question 41

Stress testing

Answer 41

Similar to scenario or sensitivity testing but it focuses only on extreme scenarios or very large changes in input assumptions.

Question 42

2 Main categories of stress tests

Answer 42

- 'top down' stress-scenario tests - 'bottom up' stress-variable tests, where, instead of looking at a particular scenario and varying all risk factors in a mutually-consistent fashion, the effect of a significant adverse change in a crucial factor (or a narrow range of crucial factors) is considered.

Question 43

Outline the advantages of stress testing

Answer 43

- the ability of supervisors to compare the impact of the same stresses on differing organisations - the explicit examination of extreme events which might not otherwise be considered, eg if a stochastic approach was adopted - use in assessing the suitability of any response strategies, by assessing the expected (gross) impact of the stress in the absence of any response, and the expected (net) impact in the presence of the proposed response

Question 44

Outline the limitations of stress testing

Answer 44

- It is subjective as to which assumptions to stress and the degree of stress(es) to consider. - It assigns no probability to the events considered. - It looks only at extreme situations, and so needs to be coupled with other techniques, eg simulation, in order to understand the full range of outcomes.

Question 45

Business Continuity Management

Answer 45

A program which exists in most businesses to ensure that they can continue to operate in the fact of disaster or extreme events, usually in the context of operational risk. The idea from BCM of simulating emergencies to test what participants' reactions would be, can be a useful way of establishing the responses to an extreme event, which scenario analysis will then use to determine their likely long-run impact.

Question 46

Back-Testing

Answer 46

A way of validating the models currently in use within an organisation. It involves running a model using historic data (so we effectively use a scenario that has already happened), and comparing the model output with what actually happened in reality. Any discrepancies can be investigated and their causes remedied. Back-testing of models is required under Basel II, and the results of this can impact on a bank's capital requirements.

Question 47

Define what is meant by a stochastic model

Answer 47

A stochastic model is used when inputs to a model are uncertain. Its key benefit is that it provides a probability distribution for the model outputs. This is achieved by running the model repeatedly, each run being known as a simulation, and accumulating the results of these simulations to give a distribution of potential outcomes. From this outcome distribution, we can estimate the mean outcome, its variance, and probabilities associated with the outcome being more or less than a certain value.

Question 48

Historical simulation

Answer 48

Each simulation is generated through direct reference to historical data, eg by random sampling

Question 49

Outline the advantages of historical simulation

Answer 49

- It is applicable to many situations, as long as suitable past data is available. - It does not require large amounts of past data, if the sampling is done "with replacement". - It does not require the specification of probability distributions for the inputs. - It reflects the characteristics of the past data (including non-linearity, non-normality, interdependencies etc) without the need for parameterisation.

Question 50

Outline the disadvantages of historical simulation

Answer 50

- It cannot be performed in the absence of any relevant past data. - It assumes that past data is indicative of the future. - It does not take into account inter-temporal links between past data items (eg auto-correlation) - It may underestimate uncertainty (as it is based only on what actually happened in the past, rather than on what potentially could have happened) and so, in practice, should generally be used with other methods (eg stress tests) so as to consider a greater range of outcomes.

Question 51

Monte Carlo simulation

Answer 51

Each simulation uses random numbers to generate input values and the model is then run using these values.

Question 52

Outline the advantages of Monte Carlo simulation

Answer 52

- Computer packages are widely available to do most of the work, these can be easily adapted and updated. - Increasing the number of simulations increases the accuracy of the output of reducing the estimation error. - It is possible to simulate the interdependence of risks. - It is a widely understood technique as it uses relatively simple mathematics. - It can be used to model complex financial instruments, such as derivatives.

Question 53

Outline the disadvantages of Monte Carlo simulation

Answer 53

- The random selection of parameter values may lead to a set of simulations which is not representative of the full range of possibilities - unless the set is sufficiently large. - Large sets of simulations may be time consuming to perform.

Question 54

Factor-based approach vs Data-based Approach

Answer 54

FACTOR-BASED: the causal links between variables are described explicitly within the model. DATA-BASED: Causality is not the main focus. Focuses more on modelling the key variables rather than the factors which drive them.

Question 55

Outline 2 advantages of the factor-based approach, when compared to the data-based approach

Answer 55

- the discipline imposed on understanding what drives the key variables - making the relationships between the drivers explicit

Question 56

Outline a disadvantage of the factor-based approach, when compared to the data-based approach

Answer 56

The additional effort required which, in some applications, may not be justifiable.

Question 57

Pseudo-random numbers

Answer 57

Numbers which appear not to follow a pattern but which are the result of an underlying mathematical process.

Question 58

Outline the properties of Pseudo-random numbers that make them appropriate for the purposes of simulation, and state one method of generation that might be used.

Answer 58

For the purposes of simulation, psuedo-random numbers should: - be replicable - to facilitate checking - repeat only after a long period - to create valid lengthy simulations - be uniformly distributed over a large number of dimensions - exhibit no serial correlation

Question 59

Describe how any differences between market values and modelled values should be considered

Answer 59

Any differences between market values and modelled values should either be: - explained, or - identified (to users of the model) as being indicative of possible error (eg model error, data error)