Discuss the extend of which of the risk we discusses before (Module 3) can be amenable to quantitative analysis
Risk aggregation and correlation
Enterprise-wide risk aggregation techniques incorporating the use of correlation
Measures of correlation and relative merits of each for modeling purpose
Use of scenario analysis and stress testing in the risk measurement process
(pros and cons of each)
To what extent which quantitative analysis might be feasible for different risk type
Basic concepts of correlation and scenario/ stress testing
\(Y_{n,t} = \beta_{0,n} + \left( \sum \limits_{i} \beta_{i,n} X_{i,t} \right) + \epsilon_{n,t}\)
\(Y_{n,t}\):
Value of the nth variable at time \(t\)
\(X_{i,t}\):
Value of the ith factor at time \(i\)
\(\beta_{i,n}\):
Weight give to the ith factor in respect of the n^th variable
\(\epsilon_{n,t}\):
Error term (difference between modeled and actual)
Factors are determined based on regression analysis
Risk has to be quantifiable for it to be assessed and analyzed through mathematical models
Ideally model should be backed up by good quality historical data and that can lead to the use of statistical analysis
Different methods of quantifying risk are applicable to different types of risk
Below are the major types of risk faced by an insurance company and the main techniques used to quantify the risks
Dynamic Financial Analysis
Modeling risk that the enterprise as a whole exposed and the relationship between these risks
Output from the model:
Typically cashflow information used to produce projected b/s and p&l accounts
e.g. Method of assessing company’s capital (Module 30)
Financial Condition Reports
Report of the current solvency position of a company and its possible future development
Key considerations (inputs?)
Risks the company is exposed
Projections of the expected level and profitability of new business (incl. unusual features it may have)
For insurers, this is dependent on the underwriting process
Underwriting Modeling or Reviews
Insurance and underwriting risk are quantitative by nature so quantitative models are used to assess
Insurance risk:
Risk arises from fluctuations in the timing, frequency and severity of insured events relative to the expectations at the time of u/w-ing or pricing (incl. mortality, morbidity, property and casualty)
Can incl. peresistency and expense risks
U/w risk is typically = to insurance risk
Example: Modeling the u/w-ing risk of a life insurance company issuing a policy to a smoker for a premium relevant to a non-smoker
First measure the loss to the insurance company as the difference in the expected PV of the premiums charged
Combine the above with a probability of occurrence will yield an underwriting model
Market and economic risk have been subject to more quantitative analysis than the other risks
Interest rate models:
Considered short-term, long-term interest rates and full yield curve
FX risk (currency movements) and basis risk (extent to which a particular position reflects the position required) also lends it selves well to quantitative analysis
Methods:
Typically use time series (Module 17) and risk measures like VaR and TVaR
Will typically also involve scenario testing
Credit Risk Models
Model typically concern with the credit risk of a single entity (rather than a credit portfolio)
Risk is usually assess using both quantitative and non-quantitative criteria (Module 23)
Can be modeled quantitatively but have not been done much in the past
Asset liability modeling:
Method of projecting both the assets and liabilities of an institution within the same model (w. consistent assumptions)
To assess how well the asset match the liabilities and to understand the probable evolution of future cashflows
Asset liability modeling for liquidity risk:
Difficult to deal with quantitatively
Potential risks are difficult to model with statistical distributions and the worst case typically involve insolvency
Internal and external loss data
Orgs. start collecting historical data on op-risk loss to further along quantitative analysis (Module 24)
Use of external loss data
Provide information of situations the company has not faced
Benchmark internal data
Scenario analysis and simulations
Quantitative analysis will never be substitute for the use of worst case scenarios
As only one is required to take down the company
Sec 3 will discuss scenario analysis and simulations
Legal; Regulatory; Agency; Reputational; Moral hazard; Political; Project; Strategic
Demographic risk: is a quantitative risk but methods where relatively crude until recently
For risk quantification we are interested in the extreme events in addition to the expected events
Black swan events:
One-off events that are very rare, hard to predict and high impact
Typically events that are only predictable with hindsight
e.g. 2008 credit crunch
Impossible to predict precisely but they are events with largest impacts and is likely to be of most interest and require mitigation
Processes that could help response to black swan events
Use previous experiences and incorporate learning points from past events into ERM strateg
Aim to become better able to react appropriately to surprising events
Develop and emerging risk register of potential future issues
Need to focus on the tails of probability distributions
Avoid simplifying assumptions (e.g. avoid using normal distribution)
Extreme value theory (Module 20)
Data might be limited in volume and/or heterogeneous
Can supplement with alternative data source
Imperative to consider the interrelationships between risks
(key to ERM)
Need to model many different risk factors and their dependencies within the same model (multivariate distribution)
Some risks are unquantifiable (many of the op-risk)
Generally a rough estimate can still be made or a range of scenarios considered
Can use risk ranges, risk buckets to analyze (due to the lack of granularity) and display the results in a risk map (Module 13)
Correlation:
Measure of how different variables relate or associate to each other
For ERM, we’re interested in how different risk respond to changes in a given risk factor
Pearson’s \(\boldsymbol{\rho}\)
\(\in [-1, 1]\)
Measure of linear dependencies between the the variables
e.g. if \(Y = aX + b\) then \(\mid\rho(X,Y)\mid = 1\)
\(\pm 1\) means perfect linear correlation (positive linear and negative linear)
Pearson’s \(\boldsymbol{\rho}\) for r.v. \(X\) and \(Y\)
\(\rho_{X,Y} = \dfrac{\sigma_{X,Y}}{\sigma_X \sigma_Y} = \dfrac{\mathrm{Cov}(X,Y)}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}\)
Sample correlation coefficient for discrete sample size \(T\):
\(r_{X,Y} = \dfrac{S_{X,Y}}{S_X S_Y} = \dfrac{\dfrac{1}{T-1} \sum \limits_{t=1}^T (X_t - \bar{X})(Y_t - \bar{Y})}{\left[\sqrt{\dfrac{1}{T-1}\sum \limits_{t=1}^T(X_t - \bar{X})^2}\right]\left[\sqrt{\dfrac{1}{T-1}\sum \limits_{t=1}^T(Y_t - \bar{Y})^2}\right]}\)
Example
| \(t\) | \(X_t\) | \(Y_t\) | \((X_t - \bar{X})^2\) | \((Y_t - \bar{Y})^2\) | \((X_t - \bar{X})(Y_t - \bar{Y})\) |
|---|---|---|---|---|---|
| 1 | 63 | 52 | 0.25 | 126.56 | -5.63 |
| 2 | 51 | 71 | 132.25 | 60.06 | -89.13 |
| 3 | 72 | 45 | 90.25 | 333.06 | -173.38 |
| 4 | 64 | 85 | 2.25 | 473.06 | -2.63 |
| \(\sum\) | 250 | 253 | 225.00 | 993.74 | -235.50 |
\(\bar{X} = 250 / 4 = 62.5\)
\(\bar{Y} = 253 / 4 = 63.25\)
\(r_{X,Y} = \dfrac{-235.5}{3} \left/ \middle(\sqrt{\dfrac{225}{3}} \times \sqrt{\dfrac{992.74}{3}}\right) = -0.498\)
Advantages
Value of the linear correlation coefficient is unchanged under the operation of strictly increasing linear transformation
i.e. \(\rho(a+bX, c+dY) = \rho(X,Y)\) where \(b, d > 0\)
Limitations
Value of the linear correlation coefficient is not unchanged under operation of a generally (non-linear) strictly increasing transformation
Only a valid measure of correlation if the marginal distribution are jointly elliptical (e.g. multivariate normal)
Linear correlation is not defined where \(\mathrm{Var}(X)\) or \(\mathrm{Var}(Y)\) is infinite
\(\therefore\) Can not be used for some heavy tailed distributions
\(\rho = 0\) does not imply that there is no relationship between variables
Only there is no linear relationship
There is no guarantee that we can put together a joint distribution for any given r.v. \(X\), \(Y\), and \(\rho\)
Some \(\rho\) might be unattainable
(i.e. incompatible with the marginal distributions)
This is a problem copula will solve
Value of the linear correlation is dependent on the joint and marginal distribution
See appendix for more
Calculate by looking at the position of each item of observed data when ordered (rather than the values of the data themselves)
The 2 main rank correlation measures are:
Spearman’s \(\rho\)
Kendall’s \(\tau\)
Advantages over linear correlation
Rank correlation of a bivariate distribution is independent of the marginal distribution
\(\therefore\) has more attractive properties
Spearman’s \(\rho\) for r.v. \(X\) and \(Y\) w/ marginal distribution functions \(F_X\) and \(F_Y\)
\(_{s}\rho(X,Y) = \rho \left( F_X(X), F_Y(Y)\right)\)
Spearman’s sample \(\rho\) for 2 r.v. \(X\) and \(Y\) and sample size \(T\):
\[_{s}r_{X,Y} = 1 - \dfrac{6}{T(T^2-1)}\sum \limits_{t=1}^T(V_t - W_t)^2\]
Examples
| \(t\) | \(X_t\) | \(Y_t\) | \(V_t\) | \(W_t\) | \((V_t-W_t)^2\) |
|---|---|---|---|---|---|
| 1 | 63 | 52 | 2 | 2 | 0 |
| 2 | 51 | 71 | 1 | 3 | 4 |
| 3 | 72 | 45 | 4 | 1 | 9 |
| 4 | 64 | 85 | 3 | 4 | 1 |
| \(\sum = 14\) |
Kendall’s \(\tau\) for r.v. \(X\) and \(Y\)
\(\tau(X,Y) = \mathrm{E} \left[ \text{sign} \left( (X - \tilde{X})(Y - \tilde{Y})\right)\right]\)
\((\tilde{X}, \tilde{Y})\)is an independent copy of \((X,Y)\)
A new pair of r.v. with the same joint distribution as \((X,Y)\) but statistically independent of \((X,Y)\)
Kendall’s sample \(\tau\) for 2 r.v. Kendall’s sample \(\tau\)
\(t_{X,Y} = \dfrac{2}{T(T-1)}(p_c - p_d)\)
Alternative definition of \(p_c - p_d\)
\(p_c - p_d = \sum \limits_{t=1}^{T-1} \sum \limits_{s=t-1}^T \left( I \left(W_s' > W_t' \right)-I \left( W_s'<W_t' \right) \right)\)
\(W'\) are the rankings of \(Y\) re-sequenced so that the corresponding rankings of \(X\) (\(V'\)) are strictly increasing
Easier to calculate but awkward to define algebraically
Example 1
| \(t\) | \(X_t\) | \(Y_t\) | vs \(t = 1\) | vs \(t=2\) | vs \(t=3\) |
|---|---|---|---|---|---|
| 1 | 63 | 52 | |||
| 2 | 51 | 71 | \(D(-,+)\) | ||
| 3 | 72 | 45 | \(D(+,-)\) | \(D(+,-)\) | |
| 4 | 64 | 85 | \(C(+,+)\) | \(C(+,+)\) | \(D(-,+)\) |
\(p_c - p_d = 2 - 4 = -2\)
\(t_{X,Y} = \dfrac{2}{4(4-1)}(-2) = -0.33\)
Example 2: Alternative Method
| \(t\) | \(X_t\) | \(Y_t\) | \(V_t\) | \(W_t\) | \(V_t'\) | \(W_t'\) | \(\sum \limits_{s=t+1}^T \left( I \left(W_s' > W_t' \right)-I \left( W_s'<W_t' \right) \right)\) |
|---|---|---|---|---|---|---|---|
| 1 | 63 | 52 | 2 | 2 | 1 | 3 | -1 |
| 2 | 51 | 71 | 1 | 3 | 2 | 2 | 0 |
| 3 | 72 | 45 | 4 | 1 | 3 | 4 | -1 |
| 4 | 64 | 85 | 3 | 4 | 4 | 1 | |
| \(\sum = -2\) |
Properties for both Kendall’s \(\tau\) and Spearman’s \(\rho\)
\(\in [-1, 1]\)
Symmetric
(i.e. \(\tau(X,Y) = \tau(Y,X)\))
0 if r.v. are independent
1 if ranks of \(X\) and \(Y\) are perfectly aligned (comonotonic)
-1 if ranks of \(X\) and \(Y\) are precisely in reverse (countermonotonic)
\(\left(\dfrac{3}{2} \tau - \dfrac{1}{2} \right) \leq \: _{s}\rho \leq \left(\dfrac{1}{2} + \tau - \dfrac{1}{2} \tau ^2 \right)\) if \(\tau \geq 0\)
\(\left(\dfrac{3}{2} \tau + \dfrac{1}{2} \right) \geq \: _{s}\rho \geq \left(-\dfrac{1}{2} + \tau + \dfrac{1}{2} \tau ^2 \right)\) if \(\tau < 0\)
We can focus on the relationships between variables only at the tail
Calculate correlation based only on the lowest and highest \(k\%\) of observation in a sample
Calculate tail dependence (Module 18)
Definition of tail is subjective
(e.g. cut off point)
Results can be highly sensitive to the cut off and potentially unreliable (due to insufficient data points)
Deterministic model use set(s) of predetermined assumptions
Each set of assumptions uniquely determines the value to be taken by each variable in the model
For each set of assumptions, the output from the model is fully determined
(i.e. no random element)
Prudence is allowed for through the particular choice of assumptions
(e.g. by adding margins to best estimate assumptions)
Varying each input assumption one at a time to quantify the effect each has independently on the model’s output
Reasones to use sensitivity analysis as part of risk assessment:
Develop an understanding of the risk faced
Important for company to have an awareness of how risks might impact it in different circumstances
Do do by changing the assumptions used
Provide insight into the dependence of the output on subjective assumptions
Most model inputs are subjective
Model is only as accurate and reliable as the data and assumptions that goes it
Quality of the model is dependent on the knowledge and experience of whose judgement it relies
Varying the model’s inputs can show how sensitive the model is to changes in those inputs
\(\therefore\) focus attention on the most important assumptions and make clear the model’s reliance upon judgement
Satisfy supervisory authority’s requirements
e.g. VaR doesn’t consider the most extreme events and so supervisory bodies may specify that companies should investigate the effect of more significant change in their assumptions to ensure that they have considered the full range of possibilities for future outcomes
Limitation
There are no probabilities assigned to each of the options used
Options are merely viewed as possibilities of what might happens on certain circumstances
Similar to sensitivity analysis except we change multiple inputs simultaneously
Concerned with looking at the results from a model under various scenarios
A scenario is a set of model inputs that represents a plausible and internally-consistent set of future conditions
4 Steps for conducting a scenario analysis
Decide top down on the scenarios to be modeled
Can be based on historical event (but not limited to)
Ask participants for the worst plausible event they can imagine
e.g. collapse of a major financial institution, EQ, oil shortage etc
Important to draw on expertise from across the business in order to generate meaningful scenarios and resultant changes in variables
Establish the impact on risk factors (i.e. model inputs) then run the models to get a feel for the overall effect
Take action based on results
Review results and put in place plans to minimize the effect
Look to identify early warning indicators that a certain scenario may become reality
Review the scenarios to ensure they remain relevant over time
Advantages
Facilitates the evaluation of the potential impact of plausible future events on an org
Not restricted to consideration of what has actually happened
Provides useful additional information to supplement traditional models based on statistical information
Facilitate the production of action plan to deal with possible future catastrophes by assessing the possible impact both pre and post implementation of a specified mitigation strategy
Disadvantages
Potentially complex process
Relies on successful generation of hypothetical extreme but plausible events
Uncertain whether the full set of scenarios considered is representative or exhaustive
Absence of any assigned probabilities to any of the scenarios (similar to sensitivity analysis)
Similar to scenario and sensitivity testing but focuses only on extreme scenarios or very large changes in input assumptions
2 main categories of stress test:
Top down stress scenario test
Bottom up stress variable tests
Advantages
Ability for supervisors to compare the impact of the same stresses on differeing org.
Explicit examination of extreme events which might not otherwise be considered (e.g. if a stochastic approach was adopted)
Use in assessing the suitability of any response strategies
Limitations
Subjective as to which assumptions to stress and the degree of stress to consider
Assigns no probability to the events considered
Only looks at extreme situations
So needs to be coupled with other techniques (e.g. simulations) in order to understand the full range of outcomes
Business Continuity Management
Program to ensure a business can continue to operate in the face of disaster or extreme events (usually in the context of operational risk)
Simulate emergencies to test what participants’ reactions would be to an extreme event then use scenario analysis to determine their likely long run impacts
Back testing
Way to validate the models currently in use within an org.
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
Discrepancies can be investigated and their causes remedied
Back testing is required under Basel II and results can impact the bank’s capital requirements
Stochastic model is used when the inputs to a model is uncertain
Key benefits:
Provides a probability distribution for the model outputs
Run the model repeatedly (each run is a simulation) and accumulating the results to give a distribution of potential outcomes
From the outcome distribution we can estimate the mean, variance, probabilities associated with the outcome being more or less than a certain value
Each simulation is generated through direct reference to historical data (e.g. random sampling)
Advantages
Applicable to many situations as long as suitable past data is available
Does no require large amounts of past data if the sampling is done with replacement
Does not require the specification of probability distribution for the inputs
Reflects the characteristics of the past data (incl. non-linearity, non-normality, interdependencies etc) without the need for parameterization
Disadvantages
Can not be performed in the absence of any relevant past data
Assumes that past data is indicative of the future
Does not take into account inter-temporal links between pasts data items (e.g. auto correlation)
May underestimate uncertainty (past might not capture what potentially can happen)
In practice should generally be used with other methods (e.g. stress test) so as to consider a greater range of outcomes
Simulation based on random numbers to generate input values and the model is run using these values
(e.g. using \(U(0,1)\) and look up \(F^{-1}(u)\))
Advantages
Computers packages is widely available and can be easily adopted and update
Increasing the number of simulation increases the accuracy of the output by reducing the estimation error
Possible to simulate the interdependence of risk
Widely understood technique as the math is simple
Can be used to model complex financial instruments (e.g. non-linear, non-normal payoff) like derivatives
Disadvantages
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 simulation may be time consuming to perform
Can use methods such as latin hypercube to reduce the calculation burden)
Factor based approach:
Data based approach:
Focuses more on modeling the key variables rather than the factors which drive them
Causality is not the main focus
Example with equity returns
Data based: model with time series with past equity returns and volatility as inputs
Factor based: need to model with a layered set of nested models (cascade of models)
Inner model produce values for inflation
Next layer model interest rates using inflation as input
Equity dividends might be modeled using the 2 above as inputs
Finally equity returns might be modeled using dividend and interest rates as inputs
Advantages of the factor based approach
Discipline imposed on understanding what drives the key variables
Making the relationship between the drivers explicit
Disadvantages of the factor based approach
Appear not to follow a pattern but are the result of an underlying mathematical process
For the purpose of simulation, the pseudo random numbers should:
Be replicable (for checking)
Repeat only after a long period (create valid lengthy simulations)
Be uniformly distributed over a** large number of dimensions**
Exhibit no serial correlations
E.g. use digits generated using a very large prime number (Mersenne twister that is based on \(2^N -1\))
Output from models can often be compared to observable values in the market
(e.g. market prices, implied volatility and implied correlations)
Differences between market values and modeled values should be
Explained, or
Identified as being indicative of possible error
(e.g. model error, data error)
One possible explanation for difference is that market values may at times not only represent the market’s long term views