Main sources of uncertainty when using models in ERM
See Module 21 Section “Sources of uncertainty”
Parameter uncertainty, model uncertainty and stochastic uncertainty
List the steps involved in developing and applying a model
List the features of the returns on individual equities that have been suggested by time series analyses
List the factors which are reflected in a credit spread and state the 3 most common measures of credit spread
Explain how the op-risk capital can be calculate under the following models
Implied capital model
See Module 24 Section “Top-down Models”
Income volatility model
See Module 24 Section “Top-down Models”
Capital asset pricing model
See Module 24 Section “Top-down Models”
Advice on actively managing operational risk in a formal way
Reasons why the company’s directors might have become interested in op-risk management
See Module 24 Section “Need to Assess Op-Risk” Point 1-5
Advantages of a more formal approach to op-risk
See Module 24 Section “Need to Assess Op-Risk” Point 6-8
Benefits that the directors could expect to see if they enforce an effective op-risk management process
See Module 24 Section “Need to Assess Op-Risk” Benfits bullet points
More op-risk related questions
Distinguish between bottom-up and top-down approaches for op-risk
Top-down: The company as a whole is considered
Bottom-up
Describe bottom-up approaches to assess op-risk
See Module 24 Section “Bottom up models” for the following key points
Model frequency and severity of day-to-day op-risk losses
Insufficient data so need to suppliment with external data
If there is sufficient data, use Monte Carlo
Model needs to cope well with the outer tail (e.g. EVT) but is not widely use due to limitation on internal data
Simpler approach can be used that assumes losses are related to the volume of transactions
Bottom up includes statistical and scenario analysis
State 4 top-down models that can be used in assessing operational risk
Main problems with the top-down models
All these models fail to capture successfully low probability, high consequence risk events
Do not consider specific individual activities / incidents
No assessment of the effect of individual events, so these models do not help in minimizing future losses by improving procedures in this area
CAPM: risk free = 2%, market risk premium = 4%, \(\beta\) = 1.25
Expected return from the market = 2% + 4% = 6%
Expected return from shares in company = 2% + 1.25 \(\times\) 4% = 7%
Cost of capital used by management in evaluting projects
Should be 7% to meet investors expectations
This assumes no tax is payable, and that a project that yields a return of 7% would offer the same return to equity shareholders
Extreme events
Examples of extreme events
See Module 20 Section “Low Freq/ High Sev Events”
Importance to consider extreme events separately from other types in ERM
See Module 20 Section “Low Freq/ High Sev Events”
Generalized Extreme Value distribution
Describe the GEV distribution
See Module 20 Section “GEV Distribution”
The distribution of the maximum value \(X_M\) in a sample on \(n\) iid r.v. when \(n \rightarrow \infty\)
Discuss the different parameters and the different classes corresponding to each \(\gamma\)
Describe how GEV can be used to model extreme events in ERM
See Module 20 Section “Generalized Extreme Value Distribution”
Extreme loss events correspond to the maximum values experienced over a period, so we might expect them to conform to the GEV distribution
We can calculate the maximum loss event from past data by dividing it into blocks and calculating the maximum within each block
We can do it 2 ways (max in a block or threshold)
Estimate the parameters with MLE or moments
Use the fitting distribution for percentiles, means and variances
Alternate approach to the GEV
Theshold approach with GPD
See Module 20 Section “Generalized Pareto Distribution”
Mean excess function
Define the mean excess function
See Module 20 Section “Mean XS Function”
Formula for mean XS loss for exponential with mean \(1 / \lambda\)
If \(X\) has exponential with mean \(1 / \lambda\):
\(\Pr(X >u) = \int \limits_u^{\infty} \lambda e^{-\lambda x} dx = e^{-\lambda u}\) The conditional probabilities:
\(\begin{align} \Pr(X > x + u \mid X >u ) &= \dfrac{\Pr(X > x + u \& X > u)}{\Pr(X > u)} \\ &= \dfrac{\Pr(X > x +u )}{\Pr(X>u)} \\ &= \dfrac{e^{-\lambda(x+u)}}{e^{-\lambda u}} \\ &= e^{-\lambda x} \\ \end{align}\)
Therefore:
\(\Pr(X - u \leq x \mid X > u) = 1 - e^{-\lambda x}\)
So the mean XS loss function has a constant value \(e(u) = \dfrac{1}{\lambda}\)
Shapes of the mean XS loss function for different loss distribution
Exponential
Constant, so graph would be horizontal
Normal
Symmetrical and tails off quicker than exponential on the right handside
Mean XS will tail off as the threshold increases (but always remains positive)
Uniform
Has finite upper limit
Once threshold hits the limit, the XS will always be 0
Before this point, the mean excess decrease linearly
Reasons why op-risk might NOT be best measured simply as a % of revenue
See Module 24 Section “Factor-based models”
1 year term insurance contracts with sum insured of $350,000 to a group of 100 lives
Probability it will pay out $1m or more for lives aged 50 where \(q_{50} = 0.002\)
Need at least 3 deaths during the year \(Binomial(100, q_x)\)
\(1 - (1-q_x)^{100} + { {100} \choose{1} } q_x(1-q_x)^{99} - { {100} \choose {2} } (q_X)^2 (1-q_x)^{98}\)
Just plug and play
For \(q_{60}\) Just plug the above
This example shows that \(q_{60} \gg q_{50}\), so a 1% change in all mortality rates has a greater effect for higher ages
Hence the volatility risk is much higher for the older lives in (b)