1
Q
List: Axioms of Coherence (4)
A
- Monotonicity - If L1<=L2 then F(L1)<=F(L2) i.e. If risk portfolio 2 exhibits greater or equal losses under all future scenarios than the losses on risk portfolio 1, then a monotonic risk measure will indicate that a greater or equal amount of capital should be help in respect of the former
- Subadditivity - F(L1+L2)<=F(L1)+F(L2) i.e. A merger of risk does not increase the overall level of risk, it may decrease the overall level of risk due to diversification
- Positive Homogeneity - F(k x L)=k x F(L) i.e. If we double the size of the loss situation, then we double the risk
- Translation Invariance - F(L + k) = F(L) + k i.e. If we add an amount to the loss, the capital requirement needed to mitigate the impact of the loss increases by the same amount
2
Q
Definition: Convex Risk Measure
A
Diversification will reduce the risk & the amount of capital needed - F(lL1 + (1-l)L2) <= lF(L1)+(1-l)F(L2)
3
Q
Definition: Deterministic Measures
A
Simplistic measures, giving a broad indication of the level of risk
4
Q
Definition: Probabilistic Measures
A
Involve applying a statistical distribution to a risk & measure a feature of that distribution
5
Q
List: Disadvantages of the Notional Approach (5)
A
- Potential undesirable use of a ‘catch all’ weighting for undefined asset classes
- Possible distortions to the market caused by increased demand for asset classes with high weightings
- Treating short positions as if they were the exact opposite of the equivalent long positions
- No allowance for concentration risk
- The probability of the changes is not quantified
6
Q
List: Disadvantages of the Factor Sensitivity Approach (3)
A
- Not assessing a wide range of risk, by focusing on a single risk factor
- Being difficult to aggregate over different risk factors
- The probability of the outcomes is not quantified
7
Q
List: Advantages of Deviation Measures (3)
A
- Simplicity of calculation
- Applicability to a wide range of financial risks
- Can be aggregated, if correlations are known i.e. V(aX+bY) = a^2V(X) + b^2V(Y) + 2ab*Cov(X,Y)
8
Q
List: Disadvantages of Deviation Measures (5)
A
- Difficulty in interpreting comparisons
- Potentially misleading if the underlying distribution is skewed
- Does not focus on tail risk, specifically, underestimates tail risk if the underlying distribution is leptokurtic
- Aggregations of deviations can be misleading
- Quantifies severity but not probability
9
Q
Definition: Value at Risk (VaR)
A
The maximum potential loss, with a given probability, a, over a given time period
VaR_a = inf{I e R: P(L>I <= 1-a)}
10
Q
List: Advantages of VaR (5)
A
- Simplicity of its expression
- Intelligibility of its units i.e. money
- Applicability over all sources of risk
- Allowance for the way in which different risks interact to cause losses
- Ease of its translation into a risk benchmark
11
Q
List: Disadvantages of VaR (5)
A
- Gives no indication of the distribution of losses greater than the VaR
- Underestimates asymmetric & fat-tail risks
- Sensitive to the choices of data, parameters
- VaR not sub-additive i.e. not a coherent risk measure
- May encourage ‘herding’
12
Q
List: Advantages of the Empirical VaR Approach (3)
A
- Simplicity
- No requirement to specify the distribution of returns
- Realism - Focuses on the largest market movements observed
13
Q
List: Disadvantages of the Empirical VaR Approach (4)
A
- Reliance on bootstrapping past data
- Implication that past data is indicative of future experiences
- Doesn’t facilitate stress testing or scenario testing
- Practical difficulties & limitations of interpolation
14
Q
List: Advantages of the Parametric VaR Approach (3)
A
- Ease of calculation
- Reduced dependence on past data
- Easy adjustment of parameters
15
Q
List: Disadvantages of the Parametric VaR Approach (6)
A
- Difficult to explain
- Reliance on past data
- Difficulty in ensuring parameters chosen are consistent
- Assumes that the parameter values remain constant
- Risk of adopting an inappropriate statistical distribution
- Difficulty in reflecting complex inter-dependencies
16
Q
List: Advantages of the Stochastic VaR Approach (3)
A
- More complex features of the underlying loss distribution
- Wider ranges of future possibilities than the empirical method
- Sensitivity testing
17
Q
List: Disadvantages of the Stochastic VaR Approach (4)
A
- Difficult to explain
- Subjective & difficult choice of distributions & parameters
- Gives a different answer each time
- Potentially high compute time
18
Q
Definition: Probability of Ruin
A
The probability that the net financial position of an org / line of business falls below 0 over a defined time horizon - Reciprocal of VaR
19
Q
Definition: TVaR / CVaR
A
The expected loss given that a loss over the specified VaR has occurred - E(L|VaR_a)
20
Q
List: Advantages of TVaR (2)
A
- Considers the losses beyond VaR
- Coherent risk measure
21
Q
List: Disadvantages of TVaR (2)
A
- Choice of distribution & parameters is subjective & difficult
- Highly sensitive to assumptions
22
Q
List: Factors Affecting the Choice of a Suitable Time Horizon (4)
A
- Contractual / Legal constraints
- Liquidity considerations
- Time to reinstate risk mitigation
- Time to recover from a loss event
23
Q
List: Risk Discount Rate Considerations (4)
A
- Sponsor’s cost of capital
- Inflation rates
- Interest rates
- Rates of returns on investments
24
Q
Definition: Multi-Factor Models
A
Modelling a response variable, Y_t, a time t, in terms of N explanatory variables X_(t,n)
25
Definition: Dynamic Financial Analysis
Modelling the risks to which the enterprise is exposed & the relationship between these risks
26
Definition: Financial Condition Reports
Reports into the current solvency position of a company & possible future developments
27
Definition: Asset-Liability Modelling
Method of projecting both the Assets & Liabilities of an institution with the same model, using consistent assumptions, to assess how well the assets match the liabilities, and to understand the probable evolution of of future CF's
28
List: Disadvantages of Linear Correlation (5)
1. Linear correlation coefficient is not unchanged under the operation of a general (non-linear) strictly increasing transformation
2. Not well defined where V(X) or V(Y) is infinite - Cannot be used on some heavy-tailed distributions
3. Independent variables are uncorrelated, but not all uncorrelated variables are independent
4. Valid measure of correlation only if the marginal distributions are jointly elliptical
5. Not necessarily the case that we will be able to put together a joint distribution to combine the info
29
Definition: Deterministic Model
Uses a set (or sets) of assumptions that are pre-determined
30
List: Advantages of Scenario Analysis (4)
1. Facilitates the evolution of potential impact of plausible events on an org
2. Not restricted to consideration of what has happened
3. Provides useful additional info to supplement traditional models based on statistical info
4. Facilitates the production of action plans to deal with possible future catastrophes by assessing the possible impact of a specified mitigation strategy
31
List: Disadvantages of Scenario Analysis (4)
1. Complex
2. Reliance on successfully generating hypothetical extreme but also plausible events
3. Uncertainty as to whether the full set of scenarios considered is representative or exhaustive
4. Absence of any assigned probabilities
32
List: Advantages of Stress Testing (3)
1. Compare the impact of the same stresses on differing orgs
2. Explicit examination of extreme events
3. Assessing the suitability of responses, by assessing the expected impact of the stress in the absence of any response, and then the expected impact in the proposed response
33
List: Disadvantages of Stress Testing (3)
1. Subjective as to which assumptions to stress & the degree of stress to consider
2. Assigns no probabilities to the events
3. Looks only at extreme situations
34
List: When Stress Testing Used (4)
1. Exercise that seeks to ID the key risks the firm is exposed to
2. Exercise seeking to understand whether the firm is being managed within the Board's risk appetite
3. Formal limit framework
4. Regulatory oversight
35
Definition: Stochastic Model
Inputs are uncertain & provides a probability distribution for the model outputs
36
Definition: Principal Component Analysis
Breaks down each variable's divergence from its mean into a weighted average of independent volatility factors
37
Definition: Univariate Time Series
A sequence of observations of a single process taken at a sequence of different times, {X_t: t = 1, 2, ..., T}
38
Definition: Strict Stationarity
A process is strictly stationary if the joint distributions X_r, X_(r+1), ..., X_s & X_(r+k), X_(r+k+1),...X_(s+k) are identical for integers r, s & k i.e. E(X_t)=E(X_(t+k))=mu & V(X_t)=V(X_(t+k))=s^2 - The statistical properties remain unchanged over time
39
Definition: Covariance Stationary (3)
1. Constant mean - E(X_t)=mu
2. Constant variance - V(X_t)=s^2
3. A covariance that depends only on the difference in time (the lag) between the observations i.e. Cov(X_t, X_(t+k)) depends only on k
40
Definition: Weakly Stationary
Moments of subsets of the process are defined & equal up to the n-th moment
41
Definition: Trend Stationary Process
Observations oscillate randomly around a steadily changing trend line which is a function of time only
42
Definition: Autoregressive
The current value of the time series, X_t, depends on past values of the time series, together with a single WN term
43
Definition: AR(P) Process
The current value of the time series is in terms of the p previous terms plus a current WN term - X_t = a_0 +a_1*x_(t-1)+....+a_p*x_(t-p)+e_t
44
Definition: Moving Average Process
Models the current value of the time series as a combination of past & present WN terms
45
Definition: MA(q) Process
Defines the current value of the time series in terms of the current & q previous WN terms - X_t = e_t+b_1*e_(t-1)+...+b_q*e_(t-q)
46
Definition: Partial Autocorrelation Function
Conditional correlation of X_t with X_(t-h) given that we know the values of the time series in between i.e. X_(t-1), X_(t-2), ..., X_(t-h) are known
47
List: Tests To See if Residuals are WN (5)
1. Plot of residuals against time
2. Turning point test - H_0 = Graph of the residuals is pattern-less
3. Plot of the sample ACF of residuals
4. Ljung & Box 'Portmanteau' test - H_0 = No correlation present between residuals
5. Durbin-Watson Statistic - H_0 = No correlation present between residuals
48
Definition: Heteroscedasticity
Processes where the variance changes over time
49
Definition: ARCH Models (2)
Models which capture:
1. Volatility capturing
2. Leptokurtosis
50
Definition: GARCH Models
Generalised ARCH model where the variance is now allowed to depend on the previous values of the the variance as well as previous squared values of the process
51
Definition: Copula
Function that defines the relationship between 2 or more variables, but it takes (marginal) probabilities as its arguments rather than 2 particular values of the variables concerned - P(X<=x, Y<=y) = F_(X,Y)(x,y)=C_(X,Y)[F_X(x), F_Y(y)]
52
Definition: Sklar's Theorem (1-2)
Let F be a joint distribution function with marginal CDF's F_1, ..., F_N then Sklar's theorem states that there exists a copula, C, such that for all x_1, ..., x_N e [-inf, inf] -> F(x_1, ..., x_n) = C(F_1(x_1), ..., F_N(x_N))
1. If the marginal CDF's are continuous, then C is unique
2. If we have a joint CDF & marginal CDF's, then these can be linked by a copula function
53
Definition: Converse of Sklar's Theorem
If C is a copula & F_1, ..., F_N are univariate CDF's, then the function F defined above is a joint CDF with marginal CDF's F_1, ..., F_N
54
Definition: Copula of a Distribution
If the vector of RV's, X, has a joint CDF F with continuous marginal cumulative distributions F_1, ..., F_N, then the copula of the distribution F is the distribution function C(F_1(x_1),...,F_N(x_N))
55
Definition: Empirical Copula Function
Describes the relationship between the marginal variables based upon their respective ranks
56
Definition: Survival Copula
For each copula there is a corresponding Survival Copula defined by the opposite relationship
57
Definition: Copula Density Function
Describes the rate of change of the copula CDF
58
List: Scarsini's Properties of a Good Measure of Concardance (7)
| SUC DICC
1. Completeness of Domain - M_XY is defined for all values of X,Y with X,Y being continuous
2. Symmetry - M_XY = M_YX
3. Coherence - If C_XY(u1,u2) > C_WZ(u1,u2) for all u1, u2 in [0,1] then M_XY > M_WZ
4. Unit Range - -1 <= M_XY <= 1 & the extremes of this range should should not be feasible
5. Independence - If X & Y independent then M_XY = 0
6. Consistency - If X=-Z then M_XY = -M_ZY
7. Convergence - If x1,...,xT & y1,...,yT are each seqs of T obs with joint distribution fuction tF(x,y) & copula tC(Fx(x),Fy(y)) then if tC(Fx(x),Fy(y)) tends to C(Fx(x),Fy(y)) as the # of obs (T) incr then we should also have tM_XY tending to M_XY
59
Definition: Fundamental Copulas
Copulas that represent the 3 basic dependencies that a set of variables can display
60
Definition: Explicit Copulas
Copulas that can be expressed in terms of a closed-form function
61
Definition: Implicit Copulas
Copulas based on well-known MV distributions, but no simple closed-form expressions exist for them
62
Definition: Least Squares Regression
Model using N independent explanatory variables - Y = XB + e
63
Definition: Ordinary Least Squares
The parameters are selected to minimise the sum of squared error terms - e'e = e_1^2 + ... + e_T^2 - which has solution - b = (b_1, ..., b_N)' = (X'X)^-1*X'Y
64
List: Assumptions of the Closed-form OLS Solution (6)
1. A linear relationship exists between variables
2. The inverse of the data exists
3. Explanatory variables should not be correlated with error terms
4. Error terms have a constant & finite variance
5. Error terms should not be correlated with one another
6. Error terms are normally distributed
65
Definition: Generalised Least Squares
The variance of the error terms is not necessarily assumed to be constant & they are also not necessarily assumed to be uncorrelated with one another
66
List: Qualitative Model Selection Tests (4)
1. QQ Plots
2. Histograms with superimposed fitted density functions
3. Empirical CDF's with superimposed fitted CDF's
4. Autocorrelation functions of time series data (ACF's)
67
Definition: GEV Family of Distributions
Describes the distribution of the standardised block maxima X_M = max(X_1, ..., X_n) when n is large
68
Definition: Generalised Pareto Distribution (GPD)
Describes the tail of the distribution above a threshold, P(X>x+u|X>u) for large values of u
