Properties and limitations of common risk measures
VaR
TVaR
Probability of ruin
Expected shortfall
Describe how to choose a suitable time horizon and risk discount rate
Consider how to evaluate risk measures: coherence and convexity
Examine a range of risk measures and discuss their benefits and limitations
Also discuss time horizon and how to choose an appropriate risk discount rate
Exam Note:
Need to be able to assess the overall corporate risk exposure from financial and non financial risk (qual. & quant.)
For quantitative, need to identify and apply the most appropriate risk metrics given a particular scenario
Coherence and convexity
Axioms of Coherence is a list of properties a good risk measure should have
Assume:
A number of risk portfolios
\(L_i\): Losses on each portfolio that follows certain probability distributions
Risk measure:
A real-valued function \(F\) satisfying certain properties (the following axioms)
It indicates the amount of capital that should be added to a risk portfolio with loss distribution \(L\) to make it acceptable to the risk-controller
If \(L_1 \leq L_2\) then \(F(L_1) \leq F(L_2)\)
\(F(L_1 + L_2) \leq F(L_1) + F(L_2)\)
Merger of risk situations does not increase the overall level of risk
Additional Notes:
Non subadditive risk measures incentivise the breaking up of organization or portfolios to reduce risk
Subadditivity makes decentralization of risk management systems possible
Since constraints can be placed on BUs and if they stay within these constraints then the overall risk level cannot exceed the sum of the parts
\(F(k \times L) = k \times F(L)\) for any constant \(k \geq 0\)
If we double the size of the loss situation
Then we double the risk
No reduction being given for non-existent diversification
\(F(L + k) = F(L) + k\) for any constant \(k\)
If we add (or subtract) and amount to (or from) the loss
Then the capital requirement needed to mitigate the impact of the loss increase (decrease) by the same amount
\(F(\lambda L_1 + (1 - \lambda) L_2) \leq \lambda F(L_1) + (1- \lambda) F(L_2)\) where \(\lambda \in [0,1]\)
Diversification can reduce risk and the amount of risk capital needed
Convexity follows from the subadditivity and positive homogeneity axioms (maybe find out more from books)
Deterministic measures:
(Pros) Simplistic;
(Cons) Gives a board indication of the level of risk
Broad-brush risk measures
Apply risk weighting to the market value of assets
Different weights for different asset classes
Weights \(\leq\) 100%
Sum total to compare to the value of liabilities to determine a notional (“risk-adjusted”) financial position
Advantage:
Disadvantages:
Potential undesirable use of a “catch all” weighting for (heterogeneous) undefined asset class
Possible distortions to the market caused by increased demand for asset classes
Treating short positions as if they where the exact opposite of the equivalent long position (in practice they might affect the capital requirements to different extents)
Non allowance for concentration risk
Probability of the changes considered (in the values of assets and liabilities) is not quantified
Determines the degree to which an org’s financial position (e.g. solvency) is affected by the impact that change in a single underlying risk factor (e.g. short term interest rates) has on the value of assets and liabilities
Advantage:
Disadvantage:
Not assessing wider range of risks (since it focus on a single risk factor)
Being difficult to aggregate over different risk factors
Probability of the changes considered (in the values of assets and/or liabilities) is not quantified
Similar to factor sensitivity
Consider the effect of changing a set of factors (in a consistent way)
e.g. Recession: low interest rate + low inflation + depressed equity returns
Disadvantage:
Probablistic measures:
(Pros) Potentially more accurate;
(Cons) More complex (greater model risk);
(Cons) Can imply inappropriate levels of confidence (esp. with tail risk estimation)
Standard deviation
Tracking Error
Deviation measured from the mean
\(SD = \sqrt{\dfrac{1}{T}\sum \limits_{t=1}^T (r_{X,t} - \mu)^2}\)
where \(\mu = \sum \limits_{t=1}^T \frac{1}{T}r_{X,t}\)
For return \(r_{X,t}\) on portfolio \(X\) in time period \(t\)
\(SD\) measured over \(T\) time period
Deviation measured relative to a benchmark other than the mean
\(TE = \sqrt{\dfrac{1}{T}\sum \limits_{t=1}^T(r_{X,t}-r_{B,t})^2}\)
For return on the benchmark portfolio (\(B\)) in time period \(t\) is \(r_{B,t}\)
Tracking error measured over \(T\) time period
Returns on a portfolio of assets:
Retrospectively (ex post):
Calculating past deviations based on actual historic asset allocations
Prospectively (en ante):
Based on current asset allocations but using either:
Observed historic covariances of the returns on different asset classes (semi prospectively)
Estimated future covariances (fully prospectively)
Mean-variance portfolio theory:
Risk and return of active investment strategy:
Judging the investment return by referencing what would have been achieved by passive investment in a benchmark portfolio
Tracking error measure the risk inherent in an active investment strategy
A risk-adjusted return measure
Considers the size of the average XS return (aka average active return) as a proportion of the risk exposure as measured by tracking error
Useful for rankings as it is dimensionless (no units)
Ex post information ratio
Using past returns on the actual and benchmark portfolios
\(IR = \dfrac{\text{XS Return}}{\text{Tracking Error}} = \dfrac{\dfrac{1}{T}\sum \limits_{t=1}^T(r_{X,t} - r_{B,t})}{\sqrt{\dfrac{1}{T}\sum \limits_{t=1}^T(r_{X,t} - r_{B,t})^2}}\)
Ex ante information ratio
Requires calculation of the potential prospective average XS return and tracking errors
Can use a factor based stochastic model
Advantages
Simplicity of calculation
Applicability to a wide range of financial risks
Can be aggregated given correlations
\(\mathrm{Var}(aX + bY) = a^2\mathrm{Var}(X) + b^2\mathrm{Var}(Y) + 2ab\mathrm{Cov}(X,Y)\)
Disadvantages
Difficult in interpreting comparisons other than in terms of simple ranking
Potentially misleading if the underlying distribution is skewed
Do not focus on tail risk
Underestimates tail risk if the underlying distribution is leptokurtic (thick tail)
Aggregations of deviations can be misleading
(e.g. if the component distributions are not normally distributed)
\(\mathrm{VaR}_{\alpha} = \inf \left \{ l \in \mathbb{R} : \Pr(L>l) \leq 1 - \alpha \right\}\)
Maximum loss which is not exceed with a given high probability (\(\alpha\)) over a given time period
Side note: \(\{x : P(x)\}\) means the set of all x for which \(P(x)\) is true; \(\inf\) basically means the largest element of the set
Time horizon selection:
Chosen to comply with any contractual and legislative constraints, or
Liquidity considerations
Typically a short period of time as the portfolio is usually only stable over short period (typically 1 yr for insurance and 1 day for banks)
Confidence level selection:
High confidence level is typically used for capital adequacy purposes
e.g. 99% over 10 days for Basel
Typically consider multiple confidence level
Advantages
Simplicity of its expression
Intelligibility of its unit
Applicable to all types of risks
Applicable over all sources of risk (easy comparison between risk)
Easy to translation into a risk benchmark (e.g. risk limit)
Disadvantages
Gives no indication of the distribution of losses XS VaR
Can underestimate asymmetric and fat tail risk
Can be very sensitive to the choices of data, parameters (e.g. \(\alpha\), time horizon) and assumptions
Not a coherent risk measure (not sub-additive)
Can encourage “herding” if used in regulation \(\Rightarrow\) increasing systemic risk
Given the following:
Value \(X_t\) of portfolio \(X\) at time \(t\)
Loss in period \(t\) for portfolio \(X\) = \(-(X_t - X_{t-1})\)
Losses observed over a total of \(T\) periods are ranked from smallest (\(L_{X,1}\)) to largest (\(L_{X,T}\))
\(VaR_{\alpha} = \begin{cases} L_{X, (T \times \alpha)} & \text{if } T \times \alpha \in \mathbb{Z} \\ \left[(T \times \alpha) - t_- \right] L_{X,t_+} + \left[t_+ - (T \times \alpha)\right]L_{X,t_-} & \text{else} \\ \end{cases}\)
Advantages
Simplicity
No requirement to specify the distribution of returns
Realism, focuses on the largest market movements observed
Disadvantages
Reliance on past data having captured all possible future scenarios
Implies past data is indicative of future experience
Does not facilitate stress or scenario testing
Practical difficulties and limitations of interpolation
Given the following:
Assumes that losses follow some specified statistical distribution
Let \(L\) be a r.v. that represents loss on a portfolio
\(F(x) = \Pr(L\leq x)\) is the CDF of \(L\)
\(VaR_{\alpha} = F^{-1}(\alpha)\)
Parametization
Past data
Implied future volatility from option prices
For very short time horizon can assume 0 return (\(\mu = 0\))
For long time horizon better to define losses w.r.t to log asset value: \(L_{X,t} = -( \ln X_t - \ln X_{t-1}) = - r_{X,t}\)
Advantages
Ease of calculation
Reduced dependence on past data
Easy adjustment of parameters initially derived from past data
Disadvantages
More difficult to explain than empirical
Parameters relies on past data
Difficult to ensure parameters are consistent
Assumes parameter remain constant
Risk of adopting an inappropriate distribution
Difficult to reflect complex inter dependencies
Same way to determine \(VaR\) as empirical but the data set used is not the past observed loss
Data set:
Simulated using a chosen distribution
Bootstrapped: random sampling of past observed returns
Advantages
Can accommodate:
More complex features of the underlying loss distn (e.g. skew, leptokurtosis)
Wider ranges of future possibilities than the empirical method
Sensitivity testing (e.g. choice of distn and parameter values)
Disadvantages
More difficult to explain than the other 2 approaches
Subjective and difficult choices of distribution and parameters
Gives a different answer each time (only slightly different ideally)
Potentially high computation time
Variants of \(VaR\):
\(VaR\) in terms of relative loss (e.g. relative to the expected loss)
\(VaR\) in terms of percentage
\(VaR\) in terms of returns rather than losses (e.g. based on log asset values)
Different choices of time horizon
Different choices of confidence level
Probability that the net financial position falls below zero over a defined time horizon
Closely linked to \(VaR\)
e.g. if net financial position is < \(VaR_{95\%}\) then the probability of ruin is > 5% over the same time horizon as the one we used in \(VaR\)
Aka conditional Value at Risk or expected shortfall
\(TVaR_{\alpha} = CVaR_{\alpha} = \mathrm{E}\left[ L \mid L > VaR_{\alpha} \right]\)
Comparing TVaR of VaR
Advantages
Consider losses beyond VaR
Coherent risk measure
Disadvantages
Choice of distribution and parameter is subjective and difficult
Highly sensitive to assumptions
There are similar variant as VaR as well as other things like tail conditional expectation (TCE) or worst conditional expectation (WCE)
\(TVaR_{\alpha} = \begin{cases} \dfrac{\sum \limits_{t = T \times \alpha}^T L_{X,t}}{\sum \limits_{t = T \times \alpha}^T I (t \geq T \times \alpha)} & \text{if } T \times \alpha \in \mathbb{Z} \\ \dfrac{\left(\sum \limits_{t = \lceil T \times \alpha \rceil}^T L_{X,t} \right) + [ t_+ - (T \times \alpha)]L_{X,t_-}}{\left(\sum \limits_{t = \lceil T \times \alpha \rceil}^T I (t \geq T \times \alpha)\right) + (\lceil T \times \alpha \rceil - T \times \alpha)} & \text{else} \\ \end{cases}\)
\(TVaR_{\alpha} = \dfrac{1}{1 - \alpha} \int_{\alpha}^1 VaR_{u} du\)
e.g. For Gaussian loss distribution \(TVaR_{\alpha} = \mu + \sigma \dfrac{\phi\big(\Phi^{-1}(\alpha)\big)}{1-\alpha}\)
Similar to the empirical approach but the data is based on simulation or bootstrap
Can use empirical, parametric, stochastic approach similar to VaR
Same advantages and disadvantages as TVaR with the additional
Disadvantage
\(TVaR_{\alpha} = \begin{cases} \dfrac{\sum \limits_{t = T \times \alpha}^T L_{X,t}}{T} & \text{if } T \times \alpha \in \mathbb{Z} \\ \dfrac{\left(\sum \limits_{t = \lceil T \times \alpha \rceil}^T L_{X,t} \right) + [ t_+ - (T \times \alpha)]L_{X,t}}{T} & \text{else} \\ \end{cases}\)
\(ES_{\alpha} = \int_{\alpha}^1 VaR_u du = (1 - \alpha) TVaR_{\alpha}\)
e.g. For Gaussian \(ES_{\alpha} = (1 - \alpha) \mu + \sigma \phi \big( \Phi^{-1}(\alpha)\big)\)
Once again similar to empirical approach but with data based on simulation or bootstrap
VaR is an industry standard for measuring and reporting market risk in trading portfolio
VaR draws a line between everyday and exceptional losses
VaR is based on 3 basic factors when quantifying market risk in trading portfolios
Exposure amount
Size of the position at risk
Price volatility factor:
Best estimate of future daily volatility of market prices
Liquidity factor:
Time in days to liquidate a position in an orderly fashion and in adverse market conditions, which may be problematic
Ratio of TVaR and VaR can use to indicate the skewness of a distribution
Switching from VaR to TVaR may not be advantageous if the underlying methodology is flawed
Rules of Thumb
Number of days that a mark-to-market loss might exceed \(VaR_{\alpha}\) might be estimated as:
\([100\% - \alpha] \times 250\)
Quick approximation of an n-day VaR:
1-day VaR \(\times\) \(\sqrt{n}\)
Only rough approximation as losses are not iid and the expected loss not zero (esp. for larger n)
Time horizon:
Broadly the length of time you are exposed to a particular risk
How long it takes or how difficult it is to reverse the impact of a decision or an event
Level of risk increase with time horizon
Larger possible outcomes
(e.g. insolvency)
What might happen in the intervening period
(e.g. liquidity problems)
Key factors in choosing time horizon
Time to recover from a loss event
Time to reinstate risk mitigation (e.g. re-establish a derivatives hedge)
Financial risk exposure
Key issue is the liquidity of positions impacted by the decision taken or risk event
Risk in highly liquid (e.g. gilts) investments can normally be reduced very quickly
On the other hand, derivatives, junk bonds, etc may take longer to sell
Operational risk exposure
Time horizon can be thought of as the time required for a company to recover from an event
(e.g. earthquake vs short-term power outage)
From Risk Analysis and Management for Projects (RAMP): a strategic framework for managing project risk and its financial implications
Size of the discount rate will affect the appraised viability of those projects to which it applied
The higher the discount rate the lower will be the PV of earnings (or benefits) arising in the future and the greater the negative impact on project viability
Discount rate is determined pragmatically by the sponsor
Should take account of the sponsor’s cost of capital, the rate of inflation, interest rates and rates of return on investments throughout the economy
Challenge of selecting discount rate (from Core Reading):
May no have suitable reference investments
e.g. no risk free asset that matches certain liabilities
Difficult to determine the risk free rate of interest
e.g. due to gov securities benefiting form a liquidity premium
Setting a discount rate that allows for the uncertainty of future asset values is problematic
e.g. projection/simulation approach may be required
Allowance for credit risk (e.g. default) should be made when setting the discount rate and default rates are linked to other risks which vary over time (e.g. inflation risk)
Practical discount rate choice (Again from RAMP)
Ultimately the choice of the discount rate will depend partly on issues such as the company’s cost of funds and hurdle rates that the company sets for its investments
Some companies may wish to use a higher/lower discount rate for projects which they regard as having a higher/lower inherent risk than for their other projects
i.e. risk which is incapable of mitigation
If this inherent risk varies significantly over different phases of the project, it may sometimes be appropriate to use different discount rates for each phase
Caveat: High discount rate should not be seen as a substitute for a detailed risk analysis
This could lead to the rejection of profitable low risk projects in favor of more profitable projects that carry unacceptable levels of risk
\(\begin{aligned} r_X &= r_f + \dfrac{\sigma_X \sigma_U \rho_{X,U}}{\sigma_U^2}(r_U - r_f)\\ &= r_f + \dfrac{\sigma_X}{\sigma_U} \rho_{X,U}(r_U - r_f)\\ &= r_f + \beta_X(r_U - r_f)\\ \end{aligned}\)
Where:
\(\beta_X = \dfrac{\sigma_X}{\sigma_U}\rho_{X,U}\)
\(r_X\): expected return on asset \(X\)
\(r_f\): risk free rate of return
\(r_U\): return on the universe of investment opportunities
Security Market Line
Graph with:
X-axis: Systematic risk (\(\beta\))
Y-axis: \(r_X\)
Security Market Line (SML):
y-axis intercept = \(r_f\)
Gradient = \(r_U - r_f\)
Project Market Line
Project Market Line (PML) is obtained by:
Re expressing \(r_X\) as the required return on project \(X\)
Re-expressing \(U\) as the existing project portfolio
PML shows that a higher returns is required of a project which, in the context of other project opportunities, exposes and org. to greater uncertainty (higher value of \(\dfrac{\sigma_X}{\sigma_U}\)) and/or has a lower diversification benefit (higher value of \(\rho_{X,U}\))
Caveat of using the PML to determine the discount rate
Return on projects and the correlation between them may not be stable over their respective lifetimes
In practice, it is difficult to find a relationship between the returns from projects anticipated by the PML and those actually achieved