Module 14: Introduction to risk measurement Flashcards
Axioms of coherence (4)
Properties that a good risk measure should have.
These are:
- monotonicity
- subadditivity - reflecting the effects of diversification
- positive homogeneity
- translation invariance
A consequence of subadditivity and positive homogeneity is convexity.
Axiom 1: Monotonicity
If risk portfolio 2 exhibits greater losses under all future scenarios than the losses on risk portfolio 1, then a monotonic risk measure will indicate that a greater amount of capital should be held in respect of the former, although how much more is not specified.
Axiom 2: Subadditivity
A merger of risk situations does not increase the overall level of risk.
Indeed, it may decrease the overall level of risk, as a consequence of diversification.
Note:
- non-subadditive risk measures incentivise the breaking up of organisations or portfolios to reduce risk
- subadditivity makes decentralisation of risk-management systems possible, since constraints can be placed on business units and if they stay within these contraints, then the overall risk level cannot exceed the sum of the parts.
Axiom 3: Positive Homogeneity
If we double the size of the loss situation we double the risk - no reduction being given for non-existent diversification.
Axiom 4: Translation Invariance
If we add (or deduct) an amount to (or from) the loss, then the capital requirement needed to mitigate the impact of the loss increases (or decreases) by the same amount.
A consequence of subadditivity and positive homogeneity
convexity
2 Major types of risk measure
- deterministic
- probabilistic
Deterministic risk measure
Gives a broad indication of the level of the risk taken.
3 Examples of deterministic risk measures
- notional approach
- factor sensitivity approach
- scenario sensitivity approach.
Probabilistic risk measure
Involves applying a statistical distribution to a risk (risks) and measuring a feature of that distribution.
5 Examples of a probabilistic risk measure
- deviation measures (eg standard deviation, tracking error, information ratio)
- Value at Risk (VaR)
- ruin probability
- Tail Value at Risk (TVaR)
- expected shortfall
A time horizon chosen may depend on any contractual / regulatory constraints.
The choice of a suitable time horizon will be influenced by expectations as to: (2)
- the time to recover from a loss event
- the time to reinstate risk mitigation (eg re-establish a derivatives hedge)
5 Factors to consider when setting a discount rate for a project
- the organisation’s cost of capital
- the level of inherent risk exposure
- inflation rates
- interest rates
- investment returns in the economy
Outline the notional approach
The notional approach is a broad-brush risk measure.
E.g. risk weightings might be applied to the market value of assets, the results then summed and this total then compared to the value of liabilities in order to determine a notional (‘risk-adjusted’) financial position.
Discuss the advantages and disadvantages of the notional approach
ADVANTAGE
- simple to implement and interpret across a diverse range of organisations
DISADVANTAGES
- potential undesirable use of a “catch all” weighting, for (possibly heterogeneous) undefined asset classes
- possible distortions to the market caused by increased demand for asset classes with high weightings
- treading short positions as if they were the exact opposite of the equivalent long position (in practices, they might affect the capital requirements to different extents).
Outline the factor-sensitivity approach
Determines the degree to which an organisation’s financial position (eg solvency or funding) is affected by the impact that a change in a single underlying risk factor (eg short-term interest rates) has on the value of assets and liabilities.
Discuss the advantages and disadvantages of the factor sensitivity approach
ADVANTAGE:
- Increased understanding of the drivers of risk
DISADVANTAGES:
- Not assessing a wider range of risks, by focusing upon a single risk factor
- Being difficult to aggregate over different risk factors
- The probability of the changes considered (in the values of assets and/or liabilities) is not quantified
Scenario sensitivity approach
Similar to that of factor sensitivity, but rather than changing a single underlying risk factor, the effect of changing a set of such factors (in a mutually consistent way) is considered.
Tracking error
Deviation is measured relative to a benchmark other than the mean.
Deviation measured on a portfolio of assets can be calculated in two ways:
Retrospectively (ex post) - calculating past deviations based on actual historic asset allocations.
Prospectively (ex ante) - based on current asset allocations by using either:
- the observed historic covariances of the returns on different asset classes (ie semi-prospectively), or
- estimated future covariances (ie fully-prospectively)
Discuss the advantages and disadvantages of deviation measures
ADVANTAGES:
- simplicity of calculation
- applicability to a wide range of financial risks
- can be aggregated, if correlations are known
DISADVANTAGES
- difficulty of interpreting comparisons, other than in terms of simple ranking
- potentially misleading if the underlying distribution(s) are skewed
- do not focus on tail risk and, specifically, underestimates tail risks if the underlying distributions are leptokurtic (thicker tails)
Value at Risk (VaR)
The maximum loss which is not exceeded with a given high probability (α) over a given time period.
Discuss the advantages and disadvantages of VaR
ADVANTAGES:
- the simplicity of its expression
- the intelligibility of its units, ie money
- its applicability to all types of risk
- its applicability over all sources of risk - facilitating easy comparisons between products and across businesses its inherent allowance for the way in which different risks interact to cause losses
- the ease of its translation into a risk benchmark, eg risk limit
DISADVANTAGES:
- it gives no indication of the distribution of losses greater than the VaR, eg does not reveal how much is likely to be lost should a loss occur that is greater than the VaR
- it can under-estimate asymmetric and fat-tail risks
- it can be very sensitive to the choices of data, parameters and assumptions
- it is not a coherent risk measure - VaR is not sub-additive
- if used in regulation, it may encourage ‘herding’ thereby increasing systemic risk.
3 General approaches to the calculation of VaR
- empirical (or historical)
- parametric (variance-covariance)
- stochastic
Discuss the advantages and disadvantages of the empirical approach to calculating VaR
ADVANTAGES:
- simplicity
- there is no requirement to specify the distribution of returns
- its realism, in that it focuses upon the largest market movements observed
DISADVANTAGES:
- its reliance on boot-strapping past data having captured all possible future scenarios - particularly significant market movements
- the implication that past data is indicative of future experience
- that it doesn’t facilitate stress or scenario testing
- the practical difficulties and limitations of interpolation.
Discuss the advantages and disadvantages of the parametric approach to calculating VaR
ADVANTAGES:
- ease of calculation
- the reduced dependence on past data
- easy adjustment of parameters initially derived from past data
DISADVANTAGES:
- more difficult to explain than the empirical approach
- its reliance on past data to the extent that parameters are derived from such data
- the difficulty in ensuring parameters chosen are consistent
- assuming that the parameter values remain constant
- the risk of adopting an inappropriate statistical distribution, eg normal distribution when many observed loss distributions are skewed and have a high kurtosis
- the difficulty in reflecting complex inter-dependencies, eg using a correlation matrix rather than a copula function.
Stochastic approach to deriving the VaR
The derivation of a stochastic VaR is the same as for the empirical approach (ie using ranked losses) but the data set used is not the full set of observed past losses.
The data set can be:
- SIMULATED - using a chosen statistical distribution
- BOOTSTRAPPED - the losses used to calculate the VaR are derived from random sampling of past observed returns.
Discuss the advantages and disadvantages of the stochastic approach to deriving VaR
ADVANTAGES
- more complex features of the underlying loss distribution, eg skew, leptokurtosis
- wider ranges of future possibilities than the empirical method
- sensitivity testing, eg choice of distribution and parameter values
DISADVANTAGES
- more difficult to explain than the other 2 approaches
- the subjective and difficult choices of distribution and parameter values
- it gives a different answer each time
- the potentially high computation time
Probability of ruin
The probability that the net financial position of an organisation or line of business falls below zero over a defined time horizon.
It is closely linked to the Value at Risk.
Tail Value at Risk (TVar) or Conditional Value at Risk (CVar)
The expected loss given that a loss over the specified VaR has occurred.
Outline the advantages and disadvantages of TVar (compared to VaR)
ADVANTAGES:
- considers the losses beyond the VaR
- it is a COHERENT risk measure, so it facilitates the aggregation of TVaR values arising from distinct parts of an organisation to determine the overall TVaR.
DISADVANTAGES:
- the choice of distribution and parameter values is subjective and difficult
- it is highly sensitive to assumptions - a significant concern as we are using (uncertain) information from further into the tail of the loss distribution.
3 Basic factors upon which VaR is based when quantifying market risk in trading portfolios
- EXPOSURE AMOUNT - the size of the position at risk
- PRICE VOLATILITY FACTOR - the “best estimate” of future daily volatility of market prices. For portfolios, this should include correlations between market movements by way of a correlation matrix.
- LIQUIDITY FACTOR - the time in days to liquidate a position in an orderly fashion and in adverse market conditions, which may be problematic.
Outline an example of when switching from VaR to TVaR may not be advantageous
Switching from VaR to TVaR may not be advantageous if the underlying methodology is flawed.
Eg based on unadjusted historical data which is not appropriate for future application.
Describe the 2 key factors influencing the choice of a suitable time horizon.
The choice of a suitable time horizon will be influenced by expectations as to:
- The time to recover from a loss event
- The time to reinstate risk mitigation
A discount rate should take account of:
- the sponsor’s cost of capital
- the rate of inflation
- interest rates
- rates of return on investments throughout the economy
Some companies wish to use a higher / lower discount rate for projects which they regard as having a higher / lower inherent risk.
REVISE THE MATHEMATICAL ASPECTS
SUMMARY OF USEFUL FORMULAE ON PG 345 / 936
