Module 14: Introduction to risk measurement

Question 1

Axioms of coherence (4)

Answer 1

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.

Question 2

Axiom 1: Monotonicity

Answer 2

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.

Question 3

Axiom 2: Subadditivity

Answer 3

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.

Question 4

Axiom 3: Positive Homogeneity

Answer 4

If we double the size of the loss situation we double the risk - no reduction being given for non-existent diversification.

Question 5

Axiom 4: Translation Invariance

Answer 5

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.

Question 6

A consequence of subadditivity and positive homogeneity

Answer 6

convexity

Question 7

2 Major types of risk measure

Answer 7

• deterministic

- probabilistic

Question 8

Deterministic risk measure

Answer 8

Gives a broad indication of the level of the risk taken.

Question 9

3 Examples of deterministic risk measures

Answer 9

• notional approach
• factor sensitivity approach
• scenario sensitivity approach.

Question 10

Probabilistic risk measure

Answer 10

Involves applying a statistical distribution to a risk (risks) and measuring a feature of that distribution.

Question 11

5 Examples of a probabilistic risk measure

Answer 11

• deviation measures (eg standard deviation, tracking error, information ratio)
• Value at Risk (VaR)
• ruin probability
• Tail Value at Risk (TVaR)
• expected shortfall

Question 12

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)

Answer 12

• the time to recover from a loss event

- the time to reinstate risk mitigation (eg re-establish a derivatives hedge)

Question 13

5 Factors to consider when setting a discount rate for a project

Answer 13

• the organisation’s cost of capital
• the level of inherent risk exposure
• inflation rates
• interest rates
• investment returns in the economy

Question 14

Outline the notional approach

Answer 14

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.

Question 15

Discuss the advantages and disadvantages of the notional approach

Answer 15

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).

Question 16

Outline the factor-sensitivity approach

Answer 16

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.

Question 17

Discuss the advantages and disadvantages of the factor sensitivity approach

Answer 17

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

Question 18

Scenario sensitivity approach

Answer 18

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.

Question 19

Tracking error

Answer 19

Deviation is measured relative to a benchmark other than the mean.

Question 20

Deviation measured on a portfolio of assets can be calculated in two ways:

Answer 20

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)

Question 21

Discuss the advantages and disadvantages of deviation measures

Answer 21

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)

Question 22

Value at Risk (VaR)

Answer 22

The maximum loss which is not exceeded with a given high probability (α) over a given time period.

Question 23

Discuss the advantages and disadvantages of VaR

Answer 23

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.

Question 24

3 General approaches to the calculation of VaR

Answer 24

• empirical (or historical)
• parametric (variance-covariance)
• stochastic

Question 25

Discuss the advantages and disadvantages of the empirical approach to calculating VaR

Answer 25

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.

Question 26

Discuss the advantages and disadvantages of the parametric approach to calculating VaR

Answer 26

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.

Question 27

Stochastic approach to deriving the VaR

Answer 27

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.

Question 28

Discuss the advantages and disadvantages of the stochastic approach to deriving VaR

Answer 28

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

Question 29

Probability of ruin

Answer 29

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.

Question 30

Tail Value at Risk (TVar) or Conditional Value at Risk (CVar)

Answer 30

The expected loss given that a loss over the specified VaR has occurred.

Question 31

Outline the advantages and disadvantages of TVar (compared to VaR)

Answer 31

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.

Question 32

3 Basic factors upon which VaR is based when quantifying market risk in trading portfolios

Answer 32

1. EXPOSURE AMOUNT - the size of the position at risk
2. 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.
3. LIQUIDITY FACTOR - the time in days to liquidate a position in an orderly fashion and in adverse market conditions, which may be problematic.

Question 33

Outline an example of when switching from VaR to TVaR may not be advantageous

Answer 33

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.

Question 34

Describe the 2 key factors influencing the choice of a suitable time horizon.

Answer 34

The choice of a suitable time horizon will be influenced by expectations as to:

1. The time to recover from a loss event
2. The time to reinstate risk mitigation

Question 35

A discount rate should take account of:

Answer 35

• 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.

Question 36

REVISE THE MATHEMATICAL ASPECTS

Answer 36

SUMMARY OF USEFUL FORMULAE ON PG 345 / 936