14. Risk measurement

Question 1

List the 4 axioms of convexity

Answer 1

1. Monotonicity
2. Sub-additivity
3. Positive homogeneity
4. Translation invariance

**Convexity

Question 2

Define monotonicity

Answer 2

If L1 < L2 then F(L1) < F(L2)
* A riskier portfolio requires a greater amount of capital to mitigate the impact of the risk.

Question 3

Define sub-additivity

Answer 3

F[L1+ L2] <= F(1) + F(L2)
* A merger of two risk situations doesn’t increase the level of risk. Or alternatively, a company cannot de-risk simply by breaking down into smaller, separate constituents business units.

Question 4

Define positive homogeneity

Answer 4

• For some constant k
• If we multiply our exposure to a certain risk by a fixed amount, k, then the amount of capital needed to mitigate the risk also increases at the same rate

Question 5

Define translation invariance

Answer 5

• For some amount k
• If we add a fixed amount to the loss then the amount of capital needed to mitigate that risk also increases by the same amount.

Question 6

Convexity

Answer 6

F(kL1 + (1-k)L2) <= kF(L1) + (1-k)F(L2) where k is between 0 and 1
* Due to positive homogeneity and sub-additivity
* States that by diversifying across different projects the amount of risk is reduced and the corresponding amount of risk capital is reduced.

Question 7

Outline

Answer 7

• Sub-additivity axiom rules out VaR as a coherent risk measure except where losses follow an elliptical distribution
• Positive homogeneity axiom may be argued against since large values of k may mean risk has actually been concentrated, i.e.
  F(kL) > k * F(L)

Question 8

What are two high level approaches to measuring risk

Answer 8

• Deterministic approaches
• Probabilistic approaches

Question 9

List deterministic approaches to measuring risk

Answer 9

• Notional approach
• Factor sensitivity
• Scenario sensitivity

Question 10

Outline the notional approach to measuring risk

Answer 10

• Broad-brush risk measure
• Apply risk weightings to asset market values
• Weights <= 100% and based on riskiness of asset class
• Same weight across asset classes
• Add results together and compare to value of liabilities to get notional risk-adjusted financial position

Question 11

Outline merits of notional approach to measuring risk

Answer 11

Pros
* Simple to implement and interpret / compare across diff orgs

Cons
* Potential undesirable use of “catch all” weighting, for (possibly heterogeneous) undefined asset classes
* Possible distortions to market caused by increased demand for asset classes with high weightings
* Treating short positions as if exact opposite of long position when in practice, might affect capital requirements to diff extent
* No allowance for concentration risk since weighting for asset class is same regardless of whether holding s in one security or many
* Probability in changes in assets and liabilities considered is not quantified

Question 12

Outline factor sensitivity as risk measure

Answer 12

• Determines degree to which org’s financial position is affected by impact a change in one underlying risk factor has on asset and/or liability values

Question 13

Outline merits of factor sensitivity to measuring risk

Answer 13

Pros
* Increased understanding of drivers of risk

Cons
* Focus on single factor = not assessing variety of risks
* Difficult to aggregate over diff risk factors
* Probability in changes in assets and/or liabilities considered is not quantified

Question 14

Outline scenario sensitivity as risk measure

Answer 14

• Considers effect of changes in multiple factors on A + L
• Probability in changes in assets and/or liabilities considered is not quantified

Question 15

List probabilistic risk measures

Answer 15

• Deviation
• VaR
• Ruin probability
• TVaR/CVaR
• Expected shortfall

Question 16

List deviation based measures

Answer 16

• SD
• Tracking error
  Information ratio

Question 17

What are merits of deviation based measures

Answer 17

Pros
* Simple calc
* Applicable to wide range of financial risks
* Can be aggregated if we know correlations e.g.,
Var(aX + bY) = a^2var(X) +b^2var(Y) + 2abcov(X,Y)
Cons
* Difficult to interpret when doing comparisons other than by simple ranking
* May be misleading if underlying distribution is skewed
* Doesn’t focus on tail risk - underestimates tail risk if underlying distribution is leptokurtic
* Aggregated deviations can be misleading, e.g. if component distributions aren’t normal

Question 18

Explain VaR

Answer 18

Maximum potetntial loss within a given probability α, over a given time period.
VaR_α=inf⁡{I ∈R:P(L>I)≤1-α

Question 19

What are the factors VaR is based on when quantifying market risk

Answer 19

• Exposure amount
• Price volatility factor – best estinate of future daily volatility of market prices. Must include correlations between market movements using a correlation matrix when dealing with a portfolio
• Liquidity factor – time in days to liquidate a position in orderly fashion and in adverse market conditions

Question 20

What are advantages of VaR

Answer 20

• Simple expression
• Intelligibility of its units, i.e., money
• Applicable to all types of risks
• It’s applicability over all risk sources- facilitating easy comparisons between products and across business its inherent allowance for way in which different risks interact to cause losses
• Ease of translation into risk benchmark, eg risk limit
  Cons
• Gives no indication of distribution of losses greater than VaR, eg doesn’t reveal how much is likely to be lost should loss occur that is greater than VaR
• Can underestimate asymmetric and fat-tail risks
• Can be very sensitive to choices of data, parameters and assumptions
• Not coherent risk measure - VaR not always sub-additive
• If used in regulation- may encourage “herding” thereby increasing systematic risk

Question 21

What are the disadvantages of VaR

Answer 21

• Gives no indication of distribution of losses greater than VaR, eg doesn’t reveal how much is likely to be lost should loss occur that is greater than VaR
• Can underestimate asymmetric and fat-tail risks
• Can be very sensitive to choices of data, parameters and assumptions
• Not coherent risk measure - VaR not always sub-additive
• If used in regulation- may encourage “herding” thereby increasing systematic risk

Question 22

Define ruin probability

Answer 22

• Probability that net financial position of org or line of business falls below zero over defined time horizon

Question 23

Define TVaR

Answer 23

Expected loss given that loss over specified VaR has occurred
TVaR_∝=CVaR_∝=E[L|L>VaR_∝]

Question 24

Compare TVaR to VaR

Answer 24

Pros (vs VaR)
* Considers losses beyond VaR
* Coherent risk measure&raquo_space; facilitates aggregation of TVaR values arising from distinct parts of org to determine overall TVaR

Cons (vs VaR)
* Choice of distribution and parameter values is subjective&raquo_space; difficult
* Highly sensitive to assumption - significant concern since using uncertain information from further into the tail of loss distribution

Question 25

Compare expected shortfall to VaR

Answer 25

Pros (vs VaR) * Considers losses beyond VaR * Coherent risk measure >> facilitates aggregation of ES values arising from distinct parts of org to determine overall ES Cons (vs VaR) * Choice of distribution and parameter values is subjective >> difficult * Highly sensitive to assumption - significant concern since using uncertain information from further into the tail of loss distribution * Little intuitive meaning * Cannot be readily linked to current valuation

Question 26

What are the 3 rules of thumb for exposure estimation

Answer 26

The number of days that a mark-to-market loss might exceed VaR(α) might be estimated as [100% - α]*250, where α is the confidence level and 250 is the number of trading days in a year Let X be loss over 1 day. If X ~ N(μ,σ^2) then the n-day loss is distributed as N(nμ,nσ^2), and n-day volatility is √n*σ. A quick approximation of an n-day var might be approximated by multiplying n day VaR by √n (assuming mu = 0)

Question 27

Define the time horizon

Answer 27

* Length of time for which org is exposed to risk or … * … time required to recover from or reverse effects of an event. * Longer duration = higher risk level by: o Outcome e.g. insolvency o Effects on intervening period e.g. liquidity problems

Question 28

What should you consider when choosing a time horizon?

Answer 28

* Contractual / legal constraints, e.g., general insurance company usually bound to 1 year contracts * Liquidity concentrations, i.e., time taken to liquidate investment portfolio * Time to reinstate risk mitigation, e.g., re-establish a derivatives hedge * Time to recover from loss event, e.g., operational risks like fire

Question 29

Methods to deriving risk discount rate

Answer 29

* RAMP * CAPM

Question 30

How can we use CAPM to determine RDR

Answer 30

* Measure of systematic risk. By examining beta we can see: o Greater the uncertainty is associated with returns on the security relative to those on the market of investments, the greater the expected return o The greater the correlation between the returns on the security and those on the market, the greater the expected return * Indicates how expected return on security or project or portfolio I, is correlated to expected return from market