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f106 enterprise risk management '24 > 14. Risk measurement > Flashcards

14. Risk measurement Flashcards

1
Q

List the 4 axioms of convexity

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

**Convexity

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2
Q

Define monotonicity

A

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

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3
Q

Define sub-additivity

A

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.

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4
Q

Define positive homogeneity

A
  • 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
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5
Q

Define translation invariance

A
  • 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.
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6
Q

Convexity

A

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.

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7
Q

Outline

A
  • 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)
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8
Q

What are two high level approaches to measuring risk

A
  • Deterministic approaches
  • Probabilistic approaches
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9
Q

List deterministic approaches to measuring risk

A
  • Notional approach
  • Factor sensitivity
  • Scenario sensitivity
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10
Q

Outline the notional approach to measuring risk

A
  • 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
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11
Q

Outline merits of notional approach to measuring risk

A

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

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12
Q

Outline factor sensitivity as risk measure

A
  • 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
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13
Q

Outline merits of factor sensitivity to measuring risk

A

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

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14
Q

Outline scenario sensitivity as risk measure

A
  • Considers effect of changes in multiple factors on A + L
  • Probability in changes in assets and/or liabilities considered is not quantified
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15
Q

List probabilistic risk measures

A
  • Deviation
  • VaR
  • Ruin probability
  • TVaR/CVaR
  • Expected shortfall
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16
Q

List deviation based measures

A
  • SD
  • Tracking error
    Information ratio
17
Q

What are merits of deviation based measures

A

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

18
Q

Explain VaR

A

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

19
Q

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

A
  • 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
20
Q

What are advantages of VaR

A
  • 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
21
Q

What are the disadvantages of VaR

A
  • 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
22
Q

Define ruin probability

A
  • Probability that net financial position of org or line of business falls below zero over defined time horizon
23
Q

Define TVaR

A

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

24
Q

Compare TVaR to VaR

A

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

25
Q

Compare expected shortfall to VaR

A

Pros (vs VaR)
* Considers losses beyond VaR
* Coherent risk measure&raquo_space; 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&raquo_space; 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

26
Q

What are the 3 rules of thumb for exposure estimation

A

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 √nσ. A quick approximation of an n-day var might be approximated by multiplying n day VaR by √n (assuming mu = 0)

27
Q

Define the time horizon

A
  • 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
28
Q

What should you consider when choosing a time horizon?

A
  • 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
29
Q

Methods to deriving risk discount rate

A
  • RAMP
  • CAPM
30
Q

How can we use CAPM to determine RDR

A
  • 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

f106 enterprise risk management '24 (35 decks)

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