14. Risk measurement Flashcards
List the 4 axioms of convexity
- Monotonicity
- Sub-additivity
- Positive homogeneity
- Translation invariance
**Convexity
Define monotonicity
If L1 < L2 then F(L1) < F(L2)
* A riskier portfolio requires a greater amount of capital to mitigate the impact of the risk.
Define sub-additivity
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.
Define positive homogeneity
- 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
Define translation invariance
- 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.
Convexity
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.
Outline
- 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)
What are two high level approaches to measuring risk
- Deterministic approaches
- Probabilistic approaches
List deterministic approaches to measuring risk
- Notional approach
- Factor sensitivity
- Scenario sensitivity
Outline the notional approach to measuring risk
- 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
Outline merits of notional approach to measuring risk
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
Outline factor sensitivity as risk measure
- 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
Outline merits of factor sensitivity to measuring risk
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
Outline scenario sensitivity as risk measure
- Considers effect of changes in multiple factors on A + L
- Probability in changes in assets and/or liabilities considered is not quantified
List probabilistic risk measures
- Deviation
- VaR
- Ruin probability
- TVaR/CVaR
- Expected shortfall
List deviation based measures
- SD
- Tracking error
Information ratio
What are merits of deviation based measures
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
Explain VaR
Maximum potetntial loss within a given probability α, over a given time period.
VaR_α=inf{I ∈R:P(L>I)≤1-α
What are the factors VaR is based on when quantifying market risk
- 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
What are advantages of VaR
- 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
What are the disadvantages of VaR
- 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
Define ruin probability
- Probability that net financial position of org or line of business falls below zero over defined time horizon
Define TVaR
Expected loss given that loss over specified VaR has occurred
TVaR_∝=CVaR_∝=E[L|L>VaR_∝]
Compare TVaR to VaR
Pros (vs VaR)
* Considers losses beyond VaR
* Coherent risk measure»_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»_space; difficult
* Highly sensitive to assumption - significant concern since using uncertain information from further into the tail of loss distribution
Compare expected shortfall to VaR
Pros (vs VaR)
* Considers losses beyond VaR
* Coherent risk measure»_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»_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
What are the 3 rules of thumb for exposure estimation
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)
Define the time horizon
- 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
What should you consider when choosing a time horizon?
- 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
Methods to deriving risk discount rate
- RAMP
- CAPM
How can we use CAPM to determine RDR
- 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
