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