Mildenhall Chapter 5: Risk Measure Properties Flashcards
(18 cards)
Probability Function P
Statistical and Information Uncertainty
Which is harder to avoid? Solution?
Statistical uncertainty: P is affected by estimation risk
Information uncertainty: P is based on limited/filtered subset of ambiguous information
Information Uncertainty is harder to avoid –> create generalized scenarios
e.g. sum the max loss of each generalized scenario, even if we have limited/filtered info, it’ll still give a good estimate of worst case scenario
Pros of Risk Measures (p_c) that satisfy coherent properties
- Intuitive and easy to communicate
- Can be used for capital and pricing
- Has properties that align with rational risk preferences
Risk Measure Properties
Translation Invariant (TI)
Also derive the Risk Margin
p(X+c) = p(X) + c
Increasing a loss by a constant c increases the risk by the same magnitude c
Mean, VaR and TVaR all are TI
Standard deviation is NOT TI: Var(X+c) = Var(X)
Risk Margin:
p(X) = E[X] + p(X - E[X])
Risk Measure Properties
Normalized (NORM), Acceptable
p(0) = 0
Risk = 0 if an outcome that has no gain or loss
Acceptable risk:
A risk is preferred over doing nothing if P(X) ≤ 0
Risk Measure Properties
Monotone (MON)
What about SD?
if X ≤ Y in all outcomes, then X ≽ Y, then p(X) ≤ p(Y)
SD is NOT monotone: assume X uniform ~[0,1] and Y = 1 fixed
SD would prefer Y even though X is always ≤ Y
Risk Measure Properties
Positive Loading
p(X) ≥ E[X]
The larger the sample set, the more likely it is that a risk measure has a positive loading
Risk Measure Properties
Monetary Risk Measure (MRM)
If the risk measure satisfies MON and TI, then it is MRM
Risk Measure is a monetary unit
Again, variance is not MRM because it fails MON and TI
Risk Measure Properties
Positive Homogeneous (PH)
p(aX) = a * p(X)
Users may be concerned if a risk measure is PH as risk is propertional to scale
Risk Measure Properties
Lipschitz Continuous
|p(X) - p(Y)| ≤ sup |X(w) - Y(w)|
The difference in “risk” between 2 random variables is at most the maximum of the absolute value of their outcomes
Risk Measure Properties
Subadditive
p(X + Y) ≤ p(X) + p(Y)
The risk of the portfolio should be less than or equal to the sum of the risks separately
Benefit if subadditive - insurer can manage the risk of the portfolio by managing the risk of each individual unit
Mean and TVaR are subadditive but VaR is NOT
Var or SD is if independent X, Y, but not always if dependent
V(X+Y) = V(X) + V(Y) +2 * Cov(X, Y)
Risk Measure Properties
Sublinear
Bid-Ask Spread
Positive Homogeneous (PH) and Subadditive (SA) both hold
Bid-Ask Spread:
0 = p(X-X) ≤ p(X) + P(-X) = p(X) - (-p(-X)) = ask - bid
Risk Measure Properties
Comonotonic Random Variables
Disadvantage & examples
If X = g(Z) and Y = h(Z) where g and h are increasing functions THEN X and Y are comonotonic
Aka X and Y both increase as Z increases
Disadvantage:
X and Y cannot hedge (offset) each other since they both increase if Z increases
Example:
e.g. X and Y belong to the same risk, X is GL and Y is legal fees
Extra info:
Given X, Y and their quantile functions qx and qy, qx(U) and qy(U) are comonotonic where U is a uniform variable
Risk Measure Properties
Comonotonic Additive (COMON) and Independent Additive
Comonotonic Additive (COMON):
p(X+Y) = p(X) + p(Y) where X and Y are comonotonic
There should be no diversification credit as X and Y do not diversify
Independent Additive:
p(X+Y) = p(X) + p(Y) where X and Y are independent
Again, there should be no diversification benefit
Risk Measure Properties
Law Invariant (LI)
p(X) is a function of the same distribution function F(X)
if X & Y have the same distribution function F(X), then p(X) = p(Y)
Appropriate for regulatory capital risk measure - risk of insolvency only depends on the distribution of future change in surplus, cause of loss is irrelevant
LI may be inappropriate for pricing
VaR, TVaR and SD are LI
Risk Measure Properties
Coherent (COH)
Deficiencies of VaR, Examples of COH
All 4 of these apply - MON, TI, PH, SA
Deficiencies of VaR:
* Ignores the severity of the deficiency in cases of insolvency
* Not subadditive
Examples of COH:
* TVaR (or average of TVaRs at different thresholds)
* Worst loss from a set of scendarios
Risk Measure Properties
Spectral Risk Measure
- Coherent (COH)
- Law Invariant (LI)
- Comonotonic additive (COMON)
Compound Risk Measures
Real world pricing bombines a pricing risk measure p and a capital risk measure a:
p_a(X) := p(X ^ a(X))
PH, Normalized, TI, Monotone
NOT subadditive
X ^ Y = min(X,Y)
Undesirable Properties of Expected Utility
- Firms do not have diminishing marginal utility of wealth; shareholders are impossible to satisfy
- Firms preference are not realtive to a wealth level
- Utility theory combines attitude to wealth and to risk
- Utility functions are not linear so expected utility is not a monetary risk measure
