Ch.4 Risk Measures Flashcards
Define a Risk Measure and a Risk Measurement.
A Risk Measure is a mathematical concept of risk. A Risk Measurement is a number capturing that conceptualization.
Why is the standard deviation of returns the most used measure of risk?
Because it is sufficient whenever we assume normality of returns.
What is the definition of Value-at-Risk?
What are three main issues of VaR?
- VaR is only a quantile.
- VaR is not a coherent risk measure.
- VaR is easy to manipulate.
What are the four axioms a risk measure must satisf y to be coherent?
- Monotonicity
- Translation invariance
- Positive homogeneity
- Subadditivity
Describe the monotonicity axiom of coherent risk measures.
Risk measures decrease monotonically with returns. Say an asset always has more negative returns than another, the risk measure must be higher.
Describe the translation invariance axiom of coherent risk measures.
Adding a positive constant to returns decreases the risk measure by that same constant.
Describe the positive homogeneity axiom of coherent risk measures.
Risk is directly proportional to the value of the portfolio. Multiplying an asset by a positive constant increases risk by that same constant.
Explain why Positive homogeneity might be violated in practice?
Increased holding of an asset increases the price impact of selling which increases risk.
Describe the Subadditivity axiom of coherent risk measures.
The Risk measure of the sum of two assets must be smaller or equal to the sum of the risk measures.
Show that variance on a 2 asset portfolio is subadditive.
When is VaR not subadditive?
When returns have very fat tails.
What are two ways of manipulating VaR.
- Picking Stocks with low VaR.
- Derivative strategies which dump risk outside of var 99 or 95%, but increase Expected Shortfall.
What is Expected Shortfall?
W
What is one backdrop of ES?
One first needs to calculate VaR, then integrate the tail, which increases estimation error.
What are two pros of ES compared to VaR?
- ES is coherent and VaR is not. In the sense that it satisfies the four axiom of a good risk measure.
- ES is harder to manipulate.
If we observe IID returns taht are normaly distributed, what is the sum of variances for T days?
Under which conditions does the VaR scaling law apply?
If returns are IID and normally distributed.
How might positive homogeneity of a risk measure be violated?
As the size of a portfolio increases, price movement might increase risk.
When is VaR always subbaditive?
In case of normaly distributed returns.
When does VaR always violate subadditivity?
In case of super-fat tailed distributions such as electricity prices, pegged currencies or commodities.
What are the two assumptions underlying the square root rule for time.
- Data are IID
- Data are normally distributed.
