Ch.4 Risk Measures

Question 1

Define a Risk Measure and a Risk Measurement.

Answer 1

A Risk Measure is a mathematical concept of risk. A Risk Measurement is a number capturing that conceptualization.

Question 2

Why is the standard deviation of returns the most used measure of risk?

Answer 2

Because it is sufficient whenever we assume normality of returns.

Question 3

What is the definition of Value-at-Risk?

Answer 3

Question 4

What are three main issues of VaR?

Answer 4

1. VaR is only a quantile.
2. VaR is not a coherent risk measure.
3. VaR is easy to manipulate.

Question 5

What are the four axioms a risk measure must satisf y to be coherent?

Answer 5

1. Monotonicity
2. Translation invariance
3. Positive homogeneity
4. Subadditivity

Question 6

Describe the monotonicity axiom of coherent risk measures.

Answer 6

Risk measures decrease monotonically with returns. Say an asset always has more negative returns than another, the risk measure must be higher.

Question 7

Describe the translation invariance axiom of coherent risk measures.

Answer 7

Adding a positive constant to returns decreases the risk measure by that same constant.

Question 8

Describe the positive homogeneity axiom of coherent risk measures.

Answer 8

Risk is directly proportional to the value of the portfolio. Multiplying an asset by a positive constant increases risk by that same constant.

Question 9

Explain why Positive homogeneity might be violated in practice?

Answer 9

Increased holding of an asset increases the price impact of selling which increases risk.

Question 10

Describe the Subadditivity axiom of coherent risk measures.

Answer 10

The Risk measure of the sum of two assets must be smaller or equal to the sum of the risk measures.

Question 11

Show that variance on a 2 asset portfolio is subadditive.

Answer 11

Question 12

When is VaR not subadditive?

Answer 12

When returns have very fat tails.

Question 13

What are two ways of manipulating VaR.

Answer 13

1. Picking Stocks with low VaR.
2. Derivative strategies which dump risk outside of var 99 or 95%, but increase Expected Shortfall.

Question 14

What is Expected Shortfall?

Answer 14

W

Question 15

What is one backdrop of ES?

Answer 15

One first needs to calculate VaR, then integrate the tail, which increases estimation error.

Question 16

What are two pros of ES compared to VaR?

Answer 16

1. ES is coherent and VaR is not. In the sense that it satisfies the four axiom of a good risk measure.
2. ES is harder to manipulate.

Question 17

If we observe IID returns taht are normaly distributed, what is the sum of variances for T days?

Answer 17

Question 18

Under which conditions does the VaR scaling law apply?

Answer 18

If returns are IID and normally distributed.

Question 19

How might positive homogeneity of a risk measure be violated?

Answer 19

As the size of a portfolio increases, price movement might increase risk.

Question 20

When is VaR always subbaditive?

Answer 20

In case of normaly distributed returns.

Question 21

When does VaR always violate subadditivity?

Answer 21

In case of super-fat tailed distributions such as electricity prices, pegged currencies or commodities.

Question 22

What are the two assumptions underlying the square root rule for time.

Answer 22

1. Data are IID
2. Data are normally distributed.