Chapter 2: Tools for measuring risk

Question 1

Why do we need a formal framework?

Answer 1

• Risk management is about estimating the different outcomes and the likelihood of these different outcomes.
• 2 possible approaches for a formal framework:
  
  1. Human intuition or heuristics
  2. Formal probability theory: statistics

Question 2

Is human intuition a good method for evaluating randomness?

Answer 2

• Human intuition is a poor evaluation of randomness, an inaccurate evaluation of uncertainty.
  
  • Eg. Kahneman & Tversky: behavioral finance
• So we need formal probability theory to systemize randomness, statistics is the input tool

Question 3

What are the probability paradigms?

Answer 3

1. Objective
   
   • Mostly used in finance/economics
   • Only observed events matter: how the world is
   • Probabilities = relative frequency in experiments:
     
     • Also called the labelled frequency type theory
   • Good for repeated events: eg. games of chance
2. Subjective:
   
   • One-time events: eg. forecasting weather
   • Statement about confidence in evidence to preduct = assesment about probability.
   • B. de Finetti
   • Every financial risk event is one-time = belief-type.
   • We should allow subjective beliefs for a richer set of information

• Both are used in stress tests of regulators, mostly the traditional approach but subjective added for more information

Question 4

How can you measure unfavorable outcomes?

Answer 4

1. Profit and loss: P&Ls
   
   • Change in value over time
   • Problem: not scale-free so it is hard to compare
   • Short horizon: capital appreciation or depreciation
   • Long horizon: also includes income = dividend/coupon
2. Returns:
   
   1. Simple return: relative rate of change over time, scale-free, but not unit less (%).
   2. Logreturns: log of the returns.

Question 5

What are logreturns and why are they used?

Answer 5

• Used often in finance
• Advantages:
  
  1. Additive relation compound returns
  2. Limited liability: bound to -100%, so no negative prices
• For a short horizon: the simple return is equal to the logreturns: eg. daily returns.

Question 6

What are returns?

Answer 6

• Returns are random variables, where the likelihood depends on outcomes of a probability distribution:
  
  1. Parametric distribution: eg. normal, t-distribution, generalized pareto distribution
  2. Empirical: eg. historical
• Difficult to decide on the functional form: trade off between:
  
  • Capturing empirical stylized facts
  • Choosing distributional simplicity

Question 7

What is the normal distribution?

Answer 7

• Most common assumptions that returns are identically, independent and normal.
• Advantages:
  
  • Only 2 moments to capture the full distribution so easy to handle
  • Stable: portfolio of normal distributed components is also normally distributed
  • Empirically: the normal distribution is a rough proxy of many financial variables
• Disadvantages:
  
  • Not stable under multiplication, difficult for multiperiod returns
  • Violates limited liability = not bound by -100% so negative prices are possible
  • No capturing of skewness or kurtosis

Question 8

What is the alternative for the normal distribution?

Answer 8

• Alternative: assume that the logreturns are IID normal.
• Advantages:
  
  • Excludes negative prices = limited liability
  • Stable under addition
• Disadvantages:
  
  • No skewness or kurtosis

Question 9

What is skewness?

Answer 9

• Skewness: gain/loss asymmetry, this is only a minor problem.
• Stock indices have a long left tail = negative skewness.
• Postive of zero skewness for individual stocks : tail on the right side

Question 10

What is kurtosis?

Answer 10

• Major problem that is not captured by the normal distribution: heavy tails.
  
  • Very large positive kurtosis for stock indices
  • Large positive kurtosis for individual stocks = large variation
• Not capturing the kurtosis = underestimation of risk
• Higher kurtosis = greater extermity of deviations or outliers. Normal kurtosis = 3

Question 11

What are solutions to capture heavy tails?

Answer 11

• Non-normal stable distribution: disadvantages
  
  • Higher moments = infinite as there is no convergence of the samble estimates as the sample size increases
  • Stability: long horizons are not-normal
    
    • Degree of non-normality depends on horizon
• Alternative distributions:
  
  1. T-distribution: easy to handle, but only modest fat tails for low degrees of freedom
  2. Generalized Pareto: seperately model tails, used for VaR
  3. Generalized extreme value distribution: full discription of distribution

Question 12

What is covariance?

Answer 12

• Lineair dependence where the unit depends on variables.
• Zero covariance does not mean independence
• Zero covariance does mean independence for normally distributed variables