Ch5 Measures of Financial Risk

1. Mean-Variance Framework
The traditional approach used to measure financial risks is the mean-variance framework. In risk management, we are concerned about outcomes in the left-hand tail.

1-1. Attractives of Normality
- central limit theorem
- straightforward formula for both cumulative probabilities and quantiles
- normal (elliptical) distribution requires only two parameters: mean & variance (expected return & risk)

1-2. Mean-Variance Efficient Frontier without a Risk-free Asset
The investor will choose some point along the upper edge of the feasible region, efficient frontier. The point chosen depend on thier risk-expected return preferences (utility or preference function).

1-3. Mean-Variance Efficient Frontier with a Risk-free Asset
*assumption: no short-selling constraints
The investor achieve any point along a straight line running from the risk-free rate through to a point or portfolio (market portfolio) just touching the top of the attainable set.

1-4. Nomality Assumption
If the distribution is skewed or has heavier tails, the normality assumption is inappropriate and the mean-variance framework can produce misleading estimates of risk. => the mean-variance framework can be applied conditionally on sets of parameters that might themselves be random. But, even with the greater flexibility, it's doubtful whether conditionally elliptical distribution can give sufficiently good fits to empirical return processes.
=> the mean-variance framework tells us to use the standard deviation as risk measure, but even with refinements such as conditionality, this is justified only in limited cases.

2. VaR
2-1. Basics of VaR