Under the mean-variance framework, it is necessary to assume that return distributions for portfolios are elliptical distributions. The most commonly known elliptical probability distribution function is the normal distribution.
The normal distribution is a continuous distribution that illustrates all possible outcomes for random variables.
The use of the standard deviation as a risk measurement is not appropriate for non-normal distributions. If the shape of the underlying return density function is not symmetrical, then the standard deviation does not capture the appropriate probability of obtaining undesirable return outcomes.
Value at risk (VaR) is interpreted as the worst possible loss under normal conditions over a specified period. Another way to define VaR is as an estimate of the maximum loss that can occur with a given confidence level.
Limitations of VaR
Risk managers may be interested in measuring risk over longer time periods, such as a month, quarter, or year. VaR can be converted from a 1-day basis to a longer basis by multiplying the daily VaR by the square root of the number of days (J) in the longer time period (called the square root rule).
For example, to convert to a weekly VaR, multiply the daily VaR by the square root of 5 (i.e., five business days in a week). We can generalize the conversion method as follows:
VaR(X%)J-days = VaR(X%)1-dayJ
The results of scenario analysis can be interpreted as coherent risk measures by first assigning probabilities to a set of loss outcomes. These losses can be thought of as tail drawings of the relevant distribution function. The expected shortfall for the distribution can then be computed by finding the arithmetic average of the losses. Therefore, the outcomes of scenario analysis must be coherent risk measurements, because ES is a coherent risk measurement.
Scenario analysis can also be applied in situations where there are numerous distribution functions involved.
If we allow R to be a set of random events and ρ(R) to be the risk measure for the random events, then Coherent risk measures should exhibit the following properties:
ρ(c+R) = ρ(R) — c
Expected shortfall (ES) is the expected loss given that the portfolio return already lies below the pre-specified worst case quantile return.
ES is more appropriate risk measure than VaR for the following reasons:
For a normal distribution with a mean equal to U and a standard deviation equal to sigma, the following equation can be used to calculate the expected shortfall.
A more general risk measure than either VaR or ES is known as the risk spectrum or risk aversion function. The risk spectrum measures the weighted averages of the return quantiles from the loss distributions. ES is a special case of this risk spectrum measure. When modelling the ES case, the weighting function is set to [1 / (1 — confidence level)] for tail losses. All other quantiles will have a weight of zero.
VaR is also a special case of spectral risk measure models. The weighting function with VaR assigns a probability of one to the event that the p-value equals the level of significance (i.e., p = α), and a probability of zero to all other events where p≠α.
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