## MEAN— VARIANCE FRAMEWORK #

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.

## MEAN—VARIANCE FRAMEWORK LIMITATIONS #

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 #

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**

- Two arbitrary parameters are used in the calculation—the confidence level and the holding period. The confidence level indicates the likelihood or probability that we will obtain a value greater than or equal to VaR. The holding period can be any pre-determined time period measured in days, weeks, months, or years.
- VaR estimates are also subject to both model risk and implementation risk. Model risk is the risk of errors resulting from incorrect assumptions used in the model. Implementation risk is the risk of errors resulting from the implementation of the model.
- Another major limitation of the VaR measure is that it does not tell the investor the amount or magnitude of the actual loss. VaR only provides the maximum value we can lose for a given confidence level.

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

## SCENARIO ANALYSIS #

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.

## COHERENT RISK MEASURES #

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:

**Monotonicity :**a portfolio with greater future returns will likely have less risk: R_{1}≥ R_{2}, then ρ(R_{1}) ≤ ρ(R_{2})**Subadditivity :**the risk of a portfolio is at most equal to the risk of the assets within the portfolio: ρ(R_{1 }+ R_{2})≤ ρ(R_{1})+ρ(R_{2})**Positive Homogeneity:**the size of a portfolio,β, will impact the size of its risk: for all β>0, ρ(βR) = βρ(R).**Translation invariance :**the risk of a portfolio is dependent on the assets within the portfolio: for all constants*c,*

ρ(c+R) = ρ(R) — c

## EXPECTED SHORTFALL (Conditional VaR) #

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:

- ES satisfies all of the properties of coherent risk measurements including subadditivity. VaR only satisfies these properties for normal distributions.
- The portfolio risk surface for ES is convex because the property of subadditivity is met. Thus, ES is more appropriate for solving portfolio optimization problems than the VaR method.
- ES gives an estimate of the magnitude of a loss for unfavourable events. VaR provides no estimate of how large a loss may be.
- ES has less restrictive assumptions regarding risk/return decision rules.

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.

## SPECTRAL RISK MEASURES #

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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