Mean-Variance Framework
The traditional mean-variance model estimates the amount of financial risk for portfolios in terms of the portfolio’s expected return (mean) and risk (standard deviation or variance). A necessary assumption for this model is that return distributions for the portfolios are elliptical distributions.
The efficient frontier is the set of portfolios that dominate all other portfolios in the investment universe of risky assets with respect to risk and return. When a risk-free security is introduced, the optimal set of portfolios consists of a line from the risk-free security that is tangent to the efficient frontier at the market portfolio.
Mean-Variance Framework Limitations
The mean-variance framework is unreliable when the underlying return distribution is not normal or elliptical. The standard deviation is not an accurate measure of risk and does not capture the probability of obtaining undesirable return outcomes when the underlying return density function is not symmetrical.
Value at Risk
Value at risk (VaR) is a risk measurement that determines the probability of an occurrence in the left-hand tail of a return distribution at a given confidence level. VaR is defined as: [μ− (z)(σ)]. The underlying return distribution, arbitrary choice of confidence levels and holding periods, and the inability to calculate the magnitude of losses result in limitations in implementing the VaR model when determining risk.
Coherent Risk Measure
The properties of a coherent risk measure are:
- Monotonicity: Y ≥ X ⇒ ρ(Y) ≤ ρ(X)
- Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y)
- Positive homogeneity: ρ(hx) = hρ(X) for h > 0
- Translation invariance: ρ(X + n) = ρ(X) − n
Subadditivity, the most important property for a coherent risk measure, states that a portfolio made up of sub-portfolios will have equal or less risk than the sum of the risks of each individual sub-portfolio. VaR violates the property of subadditivity.
Expected Shortfall
Expected shortfall is a more accurate risk measure than VaR for the following reasons:
- ES satisfies all the properties of coherent risk measurements including subadditivity.
- The portfolio risk surface for ES is convex since 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 unfavorable events. VaR provides no estimate of how large a loss may be.
- ES has less restrictive assumptions regarding risk/return decision rules.
Risk Spectrum Measure
ES is a special case of the risk spectrum measure where the weighting function is set to 1 / (1 − confidence level) for tail losses that all have an equal weight, and all other quantiles have a weight of zero. The VaR is a special case where only a single quantile is measured, and the weighting function is set to one when p-value equals the level of significance, and all other quantiles have a weight of zero.
Scenario Analysis
The outcomes of scenario analysis are coherent risk measurements, because expected shortfall is a coherent risk measurement. The ES for the distribution can be computed by finding the arithmetic average of the losses for various scenario loss outcomes.
