1. Mean-Swap Variance，Portfolio Theory and Asset Pricing
Zhan Wang
Department of Finance, College of Business and Economics, West Virginia University
MSwV (SSD) and MV𝔸 Efficiency
Mean-swap variance dominance is the necessary and sufficient condition to ensure expected utility dominance for all concave utility functions.
Example: Log-normal Distribution
Suppose returns on two securities, R1 and R2 are lognormally distributed, where r1 ~N (0.1,0.22) and r2 ~N (-0.15,0.24), respectively.
Mean-volatility-asymmetry (MV𝔸) dominance is a sufficient condition to ensure expected utility dominance for all utility functions with U’>0, U”<0 and U’’’>0.
Example: Upside Gain Preference
Example: Downside Loss Aversion
Efficient Sets of First-Degree Stochastic Dominance (FSD), MV𝔸, SSD (MSwV) and MV
Introduction
Swap variance (SwV) is superior to the variance. It summarizes the entire return variation and is unbiased to distributional asymmetry.
Mean-swap variance (MSwV) dominance is the necessary and sufficient condition for the second-degree stochastic dominance (SSD). MSwV retains the same simplicity as the mean-variance (MV) model.
The difference between SwV and variance is a proxy of asymmetries in return variation (𝔸).
The mean-volatility-asymmetry (MV𝔸) analysis, a three-dimensional extension of the classical MV portfolio theory and the CAPM, is consistent with expected utility maximization for all risk-averse investors, as well as those who are downside loss-averse but prefer the prospect of upside gains.
Asset Pricing
MSwV (SSD) and MV𝔸 equilibriums can be derived from standard optimization procedures similar to that of CAPM.
Two fund-separation holds in MSwV and MV𝔸 optimization.
Optimal portfolios of MSwV (SSD) and MV𝔸 are determined by the minimizing SwV portfolios.
MSwV is a single factor model, while MV𝔸 is a two-factor (symmetry and asymmetry) model.
The MV𝔸 model explains cross-sectional expected return and is empirically robust.
Conclusions
The stochastic dominance optimization and equilibrium can now be determined by MSwV and MV𝔸.
The efficient set of MV𝔸 is much broader than that of SSD because it also includes investors who prefer (dislike) the prospect of upside gains (downside losses).
The MV𝔸 approach expands the traditional MV model to accommodate the downside loss-aversion and the upside gain- preference in addition to risk aversion.
Quadratic Symmetry
Polynomial Asymmetries
A Comparison of Variance and Swap Variance
Applications
New Criteria of Investment Decision Making, Selection, and Optimization
Asset Allocation and Efficiency Analysis
Portfolio Performance Analysis
Investment Strategies Development
Asset Valuation and Risk Analysis
Variance
Swap Variance
Quadratic Variation
Polynomial Variation
Symmetry between upside and downside variations
Asymmetry between upside and downside variations R
-0.1
-0.06
-0.02
0.02
0.06
0.1
% Difference
7.2%
4.2%
1.4%
-1.3%
-3.8%
-6.2%