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Portfolio Variance is Not Riskby Magnus Erik Hvass Pedersen, May 17, 2014 Meanvariance portfolios are commonly believed to minimize risk for a given level of expected return, because variance is believed to measure risk. But this is incorrect as proven with a short example. Negative ReturnsLet asset A be a stochastic variable with negative returns (4%), (5%) or (6%) and let asset B be a stochastic variable with positive returns 5%, 10% or 15%. The returns have equal probability of occurring and are dependent in the order they are listed so that if asset A has return (4%) then asset B has return 5%, etc. The asset returns are perfectly anticorrelated with coefficient 1. The longonly, minimumvariance portfolio lies on the efficient frontier and is when the weight for asset A is 5/6 and the weight for asset B is 1/6. This gives a portfolio with mean (2.5%) and zero variance, that is, all possible returns of the portfolio are losses of exactly (2.5%). But an investor could instead have chosen a portfolio consisting entirely of asset B which would always give a positive return of either 5%, 10% or 15%. An investment entirely in asset B is clearly superior to an investment in the minimumvariance portfolio even though asset B has higher variance. Positive ReturnsThe above example had an asset with negative returns which could be avoided by adding the constraint that returns must be positive; but the problem also exists for assets with partly negative returns or all positive returns. To see this, change asset A's possible returns to 3%, 2% or 1%. Then the minimum variance portfolio still has asset A weight 5/6 and asset B weight 1/6 which gives a portfolio return of about 3.3% with zero variance. But asset B alone would give a higher return of either 5%, 10% or 15%. So although the minimumvariance portfolio has no return spread, it has a lower return with certainty. Overlapping Return DistributionsThe problem also exists for assets that have overlapping return distributions. Let asset A's possible returns be 6%, 5% or 4%. Then the minimum variance portfolio still has asset A weight 5/6 and asset B weight 1/6 which always gives a portfolio return of about 5.8% with zero variance. But asset B alone would give a higher return of either 10% or 15% with probability 2/3 (or about 67%) and a slightly lower return of 5% with probability 1/3 (or about 33%). Variance is Not RiskThe reason asset A is included in the efficient frontier and minimumvariance portfolio is that its return has a low (sample) standard deviation of 1% while asset B has a higher standard deviation of 5%. The two assets have negative correlation so combining them in a portfolio lowers the combined standard deviation. The meanvariance efficient frontier is optimized for low variance (and standard deviation) which gives a low spread of the possible returns from the portfolio. But the spread of possible returns is not a useful measure of risk because it does not consider the probability of loss and the probability of other assets having a higher return. So the meanvariance portfolio is not optimized for risk in the traditional sense of the word which is defined in a dictionary as "the chance of injury or loss". Further Reading
This example is taken from the paper Portfolio Optimization and Monte Carlo Simulation. 
