Mean-Variance Optimization: Pitfalls & Solutions

• Mean-Variance Optimization pitfalls and solutions

Introduction to Asset Allocation

• Asset allocation divides an investment portfolio among different asset classes like stocks, bonds, and cash.
• Strategic allocation differs from tactical allocation.
• Asset allocation and performance history show different types of asset classes.
• Asset allocation addresses how much wealth should be invested in each asset class.
• Mean-variance optimization answers this question.
• MV optimization can be applied numerically using Excel/R or analytically using matrix algebra.

Mean-Variance Optimization

• Mean-variance optimization maximizes E(rp) - (γ/2) * var(rp) with respect to portfolio weights (wi).
• This is subject to constraints like Σwi = 1.
• Returns follow an independent and identically distributed normal distribution (ri ~ iidN(μi, σi²)).
• The covariance between returns of assets i and j is denoted as σij.
• Portfolio return E(rp) is the sum of individual asset returns multiplied by their weights: E(rp) = Σwiμi.
• Portfolio variance var(rp) is calculated as the sum of weighted variances and covariances: var(rp) = Σwi²σi² + 2ΣΣwiwjσij.
• Critical assumptions and possible pitfalls exist in mean-variance optimization.

Pitfalls of Mean-Variance Optimization

• The pitfalls include investor utility, the normal distribution, garbage in/out, and the single-period framework.

Investor Utility Pitfalls

• Mean-variance utility functions are restrictive, considering only mean and variance while possibly overlooking higher moments.
• Investors may care about higher moments like skewness and kurtosis.
• The utility function is correct only with quadratic utility, where investors don't care about higher moments, or the asset distribution is multivariate normal.

Investor Utility: Taylor Expansion

• Taylor expansion of utility around the mean is expressed as U(Rp) = U(E(Rp)) + (Rp - E(Rp)) * U'(E(Rp)) + (1/2) * (Rp - E(Rp))² * U"(E(Rp)) + ... + (1/n!) * (Rp - E(Rp))^n * U^(n)(E(Rp)).

Quadratic Utility

• Expected utility is expressed as E[U(Rp)] = U(E(Rp)) + (1/2)* Var(Rp) * U''(E(Rp)).
• With quadratic utility, expected utility depends on mean and variance only, making mean-variance optimization appropriate.

Normality

• If utility is not quadratic, there is a need for restriction on the distribution
• A stable distribution remains stable under linear combinations of independent random variables with the same distribution.
• Normal distribution is an example of this
• Individual asset returns that are normally distributed depend only on the mean and variance (and have finite variance).
• Combinations (portfolios) of returns should depend on the mean and variance if they are also normally distributed.
• This implies individual asset returns are normally distributed.
• Again, expected utility relies on mean and variance and mean-variance optimisation is appropriate.

Normality implications

• If returns normality, then E[(Rp – E(Rp))ⁿ] = 0 when n is odd and E[(Rp – E(Rp))ⁿ] = (n! / ((n/2)! * 2ⁿ/²)) * Var(Rp)ⁿ/²when n is even.

Empirical Evidence Against Normality

• Returns show negative skewness.
• Probability of extreme returns is high, i.e. kurtosis > 3.

Shortcomings of Mean-Variance Utility

• Variance treats upside and downside risk the same.
• Investors prefer positive skewness and dislike negative skewness, and only the first two moments matter.
• Investors tend to overestimate the probability of disasters.
• Bad times other than low means and high variances matter, like relative wealth compared to others

Realistic Utility Functions

• Alternative functions include safety first utility, loss aversion/prospect theory, habit utility, "catching up with the Joneses," and uncertainty aversion.

Safety First Utility Framework

• Roy's utility (1952) minimizes disaster risk; ideal for agents facing crucial liabilities and holding assets until safety levels are met, before taking on risk.
• Quantile Utility Maximization focus on pessimism through intuitive measures of downside risk aversion.
• Quantiles like the 0.1 quantile (cf. VaR at 10%) are considered for the worst outcome.

Loss Aversion or Prospect Theory

• Developed by Kahneman and Tversky (1979), Prospect Theory includes loss aversion utility (losses cause more pain than gains bring joy).
• It includes probability transformation for overweighting low probability events like disasters.

Loss Aversion Utility Function

• Utility is relative to a reference point.
• People are concaved over gains
• People are risk adverse
• People are convex over losses (risk seeking)
• Investors are more sensitive to losses than to gains.

Habit Utility

• Wealth is relative to a reference point.
• The reference point is equivalent to subsistence.
• Habit utility requires portfolios' returns in relation to habit.
• Wealth closer to habit increases risk aversion.
• Wealth far above habit increases risk tolerance and holdings of equities.
• Habits can evolve over time.

Catching Up with the Joneses

• Relative utility compares wealth/actions to others.
• "Bad times" are relative to other investors.
• Performance is about being being better than peers, reflecting social status.
• Utilities exhibit externalities because they depend on both one's returns and peers.
• It leads to herding, with portfolio managers benchmarking to each other and holding same stock.

Uncertainty Aversion

• Knightian uncertainty leads to multiple probability distributions for assets.
• Liberalized emerging countries have 'good' (stable) and 'bad' (coups) distributions.
• Investors' utility depends on both risk and uncertainty.
• More precise information about distributions increases utility
• An uncertainty aversion parameter exists, similar to risk aversion.
• Ambiguity aversion includes the recognition that numerous distributions exist.

Realistic Utility Functions: Summary

• Utility functions measure bad times, such as low wealth or relative performance.
• These bad times are marginal, and consider spare dollars as precious.
• Mean-variance utility is restrictive as it only represents bad times through returns that are low, or portfolio returns that are very diverse.

Garbage In, Garbage Out (GIGO)

• The most important limitation of MV means that mean-variance frontiers are highly sensitive to estimates of means, volatilities, and correlations.
• MV assumes the the mean and variance are known.
• Small shifts in assumptions leads to potentially large changes in optimized portfolio.
• This leads to error maximizing portfolios, giving very little reliable investment value.
• Errors in mean are at least 10 times more important than errors in variance

Solutions to Garbage In, Garbage Out (GIGO)

• Adjust/change utility
• Use constraints
• Use resampling techniques
• Use robust statistics
• Don't just use historical data
• Use economic models
• Keep it simple

Change Utility (GIGO Solution)

• Don't use mean-variance utility
• Use more realistic utility functions
  • Investors are fearful of other risks
  • Investors care about relative performance
  • Investors fear losses much more than they cherish gains
• Problem: There are no commercial optimizers that compute optimal portfolios for more realistic utility functions

Use Constraints (GIGO Solution)

• Use constraints to force the portfolio to "look right"
• Shrinkage involves shrinks weights back to economically reasonable values
• Examples:
  • Constrain maximum active position sizes
  • Constrain portfolio attributes BUT, portfolio constraints often get in the way of good portfolio construction

Use Resampling Techniques (GIGO Solution)

• Resample optimal portfolios for Resampling Efficiency
• Steps;
  • Sample mean vector and covariance matrix of returns centered at initial point
  • Calculate MV efficient frontier
  • Steps are repeated with observations available in Step 4
  • Find the average of portfolio weights for the RE optimal portfolio
  • Apply investability constraints to portfolio

Use Robust Statistics (GIGO Solution)

• Bayesian shrinkage is used for expected returns.
  • Goal is to improve expected returns
  • This involves taking care of outliners and extreme values
  • Estimates are shrunk back to their original values or their economies.
• Steps
  • Start with "common sense" prior belief
  • Examine sample data
  • Weight prior belief and sample data
• Black-Litterman model shrinks expected returns to market-neutral

Use Robust Statistics for Covariance Matrix (GIGO Solution)

• Improvement on variances with respect to sample
• Estimate variance-covariance matrix and the averages are weighted to a particular structure
• Contributers where Ledoit and Wolf
• Difficulties
  • Estimates from shale intensity
    • Ledoit and Wolf formula is used
  • Estimate is shrinkage target
    • Single index factor matrix

Don't Just Use Historical Data (GIGO Solution)

• Past data is no gaurantee for future returns
• Using short samples is dangerous
  • This leads to procyliality
  • Expected returns and high - high prices are high because future returns tend to be low
  • High volatilities and high prices where risk is high
• What about long historical samples?

Use Economic Models (GIGO Solution)

• Valuation requires an economic framework
• Combine economic models with statistical models
• Black-Litterman:
  • Expected returns are implied
  • Adjust belief based on shrinkage estimator
• Factor investing
  • Asset class could be same factors
  • Factor risks has traditional asset classes
  • Diversity might be overstated

Keep It Simple (GIGO Solution)

• The main principle of mean-variance investing is to hold diversified portfolios
• Hold simple diversified portfolios and rather optimised ones
• The simplest strategy, an equally-weighted portfolio, often turns out to be one of the best performers

Dynamic Portfolio Choice

• Dynamic portfolio choice is a portfolio choice with long horizons
• Investors change their portfolio due to return and investment opportunites
• How investors change portfolios includes;
  • time-varying investment opportunites
  • horizon approaching
  • Liabilities and risk change

Long-Horizon Investing Fallacies

• Being Long isn't always optimal as markets are constantly changing
• Investing in long-term means short-term (Invest into shorter investments)
• Long investments should be short term successes

Multi-period Framework: Rebalancing

• Most basic and fundamental long-run investment strategy
• Idea: Rebalance to fixed asset positions determined in a one-period portfolio choice problem where asset weights reflect investor's attitude toward risk
• Why? Avoiding one asset (class) dominates the portfolio (given level of risk aversion)
• Investors should rebalance to stay focused

Multi-period Framework: Rebalancing Framework

• Rebalancing = Most basic and fundamental long-run investment strategy
• Idea: Rebalance to fixed asset positions determined in a one-period portfolio choice problem where asset weights reflect investor's attitude toward risk -Why? Avoiding one asset (class) dominates the portfolio (given level of risk aversion) -Rebalancing -Contingent versus calendar rebalancing
  • Account for transaction costs