Mean Variance: Optimizes a portfolio based on inputs of historical/expected returns and standard deviations. ie, given expected returns/deviations for 4 asset classes the Mean Variance method will calculate the optimal portfolio combination of the 4 assets to yield the best risk/return trade-off.
Benefits - easy/cheap to implement and understand, only 1 output given. Negatives - requires a large amount of estimated input data, static approach (one iteration), can result in concentrations due to the way the optimization works.
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Resampled Mean Variance: basically runs a bunch of Mean Variance optimizations based on different assumptions and averages the results to get an optimizes portfolio.
Benefits - more optimizations result in better diversification and a more stable efficient frontier. Negatives - no mathematical rationale behind doing this method, still a static approach, relies on estimates.
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Black-Litterman: Starts with the market portfolio and backs out the expected returns, risk premiums, covariances, etc implied by market prices, assuming market equilibrium. From there a Mean-Variance optimization is run using those inputs to generate an efficient frontier.
Benefits - high level of diversification, overcomes weakness of MV which is the variability of estimated returns. Negatives - static approach, difficult to estimate returns.
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Monte Carlo Simulation: Computer generated iterative process that incorporates different input variables (contributions/withdrawals, taxes, capital market factors, etc) to generate a range of possible outcomes.
Benefits - multiple output = not a static approach, incorporates compounding and other relevant information, generates a distribution of returns instead of a single prediction. Negatives - complex and expensive to generate, still relies on the accuracy of input data.