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## Mean-Variance Optimization

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**Mean-Variance Optimization**• Investors should choose from efficient portfolios consistent with the investor’s risk tolerance**Unconstrained Mean-Variance Optimization**• Weights can take on any value (positive or negative), only constraint is that the weights sum to one • Black (1972) provided a short cut for finding minimum variance portfolios: • Asset weights of any minimum variance portfolio is a linear combination of the asset weights of any other two minimum variance portfolios (Black’s Two-Fund Theorem) • Mathematical proof provided in the 1972 paper**Unconstrained Mean-Variance Optimization**Numerical example • Suppose we have three asset classes: equities (S), fixed income (B), and real estate(RE) • Composition of the minimum variance portfolio with E(r) = 10% is (70%, 20%, 10%) • Composition of the minimum variance portfolio with E(r) = 8% is (50%, 30%, 20%) • What is the composition of the minimum variance portfolio with E(r) = 9.5%?**Constrained M-V Optimization**• Minimum asset class weight = 0 • Most relevant for strategic asset allocation • May short assets within a class, but not the entire class • To generate the efficient frontier, use the corner portfolio theorem of Markowitz (1959, 1987) • There are infinitely many efficient portfolios, only need a limited number of “corner portfolios” to identify them all • Corner portfolios: They are located when an asset class is either added or dropped along the efficient frontier (i.e., weight going from zero to strictly positive or from strictly positive to zero)**Constrained M-V OptimizationCorner Portfolios**GMV portfolio**Constrained M-V Optimization**• Every efficient portfolio is a linear combination of the two corner portfolios immediately adjacent to it (on either side of it). Thus, by locating all corner portfolios, you can generate the entire efficient frontier • Markowitz (1959, 1987 – two books) provides the “critical line algorithm” to do this • Our textbook calls this the “Corner Portfolio Theorem”: • The asset weights of any minimum variance portfolio are a (positive) linear combination of the weights of the two adjacent corner portfolios that bracket it in terms of E(r)**Constrained M-V OptimizationNumerical Example**• Find the composition of the mean-variance efficient portfolio E(r) = 8% • Which two corner portfolios does it lie in between? • Solve for the weights: • UK equities = ? • Ex-UK equities = ? • Intermediate bonds = ? • Long term bonds = ? • International bonds = ? • Real estate = ?**Constrained M-V Optimization**• Variance of the 8% E(r) portfolio • Use the variance formula (sum of the variance-covariance matrix bordered by the weights) • In this case, there are four assets**Re-sampled Efficient Frontier Optimization**• Mean variance optimization treats the inputs as population parameters • But they are only sample estimates • Estimation errors of the inputs will distort the optimization results • Most important input in MV optimization is E(r) • Ziemba (2003) show that estimation error in E(r) is 10 times as important as estimation error in , and 20 times as important as estimation error in **Re-sampled Efficient Frontier Optimization**• Re-sampling: Take several efficient portfolio simulations using different parameters for E(r), , and as sensitivity analysis • For each level of E(r), average the weights of each asset class from different efficient portfolios (that were estimated using different inputs) • These can then be integrated into a re-sampled efficient frontier • Re-sampled efficient frontier tends to be more diversified and more stable over time**Monte-Carlo Simulation in Asset Allocation**• Most likely application: • Given an existing asset allocation, calculate terminal wealth using random draws from historical distributions of returns • Provides information concerning the range of possible results and the relative likelihood of each (e.g., “80% of the times, will have terminal wealth greater than $1 million, given current portfolio”)**Experience Based Approaches to Asset Allocation**• Rely on tradition, experience and/or rules of thumb • Inexpensive to implement • 60/40 stock bond allocation as neutral starting point • Allocation to bonds increases with risk aversion • Allocation to stocks increases with time horizon • Allocation to equities = 100 - age**Strategic Asset Allocation for Defined Benefit Plans**• Must maintain liquidity to pay current benefits • Asset/Liability Management used to control: • Shortfall risk • Volatility of pension surplus • Asset-only management seeks to minimize standard deviation relative to required return (e.g., 6%)**Rest of the chapter**• Ideas for project • Contribution to risk by different asset classes (rows of the Var-Cov matrix)**New ways of thinking about diversification**• Allocating capital across asset classes and investment styles represents superficial diversification if payoffs are exposed to the same set of risk factors. • Diversify across underlying risk factors • E.g., Bradley Jones, Rethinking Portfolio Construction and Risk Management: A Third Generation in Asset Allocation, 2012 (presentation available on the web) • 1st generation: 60/40 • 2nd generation: more asset classes**New ways of thinking about diversification**• Equity risk premium prevails only in “normal” markets • Diversify across different risk exposures, such as: • interest rate risk • commodity risk • style risk premia, such as value, momentum and volatility • liquidity risk • sovereign risk