MODERN PORTFOLIO THEORY: Dealing with Uncertainty
Philippe Jorion PAAMCO and UC-IRVINE
(c) 2018 P. Jorion
E-mail: pjorion@uci.edu
EXAMPLE OF EFFFICIENT SET: STOCKS and BONDS
Notes: Expected returns are in excess of cash [Source:Cliffwater (2018)]; optimal portfolio reflects risk aversion of 2
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PRINCIPLES OF PORTFOLIO OPTIMIZATION
MPT -- Philippe Jorion
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IGNORING UNCERTAINTY: SENSE OF FALSE PRECISION
Deutsche Bank reports Economic Capital of €35,438 million » This is a worst-loss number (Value at Risk) at a 99.98% confidence level over 1 year, across all types of risks » Common Equity Tier 1 is €42,244 m » Supposed to include operational risk--but DOJ wanted to impose a penalty of €14 billion
Such statements ignore model uncertainty and give a false sense of precision
MPT -- Philippe Jorion
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UNCERTAINTY IN PORTFOLIO OPTIMIZATION Issues: (1) Portfolio weights, outputs of optimization, are very sensitive to changes in inputs Âť i.e., expected returns, variances, correlations
(2) Inputs can be very imprecisely measured, especially expected returns, and to a lesser extent, variance and correlations ➪This implies that portfolio weights can be subject to substantial uncertainty MPT -- Philippe Jorion
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UNCERTAINTY (1): Weights are Sensitive to Exp. Ret. ď Ź
Changes in expected returns (e.g. on stocks) have a large effect on optimal weights
MPT -- Philippe Jorion
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UNCERTAINTY (2): Uncertainty in Exp.Ret. Estimates How can we confirm that the equity premium is significantly positive, based on history? T-statistic = / ( / N ), where N is number of years, risk premium, volatility of returns We require, say t-stat > 2, so solve for N
MPT -- Philippe Jorion
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EVALUATING UNCERTAINTY: Example We can evaluate the effect of uncertainty: Use for example 11 years of returns on global bond markets to infer expected returns, variances, and correlations Perform optimization with no short-sales, choosing portfolio with maximum Sharpe ratio To evaluate uncertainty in input parameters, resample from the returns data over 11 years to obtain “statistically equivalent portfolios” Evaluate distribution of these portfolios, i.e. performance and weights Source: Jorion (1992), “Portfolio Optimization in Practice,” Financial Analysts Journal MPT -- Philippe Jorion
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STATISTICALLY EQUIVALENT PORTFOLIOS: Performance True Optimal Portfolio
Period: 1978-1988 MPT -- Philippe Jorion
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ADJUSTMENTS FOR UNCERTAINTY Constraints: Impose constraints to stabilize weights (but how good is this assumption?) Shrinkage estimation: Lower the dispersion in expected returns (and covariance matrix) by shrinking expected returns toward a common value
» Bayesian approach » Black-Litterman (1990) shrinkage to implied views » Mixing multiple expert inputs MPT -- Philippe Jorion
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“RULES OF THUMB” FOR EVALUATING UNCERTAINTY
Number of assets: Effect of uncertainty worsens with larger number of assets » Asset allocation vs. stock optimization
Noise in parameters: Expected returns estimated from historical data are much more “noisy” than volatilities and correlations Type of portfolios: Long/short portfolios are more sensitive to changes in correlations
» LTCM (1998) took a highly leveraged bet on swap-Treasury correlation MPT -- Philippe Jorion
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CONCLUSIONS Making prudent portfolio decisions under uncertainty requires combining judgment with MPT model results Beware of the impression of false precision from portfolio optimization Ask how sensitive the results are to the inputs, in particular assumptions about expected returns Verify that the “optimal” portfolio is robust to changes in input parameters MPT -- Philippe Jorion
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