P jorion

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

2

PRINCIPLES OF PORTFOLIO OPTIMIZATION

MPT -- Philippe Jorion

3

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

4

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

5

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

6

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

7

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

8

STATISTICALLY EQUIVALENT PORTFOLIOS: Performance True Optimal Portfolio

Period: 1978-1988 MPT -- Philippe Jorion

9

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

10

“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

11

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

12