Lecture 6

1. Introduction
Criticism of Mean-Variance Model
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Sensitivity to Input
Robust Portfolio Construction
Robust Portfolio Construction Asset Benchmark Alpha 1 Alpha 2 True Return Alpha Weights (%) (%) Alpha (%) Std. Dev. (%) Std. Dev. (%) Portfolio Construction 1 1 Alpha 2 0.5 2.4 2.5 2.48 Alpha Robust 0.42 0.5 Benchmark Asset Alpha True Return Asset 2 2.5 0.5 Weights (%) (%) 0.5 Alpha (%) Std.2.4 Dev. (%) 2.42 Std. Dev. (%) 0.33 0.5 2.4 2.5 2.48 0.42 0.5 Table Returns and Standard for Example 1 0.5 2.5 2.41: Expected 2.42 0.33 Deviations 0.5 1.0 Table 1: Expected Returns and Standard Deviations for Example 1 A Attribute Folio A Folio B Folio C Folio D Alpha 1 2 1 2 Budget Attribute Folio A Folio B Folio C Folio D Weight 10.169 0.5253 0.5546 Alpha 1 Asset 12 20.831 Asset 2 Weight 0.831 0.169 0.7796 0.7503 Budget Asset 1 Weight 0.169 0.831 0.5253 0.5546 Table 2: Optimal Portfolios Asset 2 Weight 0.831 0.169 0.7796 0.7503 for Example 1 Table 2: Optimal Portfolios for Example 1
C D
M
B
portfolio (shown as point ’M’ in Figure 1). The estimates of expected returns and standard deviations of the two assets are given in Table 1. We assume that0.0 the correlation between the two assets * Ceria and Stubbs (2006) 0.0 Asset 1 Weight is 0.7. The feasible region of this example is illustrated in Figure 1 asdeviathe intersection of the shaded shown as point ’M’ in Figure 1). The estimates of expected returns and standard ellipsoidal region budget constraint, i.e., the feasible region of this example is simply the e two assets are given in Table 1.and Wethe assume that the correlation 3 between the two assets line segment between points A and B.
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1. Introduction
Criticism of Mean-Variance Model
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Counterintuitive Inexplicable Too Sensitive to Input Parameters Error Maximisation (Michaud, 1989) …
Bayesian Approach
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Overview
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Bayesian approach recognises the fact that the true parameter values are not observable and their estimates contain error. It estimates the distribution of expected values (returns, in our case) rather than a point estimate. The distribution is obtained by updating a base distribution with new information. Once a new distribution of asset returns is obtained, the usual portfolio optimisation procedure can be applied.
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Examples
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Black-Litterman model (Black and Litterman, 1991) Bayes-Stein estimator (Jorion, 1986)
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2. Bayesian Approach
Bayesian Approach
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Prior Distribution:
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2. Bayesian Approach
Bayes-Stein Estimator
Using r ¯, the expected return of the minimum variance portfolio is
0 1 ⌃ 0 wr ¯= 0 1⌃
r ¯ 11
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Therefore, the prior distribution is given as follows. Prior: ¯) µ ⇠ N (¯ µ, ⌃ with 10 ⌃ µ ¯= 0 1⌃ r ¯ 1, 11
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Bayesian Approach
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Account for the estimation errors in the return distribution.
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Portfolio Resampling
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Find optimal portfolios for many possible input parameters. Take average of them.
Much of the criticism can be traced back to the inherent uncertainty in the estimates of the input parameters.
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1. Introduction
Accounting for Estimation Errors
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and
1 ¯ ⌃= ⌃ ⌧
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2. Bayesian Approach
Bayes-Stein Estimator
Bayes-Stein estimator uses the sample mean of the asset returns to update the prior distribution. In the context of Black-Litterman model, the sample mean returns can be regarded as absolute views on the asset returns and the confidence of the views is represented by the covariance of the sample mean returns, ⌃/T . Hence, the view portfolio is defined by P = I , q = r ¯, and ⌦ = ⌃/T . Substituting these values into the Black-Litterman formulas, the posterior distribution is given as follows. Posterior: µ ⇠ N (ˆ µ , ⌃µ ) ⌧ T µ ¯+ r ¯, µ ˆ= T +⌧ T +⌧ and 1 ⌃µ = ⌃ T +⌧
Portfolio Management
6. Robust Optimisation
Chulwoo Han Durham University
Outline
1. Introduction 2. Bayesian Approach 3. Portfolio Resampling 4. Robust Optimisation 5. Reading List
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The prior of the expected return is a base estimate of the expected return which is believed to be the best estimate without additional information, e.g., equilibrium return.
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New Information:
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If a new information such as investor view is available, it is combined with the prior distribution to yield an updated distribution.