February 21, Purdue University Dept. of Electrical and Computer Engineering. Markowitz Portfolio Optimization. Benjamin Parsons.

Transcription

1 Purdue University Dept. of Electrical and Computer Engineering February 21, 2012

2 Outline

3 Evaluate variations of portfolio optimization Bayes-Stein error estimation Bayes-Stein error estimation with short-sales constrained Combination of naive and minimum variance portfolios

4 Evaluate variations of portfolio optimization Bayes-Stein error estimation Bayes-Stein error estimation with short-sales constrained Combination of naive and minimum variance portfolios Compare the variations to each other and a naive approach

5 Evaluate variations of portfolio optimization Bayes-Stein error estimation Bayes-Stein error estimation with short-sales constrained Combination of naive and minimum variance portfolios Compare the variations to each other and a naive approach Reproduce and expand on the results of DeMiguel et al. [1]

6 Outline

7 Naive Approach 1/N The naive portfolio is an equally weighted portfolio w = 1 N

8 Naive Approach 1/N The naive portfolio is an equally weighted portfolio w = 1 N This does not use Σ or µ, negating estimation error

9 Naive Approach 1/N The naive portfolio is an equally weighted portfolio w = 1 N This does not use Σ or µ, negating estimation error DeMiguel et al. [1] finds that this method generally performs very well.

10 Combination of 1/N and Minimum Variance Novel approach introduced by DeMiguel et al. [1] Similar to Mean-variance and Minimum-variance approach by Kan [3]

11 Combination of 1/N and Minimum Variance Novel approach introduced by DeMiguel et al. [1] Similar to Mean-variance and Minimum-variance approach by Kan [3] Uses the Minimum-variance because it does not need µ, which is harder to estimate than Σ

12 Combination of 1/N and Minimum Variance Novel approach introduced by DeMiguel et al. [1] Similar to Mean-variance and Minimum-variance approach by Kan [3] Uses the Minimum-variance because it does not need µ, which is harder to estimate than Σ Minimum-variance : min wt wt t w t, s.t. 1 N w t = 1 1/N : w t = 1/N

13 Combination of 1/N and Minimum Variance Combination portfolio: w = c 1 N 1 N + d 1 1 N, s.t.1 N w = 1

14 Combination of 1/N and Minimum Variance Combination portfolio: w = c 1 N 1 N + d 1 1 N, s.t.1 N w = 1 Combination portfolio c and d are chosen to maximize the utility for a mean-variance investor

15 Combination of 1/N and Minimum Variance Combination portfolio: w = c 1 N 1 N + d 1 1 N, s.t.1 N w = 1 Combination portfolio c and d are chosen to maximize the utility for a mean-variance investor Kan [3] shows that d 1 1 N can be used instead of the actual minimum variance weights because they are proportional

16 Bayes-Stein shrinkage portfolio Reduces the estimation error on sampled Σ and µ

17 Bayes-Stein shrinkage portfolio Reduces the estimation error on sampled Σ and µ A predictive distribution for the returns is combined with the sampled data

18 Bayes-Stein shrinkage portfolio Reduces the estimation error on sampled Σ and µ A predictive distribution for the returns is combined with the sampled data The Σ and µ produced by this method are meant to minimize the estimation risk on the portfolio, not the estimation error on Σ and µ

19 Bayes-Stein shrinkage portfolio Reduces the estimation error on sampled Σ and µ A predictive distribution for the returns is combined with the sampled data The Σ and µ produced by this method are meant to minimize the estimation risk on the portfolio, not the estimation error on Σ and µ Introduced by Jorion [2] in 1986

20 Bayes-Stein shrinkage portfolio µ bs t = (1 φ t )µ t + φ t µ t

21 Bayes-Stein shrinkage portfolio µ bs t = (1 φ t )µ t + φ t µ t µ t is the Grand Mean, the mean of the minimum variance portfolio

22 Bayes-Stein shrinkage portfolio µ bs t = (1 φ t )µ t + φ t µ t µ t is the Grand Mean, the mean of the minimum variance portfolio N+2 φ t = (N+2)+M(µ t µ t ) 1 t (µ t µ t ) N is the number of assets M is the number of data points used in the estimation

23 Bayes-Stein shrinkage portfolio µ bs t = (1 φ t )µ t + φ t µ t µ t is the Grand Mean, the mean of the minimum variance portfolio N+2 φ t = (N+2)+M(µ t µ t ) 1 t (µ t µ t ) N is the number of assets M is the number of data points used in the estimation 0 φ t 1 This shrinks the means toward the grand mean

24 Bayes-Stein shrinkage portfolio Σ bs t = Σ t (1 + 1 M ) + 1 M(M+1) 1 Σ 1 t 1 Now use the µ bs and Σ bs to calculate the mean-variance portfolio 1 N

25 Outline

26 60 month out of sample evaluation Sharpe ratio Returns Turnover

27 Outline

28 10 Farma French Industry s + US Market, reproduce from DeMiguel [1]

29 10 Farma French Industry s + US Market, reproduce from DeMiguel [1] 20 Farma French size and book to market s + Us Market, reproduce from DeMiguel [1]

30 10 Farma French Industry s + US Market, reproduce from DeMiguel [1] 20 Farma French size and book to market s + Us Market, reproduce from DeMiguel [1] S&P 500

31 Outline

32 Implementing the project in Matlab

33 Implementing the project in Matlab Data I/O code is complete

34 Implementing the project in Matlab Data I/O code is complete toolbox will be utilized

35 Questions

36 References Victor DeMiguel, Lorenzo Garlappi, and Raman Uppal. Optimal versus naive diversification: How inefficient is the 1/n portfolio strategy? Review of Financial Studies, 22(5): , Philippe Jorion. Bayes-stein estimation for portfolio analysis. Journal of Financial and Quantitative Analysis, 21(03): , September Raymond Kan and Guofu Zhou. Optimal portfolio choice with parameter uncertainty. Journal of Financial and Quantitative Analysis, 42(03): , September 2007.