optimal growth portfolio construction and effective tail risk … · 2019. 2. 26. · equity arena,...

S. J. Kusiak 6/8/2012

Optimal Growth Portfolio Construction and

Effective Tail Risk Mitigation

Steven J. Kusiak, Ph.D.

[email protected]

17th Annual Northfield Summer Seminar

June 8, 2012

S. J. Kusiak 6/8/2012

Why Go Beyond Mean-Variance Portfolio Construction &

The “New Non-Normal?”

S. J. Kusiak 6/8/2012

What Does Recent History Tell Us?

* Assets: iShares Fixed-Income ETFs ( 1= SHV, 2 = SHY, 3 = IEF, 4 = TIP, 5 = CSJ, 6 = LQD, 7 = HYG, 8 = EMB), 5 year rolling historical window.

S. J. Kusiak 6/8/2012

Tail Risk Indicators &

Large Deviation Monitoring

S. J. Kusiak 6/8/2012

A Tail Risk Indicator: ‘Systemic Risk*’

*Kritzman, Li, Page, and Rigoban (2010).

Definition of the “Absorption Ratio”:

S. J. Kusiak 6/8/2012

Representative Historical Principal Components*

(lo

g1

0)

* S&P 500 Index

Period 1

Period 2

S. J. Kusiak 6/8/2012

Representative Historical Principal Components*

K = 3 is about right!

* S&P 500 Index.

S. J. Kusiak 6/8/2012

How Does One Construct the Absorption Ratio?

S. J. Kusiak 6/8/2012

How Does One Construct the Absorption Ratio

Fact: The value of the Absorption Ratio tends to a CONSTANT with increasing historical window length. (Proof: The spectrum of a historically expanding covariance matrix tends to a stationary one.)

S. J. Kusiak 6/8/2012

How to Construct the Absorption Ratio

Fact: The value of the Absorption Ratio tends to a CONSTANT with increasing historical window length. (Proof: The spectrum of a historically expanding covariance matrix tends to a stationary one.) Consequence: All derived information from the Absorption Ratio, e.g., the “Standardized Shift”, are constant or zero-valued.

S. J. Kusiak 6/8/2012

Portfolio Construction From the Ground Up

S. J. Kusiak 6/8/2012

How to Invest/Diversify in More than 1 Asset?

Risk (Standard Deviation)

Return (mean)

Minimum Return

Maximum Return

Assets’ Historical Risk-Return

“Efficient Frontier”

How to balance between risk and reward???

S. J. Kusiak 6/8/2012

Markowitz Mean-Variance Optimization

Remark: For a list of expected returns the optimal solutions form the so-called “Efficient Frontier” – the optimal risk vs. return curve – and the optimal portfolios, i.e., the collection of weights w.

Extension: Why not go further and add portfolio skewness and kurtosis and optimize across 4 (or more) terms? Wouldn’t this be better?

S. J. Kusiak 6/8/2012

Incorporating Portfolio Skewness

Co-Skewness Tensor

Portfolio Skewness

Remark: R must be a T x N matrix with T > N squared values as the Co-Skewness Tensor is rank 3. I.e., it resembles a vector (of length N) of square matrices (of size N x N).

Sub-moments

S. J. Kusiak 6/8/2012

Incorporating Portfolio Kurtosis

Co-Kurtosis Tensor

Portfolio Kurtosis

Sub-moments

Remark: R must be a T x N matrix with T > N cubed values as the Co-Kurtosis Tensor is rank 4. I.e., it resembles a square matrix (of size N x N) of square matrices (of size N x N). Hence, for N ~ 10 assets, T ~ 1,000 months or about 100 years of historical data. Good luck!

S. J. Kusiak 6/8/2012

Factor Modeling of Co-Skewness and Co-Kurtosis*

Observations: • Cumulant tensors like co-skewness and co-kurtosis are useful, but too big in size. • Too hard to estimate with reasonable data. • Too hard to optimize. • Too hard to store on a computer.

*Morton, Lim (2009).

Result: • Must resort to approximation, need small implicit factors. • Factors? Which ones, and how many more than for covariance estimation? • PCA? Must implement a tensor-analog for positive definite matrices, PCA becomes Principal Cumulant Component Analysis (PCCA), NOT EASY STUFF!

S. J. Kusiak 6/8/2012

Optimal Growth Portfolios

What is “Optimal Growth” and why should it out-perform mean-variance portfolio construction?

A portfolio that grows “optimally” is one that, over an investment period, maximizes return, minimizes risk (e.g., asset covariance), maximizes the tendency of more positive than negative returns (e.g., asset coskewness), minimizes tail risk events (e.g., asset cokurtosis), etc.*

Optimal Growth Portfolios (OGP) are implicitly determined by finding the best portfolio of all choices that would, and will, grow “optimally” in any investment period.

* Optimal growth, a proprietary investment technology, captures and optimizes the full (infinite) extent of these moments in a novel and stable manner.

S. J. Kusiak 6/8/2012

Baseline Portfolio Construction Historical Back-Tests

S. J. Kusiak 6/8/2012

The Back-Tests

1. Use historical returns of prescribed length to generate a covariance matrix and expected returns for 6 test portfolios

2. Determine optimal investment weights and create portfolios 1-6 across the historical expected returns

3. Receive weighted returns/growth in the next period and repeat items 1 and 2 above through rolling historical data

Risk (standard deviation)

Return (mean)

Minimum Return

Maximum Return

Portfolio 1

Portfolio 2

Portfolio 3

Portfolio 4

Portfolio 5

Portfolio 6

Assets’ Historical Risk-Return

S. J. Kusiak 6/8/2012

A Simple Asset Allocation Back-Test

8 ETF Asset Universe* Short-Term Treasury Bonds

1-3 Year Treasury Bonds 7-10 Year Treasury Bonds

TIPS 1-3 Year Credit Bonds

Investment Grade Corporate Bonds High-Yield Corporate Bonds

Emerging Market Bonds

& 8 Year Rolling Historical Window

* Assets: iShares Fixed-Income ETFs (SHV, SHY, IEF, TIP, CSL, LQD, HYG, EMB). Returns are monthly-based.

S. J. Kusiak 6/8/2012

Portfolios 1* and 6*

Portfolio 1 Portfolio 6

* The optimal growth and mean-variance portfolios are identical in the extreme cases (or ends of the efficient frontier) and their values fall on the same lines.

S. J. Kusiak 6/8/2012

Portfolios 2, 3, 4 and 5



S. J. Kusiak 6/8/2012

8 Asset Universe Out-of-Sample Results* 2007- Q1 2012

* Annualized values.

S. J. Kusiak 6/8/2012


* Monthly values with 10 bps of trading costs.

8 Year Rolling Historical Window

S. J. Kusiak 6/8/2012







& 4 Year Rolling Historical Window


S. J. Kusiak 6/8/2012

Portfolios* 1, 2, 3, 4, 5 and 6 Portfolio 1 Portfolio 2 Portfolio 3

Portfolio 4 Portfolio 5 Portfolio 6

* Monthly values with 10 bps of trading costs. Maximum-Diversification replaces asset returns with volatilities and portfolio mean with portfolio volatility. Choueifaty and Coignard (2008).

S. J. Kusiak 6/8/2012


* Monthly values with 10 bps of transaction costs.


S. J. Kusiak 6/8/2012

50% Upper-Bound: Portfolios* 1, 2, 3, 4, 5 and 6



Portfolio 1 Portfolio 2 Portfolio 3

Portfolio 6

S. J. Kusiak 6/8/2012

Optimal Growth is Not Log-Utility

Recall: Logarithmic utility has no risk-aversion adjustment, it just yields an optimal portfolio maximizing

OG/MV Portfolios

Log Utility Portfolios

S. J. Kusiak 6/8/2012







& 8 and 4 Year Rolling Historical Windows

& Proprietary Predicted Returns


S. J. Kusiak 6/8/2012


* With proprietary predicted returns (Information Coefficients range from 0.4 to 0.6) and 10 bps trading costs. Monthly values.


S. J. Kusiak 6/8/2012


* With proprietary predicted returns (Information Coefficients range form 0.4 to 0.6) and 10 bps trading costs. Monthly values.


S. J. Kusiak 6/8/2012

What About Portfolio Turnover*?

Remark: Portfolio turnover in the 8 (and 4) year rolling historical window cases is 81% (and 19%) higher on average for the Optimal Growth construction versus Mean-Variance, representing a more dynamic (non-stationary) process.

8 Year Historical Window 4 Year Historical Window

* No limits beyond the standard long-only ones are placed on the assets, hence Optimal Growth tends to seek out more “optimal” portfolios and turn-over positions in larger amounts.

S. J. Kusiak 6/8/2012

A 2nd Simple Asset Allocation Back-Test

11 ETF Asset Universe* S&P 500

Dow Jones Euro Stoxx 50 MSCI Japan

MSCI Emerging Markets FTSE EPRA/NAREITs

European Government 1-3 Year Bonds European Government 7-10 Year Bonds

European Inflation Linked Bonds European Corporate Bonds

Emerging Market Bonds Physical Gold ETC

& 5 and 8 Year Rolling Historical Window

* Assets: iShares Multi-Asset Class ETFs (IUSA, EUE, IJPN, IEEM, IWDP, IBGS, IBGM, IBCI, IBCX, SEMB, SGLN). Monthly-based historical returns.

S. J. Kusiak 6/8/2012

Portfolios 1* and 6*

* The optimal growth and mean-variance for Portfolio 6 are identical in this extreme case and their values fall on the same line.


S. J. Kusiak 6/8/2012

Portfolios 2, 3, 4 and 5 Portfolio 2

Portfolio 3

Portfolio 4

Portfolio 5

S. J. Kusiak 6/8/2012

11 Asset Universe Out-of-Sample Results 2005- Q1 2012

S. J. Kusiak 6/8/2012




S. J. Kusiak 6/8/2012

A 2nd Simple Asset Allocation Back-Test

11 ETF Asset Universe* S&P 500

Dow Jones Euro Stoxx 50 MSCI Japan

MSCI Emerging Markets FTSE EPRA/NAREITs

European Government 1-3 Year Bonds European Government 7-10 Year Bonds

European Inflation Linked Bonds European Corporate Bonds

Emerging Market Bonds Physical Gold ETC

& 5 and 8 Year Rolling Historical Windows

& Proprietary Predicted Returns

* Assets: iShares Multi-Asset Class ETFs (IUSA, EUE, IJPN, IEEM, IWDP, IBGS, IBGM, IBCI, IBCX, SEMB, SGLN). Monthly-based historical returns.

S. J. Kusiak 6/8/2012

5 Year Window with Predicted Returns*

* With proprietary predicted returns, 10 bps TCs. The optimal growth and mean-variance for Portfolios 1 and 6 are identical in these extreme cases and their values fall on the same line.

OG6/MV6

OG5

MV5

OG4

MV4

MV3 OG3

1/N

S. J. Kusiak 6/8/2012

11 Asset Universe Out-of-Sample Results* 2005 - ~Q1 2012

* 5 Year historical window with proprietary predicted returns (Information Coefficients range from 0.4 to 0.6) and 10 bps trading costs.


S. J. Kusiak 6/8/2012

11 Asset Universe Out-of-Sample Results* 2009 - ~Q1 2012

* 8 Year historical window with proprietary predicted returns (Information Coefficients range from 0.4 to 0.6) and 10 bps trading costs.


S. J. Kusiak 6/8/2012

F.A.Q.s Question: Can this method deal with 100, 500, or more assets? Answer: The analytical basis of the Optimal Growth technique can accommodate hundreds of assets with individual and group constraints. In some cases the sky is the limit and thousands of assets may be used. Question: What about active equity? Answer: Optimal Growth portfolio constructions works extremely well in the active equity arena, e.g.,10 aggregate simulations with predicted returns (IC=0.2) of S&P 500 equity portfolios of size 20 stocks, long-only, max weight 50%

S. J. Kusiak 6/8/2012

F.A.Q.s Question: Can this method deal with 100, 500, or more assets? Answer: The analytical basis of the Optimal Growth technique can accommodate hundreds of assets with individual and group constraints. In some cases the sky is the limit and thousands of assets may be used. Question: What about active equity? Answer: Optimal Growth portfolio constructions works extremely well in the active equity arena, e.g., Question: Does the technique simply use a different utility function other than the quadratic one in mean-variance optimization? Answer: No, Optimal Growth is the result of a more supple and appropriate analytical framework and is not reliant upon any utility function. Question: Is there an article, or research, available discussing more background and details of Optimal Growth Portfolio Construction? Answer: Yes, in progress and available upon request.

optimal growth portfolio construction and effective tail risk … · 2019. 2. 26. · equity arena,...

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