MSC Thesis - Henry Ward

7/31/2019 MSC Thesis - Henry Ward

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Portfolio Optimization and Rebalancing as a Risk

Management Strategy

September 10th, 2011

Henry S. Ward, MSC Capital Markets - EDHEC

Abraham Lioui, Professor of Finance - EDHEC

EDHEC Business School does not express approval or disapproval concerning the opinions

given in this paper which are the sole responsibility of the author.


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Abstract

In this paper, we evaluate an investors ability to choose a specific risk profile for

their portfolio among three optimization strategies. We evaluate a mean-

variance, min-variance, and 1/N rebalancing out-of-sample strategies for

randomly constructed portfolios of liquid S&P500 assets between 2006 and

2011. We show that the reduction in variance between these three portfolio

strategies is significant providing the investor a low risk, medium risk, and high

risk portfolio with the chosen assets. Further, we show that the returns of the

three portfolios is statistically significant and confirms that higher risk leads to

higher rewards. Lastly, we show that no strategy is dominant as measured by

Sharpe Ratio because of estimation error in the mean-variance optimization.

However, there is clear dominance in risk reduction strategies. We conclude that

investors can select a risk profile optimization strategy afterchoosing the desired

assets. This has strong implications for the traditional financial services which

currently chooses assets based on the desired risk profile.


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Abstract .......................................................................................................................2Introduction.................................................................................................................5DataDescription ..........................................................................................................8TestMethodologyandConstruction ............................................................................8RebalancingTechnique ..............................................................................................10

1/N..............................................................................................................................................10

MinimumVariance ...............................................................................................................10Tangency ..................................................................................................................................10NoOptimization.....................................................................................................................11

AppendixAOut-of-SampleTestResults ..................................................................14August1st2005August1st2009....................................................................................15January1st2006January1st2010 ................................................................................15January1st2007January1st20011..............................................................................15August1st2007August1st2011....................................................................................16AcrossAllTimePeriods ......................................................................................................16

AppendixBDistributionofPortfolioReturnsByStrategy ........................................17


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NoOptimization.....................................................................................................................171/N..............................................................................................................................................17

MinimumVariance ...............................................................................................................18Tangency ..................................................................................................................................18

AppendixCDistributionofPortfolioVolatilitybyStrategy ......................................19NoOptimization.....................................................................................................................191/N..............................................................................................................................................19MinimumVariance ...............................................................................................................20Tangency ..................................................................................................................................20

AppendixDPortfolioVolatilityvsPortfolioSize ......................................................21StrategyVolatilitybyPortfolioSize ................................................................................21PortfolioVolatilityasFunctionofPortfolioSize ........................................................22


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Introduction

Investors face two decisions when constructing a portfolio. The first being which

assets to include in the portfolio. The second is how much of each asset should

comprise the total portfolio. The answer to the first question is often determined

by investor preferences, in the case of retail investors, and by prospectus in the

case of institutional investors. The general rule of thumb has been for

aggressive investors to select higher beta assets and conservative investors to

choose less risky investments.

Institutional investors often focus on variations of mean-variance optimization

with efforts to improve estimators or to include investor views (Black & Litterman

1992).

Retail investors most often randomly allocate assets and take a naive buy-and-

hold strategy. For our purposes, the investor believes they have better

information regarding N assets and they have chosen these N assets to part of

their portfolio. Given these choices, how does the investor best allocate the

portfolio in these assets to capitalize on their bias while controlling the overall

portfolio risk?

OptimalPortfolioSize

To answer this question we first establish a range for the appropriate number of

assets. We can approximate the total portfolio risk as:


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where is the portfolio risk, is the average risk of the N assets, and is the

correlation between each pair of assets.

Fig 1 is a plot of the portfolio risk where = 30% and = 0.2. It is apparent that

the diversification contribution to the portfolio of the Nth asset decreases rapidly.

Portfolio Volatility Versus Number of Assets

In the table below we see the marginal contribution to the portfolio becomes very

small after six to ten assets. For most investors, the transaction costs of a large

portfolio far exceed the diversification value. For this reason, portfolios were

created with between two and twenty individual assets with one outlier portfolio

consisting of 50 assets.


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N N + 1 Difference

2 23.248% 3 20.494% 2.79%

5 18.000% 6 17.321% 0.679%

10 15.875% 11 15.667% 0.208%

20 14.697% 20 14.639% 0.058%

StrategyDescriptions

To study the impact different strategies have on portfolio performance we have

created four out-of-sample optimization and rebalancing strategies on randomly

generated portfolios. The four strategies represent more than just a preference

in strategy. They represent four fundamental opinions of the ability statistical

equity analysis has on our ability to improve portfolio performance. The four

strategies are:

No Optimization (Buy-Hold)

This is the benchmark portfolio to compare whether any form of rebalancing

improves a completely passive approach.

1/N Rebalancing

This is nave and uniformed strategy which, when compared to a buy-hold

strategy, determines whether the act of rebalancing adds alpha to active

management of portfolio. This strategy also serves as a benchmark for informed

optimization and rebalancing strategies.

Minimum Variance

The minimum variance rebalanced portfolio accepts the risk of estimation error in

the covariance matrix in the hopes of minimizing total portfolio risk.

Tangency Portfolio


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The tangency portfolio accepts both the risk of estimation error in the covariance

matrix and the risk of estimation error in the expected returns to improve the

portfolio Sharpe Ratio.

TestMethodologyandConstruction

To test the performance of the four different rebalancing strategies, we created a

series of out-of-sample portfolio rebalancing simulations.

DataDescription

The financial data consisted of a database of daily adjusted-close stock prices for

S&P500 stocks during the 10-year period from August 1st 2001 to August 1st

2011.

TimeFrame

To reduce the impact of marketing time affecting the results, we created four

batches of tests that corresponded to four different time periods the simulations

were run against. The four time periods were identical in length, 4 years, and

were staggered by six months for each batch. The exact start-date and end-date

are shown below.

TestBatch StartDate EndDate1 1-Aug-05 1-Aug-09

2 1-Jan-06 1-Jan-10

3 1-Jan-07 1-Jan-11

4 1-Aug-07 1-Aug-11


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Within each batch, 440 portfolios were created by randomly selecting stocks from

the S&P500 index that have existed within the S&P500 during the entire period

from August 1st 2001 to August 1st 2011. We chose stocks from the S&P500 to

remove liquidity premium and bias from the simulations. However, we

acknowledge there exists a survivorship bias as we did not include stocks that

were introduced to or removed from the S&P500 in the sample set.

PortfolioConstruction

The 440 portfolios were constructed by constructing 40 random portfolios of a

given size. With in each size, 10 portfolios were run as an out-of-sample test for

each rebalancing technique. The breakdown of the number of portfolios within

each portfolio size and rebalancing technique are shown below.

NumberofRandomPortfoliosConstructed

PortfolioSize 1/N MinVariance NoOptimization Tangency

2 10 10 10 10

4 10 10 10 10

6 10 10 10 10

8 10 10 10 10

10 10 10 10 10

12 10 10 10 10

14 10 10 10 10

16 10 10 10 10

18 10 10 10 10

20 10 10 10 10

50 10 10 10 10

Total 110 110 110 110

It is important to remember that this series of tests were run within Test Batch

for a total of 1760 out-of-sample portfolio simulations across all four date-ranges.


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RebalancingTechnique

Each portfolio was initially constructed with N randomly chosen stocks from the

S&P500 with each stock holding a weighting of 1/N. On each Friday the portfolio

is rebalance according to the appropriate optimization algorithm. Each

optimization technique is described in turn.

1/N

The 1/N strategy rebalances the portfolio by reallocating the portfolio weights to

be equal across the N stocks.

MinimumVariance

The minimum variance portfolio is estimated by using the fPortfolio package (D.

Wuertz, Y. Chalabi, W. Chen and A. Ellis, 2010) in the R language. The

fPortfolio function minVariancePortfolio calculates the Efficient Frontier and the

Minimum Variance portfolio. The optimization uses the covEstimator function

which provides a standard covariance estimate of price-returns time series data.

The optimization was run as an out-of-sample estimation with a rebalance on

each Friday over the previous four-years price-returns data.

Tangency

That tangency portfolio is estimated by using the fPortfolio packages

tangencyPortfolio function. This function uses the standard covEstimator to

estimate the portfolio covariance matrix. Additionally, it estimates expected

return as the average historical return of each stock. This is run as an out-of-


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sample optimization every Friday over the previous four years data identically to

the Minimum Variance portfolio.

NoOptimization

The No Optimization strategy is a buy-and-hold strategy. After the initial 1/N

allocation of the randomly chosen stocks, the portfolio is never rebalanced.

Conclusions

The purpose of this study is to understand the effect portfolio optimization and

rebalancing techniques have on the performance of an equity portfolio. After

running an out-of-sample test across all four strategies a number of interesting

conclusions follow.

NaveRebalancingIsSuperiortoBuy-HoldinGeneratingReturns

The Buy-Hold strategy consistently underperforms nave rebalancing in the form

of 1/N for improving average return and Sharpe Ratio. This is a result of 1/N

rebalancing generating returns from regular buy-low-sell-high from rebalancing

against volatility.

Despite the additional performance, 1/N rebalancing increases portfolio volatility

over a buy-hold strategy. It is unclear what causes this additional volatility and

this warrants further research.

MinimumRiskPortfoliosareMinimumRisk

Across all portfolio sizes, time periods, and strategies, the minimum variance

portfolio was the best strategy to reduce portfolio volatility. It is clear that the


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covariance estimates are accurate enough to deliver a portfolio strategy that is

superior to all other strategies in minimizing the volatility of the affected portfolio.

TangencyPortfoliosChaseVolatility

The seminal paper 1/N (DeMiguel, Garlappi, Uppal 2006) research the effect

estimation error in expected returns has on an accurate efficient portfolio

optimization. This paper suggests that the covariance estimates are fairly robust

as the minimum variance portfolio is superior to other strategies.

However, the tangency portfolio augments the minimum variance portfolio

estimate by introducing expected returns to the optimization. The result is widely

varying outcomes of portfolios rebalanced with a tangency portfolio. Even

though all portfolios were restricted to long-only, the distribution of returns varied

widely. The volatility of the tangency portfolio was on average twice that of the

minimum variance portfolio. The average gains were greater than the other

optimization strategies however the standard deviation or returns was often three

times greater than any other strategy.

It appears the tangency optimization increases volatility rather than optimizes it

causing wild swings in performance and not significantly improving expected

performance.

MinimimumRiskPortfolioTakesBestAdvantageofDiversification

In Appendix D we study the increasing diversification benefit of adding more

stocks to a portfolio for each strategy. It is interesting to note that adding more

stocks to the tangency portfolio strategy increases the overall portfolio volatility.


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This is presumably due to introducing more estimation error and having more

opportunities to chase volatility.

However, it is very clear that the minimum risk portfolio significantly reduces

portfolio volatility and maximizes the additional diversification value of each new

asset. The diversification value of the N+1 asset is asymptotically decreasing but

the first 6 8 stocks provide a strong volatility dampener for a minimum risk

portfolio.


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References

Markowitz, H. M., 1952, Mean-Variance Analysis in Portfolio Choice and Capital

Markets, Journal of Finance, 7, 7791.

D. Wuertz, Y. Chalabi, W. Chen and A. Ellis (2010), Portfolio Optimization with

R/Rmetrics

DeMiguel, Victor, Garlappi, Lorenzo and Uppal, Raman, 1/N (June 22, 2006).

EFA 2006 Zurich Meetings. Available at SSRN: http://ssrn.com/abstract=911512

Carl, Peter and Peterson, Brian (2010), PerformanceAnalytics: Econometric tools

for performance and risk analysis


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AppendixAOut-of-SampleTestResults

August1st

2005August1st

2009

Rebalance

Method

Average

Return

Standard

Deviation

ofReturns

Average

Volatility

Standard

Deviation

of

Volatility

Average

Sharpe

Ratio

Standard

Deviation

of Sharpe

Ratio

1/N -0.95% 10.58% 34.91% 27.71% 0.00 0.14

MinVariance -1.89% 10.61% 29.33% 59.07% -0.03 0.19

NoOptimization -2.51% 4.07% 29.30% 3.91% -0.08 0.13

Tangency -0.39% 22.48% 62.29% 18.53% 0.01 0.34

GrandTotal -1.44% 13.64% 38.96% 36.52% -0.02 0.22

January1st

2006January1st

2010

Rebalance

Method

Average

Return

Standard

Deviation of

Returns

Average

Volatility

Standard

Deviation

of

Volatility

Average

Sharpe

Ratio

Standard

Deviation

of Sharpe

Ratio

1/N 2.49% 4.28% 33.09% 4.92% 0.08 0.13

MinVariance 0.73% 5.05% 24.01% 5.65% 0.05 0.20

NoOptimization -0.70% 4.02% 29.66% 4.58% -0.02 0.13

Tangency 4.48% 22.48% 61.89% 18.95% 0.08 0.32

GrandTotal 1.75% 12.00% 37.16% 17.97% 0.05 0.21

January1st

2007January1st

20011

RebalanceMethod

AverageReturn

Standard

Deviation ofReturns

AverageVolatility

Standard

Deviation

ofVolatility

Average

SharpeRatio

Standard

Deviation

of SharpeRatio

1/N 4.64% 3.78% 34.33% 6.81% 0.14 0.11

MinVariance 1.14% 4.12% 24.65% 6.02% 0.04 0.15

NoOptimization 1.47% 3.92% 30.54% 4.49% 0.05 0.12

Tangency 8.97% 22.52% 60.65% 20.10% 0.16 0.37

GrandTotal 4.05% 12.14% 37.54% 17.78% 0.10 0.22


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August1st

2007August1st

2011

Rebalance

Method

Average

Return

StandardDeviation of

Returns

Average

Volatility

Standard

Deviationof

Volatility

AverageSharpe

Ratio

Standard

Deviationof Sharpe

Ratio

1/N -2.19% 13.23% 36.08% 6.97% -0.04 0.34

MinVariance -1.99% 11.92% 26.12% 9.65% -0.02 0.31

NoOptimization -0.75% 4.96% 32.07% 6.05% -0.01 0.14

Tangency 3.89% 27.03% 63.87% 19.98% 0.07 0.36

GrandTotal -0.26% 16.50% 39.54% 18.81% 0.00 0.30

AcrossAllTimePeriods

Rebalance

Method

Average

Return

Standard

Deviation of

Returns

Average

Volatility

Standard

Deviation

of

Volatility

Average

Sharpe

Ratio

Standard

Deviation

of Sharpe

Ratio

1/N 1.00% 9.31% 34.60% 14.88% 0.05 0.21

MinVariance -0.50% 8.71% 26.03% 30.18% 0.01 0.22

NoOptimization -0.62% 4.48% 30.39% 4.92% -0.02 0.14

Tangency 4.24% 23.86% 62.18% 19.37% 0.08 0.35

GrandTotal 1.03% 13.84% 38.30% 24.12% 0.03 0.25


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AppendixBDistributionofPortfolioReturnsByStrategy

NoOptimization

1/N


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MinimumVariance

Tangency


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AppendixCDistributionofPortfolioVolatilitybyStrategy

NoOptimization

1/N


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MinimumVariance

Tangency


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AppendixDPortfolioVolatilityandPortfolioSize

StrategyVolatilitybyPortfolioSize

AverageVolatilitybyOptimization

PortfolioSize 1/N minriskPortfolio NoOptimization tangencyPortfolio

2 47.50% 50.09% 37.07% 52.19%

4 35.76% 28.55% 32.13% 55.79%

6 34.96% 28.50% 31.23% 61.09%

8 33.73% 25.16% 30.14% 60.95%

10 33.81% 24.18% 30.06% 56.86%

12 32.76% 23.14% 29.38% 60.32%

14 32.48% 22.29% 28.75% 67.95%

16 33.00% 21.81% 29.31% 66.39%

18 32.90% 22.14% 29.43% 67.61%

20 32.40% 21.34% 28.90% 61.48%

50 31.34% 19.12% 27.89% 73.33%

34.60% 26.03% 30.39% 62.18%


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PortfolioVolatilityasFunctionofPortfolioSize

MSC Thesis - Henry Ward

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