Group : Luis MAGNET, Thu-Phuong DO

PROJECT PORTFOLIO MANAGEMENT EUROPE STOCKS MINIMUM VARIANCE

28 March 2014

The recent subprime crisis in US (2008), followed by the European sovereign debt crisis (2009), has triggered the need for an alternative technique of portfolio management, which integrates also the risk management in order to protect capital and avoid potential crash in value.

The smart beta, including the risk-based and fundamental-based approach in portfolio management has been successful so far.

Accordingly, we propose an ameliorated Minimum Variance strategy for our portfolio of Europe stocks to achieve the superior return to benchmark but with low volatility.

Tools utilized: Reuters Eikon for financial data R software for optimization program Excel and VBA for performance analysis

1/ INTRODUCTION

16 Jan 2014 – 18 Feb 2014: Minimum Variance Portfolio The eligible investment list is shortlisted from the universe of stocks

from Europe Stoxx 600 by a quantitative filter; Criterion: lowest quintile of beta for each sector (i.e. 20% lowest-beta

stocks) in order to ensure a certain level of diversification

The optimization solution is referred to Roger Clarke et al (2011), Minimum Variance Portfolio Composition, The Journal of Portfolio Management, Volume 37 Number 2 (cf. slide 4 & 5)

From 18 Feb 2014 onward: Strategy “Equal Risk Allocation” in complementary with strategy Minimum Variance We continued to use the previous method to select the eligible list; The approach “Equal Risk Contribution” was then applied to

determine the optimal weight (cf. slide 6).

The objective of the latter strategy is to ameliorate the portfolio’s return, at the same time to maintain the low risk level.

2/ STRATEGY (1/4) 2.1. Introduction

According to Markowitz (1952), the optimal weight of the Global Minimum Variance is determined analytically by the following formula:

𝑤𝑀𝑉 = Ω−1𝑒

𝑒′Ω−1𝑒

Where:

Ω : NxN variance-covariance matrix e : the unit vector whose length is N

We estimated the variance-covariance matrix using the betas drawn from market model developed by Sharpe (1963):

𝑟𝑖 = 𝛼𝑖 + 𝛽𝑖 𝑟𝑚 + 𝜀𝑖

Where:

𝑟𝑖: return of i-th stock

𝑟𝑚: market return (i.e. return of the market capitalization weighted portfolio)

𝜀𝑖 represents the idiosyncratic risk, 𝜀𝑖 ~ Ɲ (0, 𝜎𝑖2)

The variance-covariance matrix is estimated as follows:

Ω = 𝛽𝛽′𝜎𝑚2 + Diag(𝜎2)

2/ STRATEGY (2/4) 2.2. Minimum Variance optimization problem

The objective is to construct a long-only portfolio of lowest variance. We need firstly to define the long-short beta (LS beta) and the long-only beta (LO beta).

LS beta is defined by the following formula:

𝛽𝐿𝑆 =

1𝜎𝑚

2 + 𝛽𝑖 2

𝜎𝑖2

𝛽𝑖 𝜎𝑖2

In contrast, there is no close formula for LO beta. Instead, it is estimated via a recursive process:

𝛽𝐿 =

1𝜎𝑚

2 + 𝛽𝑖 2

𝜎𝑖2𝛽𝑖 <𝛽𝐿

𝛽𝑖 𝜎𝑖2𝛽𝑖 <𝛽𝐿

2/ STRATEGY (3/4) 2.2. Minimum Variance optimization problem

𝑥∗ = min 𝑥𝑇Σ𝑥

𝑅𝐶𝑖∗ =

1

𝑛 ∀𝑖 ∈ 1; 𝑛

𝟏𝑇𝑥 = 1𝑥𝑖 ≥ 0 ∀𝑖 ∈ 1; 𝑛

Where:

𝑥∗: the optimal weight

Σ: variance – covariance matrix

𝑅𝐶𝑖∗: relative risk contribution, which is calculated as follows:

𝑀𝑅𝑖 = Σ𝑥 𝑖 𝑅𝐶𝑖 = 𝑀𝑅𝑖 ∗ 𝑥𝑖

𝑅𝐶𝑖∗ =

𝑅𝐶𝑖𝜎(𝑥)

2/ STRATEGY (4/4) 2.3. Equal Risk Contribution (ERC) optimization problem

3/ ALLOCATION & REBALANCING (1/4) 3.1. Initial portfolio allocation

Graph 1: Portfolio exposure by sector

3/ ALLOCATION & REBALANCING (2/4) 3.1. Initial portfolio allocation

Graph 2: Portfolio exposure by currency

Initially, we reduced the total universe of 600 stocks to the

investment list of 41 stocks, whose weight varies from 0.3%

to 8%.

Although we applied the quantitative filter by sector to attain

the “Best in class” shortlist, our fund had added exposure to

defensive sectors, such as Healthcare, Utilities, Foods &

Beverage (cf. Graph 1).

In addition, our portfolio exposed mostly to EUR, GBP and

CHF (cf. Graph 2). The large position in GBP and CHF

implies the need for exchange rate management.

3/ ALLOCATION & REBALANCING (3/4) 3.1. Initial portfolio allocation

3/ ALLOCATION & REBALANCING (4/4) 3.2. Weekly rebalancing

NEW ENTRANTS THE OUTS

12 stocks

Rangold Resources

(Materials)

Ultra Electronics Holdings

(Capital Goods)

Provident Financial plc

(Diversified finance)

etc …

16 stocks

Royal Vopak NV (Energy)

Bureau Veritas (Comm &

Prof. services)

Ryan Air (Transportation)

Dassault Systèmes (Tech)

etc …

This strategy is supposed to have an important rate of turnover due to the high sensitivity

of result to inputs. After 3 months, we observed the aforementioned change in portfolio’s

composition. 16 stocks were removed and 12 new stocks were added to our portfolio.

The following table and graph showed the comparison of our portfolio’s performance over the

course of 3 months to other strategies, like Equally weighted (EW), pure Minimum Variance (MV)

and Equal Risk Contribution (ERC).

Overall, we outperformed the benchmark (i.e. Europe Stoxx 600) and maintained a lower level of

volatility during the period.

Date Benchmark Portfolio (PF) EW MV ERC

23-Jan-2014 -4.69% -1.26% 1.41% 2.46% 1.12%

30-Jan-2014 -2.93% -0.96% -3.04% -4.21% -3.15%

6-Feb-2014 0.13% -0.06% -0.73% -0.58% -0.16%

13-Feb-2014 1.62% 2.17% 1.45% 0.71% 0.68%

20-Feb-2014 2.54% 0.62% 2.14% 2.55% 2.13%

27-Feb-2014 -3.75% 0.44% 1.09% 0.42% 1.00%

6-Mar-2014 -1.6% -0.44% 1.01% -1.28% -0.88%

13-Mar-2014 -3.47% 2.79% -0.67% 0.03% 0.17%

20-Mar-2014 1.27% 0.03% -3.05% -3.05% -2.21%

28-Mar-2014 0.88% 1.34% 1.51% 2.32% 1.41%

Volatility 2.46% 1.32% 1.8% 2.18% 1.58%

Average return -1.00% -0.09% 0.11% -0.06% 0.01%

Tracking error - 1.90% 3.03% 3.26% 2.86%

Correlation - 70.18% 11.68% 11.56% 15.55%

4/ PERFORMANCE ANALYSIS (1/2)

800,00

850,00

900,00

950,00

1000,00

1050,00

Performance comparison

.STOXX PF EW MV ERC*

4/ PERFORMANCE ANALYSIS (2/2)

Strengths:

Excess return to benchmark

Lower level of volatility

More stable variance-covariance matrix than historical ones

Weaknesses:

Loss in terms of absolute value

Complicated implementation and calculation

Important rate of turnover (>200%)

5/ CONCLUSION

Clarke R., De Silva H., Thorley S. (2011), Minimum

Variance Portfolio Composition, The Journal of Portfolio

Management, Volume 37 Number 2

Roncalli T. (2013), Introduction to Risk Parity and

Budgeting, Chapman and Hall/CRC Financial Mathematics

Series

6/ BIBLIOGRAPHY