The document describes a portfolio management strategy using minimum variance and equal risk contribution approaches for European stocks. It summarizes the implementation of the strategy from January to March 2014, which outperformed benchmarks while maintaining lower volatility. Weekly rebalancing resulted in over 200% portfolio turnover as the optimal weights were highly sensitive to small changes in inputs. Overall the strategy showed excess returns but also had weaknesses like absolute losses and complex calculations.

1. Group : Luis MAGNET,Thu-Phuong DO
PROJECT PORTFOLIO MANAGEMENT
EUROPE STOCKS MINIMUM VARIANCE
28 March 2014

2.  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 andVBA for performance analysis
1/ INTRODUCTION

3.  16 Jan 2014 – 18 Feb 2014: MinimumVariance 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 MinimumVariance
 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

4.  According to Markowitz (1952), the optimal weight of the Global Minimum Variance is
determined analytically by the following formula:
𝑤 𝑀𝑉 =
Ω
−1
𝑒
𝑒′Ω
−1
𝑒
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 𝑟 𝑚 + 𝜀𝑖
Where:
 𝑟𝑖: return of i-th stock
 𝑟 𝑚: market return (i.e. return of the market capitalization weighted portfolio)
 𝜀𝑖 represents the idiosyncratic risk, 𝜀𝑖 ~ Ɲ (0, 𝜎𝑖
2)
 The variance-covariance matrix is estimated as follows:
Ω = 𝛽𝛽′𝜎 𝑚
2 + Diag(𝜎2)
2/ STRATEGY (2/4)
2.2. Minimum Variance optimization problem

5.  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 =
1
𝜎 𝑚
2 +
𝛽𝑖
2
𝜎𝑖
2
𝛽𝑖
𝜎𝑖
2
 In contrast, there is no close formula for LO beta. Instead, it is estimated via a
recursive process:
𝛽𝐿 =
1
𝜎 𝑚
2 +
𝛽𝑖
2
𝜎𝑖
2𝛽𝑖 <𝛽 𝐿
𝛽𝑖
𝜎𝑖
2𝛽𝑖 <𝛽 𝐿
2/ STRATEGY (3/4)
2.2. Minimum Variance optimization problem

6.  𝑥∗ = min 𝑥 𝑇Σ𝑥
𝑅𝐶𝑖
∗
=
1
𝑛
∀𝑖 ∈ 1; 𝑛
𝟏 𝑇
𝑥 = 1
𝑥𝑖 ≥ 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
∗
=
𝑅𝐶𝑖
𝜎(𝑥)
2/ STRATEGY (4/4)
2.3. Equal Risk Contribution (ERC) optimization problem

7. 3/ ALLOCATION & REBALANCING (1/4)
3.1. Initial portfolio allocation
Graph 1: Portfolio exposure by sector

8. 3/ ALLOCATION & REBALANCING (2/4)
3.1. Initial portfolio allocation
Graph 2: Portfolio exposure by currency

9.  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

10. 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
 RoyalVopak NV (Energy)
 BureauVeritas (Comm &
Prof. services)
 RyanAir (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.

11.  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)

12. 800,00
850,00
900,00
950,00
1000,00
1050,00
Performance comparison
.STOXX PF EW MV ERC*
4/ PERFORMANCE ANALYSIS (2/2)

13.  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

14.  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