1. Using Smart Beta to drive returns
Domenico Mignacca
Head of Risk Management- Eurizon Capital SGR
Novara 22 Settembre 2016

2. Agenda Introduction to Smart Beta Strategies
Weighting schemes in smart beta
The Factor Risk Parity Approach
Using Factor Risk Parity for Strategic Asset Allocation

3. Introduction to Smart Beta Strategies
“AUM in Smart Beta and actively managed ETF products is projected to rise from $265 billion in 2014 to $1.1 trillion by the end of more than a 4x increase - making this the fastest growing product set in the asset management industry” (
"While the selection of a smart beta product can require almost as much due diligence as selecting an active manager, it is important to keep in mind that the underlying smart beta index is transparent and rules-based; there is no active decision as to what securities are included or excluded within the index" “The most common reasons for using smart beta products were to protect portfolios in down markets, control volatility and increase alpha”

4. Introduction to Smart Beta Strategies
“Smart beta refers to a set of rules-based investment strategies which passively follow alternative index construction rules and do not rely on traditional cap-weighted indices.”
Cap-weighted Index
Poorly diversified  High Concentration to a small number of Larg-Cap Stocks
Poorly rewarded risk factors large and growth stocks
Smart Beta
Alternative portfolio construction schemes  desire to capture particular investment factors or market inefficiencies in a rules-based and transparent way, as well as to try to obtain risk-adjusted returns above the cap-weighted indices

5. Introduction to Smart Beta Strategies
10 years performance of the SPY ETF (cap Weighted) and the RSP ETF (equally weighted) on S&P500

6. Introduction to Smart Beta Strategies
Passive Management (ETF)
overweight overvalued securities
delivering sub-optimal returns
Active Management
not transparent  come with high fees and tends to underperform the benchmark over long time periods
retain the benefits of traditional cap-weighed approaches (diversification, broad market exposure, transparency and liquidity), but at the same time have the potential to achieve results which are superior to the returns of cap-weighted benchmarks at a lower cost than active management
Smart Beta

7. Introduction to Smart Beta Strategies
Outcome Oriented
Risk Factor Oriented
Aimed at increasing diversification
Reducing risk
Increasing Sharpe ratio
Equally-weighted
Minimum variance
Maximum decorrelation
provide exposure to factor risk premium such as:
Value
Size
Low volatility
Momentum
Quality
Smart beta can be implemented for:
Equity portfolios (absolute or vs cap-weighted benchmarks);
Thematic equity portfolios (value, growth, dividend enhanced etc.);
Indices (asset allocation portfolios);
Multimanager portfolios.

8. Weighting Schemes in Smart Beta
Outcome Oriented
Risk Factor Oriented
Equally-weighted
1/N
Minimum variance
𝜔 𝑀𝑉 = 𝛴 −1 𝟏 𝑁 𝟏 ′ 𝑁 𝛴 −1 𝟏 𝑁
Inverse volatility
𝑤 𝑖 = 1 𝜎 𝑖 𝑗=1 𝑁 1 𝜎 𝑗
Use factor exposure to define the investible universe

9. Weighting Schemes in Smart Beta
As it is clear from above, the rules for portfolio construction are transparent and easy to implement.
Outcome Oriented
Equally-weighted
(optimal when uncorrelated factor have return proportional to volatility, securities are uncorrelated and all with the same volatility)
Inverse volatility
(optimal when uncorrelated factor have return proportional to volatility, and securities are uncorrelated)
Factor Risk Parity (see dedicated paragraph)
(optimal when uncorrelated factor have return proportional to volatility)

10. Weighting Schemes in Smart Beta
Risk Factor Oriented
Weighting Schemes
first quartile as factor exposure and then use, for example, equally weighted portfolio or other transparent weighting schemes
We will introduce the Factor Risk Parity Approach and then show how it is possible to us it in both Outcome Oriented and Risk Factor Oriented schemes

11. Weighting Schemes in Smart Beta
Amenc-Goltz-Lodh(2016), «Smart Beta Is not Monkey Business»

12. The Factor Risk Parity
Standard solution for portfolio’s construction: Minimum Variance Portfolio (MV) 𝜔 𝑀𝑉 = 𝛴 −1 𝟏 𝑁 𝟏 ′ 𝑁 𝛴 −1 𝟏 𝑁 Maximum Sharpe Ratio Portfolio (MSR) 𝜔 𝑀𝑆𝑅 = 𝛴 −1 (μ−𝑟) 𝟏 ′ 𝑁 𝛴 −1 (μ−𝑟) Source: Palomar, "Financial Engineering: Portfolio Optimization", HKUST. Where: w= optimal weights, r=risk-free rate, μ=vector of expected returns, Σ= (full rank)covariance matrix, 𝟏 𝑁 =unit vector. To recover the above analytical solution we do not impose the usual constraint of non negativity but just that the sum of securities weights are equal to one.

13. The Factor Risk Parity We can write the covariance matrix as:
Ʃ=𝐴 Ʃ 𝐹 𝐴′
where Ʃ 𝐹 ∈ 𝐷 + with 𝐷 + representing the set of diagonal matrices of size 𝑁 with strictly positive entries. Using the PCA technique, A is the matrix containing eigenvectors and Ʃ 𝐹 the diagonal matrix with eigenvalues. Where A is orthonormal, full rank invertible with the “nice” property that 𝐴 ′ 𝐴=𝐴 𝐴 ′ =𝐼 .
We also have that:
𝛴 −1 =𝐴 𝛴 𝐹 −1 𝐴′
N.B. for sake of simplicity, we cansider Ʃ full rank and the PCA decomposition. It is possible to handle the problem also when the eigenvalues are not «significant» and when we are willing to use a different decomposition like the Minimum Torsion Approach.

14. The Factor Risk Parity
The advantage of using uncorrelated factors is that the total variance of the portfolio can be written as the sum of the weighted contributions of each factor’s variance:
𝜎 𝑃 2 = 𝜔 ′ Ʃω= 𝜔 ′ 𝐴 Ʃ 𝐹 𝐴′ω
= 𝜔 ′ 𝐹 Ʃ 𝐹 𝜔 𝐹
= 𝑘=1 𝑁 ( 𝜎 𝐹𝑘 𝜔 𝐹𝑘 ) 2
where 𝜔 𝐹 =𝐴′ω represents a portfolio of factors. We can define the contribution of the 𝑘 𝑡ℎ factor to the total portfolio variance as 𝑝 𝑘 = ( 𝜎 𝐹𝑘 𝜔 𝐹𝑘 ) 2 𝜔 ′ Ʃω . Each 𝑝 𝑘 is positive and their sum is equal to one.

15. The Factor Risk Parity
To build a Factor Risk Parity portfolio ⇒ decompose the total variance of the portfolio as the sum of the contributions of each risk factor. The factor weights obtained with FRP portfolios are defined in order to equalize the relative contribution of each factor to the total portfolio variance: 𝑝 𝑘 = 1 𝑁 ⇒ (𝜎 𝐹𝑘 𝜔 𝐹𝑘 ) 2 = 1 𝑁 𝜔 ′ Ʃω for all k=1,..,N. We can then write 𝜔 𝐹𝑘 =±𝛾 1 𝜎 𝐹𝑘 for k=1,..,N where 𝛾 is a constant (can be: 𝛾 2 = 𝜎 𝑝 2 𝑁 ). In matrix form, the vector of factors weights is represented by: 𝜔 𝐹 = 𝛾 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑗 where the N-dimension vector 𝟏 𝑁 𝑗 contains the sign for each weight component, 𝟏 𝑁 𝑗 = ±1 ⋮ ±1 . The FRP portfolio is not uniquely defined, as there are 2 𝑁 weight vectors which satisfy the conditions. We then go back to the original constituents’ weights using the identity 𝜔=𝐴 𝜔 𝐹 . Finally, adding the budget constraint 𝟏 𝑁 ′ 𝜔=1 we end up with: 𝜔 𝐹𝑅𝑃 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑗 1 𝑁 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑗 (normalized by 1 𝑁 ′ 𝐴 𝜔 𝐹 𝛾 ). Now we are left with 2 𝑁−1 weight vectors, since both 𝜔 𝐹 and −𝜔 𝐹 lead to the same solution.

16. The Factor Risk Parity
Factor Risk Parity Minimum Volatility (FRP-MV): Among the 2 𝑁−1 FRP portfolio choose the one with the lowest volatility. It is possible to show: 𝜔 𝐹𝑅𝑃−𝑀𝑉 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑀𝑉 𝟏 𝑁 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑀𝑉 where the vector 𝟏 N MV has the same sign of its corresponding entry in A ′ 𝟏 N : 𝟏 𝑁 𝑀𝑉 =𝑠𝑖𝑔𝑛 𝐴 ′ 𝟏 𝑁 . Then 𝜔 𝐹𝑅𝑃−𝑀𝑉 has the lowest volatility among all FRP portfolios, equal to: 𝜎 𝜔 𝐹𝑅𝑃−𝑀𝑉 = 𝑁 𝟏 𝑁 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑀𝑉 .

17. The Factor Risk Parity
Factor Risk Parity Maximum Sharpe Ratio (FRP-MSR): Among the 2 𝑁−1 FRP portfolio choose the one with the highest Sharpe Ratio. It is possible to show: 𝜔 𝐹𝑅𝑃−𝑀𝑆𝑅 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑀𝑆𝑅 𝟏 𝑁 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝑁 𝑀𝑆𝑅 where the vector 𝟏 N M𝑆𝑅 has the same sign of its corresponding entry in A ′ 𝟏 N : 𝟏 𝑁 𝑀𝑆𝑅 =𝑠𝑖𝑔𝑛 𝐴 ′ μ . Then 𝜔 𝐹𝑅𝑃−𝑀𝑉 has the lowest volatility among all FRP portfolios, equal to: 𝜆 𝜔 𝐹𝑅𝑃−𝑀𝑆𝑅 = 𝑘=1 𝑁 | 𝜆 𝐹𝑘 | 𝑁
Sharpe ratio of k-th factor

18. The Factor Risk Parity
Let us start from a traditional multifactor model:

19. The Factor Risk Parity
Let us consider first the FRP-MV portfolio. We can use the following algorithm: min 𝜔 𝑁 𝜔 𝑁 ′ 𝛺 𝜔 𝑁 𝑠.𝑡. 𝑖=1 𝑁 𝜔 𝑖 =1, where 𝜔 𝑖 ≥0, 𝑖=1, . . ., 𝑁, 𝐵′ 𝜔 𝑁 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 1 𝐹 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 , If the solution is unfeasible, use only the third constraint in the above minimization and then find the minimum TEV long only solution w.r.t. the above unconstrained solution .

20. The Factor Risk Parity
From: Mignacca Domenico
From: Mignacca Domenico

21. The Factor Risk Parity
From: Mignacca Domenico
From: Mignacca Domenico

22. The Factor Risk Parity
The FRP-MV can be classified as a Smart Beta «Outcome Oriented». In what follows we are going to use the FRP-MSR solution for both «Outcome Oriented» and «Risk Factor Oriented» solution.
We will also discover how the technique can be used to remove the Black and Litterman constrain to start from a metabenchmark in order to recover expected return.
To recover expected returns we can use a reasonable prior (risk premia on orthogonal factors proportional to their risk) and the Black and Litterman framework for the posterior.
Let us consider the following steps to recover expected returns

23. The Factor Risk Parity
How to recover expected returns for FRP-MSR using views on Factors Step 1: start from a simple hypothesis: proportional risk premia on orthogonal factors: 𝜇 𝐹 𝑀𝑆𝑅 =𝜅 Ʃ 𝐹 1/2 𝟏 𝐹 𝑤ℎ𝑒𝑟𝑒 𝜅= 𝑆𝑅 𝐹 Step 2: Recover, from step 1, prior returns on standard factors:
𝜇=𝜅𝐴 Ʃ 𝐹 1/2 𝟏 𝐹

24. The Factor Risk Parity
Step 3: insert views on standard Factors (Black&Litterman): 𝜇 =𝜇+Ʃ𝐻 𝐻 ′ Ʃ𝐻 −1 (𝑞−𝐻𝜇) Step 4: Recover posterior returns on orthogonal factor and find the FRP-MSR portfolio: 𝜇 𝐹 =𝐴′ 𝜇 𝟏 𝐹 𝑀𝑆𝑅 =𝑠𝑖𝑔𝑛( 𝜇 𝐹 ) and 𝜔 𝐹 =𝛾 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅
Matrix with linear combination capturing views
views

25. The Factor Risk Parity
Step 5: recover weight on factors and expected returns on securities: 𝜔 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 1 𝐹 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 and 𝜇 𝑁 =𝐵 𝜇 . Step 6: recover securities weigths via maximization: max 𝜔 𝑁 𝜔 𝑁 ′ 𝜇 𝑁 𝑠.𝑡. 𝑖=1 𝑁 𝜔 𝑖 =1, where 𝜔 𝑖 ≥0, 𝑖=1, . . ., 𝑁 and 𝐵′ 𝜔 𝑁 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 1 𝐹 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 .

26. The Factor Risk Parity
If we use just the «prior»: 𝜇 𝐹 𝑀𝑆𝑅 =𝜅 Ʃ 𝐹 1/2 𝟏 𝐹  𝜇=𝜅𝐴 Ʃ 𝐹 1/2 𝟏 𝐹  𝜇 𝑁 =𝐵 𝜇 We can obtain the FRP-MSR portfolio (outcome oriented). We can also use the step 1-6 procedure to obtain «risk factor oriented» portfolio in a more coherent way then simply equally weight or similar weighting schemes. Let us cosider the following example where we want to build a «momentum» portfolio is a particular way: tilting the portfolio using a positive view on the spread of performance of the two MSCI sector with the highest performance difference in the last 3 months.

27. The Factor Risk Parity
We can assign views on each Risk Factor embedded in the «XXX» Factor Model
In our application we consider :
the Equity risk factors on the US Regional Model;
Main Style Factors (e.g. Growth, Size, Momentum) ;
A grouping scheme on Industry Factors allowing to assign the view at the GICS Sector aggregate level and then to allocate the tilt to the single Factor;
Absolute and Relative Views expressed as percentages of their volatility.

28. The Factor Risk Parity Portfolio Rebalance date : 01.Jan.2016
Benchmark Rebalance date: 01.Dec.2015 &
Model used: FRP-MSR with Exact Torsion

29. The Factor Risk Parity

30. The Factor Risk Parity

31. The Factor Risk Parity
We saw how to implement easily a Momentum Smart Beta portfolio. It is clear that we can adopt the same approach to tilt the portfolio in any way. For instance on traditional style factors (e.g. : Growth, Size, Momentum) or to implement Investment views even on factors/asset classes to which securities are not exposed.
In what follows, we are going to show why this approach can be useful also for asset allocation purposes and why it can be a «nice» alternative to the Black and Litterman traditional approach.

32. Using Factor Risk Parity for Strategic Asset Allocation
It is well known that in the traditional Black and Litterman Approach we have 2 degrees of freedom in order to obtain expected returns (the prior). The two degrees of freedom are:
The Sharpe Ratio Level,
The «optimal» portfolio (which can be the cap weighted portfolio or any other portfolio you may be interested in using to recover the prior)
In fact:
𝐸(𝑟)=𝑆𝑅 𝛴𝑏 𝜎
Where b is the vector containing the «optimal» portfolio’s weights. So, if we have no additional view, and we have no volatility contraints, the Maximum Sharpe Ratio Portfolio is «b». To move from it we need additional view to recover a «posterior» for returns.

33. Using Factor Risk Parity for Strategic Asset Allocation
Remember the FRP-MSR portfolio with embedded view of returns proportional to «uncorrelated factor’s» volatilities:
𝜇 𝐹 𝑀𝑆𝑅 =𝜅 Ʃ 𝐹 1/2 𝟏 𝐹  𝜇=𝜅𝐴 Ʃ 𝐹 1/2 𝟏 𝐹 𝑤ℎ𝑒𝑟𝑒 𝜅= 𝑆𝑅 𝐹
Then the FRP-MSR portfolio is:
𝜔 𝐹𝑅𝑃−𝑀𝑆𝑅 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 𝟏 𝐹 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝑀𝑆𝑅 = 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹 𝟏 𝐹 ′ 𝐴 Ʃ 𝐹 −1/2 𝟏 𝐹
The sharpe ratio of the FRP-MSR portfolio is SR.

34. Using Factor Risk Parity for Strategic Asset Allocation
The global MSR portfolio is:
𝜔 𝑀𝑆𝑅 = 𝛴 −1 𝜇 𝟏 ′ 𝑁 𝛴 −1 𝜇 ;
Given the hypothesis about expected returns proportional to orthogonal factors:
𝜇=𝜅𝐴 Ʃ 𝐹 1/2 𝟏 𝐹 ;
And using the fact that:
𝛴 −1 =𝐴 𝛴 𝐹 −1 𝐴 ′
We can show that 𝜔 𝑀𝑆𝑅 has sharpe ratio equal to SR and this implies:
𝜔 𝑀𝑆𝑅 = 𝜔 𝐹𝑅𝑃−𝑀𝑆𝑅

35. Using Factor Risk Parity for Strategic Asset Allocation
We can conclude that, in case expected returns are proportional to the risk of orthogonal factors, the factor risk parity MSR portfolio collapse to the global MSR portfolio;
With this in mind we have a framework to built strategic asset allocation without starting from a «equilibrium portfolio» as in the Black & Litterman Approach;
It is a fact that, the solution can be not feasible, as we can have positive constraint on the asset classes. To solve this point we use the expected return and maximize the MSR function or similar functions;
It is important to say that when we use securities to replicate optimal portfolio exposure the solution is not exactly on the MSR point as we have at the denominator also the idiosincratic risk of securities;
Factor Risk Parity can be used for various purposes: we illustrated Smart Beta equity portfolios as well as for Asset Allocation. It can be also used for Corporate Bonds Portfolios using CDS spreads time series.