Maxwell will present the methodologies and results behind the algorithm that has been developed by 1QBit, named Quantum Hierarchical Risk Parity, or QHRP. This is an extension of the work done by Marcos Lopez de Prado on Hierarchical Risk Parity in his paper "Building Diversified Portfolios that Outperform Out-of-Sample." QHRP tackles the problem of minimizing the risk of a portfolio of assets using a quantum-inspired approach. Although the ideas surrounding this go back to Markowitz’s mean-variance portfolio optimization of 1952’s Portfolio Selection, we have applied recent quantum-ready machine learning tools to the problem to demonstrate strong performance in terms of a variety of risk measures and lower susceptibility to inaccuracies in the input data. The quantum-ready approach to portfolio optimization is based on an optimization problem that can be solved using a quantum annealer. The algorithm utilizes a hierarchical clustering tree that is based on the covariance matrix of the asset returns. The results of real market data used to benchmark this approach against other common portfolio optimization methods will be shared in this presentation. View the White Paper: https://bit.ly/2k5xTxW.

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Quantum Hierarchical Risk Parity - A Quantum-
Inspired Approach to Portfolio Risk Minimization
Maxwell Rounds

2. Quantum Problems
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3. What Sorts of Problems Can We Solve?
• What is the fastest route from
one place to another?
• How many boxes can I fit in a
van?
• How can I arrange the paintings
in an exhibit to best engage
viewers?
Think Combinatorics
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4. What Does Solving These Problems Look Like?
Finding the Best Route
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5. Why Are They Difficult?
• We know that for a jigsaw puzzle, there is one correct
configuration of the pieces
• Fix a common size, say 500 pieces
• Say you want to program a computer to solve the puzzle
• How many different arrangements would you have to test?
Solving a Jigsaw Puzzle
5
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469658817504506926365338175055478128640000000000000000000000000000000000000000000
000000000000000000000000000000000000000000000000000000000000000000000000000000000
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6. What is the Mathematical Framework?
Binary Quadratic Programming
6
1QBit’s software allow users to:
• Frame interesting problems as binary quadratic programs
• Solve interesting problems framed as binary quadratic programs
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7. Quantum Software
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8. 1. Quantum hardware requires software that
is designed for its architecture.
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9. 9
2. Quantum software insulates users from the
complexities of quantum computers.
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10. 10
3. Quantum software allows developers to use
familiar programming languages to build
applications for quantum computers.
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11. Quantum-Ready Hierarchical Risk Parity
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12. QHRP: Introduction
• Quantum-Ready Hierarchical Risk Parity (QHRP) is a variance-minimizing algorithm that
utilizes a type of clustering from unsupervised machine learning to increase the robustness
of the out-of-sample results
• QHRP uses a quantum solver to form a tree-like hierarchy of stocks from the investable
pool
• From this hierarchy, the algorithm builds a portfolio using a minimum variance calculation
at each level
• By traversing up the tree from the lowest branches, we obtain a portfolio composed of all
of the assets in the investable pool while utilizing the data’s structure to lower risk
• Based in part on collaborative work with Marcos López de Prado, Senior Managing Director
at Guggenheim Partners and Research Fellow at Lawrence Berkeley National Laboratory,
and his paper Building Diversified Portfolios that Outperform Out-of-Sample
A Novel Approach to Minimum Variance Portfolios
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13. Classical Minimum Variance
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14. QHRP: Classic Minimum Variance Portfolio Construction
• Choose portfolio weights that minimize
the risk of the portfolio as measured by
standard deviation
• Make sure to spend all of the money
• Relies only on the covariance matrix, a
combination of asset volatiles and inter-
asset correlations
• Liked due to lack of reliance on difficult-
to-estimate expected returns
• Grown in popularity amongst many buy-
and-hold asset managers
Minimize Risk by Asking for the Least Risky Portfolio
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15. QHRP: Pitfalls of Classic Minimum Variance Portfolio Construction
Seemingly Risky Low-Risk Portfolios
15
• The algorithm input (the covariance matrix)
is essentially a combination of stock-specific
risk and stock-pair correlations
• It is these correlations that offer the
possibility of diversification
• As the number of stocks grow, however, a
classic minimum variance algorithm may
struggle to differentiate signal from noise in
these inputs
• This can lead to funny-looking portfolios that
are technically the minimum variance
portfolios in-sample but which may be
suboptimal when released into the wild
0%
10%
20%
30%
40%
AAPL
AXP
BA
CAT
CSCO
CVX
DD
DIS
GE
GS
HD
IBM
INTC
JNJ
JPM
KO
MCD
MMM
MRK
MSFT
NKE
PFE
PG
TRV
UNH
UTX
V
VZ
WMT
XOM
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16. Estimation Error from Correlations
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17. QHRP: 3 Asset Portfolio
How Many Correlations?
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18. QHRP: 3 Asset Portfolio
3 Correlations
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19. QHRP: 30 Asset Portfolio
How Many Correlations?
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20. QHRP: 30 Asset Portfolio
435 Correlations
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21. QHRP: General Asset Portfolio
• 3 Assets: 3 Correlations
• 30 Assets (DJIA): 435 Correlations
• 500 Assets (SPX): 124,750 Correlations
• n Assets: (n2
- n) / 2 Correlations
The Number of Correlations Increases Quickly
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22. QHRP: Problem to Tackle
• The number of correlations to be
estimated grows quickly with the portfolio
size
• Each correlation has its own estimation
error
• Each extra correlation does not necessarily
add extra information (KO-GE vs. PEP-GE)
• At large portfolio sizes, an optimizer can
struggle to separate signal from noise
• This can lead to results that do not perform
as well expected
Dimension & Noise
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23. QHRP Algorithm
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24. QHRP: Clustering
• QHRP defines the ‘distance’ between two
stocks based on their correlation
• The more correlated they are, the ‘closer’
they are together
• QHRP recursively divides a pool of stocks
into two, each time choosing the groups so
that the members of each group are as
close together as possible
• What emerges is an often-intuitive
perspective on how the stocks behave
Group Stocks Together
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25. QHRP: Portfolio Weights
• With the hierarchy tree, QHRP builds a
portfolio allocation
• Starting from the bottom, QHRP solves a
restricted version of the minimum variance
problem at each level, building upwards
• The result is a set of portfolio weights that
mimic a traditional minimum variance
portfolio, but with notable differences
Assigning Weight to Stocks
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26. QHRP: Portfolio Weight Comparison
• While the classical algorithm does
minimize risk, sometimes it misses
the point
• In this example, it invests over 50%
of the portfolio in just McDonalds,
Coca-Cola, and Johnson & Johnson
• The QHRP solution is more
intuitive - it blends the weight
across all 30 assets, emphasizing
some (Walmart, Procter & Gamble)
more than others (Chevron,
Microsoft)
A More Intuitive Result
26
0%
5%
10%
15%
20%
25%
30%
Classical vs. Hierarchcial Minimum Variance Portfolios
Hierarchical
Classical
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27. QHRP: Quantum Solver Benefits
• Quantum solvers are essential to the clustering portion
of QHRP
• The algorithm recursively divides each cluster into two
until each cluster has only one stock in it
• It chooses the division by finding the separation that
creates two groups whose stocks are the ‘closest
together’ internally
• This is a variant of the Max-Cut Problem, an NP-hard
problem
• For the DJIA alone, there are 1,073,741,822 ways to
divide it into two groups (and that’s just the first split)
• Quantum solvers are able to tackle this sort of
complexity better than classical computers
Where’s the Quantum?
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28. Results
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29. QHRP: Results
1-Year Realized Volatilities, Diversified Futures
29
• QHRP shows lower realized
volatility than both a minimum
variance (MV) and a risk parity
(RP) strategy on a diversified
futures asset pool.
• The results for both QHRP and
RP are also more stable when
bootstrapped than MV.
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30. QHRP: Results
Maximum Drawdown, Diversified Futures
30
QHRP shows a lower maximum
drawdown than both a
minimum variance (MV) and a
risk parity (RP) strategy on a
diversified futures asset pool.
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31. QHRP: Results
Performance, Diversified Futures
31
QHRP shows performance on par
with a risk parity (RP) strategy on a
diversified futures asset pool with
smaller drawdowns.
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32. QHRP: Results
1-Year Realized Volatilities, DJIA
32
• QHRP seems to require a diverse
correlation structure to be most
effective.
• When tested on the Dow Jones
Industrial Average, an index with
high correlations between
components, the realized volatilities
are the same or higher than a
minimum variance algorithm.
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33. 33
Maxwell Rounds
maxwell.rounds@1qbit.com
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