Presentation slide for IJCAI2020@2021/1/14

1. The views expressed here are our own and do not necessarily reflect the views of Nomura Asset management.
Any errors and inadequacies are our own.
the 29th International Joint Conference on Artificial Intelligence (IJCAI)
Jan 14-th, 2021
RM-CVaR: Regularized Multiple-𝛽 CVaR Portfolio
Innovation Lab,
Nomura Asset Management Co., Ltd.
Kei Nakagawa, Shuhei Noma and Masaya Abe

2. 1. Introduction and Motivation
2. Preliminary and Methodology
3. Experimental Results
4. Conclusion
Contents
1

3. Contents
1. Introduction and Motivation
2. Preliminary and Methodology
3. Experimental Results
4. Conclusion
2

4. 3
Introduction
Drawbacks of MVO:
(1) Risk is measured by standard deviation (SD)
(2) Error Maximization
Even small changes in the estimated returns can result in huge changes in
the whole portfolio structure.
Mean-variance optimization (MVO) has been the practical standard.
Controlling the SD leads to low deviation from the expected return
regarding both the downside and the upside.

5. 4
Introduction
12.13%
15.48%
9.23%
0.00411 0.00085 0.00487
MVO portfolio that minimizes risk
with an expected return of 13.5%.
19.5%
59.3%
21.2%
13.34%
+10%
50.3%
35.2%
14.5%
+160%
Expected Return Optimal Weight
An illustration of error maximization of MVO.
Stock A
Stock B
Bond C
Stock A
Stock B
Bond C
Annual risk of stocks is about 20%.
The 1% error is very good as an estimate…
Covariance Matrix
Stock A
Stock B
Bond C
Stock A Stock B Bond C
0.02528 0.02098 0.00411
0.02098 0.05452 0.00085

6. 5
Introduction
There is no research on Error Maximization in CVaR portfolio.
Minimum CVaR portfolio is a promising alternative to MVO.
(1) Risk is measured by conditional value at risk (CVaR)
(2) Error Maximization ?
Controlling the CVaR leads to low deviation from the expected return
regarding only the downside.
Even small changes in the probability levels can result in huge changes in
the whole portfolio structure ?

7. 6
Motivation
RM-CVaR: Regularized Multiple 𝜷-CVaR Portfolio
(3) Perform experiments to evaluate the RM-CVaR
(1) Confirm error maximization in minimum CVaR portfolio
(2) Propose a new CVaR portfolio robust to error maximization.
The purpose of our study is

8. Contents
1. Introduction and Motivation
2. Preliminary and Methodology
3. Experimental Results
4. Conclusion
7

9. 8
Preliminary
𝑟 = 𝑟1, … , 𝑟𝑛
𝑇: The return vector of stock 𝑖 (1 ≤ 𝑖 ≤ 𝑛)
𝑤 = 𝑤1, … , 𝑤 𝑛
𝑇: The portfolio weight vector for stock 𝑖
𝑝 𝑟 : The continuous probability density function of 𝑟
𝐿 𝑤, 𝑟 : The loss function, e.g., 𝐿 𝑤, 𝑟 = −𝑤 𝑇 𝑟
Φ 𝑤, 𝛼 ≔ න
𝐿 𝑤,𝑟 ≤𝛼
𝑝 𝑟 𝑑𝑟
We define the probability that the loss function is less than 𝛼 as
is fixed This is nondecreasing as a function of 𝛼.
Assume Φ 𝑤, 𝛼 is a continuous function with respect to 𝛼.

10. 9
VaR and CVaR
We define VaR and CVaR.
𝑉𝑎𝑅 𝑤|𝛽 ≔ 𝛼 𝛽(𝑤) ≔ min(𝛼: Φ 𝑤, 𝛼 > 𝛽)
𝐶𝑉𝑎𝑅 𝑤|𝛽 ≔ 𝜙 𝑤 𝛽 = 1 − 𝛽 −1 න 𝐿 𝑤, 𝑟 𝑝 𝑟 𝑑𝑟
probability level 𝐿 𝑤, 𝑟 ≥ 𝛼 𝛽(𝑤)

11. 10
Minimum CVaR Portfolio
𝐹 𝑤, 𝛼 𝛽 ≔ 1 − 𝛽 −1 න
𝑅 𝑛
𝐿 𝑤, 𝑟 − 𝛼 + 𝑝 𝑟 𝑑𝑟
𝑎 + ≔ max(𝑎, 0)
Therefore, to calculate CVaR, we also defined an auxiliary function 𝑭 as
𝜙 𝑤 𝛽 = min
𝛼
𝐹 𝑤, 𝛼 𝛽
It is difficult to directly minimize the CVaR because the integration interval
depends on VaR.
The relation between 𝜙 𝑤 𝛽 :CVaR and 𝐹 𝑤, 𝛼 𝛽 is
See [Rockafellar et al., 2000]

12. 11
Minimum CVaR Portfolio
Then, we approximated the 𝐹 𝑤, 𝛼 𝛽 by sampling 𝑟 from 𝑝(𝑟).
𝐹 𝑤, 𝛼 𝛽 ≈ 𝛼 + 𝑄 1 − 𝛽
−1
෍
𝑞=1
𝑄
−𝑤 𝑇 𝑟 𝑞 − 𝛼 +
When 𝑟 1 , … , 𝑟[𝑄] are obtained through sampling, 𝐹 𝑤, 𝛼 𝛽 is
approximated as follows.
min
𝑤∈𝑋
𝜙 𝑤 𝛽 = min
𝛼,𝑤 ∈𝑅×𝑋
𝐹 𝑤, 𝛼 𝛽
Instead of minimizing 𝜙 𝑤 𝛽 :CVaR, we can minimize 𝐹 𝑤, 𝛼 𝛽 .
See [Rockafellar et al., 2000]
𝑋: a constraint that the portfolio must satisfy

13. 12
Minimum CVaR Portfolio
min
𝑤,𝛼,𝑢1,…,𝑢 𝑞
𝛼 + 𝑄 1 − 𝛽
−1
෍
𝑞=1
𝑄
𝑢 𝑞
𝑠. 𝑡. 𝑢 𝑞 ≥ −𝑤 𝑇 𝑟 𝑞 − 𝛼
𝑢 𝑞 ≥ 0
𝑤𝑗 ≥ 0
෍
𝑗=1
𝑛
𝑤𝑗 = 1
Finally, we formulated the portfolio optimization problem
with CVaR as a LP problem which is fast and easy to optimize.
𝐹 𝑤, 𝛼 𝛽 ≈
= min
𝛼,𝑤 ∈𝑅×𝑋
𝐹 𝑤, 𝛼 𝛽
min
𝑤∈𝑋
𝜙 𝑤 𝛽
portfolio constraints
𝛼 + 𝑄 1 − 𝛽
−1
෍
𝑞=1
𝑄
−𝑤 𝑇
𝑟 𝑞 − 𝛼 +

14. 13
min
𝑤,𝐶 ∈𝑋×𝑅
𝐶
𝑠. 𝑡. 𝜙 𝑤 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
𝜙 𝑤 𝛽 𝑘 = min
𝛼 𝑘
𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘
Our Proposed: RM-CVaR Portfolio
𝐶𝛽 𝑘
: The value of CVaR
We construct the portfolio that considers the multiple CVaR.
We also defined an auxiliary function as
We minimized 𝐶, considering that 𝐶𝛽 𝑘
is a main problem of our study.
… Problem 1

15. 14
𝐶0.97
𝐶0.98
𝐶0.99
𝜙 𝑤 0.99
𝜙 𝑤 0.98
𝜙 𝑤 0.97
Minimize max margin
We want to minimize the max margin of multiple CVaR.
Value of CVaR
𝐶𝛽 𝑘
:
Illustration of Problem 1.
Our Proposed: RM-CVaR Portfolio

16. 15
Using min
𝛼 𝑘
𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 , Problem 1 can be written as follows.
min
𝑤,𝐶 ∈𝑋×𝑅
𝐶
𝑠. 𝑡. min
𝛼 𝑘
𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
We denote 𝛼 = 𝛼1, … , 𝛼 𝐾
𝑇.
We consider the following Problem 3.
min
𝑤,𝐶,𝛼 ∈𝑋×𝑅×𝑅 𝐾
𝐶
𝑠. 𝑡. 𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
… Problem 2
… Problem 3
Our Proposed: RM-CVaR Portfolio

17. 16
We proved the following relationship.
min
𝑤,𝐶 ∈𝑋×𝑅
𝐶
𝑠. 𝑡. min
𝛼 𝑘
𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
min
𝑤,𝐶,𝛼 ∈𝑋×𝑅×𝑅 𝐾
𝐶
𝑠. 𝑡. 𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
Lemma 4.1.
Problem 2 and 3 are equivalent.
(See lemma 4.1 of our paper)
… Problem 2
… Problem 3
Our Proposed: RM-CVaR Portfolio

18. 17
min
𝑤,𝐶,𝛼 ∈𝑋×𝑅×𝑅 𝐾
𝐶 + 𝜆 𝑤 − 𝑤−
1
𝑠. 𝑡. 𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
To Control the portfolio turnover, we impose the L1-regularization term.
… Problem 4
𝑤− denotes the portfolio weights before rebalancing.
𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 ≈ 𝛼 𝑘 + 𝑄 1 − 𝛽 𝑘
−1
෍
𝑞=1
𝑄
−𝑤 𝑇 𝑟 𝑞 − 𝛼 𝑘
+
Then, we approximated the𝐹 𝑤, 𝛼 𝑘 𝛽 𝑘 by sampling 𝑟 from 𝑝(𝑟).
Our Proposed: RM-CVaR Portfolio

19. 18
min
𝑤,𝐶,𝛼 ∈𝑋×𝑅×𝑅 𝐾
𝐶 + ෍
𝑖=1
𝑛
𝑢𝑖
𝑠. 𝑡. 𝑢𝑖 ≥ 𝜆𝑖|𝑤𝑖 − 𝑤𝑖
−
|
𝑢𝑖 ≥ 𝜆𝑖|𝑤𝑖 − 𝑤𝑖
−
|
𝑡 𝑞𝑘 ≥ 0
𝑡 𝑞𝑘 ≥ −𝑤 𝑇 𝑟 𝑞 − 𝛼 𝑘
𝛼 𝑘 + 𝑄 1 − 𝛽 𝑘
−1
෍
𝑞=1
𝑄
𝑡 𝑞𝑘 ≤ 𝐶 + 𝐶𝛽 𝑘
Finally, we formulated RM-CVaR portfolio optimization problem
as an LP problem which is fast and easy to optimize.
𝑤𝑗 ≥ 0, ෍
𝑗=1
𝑛
𝑤𝑗 = 1 Usual portfolio constraints
Portfolio turnover constraints
Our Proposed: RM-CVaR Portfolio

20. Contents
1. Introduction and Motivation
2. Preliminary and Methodology
3. Experimental Results
4. Conclusion
19

21. We use well-known benchmarks called Fama and French (FF) datasets.
Dataset
20
We use both datasets as monthly data from January 1989 to
December 2018.
- the FF25 dataset includes 25 portfolios formed on
the basis of size and book-to-market ratio
- the FF48 dataset contains monthly returns of 48 portfolios
representing different industrial sectors.
We use the FF25 dataset and the FF48 dataset.

22. Error Maximization
21
In the first experiment, we examine whether CVaR portfolios
suffer from error maximization.
We set 𝛽𝑖 and 𝛽𝑗 as 0.96,0.95 , 0.97,0.96 , 0.98,0.97 , 0.99,0.98 .
we defined the weight difference of two minimum CVaR portfolios
at time t.
𝐃𝐢𝐟𝐟 =
1
T
෍
𝑡=1
𝑇
𝑤𝑡
𝛽𝑖
− 𝑤𝑡
𝛽 𝑗
1
A large 𝐃𝐢𝐟𝐟 indicates that the two portfolios are different.

23. 22
Error Maximization
Only 1% difference in the 𝛽 level changes the portfolio weight on
average by 48% in FF25 and 39% in FF48.
Table 1: The weight difference of two minimum CVaR portfolios in the all period.
We can observe that the CVaR portfolio weights are highly
sensitive to the 𝜷 levels.
beta {0.96,0.95} {0.97,0.96} {0.98,0.97} {0.99,0.98} Avg
FF25 42.66% 43.72% 48.77% 57.61% 48.19%
FF48 23.95% 36.13% 25.20% 72.28% 39.39%
Avg 33.31% 39.93% 36.99% 64.95% 43.79%

24. 23
Compared to various baselines
We compare the out-of-sample performance of the portfolios bellow.
Portfolio Description
1/N Equally-weighted (1/N) portfolio
MV Minimum-variance portfolio
DRP Doubly regularized minimum variance portfolio [Shen et al., 2014].
EGO Kelly growth optimal portfolio with ensemble learning [Shen et al., 2019]
CVaR Minimum CVaR portfolio with {0.95,0.96,0.97,0.98,0.99}
ACVaR average of minimum CVaR portfolio of different beta levels
RM-CVaR our proposed
We use the data from January 2004 to December 2018 as the out-
of-sample period.
We decide the hyper-parameters to use the data from January
1989 to December 2003.
Each portfolio is updated by sliding one-month-ahead.

25. Performance Measures
・ We use the following measures that are widely used.
AR = ෑ
𝑡=1
𝑇
1 + 𝑅𝑡
12/𝑇
− 1
𝑅𝐼𝑆𝐾 =
12
𝑇 − 1
෍
𝑡=1
𝑇
𝑅𝑡 − 𝜇 𝛼
2
𝑅/𝑅 = 𝐴𝑅/𝑅𝐼𝑆𝐾
Portfolio return
𝜇 𝑅: Average of 𝑅𝑡
𝑅𝑡:
24
MaxDD = min
𝑘∈[1,𝑇]
(0,
𝑊𝑘
𝑃𝑜𝑟𝑡
max
𝑗∈ 1,𝑘
𝑊𝑗
𝑃𝑜𝑟𝑡 − 1) 𝑊𝑘
𝑃𝑜𝑟𝑡
: Cumulative return of the portfolio
TO =
12
2(𝑇 − 1)
෍
𝑡
𝑇−1
𝑤𝑡 − 𝑤𝑡−1
− 𝑤𝑡−1
−
: Portfolio weight before rebalance

26. 95% 96% 97% 98% 99% λ＝０ Best λ
AR [%] 8.27 8.45 8.48 8.58 8.48 8.46 8.36 8.35 8.73 8.42 9.03 8.95
RISK [%] 18.13 15.24 15.67 18.71 15.62 16.15 15.42 15.54 15.64 15.74 16.23 15.11
R/R 0.46 0.55 0.54 0.46 0.54 0.52 0.54 0.54 0.56 0.53 0.56 0.59
MaxDD [%] -57.63 -58.14 -61.21 -61.76 -59.44 -57.75 -56.75 -60.81 -59.54 -62.21 -54.14 -52.81
TO [%] 16.95 31.10 8.75 71.52 22.19 17.46 19.99 24.97 21.52 29.98 1,000.98 33.57
95% 96% 97% 98% 99% λ＝０ Best λ
AR [%] 8.14 8.99 9.09 11.11 11.83 11.68 10.79 11.54 11.96 12.92 15.75 17.29
RISK [%] 19.27 11.77 11.77 20.61 12.53 12.27 11.87 12.42 13.47 14.49 16.46 15.61
R/R 0.42 0.76 0.77 0.54 0.94 0.95 0.91 0.93 0.89 0.89 0.96 1.11
MaxDD [%] -59.81 -50.84 -50.25 -57.39 -47.22 -46.98 -45.21 -45.36 -48.23 -50.38 -35.29 -34.93
TO [%] 36.73 27.48 17.15 75.80 37.04 41.31 38.87 35.37 41.57 38.36 960.03 750.48
RM-CVaR
EW MV
All Out-of-sample period (from January 1989 to December 2018)
FF25 EW MV DRP EGO Avg_CVaR
CVaR RM-CVaR
DRP EGO
CVaR
Avg_CVaRFF48
Experimental Result
25
Table 2: The performance of each portfolio in out-of-sample period for FF25 dataset
(upper panel) and FF48 dataset (lower panel).

27. ・ Our study makes the following contributions:
Conclusion
26
(1) We propose RM-CVaR. We are able to prove that
the optimization problem is written as a linear programming.
(2) We demonstrate that the CVaR portfolio dramatically
changes depending on the 𝛽 level. (Error maximization)
(3) RM-CVaR exhibit a superior performance in terms of having
both higher R/R and lower maxDD.