the study of optimal portfolio selection with factor volatility models

1/40

Introduction Literature Review Methodology Empirical Analysis Conclusion

The Study of Optimal Portfolio Selectionwith Factor Volatility Models

Ta-Wei Huang

Master in Statistics,

National Taiwan University

June 15, 2017

2/40


Table of Contents1 Introduction

Markowitz’s Portfolio Selection ProblemChallenges in Markowitz’s FrameworkProposed Framework

2 Literature ReviewConditional HeteroscedasticityMultivariate Volatility ModelPortfolio Optimization

3 MethodologyProblem SettingPortfolio Derivation MethodologyPortfolio Selection MethodologyPortfolio Selection CriteriaProposed Trading Strategy

4 Empirical AnalysisEmpirical SetupForeign Exchange DataSemiconductor Stock Data

5 ConclusionConclusion

3/40


Markowitz’s Portfolio Selection Problem

Markowitz’s Portfolio Selection Problem

Let rt be the k × 1 return series. A portfolio is denoted by

wt = [wj,t]′, where wj,t is the weight to invest in asset j,

j = 1, · · · , k.

Markowitz’s Two-stage Framework

Portfolio selection is a process of determining the weight to invest

in each asset on the market at current time.

1 Future Belief: Derive µt+k ≡ E(rt+k|Ft) and

Σt+k = Cov(rt+k|Ft)2 Portfolio Weight: Determine wt+k = arg min f(µt+k,Σt+k).

4/40


Challenges in Markowitz’s Framework

Problems of Volatility Forecasting

Problems of Volatility Forecasting

Curse of Dimensionality:

For k returns, the covariance matrix has k×(k+1)2 elements. 15

stocks require analyzing 120 processes simultaneously.

Positive Definiteness:

It is not trivial to maintain positive definiteness of the

estimated volatility matrices.

Apply ”factor modeling techniques” to conquer above challenges -

transform the original assets to different portfolios and

apply univariate modeling techniques.

5/40


Proposed Framework

Proposed 3-stage Framework

We extend the Markowitz’s portfolio selection methodology to a

3-stage procedure.

Proposed Framework

1 (Portfolio Derivation)

Derive M with rt = Mrt such that Cov(rt|Ft−1) is diagonal.

2 (Future Belief Forecast)

Derive forecast of E(rj,t+1|Ft) and Var(rj,t+1|Ft) by

univariate models.

3 (Portfolio Selection)

Determine the desired portfolio weight wt+1 by some

performance measures.

6/40


Conditional Heteroscedasticity

Problem Setting of Multivariate Volatility Model

In multivariate volatility model, we decompose rt as rt = µt + et

conditional mean: µt = E(rt|Ft−1)

innovation: et = Σ1/2t εt, Cov(et|Ft−1) = Cov(rt|Ft−1) = Σt,

εt follows a general distribution D(0, Ik).

Let ρj = Cor(et, et−j) be the lag-j cross-correlation matrix of et.

We want to test

H0 : ρ1 = ρ2 = · · · = ρm = 0

H1 : ρi 6= 0 for some 1 ≤ i ≤ m.

7/40


Conditional Heteroscedasticity

Test for Multivariate ARCH Effect

Portmanteau Test:

Q?k(m) = T 2∑m

i=11

T−ib′i

(ρ−10 ⊗ ρ−10

)bi, where

bi = vec(ρ′i)

Standardized Test:

standardize the innovation by et = e′tΣ−1, et − k, then use

Q?(m) = T (T + 2)∑m

i=1 ρ2i /(T − i).

Robust Portmanteau Test:

modify Q?k by trimming the upper 95% tail of rt to get better

finite sample result

Ranked-based test: QR(m) =∑m

i=1ρi−E(ρi)V ar(ρi)

, where ρi is the

lag i rank cross-correlation.

8/40


Multivariate Volatility Model

Some Common Multivariate Volatility Models

Common Multivariate Volatility Model

1 VEC model: vectorize Σt, but may be not positive definite

2 EWMA model: vectorize Σt, but may be not positive definite

3 BEKK model: process in a quadratic form, positive definite

but many parameters

4 DCC model: modeling the correlation matrix and using

univariate volatility models

9/40


Portfolio Optimization

Commonly Used Portfolio Optimization Techniques

Global Minimum Variance Portfolio

The global minimum variance portfolio has the weight

wt+1 = argmin w′t+1Σt+1wt+1 subject to ‖w‖2 = 1.

A closed-form solution is any eigenvector associated with the smallest

eigenvalue of Σt+1.

Minimum VaR Portfolio

The minimum (1− α) VaR Portfolio has the weight

wt+1 = argmin V aRα,t+1 subject to ‖wt+1‖2 = 1. The weight can be

either derived from a distribution or computation on empirical data.

10/40


Problem Setting

Model Setting for k Asset Return Processes

Let rt = [r1t r2t · · · rkt]′ be the return vector of k assets at

time t. Denote Ft−1 be the natural filtration.

Return Structure Assumption

Decompose rt = µt + et, where µt = E(rt|Ft−1).

et = Σ1/2t εt, where Σt is the positive-definite volatility

matrix, and εt follows a general distribution D(0, Ik).

11/40


Portfolio Derivation Methodology

Portfolio Derivation Problem

We want to find non-singular Mt with et = Mtet such that

Cov(et|Ft−1) is diagonal;

components of et are pairwise independent;

only some elements of et have the ARCH effect.

Then, we trade rt = Mtrt instead of rt since we can use a

univariate volatility process on rt to predict future returns and

volatilities.

12/40




3 Portfolio Derivation Methodology

1 Principal Component Analysis:

find Mt such that Cov(bt|Ft−1) is diagonal

Var(b1,t) ≥ · · · ≥ Var(bk,t).

2 Independent Component Analysis:

find Mt such that components of bt are independent.

3 Principal Volatility Component Analysis:

find Mt such that only some of components of bt has the

ARCH effect and components of bt are at least uncorrelated.

13/40


Portfolio Selection Methodology

Portfolio Selection Problem

Each row in Mt is a weight vector on k tradable assets. So it transforms

the original k assets to k portfolios.

Portfolios with ARCH effect are called active portfolios.

Portfolios without ARCH effect are called passive portfolios.

Portfolio Selection Problem

(Single) Should we invest the best among k tradable portfolios?

(Active) Should we only invest the best active portfolio?

(Passive)Should we only invest the best passive portfolio?

(Mixed) Should we invest a mix of the best active and passive

portfolios to balance the risk?

14/40


Portfolio Selection Methodology

Active and Passive Portfolio

After deriving the investing target portfolios rt = Mtrt, one can build a

univariate time series model on each element ri,t, i = 1, · · · , k. Note

that each row of Mt is regularized with L2 norm one.

Univariate Time Series Model

If ri,t has the ARCH effect, fit an ARMA(p, q)-GARCH(1,1) model.

If ri,t has no ARCH effect, fit an ARMA(p, q) model.

Note that (p, q) is selected by Hyndman-Khandakar (2007) algorithm.

Then, we can derive the one-step prediction of return ri,t+1 and variance

σ2i,t+1 from the univariate time series model.

15/40


Portfolio Selection Criteria

Idea of Proposed Trading Strategies

If ri,t+1 > 0, long portfolio i at time t+ 1. If ri,t+1 < 0, short portfolio i

at time t+ 1. Hence, the one-step prediction of return is∣∣∣ri,t+1

∣∣∣.Idea of Proposed Trading Strategies

1 Strategy 1: Minimize (1− α) Value at Risk,

V aRα,t+1 = −(|ri,t+1|+ qασi,t+1

).

Directly use specified β instead of qα.

2 Strategy 2: Maximize the Sharpe ratio

Sharpe Ratio =|ri,t+1|σi,t+1

.

16/40


Proposed Trading Strategy

Confidence Bound and Maximum Sharpe Ratio Strategy

17/40


Empirical Setup

Rolling Training and Testing Schema

The training and testing schema is shown below. (w = rolling

window)

18/40


Empirical Setup

Parameter Tuning

We need to find the optimal confidence bound parameter β.

Greedily search β ∈ {−0.5,−0.3,−0.1, 0.1, 0.3, 0.5}.

Tuning period: week 261 to week 312 (1-year).

Testing period: week 313 to end (approximately 9 years).

19/40


Empirical Setup

Performance Evaluation Criteria

1 Average return: µp =∑Tp

t=w+1 w′trt

Tp

2 Standard Deviation: σ2p = 1Tp−1

∑Tpt=w+1

(w′trt − µp

)23 Sharpe ratio: SR =

µpσp

4 95% Value-at-Risk:

V aR95% = 95% quantitle of {w′trt}t=w+1,··· ,Tp

5 Portfolio Turnover:

TOp = 1Tp−1

∑Tp−1i=w+1

∑kj=1 |wj,t+i+1|t − wj,t+i|t|

20/40


Foreign Exchange Data

Data Description

Data: Weekly log returns, 7 exchanges data (same as Hu andTsai (2014).

Period: 2001/12/31 to 2017/03/24, totally 788 weeks.

Rolling window w = 260 weeks (i.e., 5 years)

Table: Selected 7 Exchanges

GBP 英鎊 CAD 加幣NOK 挪威克朗 SGD 新加坡元SEK 瑞典克朗 AUD 澳幣CHF 瑞士法郎

21/40



Multivariate ARCH Test

The following figure shows histograms of p-values of multivariate ARCH

test for rt−w+1, · · · , rt, t = 313, · · · , 788.

22/40



CB: β is the Risk Tolerance Measure

The smaller the β is, the less risky the portfolio is.

23/40



CB: Parameter Tuning Result

Table: Parameter Tuning Result: Values of β

PCA ICA PVCA

Single 0.1 -0.1 -0.1

Active 0.1 -0.3 -0.1

Passive 0.1 -0.1 -0.1

Mix 0.1 -0.5 -0.1

24/40



CB: Result of Confidence Bound Strategy

Fix portfolio derivation strategy,- mix strategy outperforms in average return and Sharpe ratio.- passive strategy usually has the minimum volatility and 95% VaR.

PVCA model outperforms benchmark portfolios.

Table: Performance of All CB and Benchmark Strategies (FX Dataset)

Strategy Average ReturnStandardDeviation

95% VaR Sharpe Ratio Turnover

Single-PCA-CB -5.09% 6.62% -54.06% -0.77 0.56Single-ICA-CB -0.29% 5.08% -9.95% -0.06 0.39

Single-PVCA-CB 2.69% 5.14% -22.26% 0.52 0.51Active-PCA-CB -4.82% 8.18% -37.73% -0.59 0.49Active-ICA-CB 0.74% 5.27% -25.94% 0.14 0.50

Active-PVCA-CB 3.38% 5.14% -24.08% 0.66 0.47Passive-PCA-CB -0.77% 3.55% -21.14% -0.22 0.40Passive-ICA-CB -0.35% 3.63% -24.15% -0.10 0.42

Passive-PVCA-CB 0.39% 3.03% -20.66% 0.13 0.44Mix-PCA-CB 5.53% 13.08% -151.34% 0.42 0.92Mix-ICA-CB 2.75% 8.17% -96.27% 0.34 1.34

Mix-PVCA-CB 3.87% 5.29% -57.90% 0.73 1.46Minimum Variance -0.21% 4.34% -50.99% -0.05 0.22

Equal Weight -6.96% 23.12% -276.88% -0.30 0.00

25/40



CB: Return Forecast Accuracy is Not the Key

The strategy with minimum forecast errors does not always

generate the highest return.

Table: MAD & MSD of CB Portfolios (FX Dataset)

Strategy MAD MSD Strategy MAD MSD

Single-PCA-CB 0.0035 0.000112 Pasive-PCA-CB 0.0018 0.000022

Single-ICA-CB 0.0021 0.000038 Pasive-ICA-CB 0.0019 0.000026

Single-PVCA-CB 0.0023 0.000045 Pasive-PVCA-CB 0.0019 0.000021

Active-PCA-CB 0.0035 0.000128 Mix-PCA-CB 0.0015 0.000014

Active-ICA-CB 0.0025 0.000048 Mix-ICA-CB 0.0020 0.000028

Active-PVCA-CB 0.0022 0.000043 Mix-PVCA-CB 0.0014 0.000015

26/40



CB: Cumulative Wealth

The PVCA strategy gives a stable portfolio with lower risk.

27/40



Sharpe: Result of Sharpe Strategy



Table: Performance of Sharpe and Benchmark Strategies (FX Dataset)

StrategyAverageReturn

StandardDeviation

95% VaRSharpeRatio

Turnover

Single-PCA-Sharpe 0.33% 9.99% -98.17% 0.03 2.17Single-ICA-Sharpe 4.00% 11.52% -107.60% 0.35 2.63

Single-PVCA-Sharpe 6.41% 7.20% -66.04% 0.89 2.09Active-PCA-Sharpe 3.25% 12.72% -107.83% 0.26 1.96Active-ICA-Sharpe 4.63% 11.40% -110.33% 0.41 2.53

Active-PVCA-Sharpe 2.28% 7.48% -70.54% 0.30 1.75Passive-PCA-Sharpe 3.25% 5.76% -63.71% 0.56 1.74Passive-ICA-Sharpe -1.84% 9.53% -91.74% -0.19 2.09

Passive-PVCA-Sharpe 5.71% 5.54% -61.85% 1.03 2.05Mix-PCA-Sharpe 4.05% 5.48% -55.25% 0.74 1.54Mix-ICA-Sharpe 1.78% 9.33% -92.70% 0.19 2.21

Mix-PVCA-Sharpe 4.35% 4.62% -40.31% 0.94 1.50Minimum Variance -0.21% 4.34% -50.99% -0.05 0.22

Equal Weight -6.96% 23.12% -276.88% -0.30 0.00

28/40



Sharpe: Return Forecast Accuracy is important.

The strategy with minimum forecast errors usually generates the

highest return.

Table: MAD & MSD of Sharpe Portfolios (FX Dataset)


Single-PCA-Sharpe 0.0029 0.000085 Pasive-PCA-Sharpe 0.0019 0.000024

Single-ICA-Sharpe 0.0036 0.000090 Pasive-ICA-Sharpe 0.0026 0.000062

Single-PVCA-Sharpe 0.0022 0.000037 Pasive-PVCA-Sharpe 0.0019 0.000021

Active-PCA-Sharpe 0.0035 0.000121 Mix-PCA-Sharpe 0.0017 0.000020

Active-ICA-Sharpe 0.0035 0.000087 Mix-ICA-Sharpe 0.0028 0.000060

Active-PVCA-Sharpe 0.0022 0.000039 Mix-PVCA-Sharpe 0.0015 0.000015

29/40



Sharpe: Cumulative Wealth

PVCA strategy gives a relatively stable portfolio with lower risk.

30/40


Semiconductor Stock Data

Data Description

Data: Weekly returns, 15 stocks of semiconductor companiesin Taiwan

Period: 2002/03/15 - 2017/03/24, totally 778 weeks.

Rolling window w = 260 weeks (i.e., 5 years)

Table: Selected 15 Stocks in Semiconductor Industry

2311 日月光 2408 南亞科 2454 聯發科2325 矽品 2441 超豐 2455 全新2330 台積電 5471 松翰 3006 晶豪科2351 順德 2449 京元電子 3034 聯詠2379 瑞昱 2451 創見 3035 智原

31/40



Multivariate ARCH Test

The following figure shows histograms of p-values of multivariate ARCH

test for rt−w+1, · · · , rt, t = 313, · · · , 788.

32/40



CB: Parameter Tuning Result

Table: Parameter Tuning Result: Values of β

PCA ICA PVCA

Single -0.1 0.5 -0.1

Active -0.1 0.5 -0.1

Passive -0.1 0.3 -0.1

Mix -0.1 0.5 -0.1

33/40



CB: Result of Confidence Bound Strategy

Fix portfolio selection strategy, PCA method outperforms all others inaverage return, Sharpe ratio, and portfolio turnover.

PCA and PVCA model outperform benchmark portfolios.

Table: Performance of All CB and Benchmark Strategies (SMS Dataset)

Strategy Average ReturnStandardDeviation

95% VaR Sharpe Ratio Turnover

Single-PCA-CB 27.96% 54.91% -72.78% 0.51 3.38Single-ICA-CB -0.04% 52.16% -107.65% 0.00 3.64

Single-PVCA-CB 21.32% 27.93% -50.33% 0.76 3.44Active-PCA-CB 37.19% 66.44% -101.34% 0.56 2.79Active-ICA-CB 7.48% 50.22% -100.91% 0.15 3.58

Active-PVCA-CB 10.85% 30.65% -51.83% 0.35 3.01Passive-PCA-CB 34.35% 27.12% -44.35% 1.27 3.29Passive-ICA-CB 3.50% 44.66% -74.95% 0.08 3.67

Passive-PVCA-CB 13.41% 26.74% -51.13% 0.50 3.47Mix-PCA-CB 33.72% 22.91% -37.89% 1.47 2.51Mix-ICA-CB 8.03% 35.67% -69.74% 0.23 2.73

Mix-PVCA-CB 14.69% 19.38% -35.52% 0.76 2.47Minimum Variance 10.09% 33.48% -62.61% 0.30 1.27

Equal Weight -7.84% 106.09% -224.54% -0.07 0.00

34/40



CB: Return Forecast Accuracy is Not the Key


generate the highest return.

Table: MAD & MSD of CB Portfolios (SMS Dataset)


Single-PCA-CB 0.015436 0.002214 Pasive-PCA-CB 0.00962 0.000484

Single-ICA-CB 0.018248 0.001974 Pasive-ICA-CB 0.014512 0.001307

Single-PVCA-CB 0.009953 0.000539 Pasive-PVCA-CB 0.009519 0.000476

Active-PCA-CB 0.018185 0.003098 Mix-PCA-CB 0.007887 0.000346

Active-ICA-CB 0.017406 0.001815 Mix-ICA-CB 0.011525 0.000845

Active-PVCA-CB 0.010514 0.000634 Mix-PVCA-CB 0.007168 0.000252

35/40



CB: Cumulative Wealth

Passive-PCA-CB strategy is extremely strong.

36/40



Sharpe: Result of Sharpe Strategy



Table: Performance of Sharpe and Benchmark Strategies (SMS Dataset)

StrategyAverageReturn

StandardDeviation

95% VaRSharpeRatio

Turnover

Single-PCA-Sharpe 27.98% 45.44% -53.26% 0.62 3.22Single-ICA-Sharpe 9.89% 38.79% -74.30% 0.26 3.83

Single-PVCA-Sharpe 20.95% 27.06% -47.82% 0.77 3.36Active-PCA-Sharpe 42.36% 72.20% -120.29% 0.59 2.77Active-ICA-Sharpe -10.37% 41.53% -82.20% -0.25 3.74

Active-PVCA-Sharpe 14.48% 29.32% -51.57% 0.49 3.03Passive-PCA-Sharpe 29.78% 26.09% -42.12% 1.14 3.12Passive-ICA-Sharpe -7.21% 35.10% -69.93% -0.21 3.73

Passive-PVCA-Sharpe 15.32% 26.54% -50.37% 0.58 3.40Mix-PCA-Sharpe 28.94% 23.23% -41.26% 1.25 2.44Mix-ICA-Sharpe -4.67% 27.22% -54.05% -0.17 2.85

Mix-PVCA-Sharpe 16.99% 18.79% -32.36% 0.90 2.44Minimum Variance 10.09% 33.48% -62.61% 0.30 1.27

Equal Weight -7.84% 106.09% -224.54% -0.07 0.00

37/40



Sharpe: Return Forecast Accuracy is NOT the Key


generates the highest return.

Table: MAD & MSD of Sharpe Portfolios (SMS Dataset)


Single-PCA-Sharpe 0.01260 0.00160 Pasive-PCA-Sharpe 0.00931 0.00045

Single-ICA-Sharpe 0.01381 0.00110 Pasive-ICA-Sharpe 0.01210 0.00086

Single-PVCA-Sharpe 0.00961 0.00051 Pasive-PVCA-Sharpe 0.00940 0.00047

Active-PCA-Sharpe 0.01975 0.00360 Mix-PCA-Sharpe 0.00787 0.00036

Active-ICA-Sharpe 0.01429 0.00122 Mix-ICA-Sharpe 0.00937 0.00054

Active-PVCA-Sharpe 0.01020 0.00058 Mix-PVCA-Sharpe 0.00688 0.00024

38/40



Sharpe: Cumulative Wealth

Passive-PCA strategy performs extremely well.

39/40


Conclusion

Conclusion

β is actually a risk tolerance measure. When β increases, the

portfolio volatility also grows simultaneously.

PVCA model usually results in low-risk volatility comparing

with other derivation techniques.

Forecast accuracy of returns is not the key. Volatility

estimation is important!

Our proposed strategies outperform the benchmark

minimum-variance and equally-weighted portfolios.

40/40


Thank you for your attention!

the study of optimal portfolio selection with factor volatility models

Data & Analytics