Portfolio Optimization on Multivariate RegimeSwitching GARCH Model with Normal

Tempered Stable Innovation

Cheng Peng∗

Young Shin Kim†

College of Business,

Stony Brook University,

New York, USA

December 1, 2020

Abstract

We propose a Markov regime switching GARCH model with mul-tivariate normal tempered stable innovation to accommodate fat tailsand other stylized facts in returns of financial assets. The model isused to simulate sample paths as input for portfolio optimization withrisk measures, namely, conditional value at risk and conditional draw-down. The motivation is to have a portfolio that alleviates left tailevents by combining models that incorporates fat tail with optimiza-tion that focuses on tail risk. In-sample test is conducted to demon-strate goodness of fit. Out-of-sample test shows that our approachyields higher performance measured by performance ratios than themarket and equally weighted portfolio in recent years which includessome of the most volatile periods in history. We also find that subop-timal portfolios with higher return constraints tend to be outperform-ing.

∗Cheng.peng.1@stonybrook.edu†aaron.kim@stonybrook.edu

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1 Introduction

Empirical studies have found in the return of various financial instrumentsskewness and leptokurtotic—asymmetry, and higher peak around the meanwith fat tails. Normal distribution has long been recognized as insufficient toaccommodate these stylized fact, relying on which could drastically underes-timate the tail risk of a portfolio. However, few has adequately incorporatethis fact into modeling and decision-making. The α-Stable distribution hasbeen proposed to accommodate them. The lack of existence of moments ofstable distributions could sometimes cause difficulties, to which temperedstable distribution presents a potential solution. The class of tempered sta-ble distributions is derived from the α-stable distribution by tempering tails.Recent findings from Kim et al. (2012), Kurosaki and Kim (2018), Anandet al. (2016) use the normal tempered stable (NTS) distribution and suc-cessfully model the stock returns with high accuracy. As is pointed out, theabove mentioned properties not only exist in raw returns but also in GARCH-filtered residuals of time series model. This motivates us to use multivariateNTS distribution to accommodate the asymmetry, interdependence, and fattails of the joint innovations in our model. Other distributions like Pearsondistribution and generalized hyperbolic distribution have also been proposedto this end, both having t-distribution as a special case. The estimationon the degree of freedom parameter of t-distribution in GARCH model isfound to be unreliable. Study in Kim et al. (2011) finds that normal andt-distribution are rejected and NTS distribution is favored on the residu-als of time series model. Another study in Shi and Feng (2016) found thattempered stable distribution with exponential tempering is a better choicethan t or GED distribution for innovation distribution of Markov-Regime-Switching (MRS) GARCH model.. While generalized hyperbolic distributionis very flexible, it has 5 parameters which might create difficulty in estima-tion. NTS distribution has only 3 standardized parameters and is flexibleenough to serve the purpose.

One of the drawbacks of GARCH model is the difficulty in dealing withvolatility spikes, which could indicate that the market is switching withinregimes. Various types of specification of MRS-GARCH models has been pro-posed, for example in Hamilton (1996), many of which suggest that regimeswitching GARCH model achieves a better fit for empirical data. Natu-rally, the most general model allows all parameters to switch among regimes.However, as is shown in Henneke et al. (2011), the sampling procedure in

2

MCMC method for the estimation of such model is time-consuming andrenders it improper for practical use. Recent developments in algorithm inBillioa et al. (2016) improves the estimation speed. In our model, we use theregime switching GARCH model specified in Haas et al. (2004) for simplicity.This model circumvents the path dependence problem in Markov model byspecifying parallel GARCH models.

It’s a recognized fact that the correlation of financial assets is time-varying. Specification of multivariate GARCH models with both regime-specific correlations and variance dynamics involves a balance between flexi-bility and tractability. The model in Haas et al. (2004) has been generalizedin Haas and Liu (2004) to a multivariate case. Unfortunately, it suffers fromthe curse of dimensionality and thus is unsuitable for a high dimensionalempirical study. In our model, we decompose variance and correlation sothat the variance of each asset evolves independently according to a univari-ate MRS-GARCH model. The correlation is incorporated in the innovationsmodeled by a flexible Markov swiching multivariate NTS distribution.

Modern portfolio theory is usually framed as a trade-off between returnand risk. Classical Markowitz Model intends to find the portfolio with thehighest Sharpe ratio. However, variance has been criticized for not containingenough information on the tail of distribution. Moreover, the correlationis not sufficient to describe the interdependence of asset return with non-elliptical distribution.

Current regulations for financial business utilize the concept of Value atRisk(VaR), which is the percentile of the loss distribution, to model the risk ofleft tail events. However, there are several undesired properties that renderedit an insufficient criterion. First, it’s not a coherent risk measure due to lackof sub-additivity. Second, VaR is difficult to to optimize for non-normaldistributions due to non-convex and non-smooth as a function of positionswith multiple local extrema, which causes difficulty in developing efficientoptimization techniques. Third, a single percentile is insufficient to describethe tail behavior of a distribution, which might lead to an underestimationof risk.

Theory and algorithm for portfolio optimization with CVaR measure isproposed in Krokhmal et al. (2001) Rockafellar and Uryasev (2000) to addressthese issues. For continuous distributions, CVaR is defined as a conditionalexpectation of losses exceeding a certain VaR level. For discrete distributions,CVaR is defined as a weighted average of some VaR levels that exceed aspecified VaR level. In this way, CVaR concerns both VaR and the losses

3

exceeding VaR. As a convex function of asset allocation, a coherent riskmeasure and a more informative statistic, CVaR serves as an ideal alternativeto VaR. A study on the comparison of the two measure can be found inSarykalin et al. (2014).

Drawdown has been a constant concern for investors. It’s much moredifficult to climb out of a drawdown than drop into one, considering thatit takes 100% return to recover from 50% relative drawdown. Thus maxi-mum drawdown is often used in evaluation of performance of a portfolio. Asis mentioned in Checkhlov et al. (2005), a client account of a CommodityTrading Advisor(CTA) will be issued a warning or shut down with a draw-down higher than 15% or longer than 2 years, even if small. Investors arehighly unlikely to tolerate a drawdown greater than 50%. However, maxi-mum drawdown only considers the worst case which may only occur undersome very special circumstances. It’s also very sensitive to the testing periodand asset allocation. On the other hand, small drawdowns included in aver-age drawdown are acceptable and might be caused by pure noise, of whichminimization might not make sense. For instance, a Brownian motion wouldhave drawdowns in a simulated sample path.

A relevant criterion to CVaR, conditional drawdown(CDaR) is proposedin Checkhlov et al. (2005) to address these concerns. While CVaR onlyconcerns the distribution of return, CDaR concerns the sample path. CDaRis essentially a certain CVaR level of the drawdowns. By this, we overcomethe drawbacks of average drawdown and maximum drawdown. CDaR notonly take the depth of drawdowns into consideration, but also the lengthof them, corresponding to the concerns by investors we mentioned earlier.Since the CDaR risk measure is the application of CVaR in a dynamic case,it naturally holds nice properties of CVaR such as convexity with respectto asset allocation. Optimization method with constraint on CDaR has alsobeen developed in relevant papers.

For an optimization procedure to lead to desired optimal result, the inputis of critical importance. With careless input, the portfolio optimizationcould magnify the impact of error and lead to unreasonable result. Usinghistorical sample path like in Krokhmal et al. (2002) means that we assumewhat happened in the past will happen in the future, which is an assumptionthat needs careful examination. It’s found in Lim et al. (2011) that estimationerrors of CVaR and the mean is large, resulting in unreliable allocation. Analternative way is to use multiple simulated sample paths. This motivatesus to propose a model that incorporates fat tails to generate simulation as

4

input for the optimization with tail risk measures.To this end, we propose a Markov regime switching GARCH model with

multivariate standard normal tempered stable innovations. In-sample testsare performed to examine the goodness of fit. Simulation is performed withthe model to generate multiple sample paths. To demonstrate the effective-ness of the proposed model as well as the tail risk measures, we conduct anou-of-sample study on the performance. The tested period includes some ofthe most volatile time in history caused by COVID-19 pandemic and inter-national trade tensions.

The remainder of the paper is organized as follows. Section 2 introducesthe preliminaries on NTS distribution and GARCH model. Section 3 specifiesour model, methods for estimation and simulation. Section 4 is an empiricalstudy on in-sample goodness of fit and out-of-sample performance in recentyears.

2 Preliminaries

This section reviews background knowledge about Normal Tempered Sta-ble (NTS) Distribution, GARCH and regime switching GARCH model, andvarious risk measures.

2.1 Risk Measures

In this section we discuss and summarize popular risk measures in financesuch as Value at Risk (VaR), Conditional VaR (CVaR), and conditionaldrawdown at risk (CDaR). Definitions of CDaR, and related properties ofdrawdown are following Checkhlov et al. (2005).

Consider a probability space (Ω,F∞,P) and a portfolio with N assets.Suppose that Pn is a stochastic process of price for the n-th asset in theportfolio on the space

Pn : [0,∞)× Ω −→ R, n = 1, 2, · · · , N

with the real initial value Pn(0, ·) = Pn,0 > 0. Let Rn be a stochastic processof the return for the n-th asset between time 0 to time t > 0 as

Rn(t, ω) =Pn(t, ω)− Pn,0

Pn,0, t > 0,

5

with Rn(0, ·) = 0. Let R be a N -dimensional stochastic process

R : [0,∞)× Ω −→ RN

withR(t, ω) = (R1(t, ω),R2(t, ω), · · · ,RN(t, ω))T .

Let x = (x1, x2, · · · , xN)T ∈ RN be a capital allocation vector of the portfoliosatisfying

∑Nn=1 xn = 1 and xn ∈ [0, 1] for all n ∈ 1, 2, · · · , N1. Then the

portfolio return R(x, t, ω) at time t > 0 is equal to

R(x, t, ω) =N∑n=1

xnRn(t, ω).

The definition of VaR for R(x, t, ·) with the confidence level η is

VaRη(R(x, t, ·)) = − infx|P(R(x, t, ·) ≤ x) > 1− η.

If R(x, t, ·) has a continuous cumulative distribution, then we have

VaRη(R(x, t, ·)) = −F−1R(x,t,·)(1− η),

where F−1R(x,t,·) is the inverse cumulative distribution function of R(x, t, ·).

The definition of CVaR for R(x, t, ·) with the confidence level η is

CVaRη(R(x, t, ·)) =1

1− η

∫ 1−η

0

VaR(1−x)(R(x, t, ·))dx.

If R(x, t, ·) has a continuous cumulative distribution, then we have

CVaRη(R(x, t, ·)) = −E [R(x, t, ·)|R(x, t, ·) < −VaRη(R(x, t, ·))] .

Let T > 0, and ω ∈ Ω, then (R(x, t, ω))t∈[0,T ] = R(x, t, ω)|t ∈ [0, T ] isone sample path of portfolio return with the capital allocation vector x fromtime 0 to T . The drawdown (DD(·, ω))t∈[0,T ] of the portfolio return is definedby

DD(t, ω) = maxs∈[0,t]

R(x, s, ω)−R(x, t, ω), for t ∈ [0, T ] (1)

1We consider a long only portfolio in this paper.

6

is a risk measure assessing the decline from a historical peak in some variable,and DD(t, ω)|t ∈ [0, T ], ω ∈ Ω is a stochastic process of the drawdown. Theaverage drawdown (ADD) up to time T for ω ∈ Ω is defined as

ADD(T, ω) =1

T

∫ T

0

DD(t, ω)dt (2)

that is the time average of drawdowns observed between time 0 to T . TheMaximum drawdown (ADD) up to time T for ω ∈ Ω is defined as

MDD(T, ω) = maxt∈(0,T )

DD(t, ω) (3)

that is the maximum of drawdowns that have occurred up to time T forω ∈ Ω.

Let

FDD(z, ω) =1

T

∫ T

0

1DD(t,ω)≤zdt (4)

and

ζη(ω) =

infz|FDD(z, ω) ≥ η if η ∈ (0, 1]

0 if η = 0. (5)

Then the conditional drawdown at Risk (CDaR) is defined as

CDaRη(T, ω)

=

(FDD(ζη(ω), ω)− η

1− η

)ζη(ω) +

1

(1− η)T

∫ T

0

DD(t, ω)1DD(t,ω)>ζη(ω)dt. (6)

According to Checkhlov et al. (2005), the CDaR also obtained by the follow-ing optimization formula

CDaRη(T, ω) = minζ

ζ +

1

ηT

∫ T

0

max (DD(t, ω)− ζ, 0) dt

Note that CDaR0(T, ω) = ADDω(T ) and CDaR1(T, ω) = MDDω(T ).

Let

FDD(z) =1

TE

[∫ T

0

1DD(t,·)≤zdt

](7)

and

ζη = F−1DD(η) =

infz|FDD(z) ≥ η if η ∈ (0, 1]

0 if η = 0. (8)

7

Considering all the sample path R(x, t, ω)|t ∈ [0, T ], ω ∈ Ω, we defineCDaR at η as

CDaRη(T )

=

(FDD(ζ(η))− η

1− η

)ζ(η) +

1

(1− η)TE

[∫ T

0

DD(t, ·)1DD(t,·)>ζ(η)dt

]. (9)

2.2 Scenario based estimation

We discuss estimating risk measures presented in the previous section usinggiven scenarios (historical scenario or simulation). Suppose that the timeinterval [0, T ] is divided by a partition 0 = t0 < t1 < · · · < tM = T, anddenote

Rm(x) = R(x, tm, ·).

We select ωs ∈ Ω for s ∈ 1, 2, · · · , S for number of scenarios S. Then weobtain scenarios of the portfolio return process R at time tm as

Rsm(x) = R(x, tm, ωs)

where m ∈ 0, 1, 2, · · · ,M, and s ∈ 1, 2, · · · , S. We can calculate VaR,CVaR, DD, and CDaR under the simulated scenarios. The VaR with signif-icant level η under the simulated scenario is estimated as

VaRη(Rm(x)) = − inf

u∣∣ 1

S

S∑s=1

1Rsm(x)<u > 1− η

. (10)

Let R(k)m (x) be the k-th smallest value in Rs

m(x) | s = 1, 2, · · · , S, thenCVaR at the significant level η is estimated according to the formula

CVaR1−η(Rm(x))

= − 1

1− η

1

S

dS(1−η)e−1∑k=1

R(k)m (x) +

((1− η)− dS(1− η)e − 1

S

)R(dS(1−η)e)m (x)

,

(11)

8

where dxe denotes the smallest integer larger than x (See Rachev et al. (2008)for the details).

Before we discuss the scenario based estimation on CDaR, we need todefine the accumulated uncompounded return. Let rn be a random variableof the return for the n-th asset between time ti−1 to time ti as

rn(ti, ω) =Pn(ti, ω)− Pn(ti−1, ω)

Pn(ti−1, ω), t > 0,

with rn(0, ·) = 0. Let Q be a N -dimensional random variable

Q(ti, ω) = (r1(ti, ωs), r2(ti, ωs), · · · , rN(ti, ωs))T .

Let x = (x1, x2, · · · , xN)T ∈ RN be a capital allocation vector of the portfoliosatisfying

∑Nn=1 xn = 1 and xn ∈ [0, 1] for all n ∈ 1, 2, · · · , N2. Then the

portfolio return Q(x, ti, ωs) between time ti−1 and time ti is equal to

Q(x, ti, ωs) =N∑n=1

xnQn(ti, ωs).

The scenario based estimation on accumulated uncompounded return at timeti is

U(x, ti, ωs) =t∑t=1

Qn(x, ti, ωs).

Denote U(x, ti, ωs) as U si (x). Using (1), (2), and (3), DD, ADD, and

MDD on a single simulated scenario are estimated by

DDm,s := DD(tm, ωs) = maxj∈0,1,2,··· ,m

U sj (x)−U s

m(x),

ADDM,s := ADD(tM , ωs) =1

M

M∑m=1

DDm,s,

and MDDM,s := MDD(tM , ωs) = maxm∈1,2,··· ,M

DDm,s,

respectively. Moreover, using (4), (5), (6), CDaR is estimated on a singlesimulated scenario by

CDaRη(T, s) := CDaRη(T, ωs) =1

(1− η)M

M∑m=1

DDm,s1DDm,s>ζsη ,

2We consider a long only portfolio in this paper.

9

where

ζsη := ζη(ws) =

infz|FDD(z, s) ≥ η if η ∈ (0, 1]

0 if η = 0.

with

FDD(z, s) =1

M

M∑m=1

1DDm,s≤z.

Or it is estimated by the optimization formula

CDaRη(T, s) = minζ

ζ +

1

(1− η)M

M∑m=1

max (DDm,s − ζ, 0)

.

Using (7), (8), (9), DD and CDaR on multiple scenarios are estimated by

DD(m) =1

S

S∑s=1

DDm,s

and

CDaRη(T )

=

(FDD(ζ(η))− η

1− η

)ζ(η) +

1

(1− η)M

S∑s=1

M∑m=1

1

SDDm,s1DDm,s>ζ(η), (12)

respectively, where

FDD(z) =1

S

S∑s=1

FDD(z, s)

and

ζη =

infz∣∣FDD(z) ≥ η

if η ∈ (0, 1]

0 if η = 0.

When we generate scenario for each asset return in the portfolio, weuse the uncompounded return. The portfolio return Rs

m(x) for the capitalallocation vector x is obtained by

Rsm(x) =

N∑n=1

xnRn(tm, ws) =N∑n=1

xn

m∑k=1

rn(k, s) =N∑n=1

m∑k=1

xnrn(k, s).

In this paper, we will apply time series models to generate the scenario ofrn(m, s) for m ∈ 1, 2, · · · ,M, s ∈ 1, 2, · · · , S and n ∈ 1, 2, · · · , N.

10

2.3 NTS Distribution

Let T be a strictly positive random variable defined by the characteristicfunction for λ ∈ (0, 2) and θ > 0

φT (u) = exp

(−2θ1−λ

2

λ((θ − iu)

λ2 − θ

λ2 )

).

Then T is referred to as a tempered stable subordinator with parameters λand θ. Suppose that ξ = (ξ1, · · · , ξN)T is a N -dimensional standard normaldistributed random vector with a N×N covariance matrix Σ i.e. ξ ∼ Φ(0,Σ).The N -dimensional NTS distributed random vector X = (X1, ..., XN)T isdefined as

X = µ+ ν(T − 1) +√Tdiag(γ)ξ,

where µ = (µ1, · · · , µN)T ∈ RN , ν = (ν1, · · · , νN)T ∈ RN , γ = (γ1, · · · , γN)T ∈RN

+ and T is independent of ξ. The N -dimensional multivariate NTS distri-bution specified above is denoted as

X ∼ MNTSN(λ, θ,ν,γ,µ,Σ).

Let µn = 0, and γn =√

1− ν2n(2−λ

2θ) with |νn| <

√2θ

2−λ for n ∈ 1, · · · , N.This yields a random variable having zero mean and unit variance. In thiscase, the random variable is referred to as standard NTS distributed randomvariable, and denoted X ∼ stdMNTSN(λ, θ,ν,Σ). The covariance matrix ofX is given by

ΣX = diag(γ)Σdiag(γ) +2− λ

2θν>ν.

2.4 GARCH and Regime Switching GARCH Model

GARCH(p, q) model has been studied intensively as a model for volatility.

rt = η + σtεt,

ut = σtεt,

σ2t = ω +

p∑i=1

αiu2t−i +

q∑i=1

βiσ2t−i.

where rt is the return at time t, σt is the variance at time t, η, ω, αi, βi areparameters. We will use GARCH(1,1) in the paper.

11

Before specifying our model, it would be clear if we first specify the uni-variate Markov regime switching model in Haas et al. (2004).

rt = η∆t + σ∆t,tεt,

ut = σ∆t,tεt,

σ2t = ω +αu2

t−1 + β σ2t−1,

εtiid∼ N(0, 1).

(13)

where ∆t is a Markov chain that determines the regime at time t. For amodel with a number of k regimes, ∆t has finite state space S = 1, ..., kand an irreducible and primitive k × k transition matrix P, with elementpij = P (∆t = j | ∆t−1 = i).

rt is the return at time t, η∆t is a regime specific mean return in regime∆t at time t, σ∆t,t is the standard deviation in regime ∆t at time t. Notethat in a univariate model there are k parallel GARCH models evolving atthe same time. σ2

t = (σ21,t, ...., σ

2k,t),ω = (ω1, ...., ωk),α = (α1, ...., αk),β =

(β1, ...., βk), denotes element-wise product.In this model, a number of k GARCH processes are evolving simultane-

ously. The Markov chain ∆t determines which GARCH model is realized atnext moment.

To understand (13), see the following matrix

[σ∆t,t]∆t=1,...,k, t=1,...T =

σ1,1 σ1,2 . . . σ1,t . . .

σ2,1 σ2,2 . . . σ1,t . . ....

. . . . . . . . .

σk,1 σk,2 . . . σk,t . . .

Each column is a total of k parallel standard deviations at time t. Each rowis the process of one regime through time 1 to T . The process is as followsStep 1: At t = 1, initial column is generated.Step 2: ∆1 decides which row is realized.

For example, ∆1 = 2. That is, σ2,1 is realized in real world at t = 1.Step 3: By vector calculation, 2nd column is generated.

For example, if σ2,1 is realized at t = 1, we calculate

12

σ1,2

σ2,2...

σk,2

=

ω1

ω2...

ωk

+ σ2,1ε1

α1

α2...

αk

+

β1

β2...

βk

σ1,1

σ2,1...

σk,1

Step 4: ∆2 Decides which row is realized.

For example, ∆2 = 1. That is, σ1,2 is realized in real world at t = 2. Aregime switch from regime 2 to regime 1 takes place.Step 5: Repeat Step 1 to Step 4.

It is important to note that, in the example, the realized variance at timet = 2, σ1,2, is calculated with both the realized variance at time t = 1, σ2,1,and σ1,1. This is obvious from the vector equation shown above. σ1,1 isparallel and does not exist in real world.

More generally, when there is a regime switch, from the vector equationσ2t = ω + αut

2 + β σ2t−1, we know that the variance σ∆t,t is determined

by σ∆t,t−1 rather than σ∆t−1,t−1. σ∆t,t−1 is a variance in regime ∆t at timet, which is generated simultaneously with σ∆t−1,t−1, but is not existed inreality at time t or t − 1. This makes the model different from the one inHenneke et al. (2011) that requires all parameters to switch according to aMarkov chain. Because in that way, there is no parallel process, and σ∆t,t isdetermined by σ∆t−1,t−1.

The stationary conditons require a definition of matrices

Mji = pji(β + αe>i ), i, j = 1, ..., k

and block matrixM = [Mji], i, j = 1, ...., k.

The necessary and sufficient condition for stationarity is ρ(M) < 1, whereρ(M) denotes the largest eigenvalue of matrix M . In this model, a num-ber of k GARCH processes are evolving simultaneously. The Markov chaindetermines which GARCH model is realized at next moment. In principle,regime-specific mean can be included in this model, though it may cause lossin property of zero auto-correlation.

3 MRS-MNTS-GARCH Model

This section defines the MRS-MNTS-GARCH model and introduces the pro-cedures for model fitting and sample path simulation.

13

3.1 Model Specification

To model a multivariate process, we assume the variance of each individualasset evolves independently according to univariate regime-switching GARCHmodel specified previously with possibly different number of regimes, whilethe correlation is reflected separately in the joint standard residuals mod-eled with stdMNTS distribution. The regime of correlation matrix follows aMarkov process. The pair of words regime and state, market and a marketindex, innovations and joint residuals are used interchangeably as follows:

Let ∆(i)t be the regime of the ith asset at time t defined by a Markov chain

with finite state space S = 1, . . . , k(i) and an irreducible and primitive k(i)×k(i) transition matrix, where k(i) is the number of regimes of the ith asset.Suppose the innovation εt = (ε

(1)t , . . . , ε

(N)t ) follows stdMNTS distribution of

which the parameters shift in regimes determined by a Markov chain ∆Dt . D

stands for a market index. The multivariate distribution of joint residualsfollows a Markov process with the same transition matrix as the marketindex. We use DJIA index for the market index D in this paper. Then wedefine portfolio return as

rt = ηt + σt εtut = σt εt(

σ(i)t

)2

= ω(i) +α(i)(u

(i)t−1

)2

+ β(i) (σ

(i)t−1

)2

, i = 1, . . . , N

εtiid∼ stdMNTS(λ∆D

t, θ∆D

t,ν∆D

t,Σ∆D

t),

where

• rt = (r(1)t , . . . , r

(N)t ) is the vector return of N assets at time t.

• ηt = (η(1)

∆(1)t

, . . . , η(N)

∆(N)t

) is the vector mean return at time t,

• η(i)

∆(i)t

is the mean return of the ith asset at time t, η(i)

∆(i)t

is determined

by the regime of the ith asset at time t, ∆(i)t ,

• σt =(σ

(1)

∆(1)t ,t

, . . . , σ(N)

∆(N)t ,t

)is the vector of standard deviation of N

assets at time t

14

• σ(i)t =

(σ

(i)1,t, . . . , σ

(i)

k(i),t

)is the standard deviation vector of the ith asset

with(σ

(i)t

)2

=

((σ

(i)1,t

)2

, . . . ,(σ

(i)

k(i),t

)2)

,

• u(i)t−1 is the ith element of ut, with ut =

(u

(1)t , . . . , u

(N)t

),

• ω(i) = (ω(i)1 , . . . , ω

(i)

k(i)),

• α(i) = (α(i)1 , . . . , α

(i)

k(i)),

• β(i) = (β(i)1 , . . . , β

(i)

k(i)),

• denotes element-wise product,

• and εt = (ε(1)t , . . . , ε

(N)t ) is the joint resuduals.

For each asset, the volatility process is the same as that of univariateregime-switching GARCH model in (13). It is a classic GARCH process whenit does not shift within regimes. Unlike the most general model in which thevariance at time t is calculated with the variance at time t − 1 in the lastregime, when a regime shift takes place at time t, the variance in our modelat time t is determined by the variance at time t− 1 within the new regime.The regimes of each asset and joint innovations evolve independently, alle-viating the estimation difficulty caused by all parameters switching regimessimultaneously. Intuitively, the correlation of assets change drastically whena market regime shift takes place. Thus here for simplicity we assume thatthe multivariate innovations switches regimes with the market.

3.2 Model Estimation

Though flexible enough to accommodate the many stylized facts of assetreturn, the lack of an analytical form of NTS distribution presents somedifficulties in estimation. Our estimation methodology is adapted from Kimet al. (2011), Kim et al. (2012), which is among the first works to incorporateNTS distribution in portfolio optimization. Another estimation of stdMNTSdistribution with EM algorithm combined with FFT is studied in Bianchiet al. (2016).

15

Denote the parameters of MNTS distribution in each regime as λk, θk,γk =(γ

(1)k , . . . , γ

(N)k

),νk =

(ν

(1)k , . . . , ν

(N)k

),Σk, where k is the regime, super-

script is the asset. k = 1, . . . , N . To fit the model, we follow the proceduresbelow.

First, we fit univariate Markov regime switching GARCH model with tinnovation on the index return, estimate the regime path and extract theresiduals. The regime path of the joint residuals is assumed to be the sameas this regime path. The residuals are used to estimate tail parameters λk, θkfor later input. Then, we fit univariate model on each asset and extractthe residuals for subsequent estimation of stdMNTS distribution. In eachregime, the stdMNTS distribution has two tail parameters λk, θk and oneskewness parameter vector νk. Common tail parameters λk, θk are assumedfor individual constituents, which are estimated from the DJIA index in eachregime respectively. This leaves the skewness parameters νk to be calibratedby curve fitting for each asset. Finally, to compute Σk with the formula Σk =diag(γk)

−1(ΣXk− 2−λk2θkν>k νk)diag(γk)

−1, we explicitly estimate the covariancematrix of the innovations in each regime ΣXk. Note that Σk by definitionis supposed to be positive definite matrix. To achieve this, we can eithersubstitute νk with the closest vector to νk in L2 norm which renders ΣXk −2−λk2θkν>k νk positive definite matrix, or directly find the closest positive definite

matrix to Σk. The latter method is applied in our empirical study. It’sknown that historical correlation matrix is unreliable. We find that the out-of-sample performance is enhanced with denoising techniques in Laloux et al.(1999), Juliane and Korbinian (2005). The method in Laloux et al. (1999) isapplied in our empirical study.

To summarize, the procedure is as follows.Step 1: Fit univariate Markov regime switching GARCH model with t inno-vation on the index return, estimate the regime path and extract the resid-uals.Step 2: Estimate common tail parameters λk, θk in each regime.Step 3: Fit univariate model on each asset and extract the residuals.Step 4: Estimate skewness parameters νk for assets in each regime, calculate

γk =√

1− νk2(2−λk2θk

).

Step 5: Calculate the denoised correlation matrices of joint residuals ΣXk

in each regime.Step 6: Calculate the matrix diag(γk)

−1(ΣXk − 2−λk2θkν>k νk)diag(γk)

−1, andfind the closest positive definite matrix Σk.

16

The joint residuals are assumed to follow a stdMNTS Hidden Markovprocess. In our approach, the stdMNTS distribution is estimated withineach market regime.

The number of regimes of each asset is set as equal to that of DJIA index.We find that univariate Markov switching model with more than 3 states of-ten has one unstable state that lasts for a very short period and switchesfrequently and sometimes has one state with clearly bimodal residuals dis-tribution. Thus it’s desirable to limit the number of states smaller than 4,check unimodal distriution with Dip test and choose the one with highestBIC value.

3.3 Simulation

To conduct simulation with the model, we use the following procedures.Step 1: Simulate S sample paths of standard deviation of N assets for Tdays with the fitted model.Step 2: Simulate S sample paths of a Markov chain ∆D with the market’stransition matrix. Calculate the total number of each regime in all samplepaths.Step 3: Draw i.i.d. samples from stdMNTS distribution in different regimes.The number of samples is determined by the calculation in Step 2. To drawone sample, we follow the procedure below.Step 3.1: Sample from multivariate normal distribution N(0,Σk).Step 3.2: Sample from the tempered stable subordinator T with parametersλk and θk.Step 3.3: Calculate X = νk(T − 1) +

√T (γk ξk).

Step 4: Multiply the standard deviation in step 1 with standard joint resid-uals in Step 3 accordingly. Add regime-specific mean ηk to get simulatedasset returns.

17

4 Portfolio Optimization

Conditioned on given sample path(s), classic portfolio optimization is formu-lated as a trade-off between risk and return

minxW (x, T,Ω)

s.t. Eω [R(x, T, ω)] ≥ d,

x ∈ V,

where Ω is the set of all sample paths, R(x, T, ω) is the portfolio return atend time T andW (x, T,Ω) is a risk measure, x = (x1, . . . , xn) is the allocationvector of n assets. Since there are multiple sample paths in the case ofsimulation, the expected value is used. A typical setting of V is x|

∑Ni=1 xi =

1, xi ≥ 0, meaning that short selling is not allowed. performance ratio canbe defined by substituting standard deviation in Sharpe ratio with other riskmeasures.

Similar to Markowitz Model, efficient frontier can be derived by changingd. Each portfolio on the efficient frontier is optimal in the sense that noportfolio has higher return with the same value of risk measure. For clarifi-cation in this paper, we refer to the portfolio with the highest performanceratio W (x,T,Ω)

Eω [R(x,T,ω)]as optimal portfolio and the others on efficient frontier as

suboptimal portfolios.Some problems regarding the risk measures needs further clarification. If

we use historical sample paths as input for CVaR optimization, each assethas only one path, and we will minimize the CVaR of daily portfolio returnfrom time t = 1 to t = T . However, this approach implicitly assumes thatthe daily returns are i.i.d. Since GARCH model leads to non-i.i.d. simulateddaily returns, we only minimize the CVaR of return at time t = T when weuse simulated sample paths as input.

In terms of drawdown, the common definition of absolute drawdown (ab-breviated as drawdown) is

DD(x, t, ω) = maxs∈[0,t]

P (x, s, ω)− P (x, t, ω), for t ∈ [0, T ]

The common definition of relative drawdown is

DD(x, t, ω) =maxs∈[0,t] P (x, s, ω)− P (x, t, ω)

maxs∈[0,t] P (x, s, ω), for t ∈ [0, T ]

18

However, the algorithm developed in Checkhlov et al. (2005) aims tominimize absolute drawdown due to its good properties like convexity. Whilethe properties of absolute drawdown facilitates the computation, a naturalconcern to this approach is that when we consider a long period, drawdownsat different times may differ greatly in absolute value but close in relativevalue, of which the latter we may care more about. Using uncompoundedaccumulated return in the setting could alleviate this problem. From (1), weeasily see that

DD(x, t, ω) =t∑

u=tx

Q(x, u, ω)

where tx is the value of t that gives maxu∈1,2,...,tQ(x, u, ω). It is not af-fected by the absolute value of the asset through time. It can also beviewed as an approximation to relative drawdown, since

∑tu=tx

Q(x, u, ω) ≈∏tu=tx

(1 + Q(x, u, ω)) when Q(x, u, ω) is small. This is the reason we useuncompounded accumulated return as input to CDaR optimization in thefollowing empirical study.

5 Empirical Study

In this section, we perform model fitting, simulation and portfolio optimiza-tion to demonstrate the superiority of our model.

For diversification, we set the range of weights as [0.01, 0.15]. A casestudy in Krokhmal et al. (2002) found that without constraints on weight,the optimal portfolio only consists of a few assets among hundreds. We limitthe number of states to be smaller than 4. Dip test is conducted on residualsto ensure unimodal residual distribution. When the p-value is lower than 0.1,model with fewer states is used instead. With this condition satisfied, theone with highest BIC is selected. To be concise in notation, we denote theportfolios in a convenient manner. For example, 0.9-CDaR portfolio denotesthe optimal portfolio derived from the optimization with 0.9-CDaR as riskmeasure. The standard deviation optimal portfolio is the optimal portfolio inclassic Markowitz model that maximizes Sharpe ratio, which we also denotedas mean-variance(MV) optimal portfolio.

19

5.1 Data

The data comprises of the adjusted daily price of DJIA index, 29 of the con-stituents of DJIA Index and 3 ETFs TMF, SDOW and UGL, from January2010 to September 2020. One constituent DOW is removed for that it’s notincluded in the index in all tested period. Since we use 1764 trading days’(about 7 years) data to fit the model, the actual tested period is from January2017 to September 2020, which includes some of the most volatile periods inhistory caused by the COVID-19 pandemic and trade tensions.

5.2 In-sample Test

We present an in-sample test result. The time period used in fitting is from2013-07-19 to 2020-07-22. 2-regime model is favored in this period. 1075days is classified as regime 1 and 690 days is classified as regime 2.

5.2.1 Kolmogorov–Smirnov Test

Note that the marginal distribution of stdMNTS distribution is still NTSdistribution. To examine the goodness of fit of stdMNTS distrbution on thejoint residuals, we report in Table 1 the p value and KS statistics of KS tests.As a comparison, we fit multivariate t-distribution on the joint residuals andconducted KS test on marginal distribution similarly. We also report theβ of each asset in different regimes. The degree of freedom of multivariatet-distribution in 2 regimes is 5.6 and 16.1 respectively. The data are roundedto 3 decimals place.

We can observe that β varies significantly in different regimes. Generally,for all assets, β in regime 2 has much larger absolute values, indicating asignificant skewness. NTS fits very well for all residuals in regime 1 dueto its ability to model distribution with a high peak, skewness and fat tails,where t-distribution are frequently rejected with p value close to 0. In regime2, NTS distribution performs similarly to t-distribution.

5.2.2 Residuals

Transition matrix of the residuals is shown in Table 2. Regime 2 whichcorresponds higher volatility has a lower self-transition probability.

We also provide the denoised correlation matrices of stdMNTS residualsin 2 regimes in Table 3. Since there are 2 regimes identified in this period, we

20

Table 1: KS Test

Regime β KS Statistics of NTS p value KS Statistics of t p valueAAPL regime 1 -0.078 0.021 0.654 0.08 0

regime 2 41.938 0.123 0.081 0.12 0.093AXP regime 1 0.063 0.014 0.975 0.078 0

regime 2 38.155 0.097 0.271 0.09 0.347BA regime 1 0.029 0.015 0.945 0.068 0

regime 2 70.137 0.077 0.553 0.084 0.438CAT regime 1 -0.063 0.041 0.041 0.069 0

regime 2 83.58 0.153 0.014 0.157 0.011CSCO regime 1 0.055 0.02 0.752 0.066 0

regime 2 42.562 0.079 0.511 0.078 0.538CVX regime 1 -0.062 0.042 0.034 0.062 0

regime 2 66.65 0.138 0.035 0.13 0.056DDG regime 1 0.053 0.045 0.018 0.092 0

regime 2 -67.095 0.16 0.009 0.148 0.02DIS regime 1 0.082 0.01 1 0.072 0

regime 2 52.844 0.134 0.046 0.132 0.051DOG regime 1 -0.004 0.046 0.016 0.031 0.2

regime 2 -51.012 0.099 0.251 0.092 0.323GS regime 1 -0.044 0.037 0.077 0.055 0.002

regime 2 46.111 0.123 0.081 0.117 0.112HD regime 1 0 0.031 0.217 0.044 0.021

regime 2 49.479 0.088 0.374 0.084 0.445IAU regime 1 0 0.04 0.045 0.047 0.012

regime 2 -69.031 0.029 1 0.042 0.99IBM regime 1 0.107 0.01 1 0.087 0

regime 2 68.142 0.083 0.455 0.089 0.366INTC regime 1 -0.024 0.016 0.926 0.06 0.001

regime 2 82.535 0.134 0.045 0.145 0.024JNJ regime 1 0.036 0.017 0.877 0.058 0.001

regime 2 93.757 0.043 0.986 0.083 0.448JPM regime 1 -0.027 0.038 0.073 0.063 0

regime 2 53.155 0.132 0.051 0.125 0.074KO regime 1 0.032 0.02 0.733 0.06 0

regime 2 84.339 0.031 1 0.065 0.744MCD regime 1 0.028 0.005 1 0.071 0

regime 2 43.145 0.068 0.696 0.066 0.732MMM regime 1 -0.163 0.022 0.635 0.099 0

regime 2 58.869 0.125 0.071 0.128 0.064MRK regime 1 0.022 0.016 0.923 0.058 0.001

regime 2 73.085 0.102 0.22 0.101 0.233MSFT regime 1 0.047 0.019 0.803 0.06 0

regime 2 28.885 0.085 0.429 0.082 0.463NKE regime 1 0.006 0.017 0.888 0.065 0

regime 2 71.82 0.081 0.478 0.091 0.338PFE regime 1 -0.056 0.035 0.108 0.061 0

regime 2 75.966 0.111 0.148 0.111 0.146PG regime 1 -0.06 0.017 0.868 0.071 0

regime 2 54.857 0.073 0.614 0.079 0.521REK regime 1 0.039 0.055 0.002 0.045 0.019

regime 2 -34.95 0.072 0.63 0.063 0.779RTX regime 1 0.025 0.025 0.447 0.06 0

regime 2 28.971 0.093 0.318 0.087 0.399TRV regime 1 0.009 0.027 0.363 0.071 0

regime 2 70.286 0.036 0.999 0.048 0.962UNH regime 1 -0.026 0.022 0.602 0.063 0

regime 2 51.7 0.078 0.534 0.075 0.588V regime 1 -0.012 0.031 0.211 0.056 0.001

regime 2 34.036 0.091 0.336 0.087 0.394VGLT regime 1 -0.016 0.062 0 0.031 0.208

regime 2 -51.37 0.047 0.968 0.043 0.988VZ regime 1 -0.029 0.028 0.33 0.05 0.006

regime 2 52.603 0.073 0.61 0.074 0.596WBA regime 1 0.003 0.007 1 0.07 0

regime 2 47.054 0.07 0.674 0.072 0.638WMT regime 1 -0.096 0.019 0.797 0.085 0

regime 2 86.902 0.069 0.686 0.081 0.481XOM regime 1 0.023 0.039 0.062 0.043 0.026

regime 2 42.583 0.12 0.094 0.115 0.12

Table 2: Transition Matrix

Regime 1 Regime 2

Regime 1 0.8964 0.1036

Regime 2 0.2069 0.7931

21

stack the upper tridiagonal part of the matrix with elements in the 1st regimeand the lower tridiagonal part with the 2nd together for easier comparison.We find that the residuals have distinctively different correlation matricesin 2 regimes, validating the regime switching assumption. The residuals arehighly correlated in the 2nd regime that corresponds to more volatile periods.

5.3 Out-of-sample Test

Out-of-sample test with real market data is an informative way to test theeffectiveness of a model. We use 3 types of risk measures with different con-fidence levels in the optimization, namely, maximum drawdown, 0.7-CDaR,0.3-CDaR, average drawdown, 0.5-CVaR, 0.7-CVaR, 0.9-CVaR and standarddeviation. Note that 0-CVaR is equal to the expected return, which does notmake sense as a risk measure. Minimizing α-CVaR with α smaller than 0.5means that we include the right part of the return distribution with manypositive values. This is a flaw of standard deviation and thus we set α at 0.5,0.7,and 0.9 to demonstrate the superiority.

Rolling window technique is employed with 1764 days forward movingtime window for biweekly estimation of the parameters. We simulate 1000sample paths of length 10 every two weeks with the fitted model. The simula-tion is used as input for portfolio optimization. The portfolio is hold for twoweeks until rebalance. The portfolio optimization is performed with softwarePSG with precoded numerical optimization procedure.

5.3.1 Optimal Portfolios

We report the performance of the optimal portfolios in various forms.

Ratio Measurement We use performance ratios, i.e. the mean return ofrealized path divided by a risk measure of realized path, to measure perfor-mance of the optimal portfolios. The result is reported in Table 4. The rownames denote the risk measures in the portfolio optimization. The columnsare the performance measures. For example, 10-day 0.5-CVaR portfolio isoptimized every 10 days to maximize the ratio of expected return to 10-day 0.5-CVaR, i.e. 0.5-CVaR of the return at day 10, the performance ismeasured with the ratio of mean return to risk of the whole realized path.Note that 0-CDaR is essentially average drawdown while 1-CDaR is maxi-mum drawdown. The risk measure of standard deviation is the sutdied in

22

Tab

le3:

Den

oise

dC

orre

lati

onM

atri

xof

Joi

nt

Res

idual

sin

2R

egim

es

AAPLAM

GN

AXP

BA

CAT

CRM

CSCO

CVX

DIS

GS

HD

HON

IBM

INTC

JNJ

JPM

KO

MCD

MM

MM

RK

MSFT

NKE

PG

SDOW

TM

FTRV

UGLUNH

VVZ

WBA

WM

T

AAPL

1-0.01

0.1

0.02

0.1

0.2

0.1

-0.05

0.03

0.1

0.1

0.1

0.04

0.2

-0.03

0.1

-0.01

0.1

0.1

-0.1

0.2

0.05

-0.1

-0.1

-0.040.005-0.02

0.01

0.2

-0.1

-0.004

-0.01

AM

GN

0.4

10.1

0.1

0.1

0.01

0.1

0.04

0.1

0.1

0.1

0.1

0.1

0.02

0.1

0.1

0.1

0.1

0.1

0.1

0.002

0.1

0.1

-0.2

0.03

0.1

0.01

0.1

0.1

0.1

0.1

0.1

AXP

0.4

0.4

10.1

0.2

0.1

0.2

0.1

0.1

0.3

0.2

0.2

0.2

0.2

0.02

0.3

0.04

0.1

0.2

0.1

0.2

0.1

-0.04

-0.4

-0.2

0.1

-0.1

0.1

0.2

0.002

0.1

0.04

BA

0.4

0.4

0.5

10.1

0.01

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.1

0.03

0.2

0.03

0.02

0.1

0.1

0.02

0.1

0.001

-0.2

-0.1

0.1

-0.1

0.1

0.1

0.03

0.05

0.03

CAT

0.5

0.4

0.5

0.5

10.1

0.1

0.1

0.1

0.3

0.1

0.2

0.1

0.1

0.01

0.3

0.02

0.02

0.1

0.1

0.1

0.1

-0.04

-0.3

-0.2

0.1

-0.1

0.1

0.1

0.003

0.1

0.03

CRM

0.4

0.4

0.4

0.4

0.4

10.1

-0.1

0.030.02

0.1

0.1

0.04

0.2

0.02

0.02

0.05

0.1

0.1

-0.02

0.3

0.1

0.002

-0.1

0.03

0.03

0.03

0.01

0.2

-0.02

0.005

0.02

CSCO

0.4

0.4

0.5

0.5

0.5

0.4

1-0.001

0.1

0.1

0.1

0.2

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.04

0.2

0.1

0.02

-0.3

-0.02

0.1

-0.01

0.1

0.2

0.02

0.04

0.05

CVX

0.4

0.4

0.5

0.4

0.5

0.4

0.5

10.1

0.2

0.05

0.1

0.1

-0.02

-0.000

0.2

-0.01-0.04

0.1

0.1

-0.1

0.05

-0.03

-0.2

-0.1

0.1

-0.1

0.1

-0.03

0.02

0.04

0.01

DIS

0.4

0.4

0.5

0.4

0.5

0.4

0.5

0.4

10.2

0.1

0.1

0.1

0.1

0.04

0.2

0.05

0.03

0.1

0.1

0.04

0.1

0.01

-0.2

-0.1

0.1

-0.1

0.1

0.1

0.04

0.05

0.04

GS

0.5

0.5

0.6

0.5

0.6

0.5

0.6

0.5

0.5

10.2

0.2

0.2

0.2

-0.1

0.5

-0.1

-0.1

0.2

0.02

0.1

0.1

-0.2

-0.5

-0.3

0.1

-0.3

0.1

0.1

-0.1

0.1

-0.02

HD

0.4

0.4

0.5

0.5

0.5

0.4

0.5

0.4

0.5

0.5

10.2

0.1

0.1

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.1

0.03

-0.3

-0.1

0.1

-0.04

0.1

0.1

0.04

0.1

0.1

HON

0.5

0.5

0.6

0.5

0.6

0.5

0.6

0.5

0.5

0.6

0.5

10.2

0.2

0.1

0.3

0.1

0.1

0.2

0.1

0.1

0.2

0.1

-0.4

-0.1

0.2

-0.1

0.1

0.2

0.1

0.1

0.1

IBM

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.4

0.4

0.5

0.4

0.5

10.1

0.1

0.2

0.1

0.1

0.1

0.1

0.1

0.1

0.03

-0.3

-0.1

0.1

-0.1

0.1

0.1

0.05

0.1

0.1

INTC

0.4

0.4

0.4

0.4

0.5

0.4

0.5

0.4

0.4

0.5

0.4

0.5

0.4

10.002

0.2

0.02

0.1

0.1

-0.02

0.3

0.1

-0.04

-0.2

-0.1

0.05

-0.04

0.05

0.2

-0.04

0.02

0.02

JNJ

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.4

0.4

1-0.1

0.2

0.1

0.1

0.2

-0.004

0.1

0.2

-0.2

0.2

0.1

0.1

0.1

0.1

0.2

0.1

0.1

JPM

0.5

0.5

0.6

0.5

0.6

0.5

0.6

0.5

0.5

0.6

0.5

0.6

0.5

0.5

0.5

1-0.1

-0.04

0.2

0.04

0.1

0.1

-0.1

-0.5

-0.3

0.1

-0.3

0.1

0.1

-0.05

0.1

-0.01

KO

0.3

0.4

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.3

0.4

0.4

0.4

0.3

0.4

0.4

10.2

0.1

0.2

0.03

0.1

0.2

-0.2

0.2

0.1

0.1

0.1

0.1

0.2

0.1

0.1

MCD

0.3

0.4

0.4

0.4

0.4

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

10.1

0.1

0.1

0.1

0.1

-0.1

0.1

0.1

0.1

0.04

0.1

0.1

0.05

0.1

MM

M0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.5

0.6

0.5

0.6

0.5

0.5

0.5

0.6

0.4

0.4

10.1

0.1

0.1

0.05

-0.3

-0.1

0.1

-0.1

0.1

0.1

0.1

0.1

0.1

MRK

0.4

0.4

0.4

0.4

0.4

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.4

0.4

0.4

0.4

0.4

0.4

0.5

1-0.1

0.1

0.2

-0.2

0.1

0.1

0.05

0.1

0.03

0.2

0.1

0.1

MSFT

0.4

0.4

0.5

0.5

0.5

0.4

0.5

0.5

0.5

0.5

0.5

0.6

0.5

0.4

0.5

0.5

0.4

0.4

0.5

0.4

10.1

-0.04

-0.2

-0.01

0.02

0.01

0.02

0.3

-0.1

0.001

0.01

NKE

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.4

0.5

0.4

0.4

0.4

0.5

0.3

0.3

0.5

0.4

0.4

10.1

-0.3

-0.02

0.1

-0.02

0.1

0.1

0.1

0.1

0.1

PG

0.3

0.4

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.3

0.4

0.4

0.4

0.3

0.5

0.4

0.5

0.4

0.4

0.4

0.4

0.3

1-0.1

0.2

0.1

0.2

0.04

0.03

0.2

0.1

0.1

SDOW

-0.6

-0.6

-0.7

-0.6

-0.7

-0.6

-0.7

-0.6

-0.6

-0.7

-0.6

-0.8

-0.6

-0.6

-0.6

-0.7

-0.5

-0.5

-0.7

-0.6

-0.7

-0.6

-0.5

10.2

-0.3

0.1

-0.2

-0.3

-0.1

-0.1

-0.1

TM

F-0.3

-0.2

-0.4

-0.3

-0.4

-0.3

-0.3

-0.3

-0.3

-0.4

-0.3

-0.4

-0.3

-0.3

-0.2

-0.4

-0.1

-0.2

-0.3

-0.2

-0.3

-0.3

-0.1

0.4

10.03

0.2

-0.04-0.003

0.1

0.01

0.1

TRV

0.4

0.4

0.5

0.5

0.5

0.4

0.5

0.4

0.5

0.5

0.5

0.5

0.5

0.4

0.5

0.5

0.4

0.4

0.5

0.5

0.5

0.4

0.5

-0.6

-0.2

10.02

0.1

0.1

0.1

0.1

0.1

UGL

-0.1

-0.1

-0.2

-0.1

-0.2

-0.1

-0.1

-0.1

-0.1

-0.2

-0.1

-0.2

-0.1

-0.1

-0.1

-0.2

0.001-0.05

-0.1

-0.1

-0.1

-0.1

0.01

0.2

0.2

-0.1

1-0.04

0.004

0.1

0.001

0.1

UNH

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.4

0.5

0.4

0.4

0.4

0.5

0.4

0.4

0.5

0.4

0.4

0.4

0.4

-0.6

-0.3

0.4

-0.1

10.1

0.1

0.1

0.05

V0.5

0.4

0.5

0.5

0.5

0.4

0.5

0.5

0.5

0.6

0.5

0.6

0.5

0.5

0.5

0.6

0.4

0.4

0.6

0.4

0.5

0.4

0.4

-0.7

-0.3

0.5

-0.2

0.5

10.02

0.05

0.1

VZ

0.3

0.4

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.3

0.5

-0.5

-0.1

0.4

-0.01

0.4

0.4

10.1

0.1

WBA

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.4

0.5

0.4

0.4

0.4

0.5

0.4

0.3

0.5

0.4

0.4

0.4

0.4

-0.6

-0.2

0.4

-0.1

0.4

0.4

0.4

10.1

WM

T0.3

0.3

0.3

0.3

0.3

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.3

0.3

0.4

0.3

0.4

0.3

0.4

0.4

0.3

0.3

0.4

-0.4

-0.1

0.4

-0.01

0.3

0.3

0.4

0.3

1

23

the classical Markowitz model, of which the optimziation is also called MVoptimization.

The outperformance of our model and the two tail risk measures in termsof the ratios is apparent. The performance measures are consistent in thesense that a portfolio outperforming in one measure also outperforms inother measures as well in most cases. We find that 2 tail risk measures leadto similar performance. The confidence level only has a slight impact. BothCVaR and CDaR optimal portfolios slightly but consistently outperform MVoptimal portfolio.

Accumulated Return To visualize the performance, we report in Figure 1the performance of the optimal portfolios with different risk measures in out-of-sample tests. The legend shows colors that represent different risk mea-sures used in each portfolio optimization. We use log base 10 compoundedaccumulated return as vertical axis so that the scale of relative drawdown canbe easily compared in graphs by simply counting the grids. The performancewith different risk measures is reported separately. The labels in the legendshows the confidence level of risk measure. The names of the subfigures indi-cate the risk measures used in the optimization. DJIA index and the equallyweighted portfolio are included in each graph for comparison.

As can be observed in the graphs, all the optimal portfolios follow asimilar trend, though differ in overall performance. They tend to alleviateboth left tail and right tail events. For all CDaR optimal portfolios withdifferent input, confidence level have only a small impact on the realizedpath, which often overlap. It’s possibly because the simulation length is only10 days and the drawdown distribution is far from continuous, leading toclose calculation of drawdowns with different confidence level.

Relative Drawdown Series, Return Distribution and AllocationSince the CVaR and CDaR risk measures concern the tail behavior, we plotthe relative drawdown series in Figure 2 and the return distribution in Fig-ure 3 to demonstrate the ability to alleviate left tail events. We intend toconsolidate the conclusion drawn from last section that

(1) Optimization with the simulated sample paths by our model leads togood optimal portfolios with higher performance than the index and equallyweighted portfolio.

(2) Tail risk measure CDaR and CVaR is superior to standard deviation

24

Tab

le4:

Per

form

ance

ofO

pti

mal

Por

tfol

ios

Mea

nR

etu

rn0-C

DaR

Mea

nR

etu

rn0.3

-CD

aR

Mea

nR

etu

rn0.7

-CD

aR

Mea

nR

etu

rn1-C

DaR

Mea

nR

etu

rn0.5

-CV

aR

Mea

nR

etu

rn0.7

-CV

aR

Mea

nR

etu

rn0.9

-CV

aR

Mea

nR

etu

rnS

tan

dard

Dev

iati

on

0-C

DaR

0.07

20.0

510.0

270.0

080.2

580.1

560.0

770.1

17

0.3-

CD

aR

0.07

40.0

520.0

270.0

080.2

620.1

590.0

780.1

19

0.7-

CD

aR

0.07

30.0

520.0

270.0

080.2

600.1

580.0

770.1

18

1-C

DaR

0.07

50.0

530.0

280.0

070.2

590.1

570.0

780.1

17

10-d

ay0.

5-C

VaR

0.0

76

0.0

540.0

280.0

070.2

590.1

580.0

760.1

20

10-d

ay0.7

-CV

aR

0.0

720.0

510.0

270.0

070.2

570.1

570.0

770.1

21

10-d

ay0.9

-CV

aR

0.0

710.0

500.0

270.0

070.2

430.1

470.0

720.1

16

Sta

nd

ard

Dev

iati

on0.0

64

0.0

450.0

240.0

060.2

380.1

450.0

710.1

09

DJIA

0.011

0.0

080.0

040.0

010.0

640.0

400.0

190.0

33

Equ

al

Wei

ght

0.0

320.0

230.0

110.0

030.1

360.0

830.0

370.0

65

25

0.0

0.1

0.2

0.3

0.4

2017 2018 2019 2020Date

log

cum

ulat

ive

retu

rnEqual WeightDJIA0−CDaR0.3−CDaR0.7−CDaR1−CDaR0.5−CVaR0.7−CVaR0.9−CVaRstandard deviation

Figure 1: Performance of Optimal Portfolios

in optimization.(3) Using our model and CDaR measure together leads to best perfor-

mance, while model has a greater impact than the risk measure.Due to the large number of optimal portfolios with different risk measures,

we only study 0-CDaR, 0.5-CVaR and MV optimal portfolios here. Theothers have similar results.

As is shown in Figure 2, the 0-CDaR, 0.5-CVaR and MV optimal portfo-lios have significantly smaller relative drawdown in most period in the out-of-sample test, while 0-CDaR and 0.5-CVaR optimal portfolios are slightlybetter than standard deviation portfolio.

We plot the kernel density estimation on the return distribution in Figure3. To better compare the tails of return distribution, we also plot the logbase 10 density. We can easily observe that the optimal portfolios are veryclose and alleviate both right and left tail events. They have thinner tails onboth sides than DJIA and equally weighted portfolio and slightly skewed tothe right. The optimal portfolios have lower peak than the equally weightedportfolio but still much higher peak than DJIA. Among the 3 optimal port-folios, the 10-day 0.5-CVaR portfolio has the thinnest left tail and a fatterright tail than the other 2 optimal portfolios.

26

0.0

0.1

0.2

0.3

2017 2018 2019 2020Date

Rel

ativ

e D

raw

dow

n0−CDaR10−day 0.5−CVaRDJIAEqual WeightStandard Deviation

Figure 2: Relative drawdown series of 0-CDaR, 10-day 0.5-CVaR, MV opti-mal portfolios, DJIA and equally weighted portfolio

We also report the series of allocation weights among the DJIA con-stituents and the ETFs in Figure 4. It’s reasonable that when the weightson DJIA constituents are high, the weights on shortselling ETF is low.

5.3.2 Suboptimal Portfolios

So far we have been studying the optimal portfolios that maximize per-formance ratios. In this section, we study whether the optimal portfoliooutperforms suboptimal ones.

We use 9 portfolios with different constraint on return to approximate theefficient frontier for each risk measure. For example, in each 0.3-CDaR op-timization, we perform 10 optimization with different constraints on return.The portfolio with the highest return-risk ratio is the approximated optimalportfolio. We denote the portfolio that has n lower level of constraint onreturn as level Ln suboptimal portfolio, the one that has n higher level ofconstraint on return as level Hn suboptimal portfolio. When a certain opti-mization is unfeasible, we set the allocation the same as that of a lower level.The constraints on return are (0.002, 0.010, 0.020, 0.030, 0.035, 0.040, 0.045,0.050, 0.060, 0.065). For example, when the portfolio with constraint 0.050

27

0

10

20

30

40

50

−0.10 −0.05 0.00 0.05 0.10return

dens

ity

0−CDaR10−day 0.5−CVaRStandard DeviationEqual WeightDJIA

(a) Kernel Density Estimation

1e−12

1e−07

1e−02

−0.10 −0.05 0.00 0.05 0.10return

log

dens

ity

0−CDaR10−day 0.5−CVaRStandard DeviationEqual WeightDJIA

(b) log Kernel Density Estimation

Figure 3: Kernel Density Estimation on 0-CDaR, 10-day 0.5-CVaR, MVoptimal portfolios, DJIA and equally weighted portfolio

on return has the highest performance ratio in simulation, it’s labeled theoptimal portfolio in the table, which is an approximation to the true optimalone. The portfolio with constraint 0.060 is labeled H1, while H2 to H4 havethe same constraint 0.065. Note that at each rebalance, the optimal portfoliochanges, so does portfolios of other levels, i.e. constrait 0.060 may no longerbe optimal. The performance of each label is measured with portfolios ofcorresponding level determined at each period.

Due to the size of data, we only report suboptimal portfolios with riskmeasures of 0-CDaR, 0.5-CVaR and standard deviation in Figure 5 and Ta-ble 5. We find that the optimal portfolio is consistently too conservative forall risk measures and input. That is, with the same risk measure, subopti-mal portfolios with a higher constraint on return achieve higher performanceratios than the optimal portfolio. The performance does not follow the ex-pected trend that it would first monotonically increase until reaching thepeak then monotonically decrease.

The portfolios have increasing return as well as risk from L1 to H4. Therealized paths follow similar trend with rare cross. The optimal portfolios,as expected, are in the medium part of all paths. The 10-day 0.5-CVaRsuboptimal portfolios have almost identical performance. Some paths arenot visible due to overlap. For example, the constraints on return of H4 aresometimes unfeasible, leading to same performance with H3 portfolio.

28

Tab

le5:

Per

form

ance

ofSub

opti

mal

Por

tfol

ios

Mean

Return

0-C

DaR

Mean

Return

0.3

-CD

aR

Mean

Return

0.7

-CD

aR

Mean

Return

1-C

DaR

Mean

Return

0.5

-CVaR

Mean

Return

0.7

-CVaR

Mean

Return

0.9

-CVaR

Mean

Return

Standard

Devia

tio

n

0-C

DaR

L4

0.061

0.043

0.023

0.008

0.239

0.146

0.074

0.113

L3

0.058

0.041

0.021

0.007

0.232

0.142

0.071

0.108

L2

0.061

0.043

0.023

0.007

0.236

0.144

0.072

0.110

L1

0.062

0.044

0.023

0.007

0.236

0.144

0.072

0.110

Optimal

0.072

0.051

0.027

0.008

0.258

0.156

0.077

0.117

H1

0.074

0.052

0.028

0.008

0.251

0.151

0.073

0.115

H2

0.082

0.058

0.030

0.008

0.259

0.157

0.076

0.120

H3

0.087

0.061

0.032

0.009

0.267

0.161

0.078

0.124

H4

0.087

0.062

0.032

0.009

0.268

0.162

0.079

0.125

10-d

ay

0.5-C

VaR

L4.

0.073

0.052

0.027

0.006

0.252

0.154

0.074

0.114

L3

0.074

0.052

0.027

0.007

0.253

0.155

0.075

0.115

L2

0.074

0.052

0.027

0.007

0.253

0.155

0.075

0.116

L1

0.075

0.053

0.027

0.007

0.256

0.156

0.076

0.118

Optimal

0.076

0.054

0.028

0.007

0.259

0.158

0.076

0.120

H1

0.075

0.053

0.027

0.007

0.255

0.155

0.075

0.118

H2

0.075

0.053

0.027

0.007

0.256

0.156

0.075

0.119

H3

0.073

0.052

0.027

0.007

0.256

0.155

0.075

0.119

H4

0.071

0.050

0.026

0.007

0.254

0.154

0.074

0.117

Sta

ndard

Deviation

L4

0.041

0.029

0.015

0.004

0.176

0.110

0.055

0.084

L3

0.042

0.030

0.016

0.004

0.177

0.111

0.056

0.083

L2

0.049

0.035

0.019

0.005

0.196

0.122

0.062

0.092

L1

0.059

0.042

0.022

0.006

0.225

0.138

0.069

0.103

Optimal

0.064

0.045

0.024

0.006

0.238

0.145

0.071

0.109

H1

0.069

0.049

0.026

0.007

0.232

0.141

0.069

0.108

H2

0.071

0.050

0.027

0.007

0.232

0.141

0.069

0.110

H3

0.074

0.052

0.028

0.007

0.237

0.144

0.070

0.113

H4

0.073

0.052

0.027

0.007

0.234

0.142

0.069

0.112

DJIA

0.011

0.008

0.004

0.001

0.064

0.040

0.019

0.033

EqualW

eight

0.032

0.023

0.011

0.003

0.136

0.083

0.037

0.065

29

Figure 4: Weights on 0-CDaR Optimal Portfolio on DJIA Constituents and3 ETFs

0.00

0.25

0.50

0.75

1.00

2017 2018 2019 2020Date

Wei

ght

DJIA Constituents SDOW TMF UGL

6 Conclusion

We propose the MRS-MNTS-GARCH model by combining Markov RegimeSwitching Model with the multivariate NTS-GARCH model. The model ac-commodates the regime switch and addresses tail risk in financial portfolioreturn data. We simulate sample path of portfolio returns using the modeland find optimal portfolio with CDaR, CVaR and standard deviation riskmeasure. We conduct in-sample and out-of-sample tests to examine the ef-fectiveness of the model. In in-sample test, the model fits the residuals withhigh accuracy. In out-of-sample tests, we show that the proposed modelsignificantly raises the performance of optimal portfolios measured by per-formance ratios, and that tail risk measures are slightly better than standarddeviation. By combining MRS-MNTS-GARCH model and optimization withtail risk measures, the optimal portfolios successfully alleviate extreme lefttail events. We also find that optimal portfolios tend to be conservative. Thatis, with the same risk measure, suboptimal portfolios with a higher constrainton return achieve higher performance ratios in the empirical study than theoptimal portfolio.

30

0.0

0.1

0.2

0.3

0.4

2017 2018 2019 2020Date

log

cum

ulat

ive

retu

rn

Equal WeightH1H2H3H4L1L2L3L4Optimal

(a) 0-CDaR Suboptimal Portfolios

0.0

0.1

0.2

0.3

0.4

2017 2018 2019 2020Date

log

cum

ulat

ive

retu

rn

Equal WeightH1H2H3H4L1L2L3L4Optimal

(b) 10-day 0.5-CVaR SuboptimalPortfolios

0.0

0.1

0.2

0.3

0.4

2017 2018 2019 2020Date

log

cum

ulat

ive

retu

rn

Equal WeightH1H2H3H4L1L2L3L4Optimal

(c) Standard Deviation SuboptimalPortfolios

Figure 5: Suboptimal Portfolios

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33