upload_ty3zw3hblftyvhloiiihzy7gbx.pdf

Ann.

Acad.

Rom.

Sci.

Ser.

Math.

Appl.

ISSN

2066-6594

Vol.

18,

No.

4/2026

CONSISTENCY

OF

PORTFOLIO

ALLOCATION

STRATEGIES

ACROSS

INDEPENDENT

ASSET

UNIVERSES:

EVIDENCE

FROM

DJIA

BACKTESTING

∗

Nazish

Shahid

†

Jiawei

Guo

‡

Haoyang

Liu

§

DOI 10.56082/annalsarscimath.2026.4.305

Abstract

To

determine

whether

portfolio

strategy

outcomes

are

dataset-de-

pendent

or

reﬂect

broader

trends,

this

study

examines

two

distinct

as-

set

sets.

We

evaluate

ﬁve

allocation

models—Equal

Weighting,

Maxi-

mum

Sharpe

Ratio,

Inverse

Variance,

Markowitz

Mean-Variance,

and

Robust

Optimization—using

two

15-stock

groups

derived

from

the

Dow

Jones

Industrial

Average.

Strategy

eﬀectiveness

is

measured

through

a

standardized

backtest

incorporating

a

40-day

warm-up

win-

dow

and

regular

rebalancing

schedules.

Evaluation

criteria

include

historical

returns,

risk

metrics,

turnover

frequency,

drawdowns,

and

execution

fees.

Rather

than

relying

on

a

single

backtest,

this

work

speciﬁcally

tests

whether

strategic

performance

remains

uniform

across

∗

Accepted

for

publication

on

October

08,

2026

†

nazishshahid@fccollege.edu.pk

,

Department

of

Mathematics,

Forman

Chris-

tian

College

(A

Chartered

University),

Lahore,

Pakistan;

School

of

Statistics

and

Data

Science,

Capital

University

of

Economics

and

Business,

Beijing,

China;

nazishshahid@arizona.edu

,

Department

of

Mathematics,

The

University

of

Arizona,

Tucson,

United

States

‡

32023230041@cueb.edu.cn

,

School

of

Statistics

and

Data

Science,

Capital

University

of

Economics

and

Business,

Beijing,

China

§

32023230067@cueb.edu.cn

,

School

of

Statistics

and

Data

Science,

Capital

University

of

Economics

and

Business,

Beijing,

China

305

Consistency

of

portfolio

allocation

strategies

306

diﬀerent

asset

pools.

The

ﬁndings

demonstrate

broad

structural

con-

sistency

across

both

tested

groups,

though

localized

shifts

occur

in

sec-

ondary

strategy

return

rankings.

Although

the

Equal

Weighted

model

produces

the

top

cumulative

return,

the

Inverse

Variance

approach

re-

liably

delivers

superior

downside

protection

by

minimizing

volatility,

curtailing

drawdowns,

and

maintaining

steady

position

weights.

Con-

versely,

optimization-driven

techniques

incur

greater

rebalancing

ac-

tivity

and

transaction

expenses

without

yielding

better

out-of-sample

performance.

These

results

indicate

that

risk-focused,

resilient

alloca-

tion

frameworks

oﬀer

more

dependable

portfolio

behavior

than

models

dependent

on

historical

return

forecasts.

Keywords:

portfolio

allocation,

backtesting,

portfolio

optimization,

equal

weighting,

inverse

variance

strategy,

maximum

Sharpe

ratio,

robust

optimization,

risk

management,

diversiﬁcation,

Dow

Jones

Industrial

Aver-

age

(DJIA).

MSC:

91G10,

90C90,

62P05.

1

Introduction

Portfolio

allocation

is

one

of

the

most

widely

studied

problems

in

ﬁnance

because

investment

performance

depends

not

only

on

the

selection

of

as-

sets

but

also

on

how

capital

is

distributed

among

them.

The

foundations

of

modern

portfolio

theory

were

established

by

Markowitz

[

10

],

who

formalized

the

trade-oﬀ

between

expected

return

and

risk

through

mean-variance

opti-

mization.

Subsequent

developments

by

Sharpe

[

20

,

21

]

extended

the

analysis

of

risk-adjusted

performance

and

provided

a

theoretical

framework

linking

expected

returns

to

systematic

risk.

These

contributions

continue

to

serve

as

the

foundation

for

modern

portfolio

construction

methods.

Over

time,

portfolio

allocation

research

has

expanded

far

beyond

the

original

mean-variance

framework.

Important

developments

include

im-

proved

covariance

estimation

techniques,

robust

optimization

models,

diver-

siﬁcation-based

allocation

methods,

and

Bayesian

approaches

that

incorpo-

rate

investor

views

into

the

asset

allocation

process

[

1

,

5

,

7

,

8

].

More

recently,

comprehensive

reviews

of

portfolio

optimization

have

documented

the

evo-

lution

of

the

ﬁeld

from

classical

ﬁnancial

models

toward

more

sophisticated

approaches

that

consider

multiple

objectives,

uncertainty,

and

practical

im-

plementation

constraints

[

16

,

17

].

Despite

their

theoretical

appeal,

optimization-based

portfolio

models

re-

main

highly

sensitive

to

estimation

error.

The

quality

of

portfolio

alloca-

tions

depends

heavily

on

expected

return

and

covariance

estimates,

which

N.

Shahid,

J.

Guo,

H.

Liu

307

are

inevitably

subject

to

uncertainty.

Michaud

[

14

]

argued

that

optimized

portfolios

often

exhibit

instability

due

to

estimation

error,

while

DeMiguel

et

al.

[

4

]

demonstrated

that

simple

diversiﬁcation

rules

can

frequently

outper-

form

complex

optimized

portfolios

in

out-of-sample

settings.

These

ﬁndings

have

raised

important

questions

regarding

whether

the

additional

complex-

ity

of

optimization

models

consistently

translates

into

superior

realized

in-

vestment

performance.

As

a

consequence,

simple

diversiﬁcation

methods

have

received

increas-

ing

attention

in

both

academic

and

practical

portfolio

management.

Equal-

weighted

portfolios

avoid

estimation

error

entirely

and

provide

broad

diver-

siﬁcation

across

securities.

Previous

research

has

shown

that

equal

weight-

ing

often

performs

competitively

relative

to

more

sophisticated

optimization

techniques

[

4

,

15

].

Similarly,

diversiﬁcation

and

risk-allocation

studies

have

shown

that

portfolio

performance

can

often

be

improved

by

focusing

on

risk

distribution

rather

than

return

forecasting

[

3

,

9

,

13

].

These

approaches

have

become

increasingly

attractive

because

of

their

transparency,

simplicity,

and

implementation

stability.

Among

risk-based

approaches,

inverse

variance

and

related

risk-parity

strategies

allocate

capital

according

to

asset

risk

characteristics

rather

than

expected

returns.

Such

methods

seek

to

reduce

exposure

to

highly

volatile

assets

while

maintaining

diversiﬁcation

beneﬁts.

Research

on

risk-based

portfolio

construction

has

shown

that

these

approaches

can

improve

downside-

risk

control

and

achieve

attractive

risk-adjusted

performance,

particularly

when

return

forecasts

are

unreliable

[

3

,

9

].

In

many

practical

settings,

port-

folios

constructed

using

simple

risk

measures

have

exhibited

performance

that

is

comparable

to,

or

better

than,

portfolios

derived

from

more

complex

optimization

procedures.

To

address

the

limitations

of

classical

optimization,

robust

portfolio

opti-

mization

methods

have

been

developed

to

explicitly

account

for

uncertainty

in

model

parameters.

Rather

than

relying

on

a

single

point

estimate

of

expected

returns,

robust

approaches

seek

allocations

that

remain

eﬀective

under

a

range

of

plausible

market

conditions

[

5

].

Recent

studies

continue

to

demonstrate

that

robust

optimization

can

improve

allocation

stability,

mit-

igate

estimation

risk,

and

produce

stronger

out-of-sample

performance.

Ex-

amples

include

distributionally

robust

optimization

frameworks,

robust

re-

turn

prediction

models,

sparse

robust

portfolio

selection

approaches,

and

hy-

brid

optimization

techniques

that

combine

robustness

with

decision-making

algorithms

[

6

,

23

–

25

].

The

rapid

growth

of

computational

ﬁnance

has

further

expanded

port-

folio

optimization

research.

Modern

studies

increasingly

employ

machine

Consistency

of

portfolio

allocation

strategies

308

learning,

artiﬁcial

intelligence,

and

reinforcement

learning

techniques

to

sup-

port

dynamic

asset

allocation

decisions

[

2

,

22

].

While

these

methods

oﬀer

new

opportunities

for

portfolio

construction,

recent

literature

reviews

indi-

cate

that

issues

of

robustness,

diversiﬁcation,

transparency,

and

implemen-

tation

costs

remain

central

considerations

in

practical

investment

manage-

ment

[

16

,

17

,

22

].

Interestingly,

many

empirical

studies

continue

to

report

that

simple

allocation

strategies

remain

highly

competitive

despite

advances

in

optimization

technology

[

15

].

Recent

work

has

also

highlighted

the

importance

of

understanding

asset

risk

characteristics

before

portfolio

construction.

In

[

18

],

a

volatility-based

analysis

of

KSE-30

equities

was

conducted

using

a

one-factor

credit

cop-

ula

framework

and

portfolio

optimization

techniques.

The

study

showed

that

highly

volatile

assets

tended

to

be

excluded

from

optimized

portfo-

lios,

whereas

assets

with

moderate

volatility

contributed

more

signiﬁcantly

to

ﬁnal

allocations.

In

[

19

],

portfolio

optimization

was

investigated

using

the

Black-Litterman

framework

for

the

Pakistan

Stock

Exchange

KSE-30

index.

By

combining

market

equilibrium

returns

with

analysts’

views,

the

study

demonstrated

how

forward-looking

information

can

substantially

alter

portfolio

composition

and

asset

selection

decisions.

While

[

18

]

and

[

19

]

examined

optimized

portfolio

composition–demons

trating

that

highly

volatile

assets

are

systematically

excluded

and

showing

how

forward-looking

methods

can

recalibrate

diversiﬁed

allocations—both

studies

relied

exclusively

on

expected-return-based

optimization.

Conse-

quently,

they

lacked

a

comprehensive

performance

evaluation

and

did

not

oﬀer

a

direct

comparison

between

expected-return

optimization

and

explicit

risk-based

weighting

schemes.

Although

substantial

research

exists

on

portfolio

optimization

and

port-

folio

allocation,

the

majority

of

comparative

studies

evaluate

competing

strategies

within

a

single

investment

universe.

Whether

considering

clas-

sical

mean-variance

optimization,

equal

weighting,

risk-parity

methods,

ro-

bust

optimization,

or

modern

artiﬁcial

intelligence

approaches,

conclusions

are

often

drawn

from

a

single

set

of

assets

and

a

single

portfolio

experiment.

Consequently,

it

becomes

diﬃcult

to

determine

whether

observed

perfor-

mance

diﬀerences

reﬂect

genuine

characteristics

of

the

allocation

method

or

whether

they

are

largely

driven

by

the

particular

securities

selected

for

analysis.

A

strategy

that

performs

well

in

one

portfolio

universe

may

not

necessarily

exhibit

similar

performance

when

applied

to

a

diﬀerent

set

of

assets.

The

present

study

addresses

this

limitation

by

comparing

ﬁve

portfo-

lio

allocation

strategies,

namely

Equal

Weighted,

Maximum

Sharpe

Ratio,

N.

Shahid,

J.

Guo,

H.

Liu

309

Inverse

Variance,

Markowitz

Mean-Variance

Optimization,

and

Robust

Op-

timization,

across

two

independent

asset

universes

constructed

from

Dow

Jones

Industrial

Average

(DJIA)

constituents.

Using

a

uniﬁed

backtest-

ing

framework,

portfolio

performance

is

evaluated

through

realized

return,

volatility,

drawdown,

turnover,

and

transaction-cost

measures.

Unlike

many

previous

studies

that

rely

on

a

single

investment

universe,

the

current

re-

search

focuses

on

the

consistency

of

strategy

behaviour

across

two

indepen-

dent

asset

universes.

The

study

therefore

provides

evidence

on

the

robust-

ness

of

strategy

rankings

and

oﬀers

practical

insight

into

which

portfolio

allocation

methods

provide

the

most

reliable

balance

between

return

gener-

ation,

risk

control,

diversiﬁcation,

and

allocation

stability.

Throughout

this

paper,

we

use

the

term

consistency

to

refer

speciﬁcally

to

the

preservation

of

the

relative

ranking

of

strategies

on

a

given

performance

criterion—return,

volatility,

drawdown,

or

turnover—across

the

two

independent

asset

uni-

verses,

rather

than

to

the

numerical

stability

of

the

criterion

values

them-

selves.

2

Materials

and

Methods

2.1

Dataset

and

asset

universes

The

performance

of

investment

strategies

is

evaluated

and

compared

using

historical

data

through

backtesting.

In

this

study,

one

year

of

asset

price

data

from

the

Dow

Jones

Industrial

Average

(DJIA)

market

is

considered.

Speciﬁcally,

we

analyze

the

daily

closing

prices

of

the

following

DJIA

con-

stituent

assets

to

examine

the

performance

of

diﬀerent

investment

strategies:

Apple

(AAPL),

Amgen

(AMGN),

Amazon

(AMZN),

American

Express

(AXP),

Boeing

(BA),

Caterpillar

(CAT),

Salesforce

(CRM),

Cisco

(CSCO),

Chevron

(CVX),

Walt

Disney

(DIS),

Goldman

Sachs

(GS),

Home

Depot

(HD),

Honeywell

International

(HON),

Johnson

&

Johnson

(JNJ),

Interna-

tional

Business

Machines

(IBM),

JPMorgan

(JPM),

Coca-Cola

(KO),

Mc-

Donald’s

(MCD),

3M

(MMM),

Merck

(MRK),

Microsoft

(MSFT),

NIKE

(NKE),

NVIDIA

(NVDA),

Procter

&

Gamble

(PG),

Travelers

(TRV),

Unit-

edHealth

(UNH),

Visa

(V),

Sherwin-Williams

(SHW),

Verizon

(VZ),

and

Walmart

(WMT).

The

dataset

comprises

the

DJIA

index

and

252

trading

days

of

daily

closing

prices

for

these

constituent

assets.

The

comparison

of

investment

strategies

is

conducted

separately

on

two

subsets

of

assets:

the

ﬁrst

15

and

the

last

15

asset

price

series.

This

split

is

based

purely

on

ticker

order

among

the

30

DJIA

constituents

and

is

not

Consistency

of

portfolio

allocation

strategies

310

stratiﬁed

by

sector

or

market

capitalization;

consequently,

the

two

subsets

may

diﬀer

somewhat

in

sector

composition,

which

can

inﬂuence

covariance

structure

and

the

behavior

of

variance-based

strategies.

The

analysis

based

on

the

assets

AAPL,

AMGN,

AMZN,

AXP,

BA,

CAT,

CRM,

CSCO,

CVX,

DIS,

GS,

HD,

HON,

IBM,

and

JNJ

is

referred

to

as

Analysis

A

.

The

analysis

based

on

the

assets

JPM,

KO,

MCD,

MMM,

MRK,

MSFT,

NKE,

NVDA,

PG,

SHW,

TRV,

UNH,

V,

VZ,

and

WMT

is

referred

to

as

Analysis

B

.

2.2

Portfolio

allocation

strategies

In

this

study,

ﬁve

distinct

investment

strategies

are

examined.

Each

strategy

determines

portfolio

weights

according

to

a

speciﬁc

allocation

rule,

enabling

a

comparative

analysis

of

their

performance

under

a

common

backtesting

framework.

Within

this

framework,

each

strategy

is

evaluated

at

predeter-

mined

points

in

time,

where

it

is

allowed

to

update

its

portfolio

composition

using

information

derived

from

recent

market

data.

This

study

evaluates

ﬁve

portfolio

allocation

strategies

that

translate

historical

information

on

asset

prices

into

portfolio

weights

under

a

uni-

ﬁed

backtesting

framework.

Let

N

denote

the

number

of

assets,

and

let

ω

=

(

ω

1

,

ω

2

,

.

.

.

,

ω

N

)



represent

the

vector

of

portfolio

weights,

where

ω

i

denotes

the

fraction

of

total

capital

allocated

to

asset

i

.

All

strategies

satisfy

the

full-investment

constraint



N

i

=1

ω

i

=

1

and

impose

non-negativity

con-

straints

to

prohibit

short

selling.

These

full-investment

and

non-negativity

(long-only)

constraints

are

enforced

uniformly

across

all

ﬁve

strategies

con-

sidered

in

this

study—including

the

Maximum

Sharpe

Ratio

and

Markowitz

Mean-Variance

strategies—even

though

these

two

formulations

are

some-

times

implemented

with

short

selling

permitted

in

the

broader

literature;

no

short

positions

were

allowed

in

any

of

the

backtests

reported

here.

2.2.1

Equal-weighted

strategy

The

equal-weighted

strategy

allocates

capital

uniformly

across

all

assets

in

the

investment

universe.

Each

asset

receives

the

same

weight,

independent

of

its

expected

return

or

risk

characteristics.

Formally,

the

portfolio

weights

are

deﬁned

as

[

11

,

12

]

ω

i

=

1

N

,

i

=

1

,

.

.

.

,

N.

(1)

This

strategy

serves

as

a

simple

benchmark,

as

it

avoids

estimation

error

and

does

not

rely

on

model-based

assumptions

regarding

asset

behaviour.

N.

Shahid,

J.

Guo,

H.

Liu

311

2.2.2

Maximum

Sharpe

ratio

strategy

The

maximum

Sharpe

ratio

strategy

constructs

a

portfolio

by

explicitly

opti-

mizing

risk-adjusted

performance.

The

Sharpe

ratio

measures

the

expected

excess

portfolio

return

per

unit

of

total

risk,

where

risk

is

quantiﬁed

by

the

standard

deviation

of

portfolio

returns.

Let

r

denote

the

vector

of

expected

asset

returns

and

Q

denote

the

covariance

matrix

of

asset

returns

estimated

from

historical

data.

The

portfolio

weights

are

obtained

by

solving

the

op-

timization

problem

[

12

]

ω

SR

=

arg

max

ω

r



ω



ω



Q

ω

(2)

subject

to

the

constraints

ω

≥

0

and



N

i

=1

ω

i

=

1,

and

upper

bounds

on

individual

weights

to

prevent

excessive

concentration

in

a

single

asset.

This

formulation

selects

the

portfolio

that

lies

on

the

eﬃcient

frontier

with

the

highest

slope

when

measured

relative

to

the

risk-free

asset.

In

practice,

the

strategy

emphasizes

assets

with

high

expected

returns

rela-

tive

to

their

contribution

to

overall

portfolio

volatility.

While

theoretically

appealing,

the

maximum

Sharpe

ratio

solution

is

known

to

be

sensitive

to

estimation

error,

especially

in

expected

returns,

which

may

lead

to

unstable

or

highly

concentrated

allocations

when

implemented

using

ﬁnite

samples.

2.2.3

Inverse

variance

strategy

The

inverse

variance

strategy

determines

asset

weights

using

individual

risk

measures

rather

than

expected

returns.

Let

σ

ii

denote

the

variance

of

as-

set

i

,

corresponding

to

the

diagonal

elements

of

the

covariance

matrix

Q

.

Portfolio

weights

are

assigned

inversely

proportional

to

these

variances

and

normalized

to

satisfy

the

budget

constraint

[

12

]

ω

i

=

σ

−

1

ii



N

j

=1

σ

−

1

jj

,

i

=

1

,

.

.

.

,

N.

(3)

By

emphasizing

relatively

low-variance

assets

and

ignoring

cross-asset

co-

variances,

this

approach

produces

portfolios

that

favour

stability

and

are

less

sensitive

to

estimation

noise

in

expected

returns.

2.2.4

Markowitz

mean–variance

optimization

strategy

The

Markowitz

mean–variance

strategy

formalizes

portfolio

allocation

as

a

trade-oﬀ

between

expected

return

and

risk.

Instead

of

maximizing

a

ratio,

Consistency

of

portfolio

allocation

strategies

312

this

approach

maximizes

a

quadratic

objective

function

in

which

expected

portfolio

return

is

reduced

by

a

penalty

proportional

to

portfolio

variance.

Using

the

same

notation

for

expected

returns

and

covariance,

the

optimiza-

tion

problem

is

given

by

[

12

]

ω

MV

=

arg

max

ω



r



ω

−

λ

ω



Q

ω



,

(4)

subject

to

non-negativity

and

full-investment

constraints.

The

parameter

λ

>

0

represents

the

investor’s

degree

of

risk

aversion.

Larger

values

of

λ

place

greater

emphasis

on

variance

reduction,

leading

to

more

conservative

portfolios,

while

smaller

values

allow

higher

exposure

to

risky

assets

in

pursuit

of

increased

expected

return.

Unlike

the

max-

imum

Sharpe

ratio

strategy,

which

produces

a

single

tangency

portfolio,

the

Markowitz

framework

generates

a

family

of

optimal

portfolios

along

the

eﬃcient

frontier,

each

corresponding

to

a

diﬀerent

level

of

risk

tolerance.

Although

mean–variance

optimization

provides

a

foundational

frame-

work

for

portfolio

construction,

its

eﬀectiveness

depends

critically

on

the

quality

of

the

estimated

input

parameters.

In

particular,

inaccuracies

in

ex-

pected

return

estimates

can

signiﬁcantly

aﬀect

the

resulting

optimal

weights,

motivating

the

consideration

of

alternative

methods

that

incorporate

robust-

ness

to

estimation

uncertainty.

2.2.5

Robust

optimization

with

uncertain

expected

returns

The

robust

optimization

strategy

extends

the

mean–variance

framework

by

explicitly

accounting

for

uncertainty

in

expected

returns.

Rather

than

as-

suming

that

the

expected

return

vector

is

known

exactly,

it

is

assumed

to

lie

within

a

predeﬁned

uncertainty

set

centered

around

a

nominal

estimate

r

0

.

In

this

formulation,

portfolio

weights

are

selected

to

maximize

perfor-

mance

under

the

least

favourable

realization

of

expected

returns

within

the

uncertainty

set

[

12

]

max

ω

min

r

∈S

(

r

0

)



r



ω

−

λ

ω



Q

ω



.

(5)

In

this

study,

the

uncertainty

set

S

(

r

0

)

is

speciﬁed

as

an

ellipsoid

char-

acterized

by

an

uncertainty-aversion

parameter

κ

and

a

matrix

capturing

estimation

error

in

expected

returns

[

12

]

S

(

r

0

)

=



r

:

(

r

−

r

0

)



Σ

−

1

r

(

r

−

r

0

)

≤

κ

2



.

(6)

N.

Shahid,

J.

Guo,

H.

Liu

313

This

formulation

leads

to

a

robust

counterpart

of

the

Markowitz

optimiza-

tion

problem,

in

which

an

additional

penalty

term

is

introduced

to

control

sensitivity

to

return

uncertainty.

The

resulting

allocations

aim

to

improve

stability

and

performance

robustness

when

return

estimates

are

imprecise.

2.3

Analysis

A

For

this

part

of

the

analysis,

the

closing

prices

of

AAPL,

AMGN,

AMZN,

AXP,

BA,

CAT,

CRM,

CSCO,

CVX,

DIS,

GS,

HD,

HON,

IBM,

and

JNJ

are

used

to

compute

the

portfolio

weights

corresponding

to

diﬀerent

investment

strategies.

2.3.1

Backtesting

framework

and

strategy

implementation

The

portfolio

strategies

considered

in

this

study

are

dynamic,

meaning

that

portfolio

weights

are

periodically

revised

as

new

market

information

becomes

available.

At

each

rebalancing

date,

the

existing

portfolio

composition

is

compared

with

the

allocation

suggested

by

the

strategy’s

decision

rule,

and

the

portfolio

is

adjusted

accordingly.

This

process

allows

the

strategies

to

respond

to

changing

market

conditions

while

maintaining

practical

invest-

ment

constraints

such

as

full

capital

allocation

and

the

absence

of

short

selling.

By

distinguishing

changes

caused

by

market

movements

from

those

resulting

from

deliberate

reallocation

decisions,

the

framework

provides

a

realistic

setting

for

evaluating

portfolio

performance

over

time.

Before

the

evaluation

period

begins,

an

initial

calibration

stage

is

re-

quired

to

place

each

strategy

in

a

meaningful

starting

state.

Without

this

step,

portfolios

would

start

fully

in

cash

and

remain

uninvested

until

the

ﬁrst

scheduled

rebalancing

date,

which

could

distort

early

performance

re-

sults.

To

address

this

issue,

the

ﬁrst

40

trading

days

of

historical

price

data

are

used

as

a

warm-up

period.

During

this

stage,

each

strategy

estimates

the

quantities

needed

for

portfolio

construction,

including

return

and

risk

measures.

The

resulting

allocations

approximate

the

positions

the

strate-

gies

would

have

held

had

they

already

been

operating

before

the

start

of

the

evaluation

window.

To

illustrate

the

eﬀect

of

this

initialization

process

and

to

highlight

diﬀer-

ences

in

initial

capital

allocation

(Assets

A)

across

strategies,

Fig.

1

reports

the

portfolio

weights

obtained

at

the

end

of

the

40-day

warm-up

period.

Although

all

strategies

use

the

same

historical

price

data,

they

produce

dif-

ferent

starting

allocations

because

of

their

distinct

allocation

rules.

The

equal-weighted

strategy

distributes

capital

evenly

across

all

assets,

whereas

Consistency

of

portfolio

allocation

strategies

314

Figure

1:

Initial

portfolio

weights

(Assets

A)

after

the

40-day

warm-up

period

the

remaining

strategies

allocate

capital

according

to

their

individual

ob-

jectives.

The

inverse

variance

strategy

places

greater

weight

on

assets

with

relatively

lower

estimated

risk,

such

as

JNJ,

CSCO,

and

AAPL,

while

the

optimization-based

strategies

produce

more

concentrated

allocations

based

on

their

optimization

criteria.

These

weights

represent

the

initial

port-

folio

positions

used

at

the

start

of

the

backtest

and

should

be

viewed

as

calibration-based

starting

allocations

rather

than

indicators

of

future

per-

formance.

After

the

warm-up

period,

the

backtest

is

performed

using

the

remaining

data.

Portfolio

allocations

are

updated

at

discrete

rebalancing

dates

rather

than

continuously,

reﬂecting

common

portfolio

management

practice.

In

this

study,

portfolios

are

rebalanced

every

21

trading

days,

approximately

corre-

sponding

to

a

monthly

adjustment

cycle.

Allocation

decisions

are

based

on

a

rolling

look-back

window

of

recent

historical

data,

allowing

the

strategies

to

adapt

to

changing

market

conditions

while

avoiding

reliance

on

outdated

information.

All

strategies

are

evaluated

within

a

uniﬁed

backtesting

framework

that

applies

the

same

market

data,

rebalancing

schedule,

and

performance

crite-

N.

Shahid,

J.

Guo,

H.

Liu

315

ria

to

every

strategy.

To

ensure

meaningful

comparisons,

the

initial

portfo-

lio

value

and

risk-free

rate

are

speciﬁed

at

the

framework

level

and

remain

identical

across

all

simulations.

The

annualized

risk-free

rate

is

adjusted

internally

to

match

the

daily

frequency

of

the

asset

price

series.

By

stan-

dardizing

these

settings,

any

diﬀerences

in

performance

can

be

attributed

to

the

allocation

methodology

rather

than

implementation

choices.

The

simulation

begins

only

after

the

warm-up

period

has

been

completed

so

that

every

portfolio

decision

is

based

solely

on

information

available

at

the

time

it

is

made.

This

prevents

look-ahead

bias

and

ensures

that

the

reported

results

reﬂect

realistic

investment

conditions.

Throughout

the

evaluation

pe-

riod,

the

framework

records

portfolio

values,

asset

holdings,

realized

returns,

turnover,

drawdowns,

and

trading

costs.

These

outputs

provide

the

basis

for

the

subsequent

comparison

of

portfolio

performance

across

the

competing

investment

strategies.

2.3.2

Evaluation

of

backtest

performance

outcomes

(Assets

A)

Following

backtest

execution,

Table

1

outlines

the

empirical

performance

of

each

portfolio

strategy

across

risk,

return,

and

transaction

metrics.

Properly

evaluating

these

outcomes

requires

accounting

for

the

strictly

out-of-sample

nature

of

the

decision

process,

as

portfolio

weights

were

computed

using

only

historical

data

available

at

each

rebalance

date.

For

example,

the

Maximum

Sharpe

Ratio

approach

selects

weights

that

maximize

historical

risk-adjusted

returns

over

the

preceding

look-back

window.

Once

ﬁxed,

these

weights

are

evaluated

against

subsequent,

unobserved

price

trajectories.

Consequently,

allocations

optimized

on

historical

windows

can

underperform

during

sub-

sequent

periods.

This

in-sample

optimization

drag

is

reﬂected

in

the

lower

realized

returns

and

Sharpe

ratios

of

the

Maximum

Sharpe

Ratio

strategy,

reinforcing

that

optimized

historical

performance

does

not

ensure

superior

out-of-sample

results.

Table

1

compares

the

performance

of

the

ﬁve

portfolio

strategies

across

return,

risk,

and

trading

characteristics.

In

terms

of

return,

the

Equal-

Weighted

strategy

achieves

the

highest

total

(0.15462)

and

average

return

(0.00075),

followed

by

Inverse

Variance

(total

return

0.13567)

and

Robust

Optimization

(total

return

0.12119).

In

contrast,

the

Maximum

Sharpe

Ratio

(0.11600)

and

Markowitz

Mean-Variance

strategies

(0.10239)

yield

comparatively

lower

realized

performance.

This

demonstrates

that

strategies

explicitly

optimized

on

historical

data

do

not

necessarily

deliver

superior

out-of-sample

outcomes.

From

a

risk

perspective,

the

Inverse

Variance

strategy

exhibits

the

lowest

Consistency

of

portfolio

allocation

strategies

316

volatility

(0.01162)

and

the

smallest

maximum

drawdown

(0.18125),

proving

its

ability

to

stabilize

ﬂuctuations

by

allocating

greater

weight

to

lower-

variance

assets.

Robust

Optimization

achieves

a

similarly

low

volatility

(0.01194)

and

drawdown

(0.19528).

Conversely,

the

Markowitz

strategy

records

the

highest

volatility

(0.01439)

and

the

largest

maximum

drawdown

(0.23935),

reﬂecting

its

sensitivity

to

parameter

estimation

error.

Trading

activity

and

turnover

further

distinguish

the

performance

across

strategies.

The

Maximum

Sharpe

Ratio

strategy

records

the

highest

average

turnover

(0.00776)

and

buy/sell

costs

(0.40795),

driven

by

frequent

position

reallocations

from

unstable

parameter

estimates.

The

Markowitz

strategy

exhibits

a

similarly

high

average

turnover

(0.00726).

In

contrast,

the

Equal-

Weighted

(average

turnover

0.00115)

and

Inverse

Variance

strategies

(av-

erage

turnover

0.00310)

maintain

substantially

lower

rebalancing

activity,

resulting

in

superior

allocation

stability

over

time.

In

terms

of

portfolio

structure,

Equal

Weighting

guarantees

uniform

ex-

posure

across

all

constituent

assets,

whereas

Inverse

Variance

achieves

diver-

siﬁcation

through

a

risk-balanced

allocation.

Robust

Optimization

avoids

extreme

weight

concentration

while

generating

moderate

returns.

Over-

all,

the

Inverse

Variance

strategy

provides

the

most

balanced

combination

of

risk

control,

reduced

drawdown,

and

allocation

stability,

whereas

the

Equal-Weighted

strategy

stands

out

for

its

superior

cumulative

return

and

implementation

simplicity.

Results

Equal

Weighted

Max

Sharpe

Ratio

Inverse

Variance

Markowitz

Opt.

Robust

Opt.

Total

Return

0.15462

0.116

0.13567

0.10239

0.12119

Sharpe

Ratio

0.057194

0.044587

0.053834

0.036221

0.04764

Volatility

0.012483

0.012399

0.011624

0.014396

0.011948

Average

Turnover

0.0011468

0.0077634

0.0030982

0.0072563

0.0027509

Max

Turnover

0.032336

0.30743

0.16096

0.30629

0.11499

Average

Return

0.00075239

0.00059168

0.00066444

0.00056033

0.00060797

Max

Drawdown

0.19851

0.2026

0.18125

0.23935

0.19528

Average

Buy

Cost

0.060207

0.40795

0.16084

0.36997

0.14165

Average

Sell

Cost

0.060207

0.40795

0.24126

0.36997

0.14165

Table

1:

Strategy

performance

(Assets

A)

2.3.3

Evolution

of

portfolio

value

(Assets

A)

To

illustrate

the

dynamic

behavior

of

each

allocation

strategy

over

time,

Fig.

2

displays

the

equity

curves

tracking

portfolio

value

evolution

across

N.

Shahid,

J.

Guo,

H.

Liu

317

the

backtesting

window.

These

curves

capture

the

cumulative

eﬀect

of

pe-

riodic

rebalancing

and

subsequent

market

movements,

oﬀering

clear

visual

insight

into

wealth

accumulation

patterns,

growth

trajectories,

and

draw-

downs.

Speciﬁcally,

the

trajectories

in

Fig.

2

highlight

that

the

Equal-

Weighted

portfolio

achieves

the

steepest

upward

growth

trajectory,

while

the

Inverse

Variance

curve

demonstrates

the

smoothest,

most

stable

trajectory

with

minimal

visible

pullbacks.

Conversely,

the

Markowitz

and

Maximum

Sharpe

Ratio

curves

exhibit

noticeable

structural

dips

and

heightened

ﬂuc-

tuations,

visually

conﬁrming

their

higher

drawdown

vulnerability

and

lower

out-of-sample

stability

compared

to

non-optimized

approaches.

Figure

2:

Evolution

of

portfolio

value

under

competing

investment

strategies

(Assets

A)

The

average

turnover

across

strategies

is

shown

in

Fig.

3

,

providing

insight

into

the

frequency

of

portfolio

rebalancing

and

associated

trading

activity.

As

illustrated,

the

Maximum

Sharpe

Ratio

and

Markowitz

strate-

gies

exhibit

the

highest

turnover

levels,

reﬂecting

frequent

weight

realloca-

tions

driven

by

sensitivity

to

historical

parameter

estimates.

Conversely,

the

Equal

Weighted

and

Inverse

Variance

strategies

maintain

signiﬁcantly

lower

turnover,

demonstrating

superior

allocation

stability

over

time.

From

a

practical

standpoint,

lower

rebalancing

activity

makes

these

passive

and

risk-

balanced

approaches

more

favorable

because

it

minimizes

execution

drag

and

Consistency

of

portfolio

allocation

strategies

318

preserves

net

returns.

However,

turnover

alone

is

not

the

sole

benchmark

for

evaluating

portfolio

performance;

it

must

be

assessed

alongside

net

real-

ized

returns,

maximum

drawdowns,

and

volatility

to

capture

the

complete

trade-oﬀ

between

implementation

costs

and

risk-adjusted

eﬃciency.

Figure

3:

Average

portfolio

turnover

across

investment

strategies

(Assets

A)

To

examine

how

portfolio

allocations

evolve

over

time,

the

dynamic

asset

positions

implied

by

each

strategy

are

visualized

using

area

charts

of

daily

portfolio

weights.

Fig.

4

through

Fig.

8

present

the

evolution

of

asset-level

portfolio

weights

over

time

for

the

ﬁve

allocation

strategies

considered

in

Analysis

A.

Each

ﬁgure

is

displayed

as

a

stacked

area

plot,

where

individual

components

represent

the

proportion

of

total

portfolio

capital

allocated

to

each

as-

set

at

a

given

point

in

time,

and

the

aggregate

height

reﬂects

the

full-

investment

constraint.

The

temporal

variation

in

these

stacked

proﬁles

captures

both

passive

changes

induced

by

asset

price

movements

between

rebalancing

dates

and

active

adjustments

arising

from

periodic

portfolio

re-

optimization.

Across

all

ﬁgures,

localized

contractions

in

individual

layers

correspond

to

periods

where

the

associated

asset

receives

a

reduced

alloca-

tion,

highlighting

the

dynamic

redistribution

of

capital

across

the

investment

universe.

N.

Shahid,

J.

Guo,

H.

Liu

319

Equal-weighted

strategy

(A)—Asset

allocation

over

time

Fig.

4

presents

the

dynamic

allocation

path

of

the

equal-weighted

strategy,

where

asset

weights

evolve

solely

due

to

price

ﬂuctuations

between

rebalance

dates.

Capital

remains

evenly

distributed

across

all

constituent

stocks,

displaying

smooth

weight

adjustments

over

time.

This

continuous

stability

accounts

for

the

low

trading

turnover

observed

in

Table

1

and

conﬁrms

its

role

as

a

baseline

benchmark

for

Analysis

A.

Figure

4:

Asset-level

portfolio

positions

over

time

(Equal-weighted

strategy,

A)

Maximum

Sharpe

ratio

strategy

(A)—Asset

allocation

over

time

Fig.

5

highlights

the

portfolio

allocation

dynamics

under

the

maximum

Sharpe

ratio

strategy,

where

weight

adjustments

are

driven

by

periodic

re-

calculations

of

risk-adjusted

performance.

The

stacked

area

plot

exhibits

pronounced

reallocation

spikes,

as

capital

is

dynamically

shifted

toward

top-

performing

assets

while

other

position

sizes

contract

or

vanish

completely.

These

notable

weight

swings

align

with

the

elevated

turnover

and

higher

trading

costs

reported

in

Table

1

,

reﬂecting

the

sensitivity

of

tangency

op-

timization

to

input

parameters.

Inverse

variance

strategy

(A)—Asset

allocation

over

time

Fig.

6

illustrates

the

weight

evolution

for

the

inverse

variance

strategy,

demonstrat-

Consistency

of

portfolio

allocation

strategies

320

Figure

5:

Asset-level

portfolio

positions

over

time

(Maximum

Sharpe

ratio

strategy,

A)

ing

the

persistent

allocation

structure

derived

from

risk-based

weighting.

The

portfolio

maintains

stable

position

sizes

in

low-volatility

constituents

(such

as

JNJ,

CSCO,

and

AAPL),

with

layer

thickness

adjusting

gradu-

ally

across

rebalance

points.

This

defensive

asset

distribution

directly

con-

tributes

to

the

lowest

volatility

and

smallest

maximum

drawdown

recorded

in

Table

1

.

Markowitz

optimization

strategy

(A)—Asset

allocation

over

time

Fig.

7

tracks

the

portfolio

composition

over

time

for

the

Markowitz

mean–

variance

strategy,

capturing

the

ongoing

trade-oﬀ

between

return

and

risk.

The

allocation

paths

exhibit

sharper

position

adjustments

across

assets

as

updated

parameter

estimates

modify

the

optimal

weighting

decision.

These

variable

allocation

blocks

help

explain

the

strategy’s

increased

turnover,

higher

trading

expenditure,

and

larger

drawdown

in

Table

1

.

Robust

optimization

strategy

(A)—Asset

allocation

over

time.

Fig.

8

depicts

the

allocation

path

generated

by

the

robust

optimization

strategy,

showing

how

asset

weights

adapt

over

time

while

incorporating

parameter

uncertainty.

Compared

to

the

standard

Markowitz

approach,

the

robust

formulation

leads

to

smoother

and

more

balanced

position

tran-

N.

Shahid,

J.

Guo,

H.

Liu

321

Figure

6:

Asset-level

portfolio

positions

over

time

(Inverse

variance

strategy,

A)

Figure

7:

Asset-level

portfolio

positions

over

time

(Markowitz

optimization

strategy,

A)

Consistency

of

portfolio

allocation

strategies

322

sitions.

By

penalizing

return

estimation

errors,

it

mitigates

extreme

cap-

ital

shifts

while

preserving

portfolio

ﬂexibility

and

maintaining

controlled

volatility

(Table

1

).

Figure

8:

Asset-level

portfolio

positions

over

time

(Robust

optimization

strategy,

A)

2.4

Analysis

B

For

this

part

of

the

analysis,

the

closing

prices

of

JPM,

KO,

MCD,

MMM,

MRK,

MSFT,

NKE,

NVDA,

PG,

SHW,

TRV,

UNH,

V,

VZ,

and

WMT

are

used

to

compute

the

weights

of

the

diﬀerent

investment

strategies.

To

highlight

diﬀerences

in

initial

capital

allocation

across

strategies,

Fig.

9

displays

the

portfolio

weights

computed

at

the

end

of

the

warm-up

period,

where

each

strategy

translates

the

same

historical

price

informa-

tion

into

a

starting

allocation.

The

equal-weighted

strategy

gives

uniform

exposure

to

the

full

asset

set

and

therefore

serves

as

the

broad

diversiﬁca-

tion

benchmark.

Under

the

inverse

variance

strategy,

assets

such

as

KO,

MCD,

PG,

V,

VZ,

and

WMT

receive

comparatively

larger

weights,

indicat-

ing

that

their

estimated

variances

over

the

calibration

window

were

relatively

low

compared

to

other

assets.

The

maximum

Sharpe

ratio

and

Markowitz

strategies

are

more

selective,

assigning

substantial

weight

to

a

narrower

set

of

assets

while

giving

very

small

positions

to

others.

The

robust

optimiza-

N.

Shahid,

J.

Guo,

H.

Liu

323

tion

strategy

keeps

a

wider

spread

of

exposures

than

the

more

aggressive

optimization

rules,

with

visible

emphasis

on

assets

that

combine

moderate

risk

with

stable

historical

performance.

The

strategies

diﬀer

mainly

in

how

they

balance

concentration,

defensive

exposure,

and

sensitivity

to

estimated

returns.

These

initial

weights

should

therefore

be

interpreted

as

calibration-

period

positions

rather

than

evidence

of

long-term

asset

performance.

Figure

9:

Initial

portfolio

weights

(Assets

B)

after

the

40-day

warm-up

period

2.4.1

Evaluation

of

backtest

performance

outcomes

(Assets

B)

After

running

the

backtest,

Table

2

summarizes

the

performance

of

each

strategy

for

Assets

B

using

standard

evaluation

metrics:

realized

return,

volatility,

drawdown,

turnover,

and

trading

cost.

The

market

conditions

and

testing

rules

were

kept

the

same

for

all

methods.

The

interpretation

of

Table

2

must

account

for

the

fact

that

portfolio

de-

cisions

are

made

using

only

information

available

before

each

rebalance

date.

This

point

is

especially

relevant

for

Analysis

B

because

the

optimization-

based

strategies

often

form

concentrated

allocations

from

a

limited

estima-

tion

window.

Their

selected

portfolios

may

appear

eﬃcient

at

the

time

of

construction,

yet

their

realized

performance

depends

on

price

movements

Consistency

of

portfolio

allocation

strategies

324

Results

Equal

Weighted

Max

Sharpe

Ratio

Inverse

Variance

Markowitz

Opt.

Robust

Opt.

Total

Return

0.07382

0.056951

0.046644

0.048307

0.052193

Sharpe

Ratio

0.035303

0.028645

0.023888

0.023782

0.026636

Volatility

0.0096722

0.0091613

0.0089821

0.009571

0.0089964

Average

Turnover

0.0011811

0.0056974

0.0027293

0.0066467

0.0022865

Max

Turnover

0.032776

0.21149

0.10211

0.21104

0.090746

Average

Return

0.00038078

0.00030194

0.00025419

0.00026722

0.0002792

Max

Drawdown

0.11513

0.12427

0.10273

0.12852

0.10863

Average

Buy

Cost

0.061486

0.30048

0.14273

0.34909

0.1195

Average

Sell

Cost

0.061486

0.30048

0.21409

0.34909

0.1195

Table

2:

Strategy

performance

(Assets

B)

that

occur

after

the

weights

are

ﬁxed.

The

empirical

results

show

that

greater

optimization

complexity

can

fail

to

translate

into

stronger

realized

outcomes

in

this

asset

universe.

In

terms

of

return,

the

Equal-Weighted

strategy

achieves

the

highest

to-

tal

(0.07382)

and

average

return

(0.00038),

demonstrating

that

simple,

broad

capital

distribution

eﬀectively

captures

market

recovery

without

relying

on

active

selection

or

frequent

reoptimization.

The

Maximum

Sharpe

Ratio

(0.05695)

and

Robust

Optimization

strategies

(0.05219)

follow,

whereas

the

Markowitz

(0.04831)

and

Inverse

Variance

strategies

(0.04664)

yield

weaker

cumulative

performance.

Inverse

Variance

deliberately

forfeits

upside

po-

tential

by

concentrating

on

low-risk

assets,

while

the

Markowitz

strategy

is

impacted

by

parameter

instability

and

unfavorable

reallocations.

From

a

risk

perspective,

the

Inverse

Variance

strategy

provides

the

stron-

gest

downside

protection,

recording

the

lowest

volatility

(0.00898)

and

the

smallest

maximum

drawdown

(0.10273).

This

conﬁrms

that

the

Assets

B

universe

contains

suﬃcient

low-risk

constituents

for

a

variance-based

rule

to

mitigate

portfolio

stress.

Robust

Optimization

exhibits

a

similarly

defensive

proﬁle

with

low

volatility

(0.00899)

and

a

controlled

maximum

drawdown

(0.10863).

The

Equal-Weighted

strategy

incurs

slightly

higher

volatility

(0.00967)

but

maintains

a

moderate

drawdown

(0.11513)

due

to

its

broad

asset

distribution,

whereas

the

Markowitz

strategy

experiences

the

largest

maximum

drawdown

(0.12852).

Trading

activity

and

costs

further

separate

the

strategies.

The

Markowitz

strategy

records

the

highest

average

turnover

(0.00665)

and

trading

costs

(0.34909),

followed

by

the

Maximum

Sharpe

Ratio

strategy

(average

turnover

0.00570).

These

high

turnover

rates

highlight

the

sensitivity

of

classical

op-

timization

to

shifting

historical

estimates,

where

additional

trading

activity

N.

Shahid,

J.

Guo,

H.

Liu

325

fails

to

yield

a

performance

advantage.

Conversely,

the

Equal-Weighted

(average

turnover

0.00118)

and

Robust

Optimization

strategies

(average

turnover

0.00229)

maintain

high

allocation

stability

and

minimal

transaction

drag.

Overall,

the

results

demonstrate

a

clear

separation

between

broad

capital

distribution

and

risk-balanced

diversiﬁcation.

The

Equal-Weighted

strategy

oﬀers

the

strongest

total

return

and

risk-adjusted

eﬃciency

through

sim-

ple

diversiﬁcation,

while

the

Inverse

Variance

strategy

remains

the

most

eﬀective

choice

for

capital

preservation

and

downside

control.

Robust

Opti-

mization

serves

as

a

practical

middle

ground

among

optimized

methods

by

maintaining

allocation

stability

while

capturing

higher

returns

than

purely

defensive

approaches.

2.4.2

Evolution

of

portfolio

value

(Assets

B)

Fig.

10

traces

the

portfolio

value

trajectories

for

Assets

B,

illustrating

how

each

strategy

accumulates

wealth

over

the

backtest

period.

Consistent

with

the

trends

in

Analysis

A,

the

Equal-Weighted

strategy

maintains

the

high-

est

equity

curve

throughout

the

horizon,

conﬁrming

its

superior

cumulative

return

capability.

The

Inverse

Variance

curve

displays

the

most

consistent,

low-volatility

progression,

maintaining

a

smooth

upward

slope

even

during

market

pullbacks.

In

contrast,

the

Markowitz

and

Maximum

Sharpe

Ratio

curves

lag

behind,

exhibiting

sharp

drawdowns

and

noticeable

stagnation

that

visually

illustrate

their

out-of-sample

instability

and

sensitivity

to

es-

timation

errors.

The

average

turnover

across

strategies

is

shown

in

Fig.

11

,

providing

insight

into

the

frequency

of

portfolio

rebalancing

and

associated

trading

activity.

Mirroring

the

behavior

observed

in

Analysis

A,

the

results

for

Assets

B

conﬁrm

that

the

Maximum

Sharpe

Ratio

and

Markowitz

strate-

gies

generate

the

highest

turnover,

whereas

the

Equal

Weighted

and

Inverse

Variance

models

maintain

the

lowest

rebalancing

requirements.

This

con-

sistency

reinforces

that

parameter-dependent

optimizations

persistently

in-

cur

higher

trading

frequency

across

diﬀerent

asset

pools,

while

risk-weighted

and

static

allocations

oﬀer

greater

operational

stability.

Consequently,

while

low

turnover

makes

Equal

Weighted

and

Inverse

Variance

allocations

struc-

turally

more

favorable

for

cost

management,

rebalancing

frequency

remains

just

one

dimension

of

strategy

selection

and

must

be

weighed

against

draw-

down

control

and

net

realized

returns.

To

examine

how

portfolio

allocations

evolve

over

time,

the

dynamic

asset

positions

implied

by

each

strategy

are

visualized

using

area

charts

of

daily

Consistency

of

portfolio

allocation

strategies

326

Figure

10:

Evolution

of

portfolio

value

under

competing

investment

strate-

gies

(Assets

B)

Figure

11:

Average

portfolio

turnover

across

investment

strategies

(Assets

B)

N.

Shahid,

J.

Guo,

H.

Liu

327

portfolio

weights.

Fig.

12

through

Fig.

16

present

the

evolution

of

asset-level

portfolio

weights

over

time

for

the

allocation

strategies

considered

in

Analysis

B.

The

stacked

area

plots

show

how

each

method

distributes

capital

across

the

same

asset

universe

and

how

those

distributions

change

as

prices

move

and

rebalancing

decisions

occur.

Equal-weighted

strategy

(B)—Asset

allocation

over

time

Fig.

12

illustrates

the

time-varying

portfolio

composition

of

the

equal-weighted

strat-

egy.

The

allocation

remains

broadly

distributed,

and

changes

in

the

stacked

areas

mainly

reﬂect

asset

price

movements

between

rebalancing

dates.

This

pattern

explains

the

strategy’s

low

turnover

and

shows

why

it

functions

as

a

stable

benchmark

in

Analysis

B.

Figure

12:

Asset-level

portfolio

positions

over

time

(Equal-weighted

strategy,

B)

Maximum

Sharpe

ratio

strategy

(B)—Asset

allocation

over

time

Fig.

13

depicts

the

evolution

of

asset

allocations

under

the

maximum

Sharpe

ratio

strategy.

The

area

chart

displays

more

abrupt

changes

than

the

equal-

weighted

and

inverse

variance

strategies,

with

capital

shifting

toward

se-

lected

assets

as

risk-adjusted

return

estimates

are

updated.

Several

layers

Consistency

of

portfolio

allocation

strategies

328

expand

or

contract

sharply

at

rebalancing

points,

and

some

asset

exposures

become

temporarily

very

small

or

nearly

disappear

from

the

stacked

area

plot.

These

visible

reallocations

are

consistent

with

the

higher

turnover

re-

ported

in

Table

2

and

with

the

sensitivity

of

Sharpe-ratio

optimization

to

estimated

returns

and

covariance

inputs.

Figure

13:

Asset-level

portfolio

positions

over

time

(Maximum

Sharpe

ratio

strategy,

B)

Inverse

variance

strategy

(B)—Asset

allocation

over

time

Fig.

14

presents

the

allocation

dynamics

of

the

inverse

variance

strategy.

The

port-

folio

keeps

relatively

persistent

exposure

to

lower-volatility

assets,

and

its

weights

adjust

more

gradually

over

time.

This

stable

structure

supports

the

low

volatility

and

drawdown

results

observed

in

Table

2

.

Markowitz

optimization

strategy

(B)—Asset

allocation

over

time

Fig.

15

shows

the

evolution

of

portfolio

weights

for

the

Markowitz

strategy.

The

allocation

path

contains

sharper

shifts

across

assets,

reﬂecting

the

strat-

egy’s

sensitivity

to

estimated

expected

returns

and

covariance

relationships.

This

behaviour

helps

explain

its

higher

turnover,

larger

trading

cost,

and

weaker

drawdown

control.

N.

Shahid,

J.

Guo,

H.

Liu

329

Figure

14:

Asset-level

portfolio

positions

over

time

(Inverse

variance

strat-

egy,

B)

Figure

15:

Asset-level

portfolio

positions

over

time

(Markowitz

optimization

strategy,

B)

Consistency

of

portfolio

allocation

strategies

330

Robust

optimization

strategy

(B)—Asset

allocation

over

time

Fig.

16

illustrates

the

asset

allocation

path

generated

by

the

robust

optimization

strategy.

Compared

with

the

standard

Markowitz

allocation,

the

robust

strategy

produces

smoother

and

more

balanced

position

changes.

Its

al-

location

path

reﬂects

the

eﬀect

of

accounting

for

uncertainty

in

expected

returns,

which

reduces

extreme

reallocations

while

preserving

adaptability.

Figure

16:

Asset-level

portfolio

positions

over

time

(Robust

optimization

strategy,

B)

Across

these

ﬁgures,

the

deﬁning

feature

of

Analysis

B

is

the

contrast

be-

tween

stable

broad

exposure

and

concentrated

optimization.

Equal

weight-

ing

produces

the

smoothest

benchmark

allocation,

inverse

variance

delivers

a

defensive

risk-based

proﬁle,

and

robust

optimization

moderates

the

insta-

bility

associated

with

standard

mean–variance

optimization.

The

maximum

Sharpe

ratio

and

Markowitz

strategies

show

more

visible

reallocation

blocks,

which

match

their

higher

turnover

and

less

stable

realized

performance.

For

Assets

B,

allocation

stability

and

controlled

risk

exposure

contribute

more

reliably

to

long-term

performance

than

frequent

re-optimization

driven

by

unstable

historical

parameter

estimates.

N.

Shahid,

J.

Guo,

H.

Liu

331

Criterion

Equal

Weighted

(EW)

Inverse

Variance

(IV)

Robust

Optimi-

zation

(RO)

Max

Sharpe

Ratio

(MSR)

Markowitz

Mean-

Variance

(MV)

Return

performance

Highest

in

A

and

B

Second

in

A,

lowest

in

B

Moderate

Moderate-

low

Lowest

in

A,

second

lowest

in

B

Volatility

control

Moderate

Lowest

Near-lowest

Moderate

Highest

Maximum

drawdown

Moderate

Lowest

Second

lowest

Higher

Highest

Turnover

Lowest

Low-

moderate

Low

Highest

Highest

Transaction

costs

Lowest

Low

Low

Highest

Highest

Allocation

stability

Highest

High

High

Low

Low

Diversiﬁ-

cation

Maximum

Risk-based

Balanced

Concen-

trated

Concen-

trated

Sensitivity

to

estimation

error

None

Low

Controlled

High

Very

high

Practical

portfolio

construction

Excellent

Excellent

Good

Weak

Weak

Table

3:

Comparative

backtesting

summary

from

Analysis

A

and

Analysis

B

Consistency

of

portfolio

allocation

strategies

332

3

Discussion

The

results

in

Table

3

demonstrate

broad

structural

consistency

across

both

asset

universes

A

and

B.

While

the

overall

hierarchy

between

heuristic

and

optimization-driven

models

holds,

speciﬁc

return

rankings

vary

slightly

be-

tween

the

two

pools

(e.g.,

Inverse

Variance

yields

the

second-highest

return

in

Analysis

A

but

ranks

lower

in

Analysis

B,

whereas

Maximum

Sharpe

Ratio

improves

its

return

rank

in

Analysis

B).

Nevertheless,

key

perfor-

mance

characteristics—such

as

volatility

control

and

turnover

behavior—

remain

stable.

While

the

Equal

Weighting

method

yielded

the

top

cumula-

tive

return

in

both

groups,

the

Inverse

Variance

framework

systematically

achieved

the

lowest

volatility,

minimal

maximum

drawdowns,

and

modest

turnover

while

preserving

competitive

return

levels,

making

it

the

most

bal-

anced

allocation

overall.

Conversely,

despite

their

theoretical

foundation,

the

Markowitz

and

Maximum

Sharpe

Ratio

strategies

exhibited

elevated

turnover

rates,

greater

transaction

drag,

and

lower

realized

returns.

Ro-

bust

Optimization

enhanced

weighting

stability

relative

to

classical

mean–

variance

models,

yet

its

performance

remained

less

dependable

than

Inverse

Variance.

While

the

Inverse

Variance

strategy

appears

to

oﬀer

the

most

balanced

overall

performance,

the

results

also

demonstrate

that

portfolio

allocation

decisions

depend

strongly

on

the

investor’s

objective.

The

Equal

Weighted

strategy

consistently

generated

the

highest

realized

returns

in

both

analy-

ses,

making

it

attractive

when

return

maximization

is

the

primary

goal.

In

contrast,

the

Inverse

Variance

strategy

delivered

the

lowest

volatility

and

smallest

maximum

drawdowns,

indicating

superior

downside-risk

control.

It

also

maintained

relatively

low

turnover

and

transaction

costs

compared

with

the

Maximum

Sharpe

Ratio

and

Markowitz

strategies,

whose

frequent

reallocations

increased

implementation

costs

without

producing

superior

re-

alized

outcomes.

These

ﬁndings

suggest

that

investors

seeking

growth

may

prefer

Equal

Weighting,

whereas

investors

emphasizing

risk

management,

capital

preservation,

and

allocation

stability

would

likely

favor

the

Inverse

Variance

approach.

Overall,

the

latter

provides

the

strongest

combination

of

risk

control,

turnover

management,

and

drawdown

reduction,

making

it

a

compelling

choice

for

practical

portfolio

allocation.

An

additional

implication

of

the

comparative

analysis

concerns

the

iden-

tiﬁcation

of

assets

suitable

for

subsequent

portfolio

optimization.

The

volati-

lity-based

investigation

[

18

]

of

KSE-30

equities

showed

that

highly

volatile

stocks

were

generally

excluded

from

optimized

portfolios,

whereas

assets

with

lower

or

moderate

sensitivity

to

market

movements

contributed

more

N.

Shahid,

J.

Guo,

H.

Liu

333

meaningfully

to

portfolio

construction.

Similarly,

the

Black-Litterman

anal-

ysis

[

19

]

retained

only

a

limited

set

of

assets

with

favorable

return

and

risk

characteristics

after

blending

market

equilibrium

returns

with

analyst

views.

The

appearance

of

assets

such

as

HBL,

LUCK,

and

POL—constituents

of

the

Pakistan

Stock

Exchange

KSE-30

index

examined

in

[

19

]—within

the

ﬁnal

Black-Litterman

allocation

suggests

that

securities

exhibiting

both

ac-

ceptable

volatility

characteristics

and

favorable

expected-return

proﬁles

may

represent

promising

candidates

for

future

portfolio

construction.

Although

a

formal

combined

optimization

framework

would

be

required

to

conﬁrm

this

observation,

the

current

ﬁndings

provide

a

preliminary

indication

that

infor-

mation

from

both

analyses

may

assist

in

screening

equities

for

subsequent

optimal

portfolio

development

and

future

Black-Litterman

applications.

Furthermore,

the

comparison

highlights

an

important

limitation

of

his-

torically

popular

optimization-based

investment

methods.

In

portfolio

the-

ory,

the

Maximum

Sharpe

Ratio

strategy

is

frequently

regarded

as

an

at-

tractive

allocation

rule

because

it

seeks

the

highest

risk-adjusted

return

on

the

eﬃcient

frontier.

However,

the

backtesting

results

obtained

in

both

analyses

indicate

that

this

theoretical

advantage

did

not

translate

into

con-

sistently

superior

realized

performance.

The

strategy

exhibited

substantially

higher

turnover

and

transaction

costs

than

simpler

allocation

rules,

while

its

realized

returns

and

risk-adjusted

outcomes

remained

comparatively

mod-

est.

Similar

behavior

was

observed

for

the

Markowitz

strategy.

These

ﬁnd-

ings

emphasize

that

allocations

derived

from

historical

optimization

may

be

highly

sensitive

to

estimation

error

and

changing

market

conditions.

Conse-

quently,

relying

exclusively

on

historically

estimated

inputs

may

not

always

be

suﬃcient

for

index

investment

decisions,

particularly

when

implementa-

tion

costs

and

allocation

stability

are

important

considerations.

Ultimately,

this

analysis

demonstrates

that

the

Inverse

Variance

strategy

provides

the

most

balanced

overall

proﬁle,

while

revealing

broadly

consistent

structural

performance

patterns

across

two

independent

asset

universes.

Al-

though

asset-universe

composition

introduces

slight

variations

in

total

return

rankings

for

speciﬁc

strategies,

both

evaluations

yield

identical

overarching

conclusions

regarding

return

generation,

risk

control,

turnover

behavior,

and

allocation

stability.

Speciﬁcally,

simpler

diversiﬁcation

and

risk-aware

ap-

proaches

consistently

match

or

outperform

complex

optimization

methods

on

realized

portfolio

outcomes.

This

indicates

that

the

structural

stability

of

the

allocation

mechanism

is

more

vital

to

performance

than

aggressive

op-

timization

based

on

historical

estimates.

Because

this

pattern

holds

across

multiple

criteria

and

independent

datasets,

the

observed

behavior

reﬂects

a

robust

portfolio

allocation

principle

rather

than

a

dataset-speciﬁc

artifact.

Consistency

of

portfolio

allocation

strategies

334

4

Conclusion

This

study

was

conducted

to

examine

whether

diﬀerent

portfolio

allocation

strategies

produce

consistent

investment

outcomes

when

applied

to

diﬀer-

ent

groups

of

equities

within

the

same

market

environment.

Using

one

year

of

daily

closing

price

data

from

the

Dow

Jones

Industrial

Average

(DJIA)

constituents,

two

independent

analyses

were

performed

on

separate

sets

of

ﬁfteen

assets.

Five

portfolio

allocation

strategies,

namely

Equal

Weighted,

Maximum

Sharpe

Ratio,

Inverse

Variance,

Markowitz

Mean-Variance

Op-

timization,

and

Robust

Optimization,

were

evaluated

under

an

identical

backtesting

framework

with

periodic

rebalancing.

The

main

contribution

of

this

work

is

the

direct

comparison

of

simple

diversiﬁcation

approaches,

risk-based

allocation

methods,

and

optimization-

based

portfolio

strategies

across

two

independent

asset

universes

using

the

same

evaluation

methodology.

This

allows

the

robustness

of

strategy

per-

formance

to

be

assessed

rather

than

relying

on

results

from

a

single

asset

set.

The

evidence

obtained

from

both

analyses

shows

that

strategy

per-

formance

is

highly

consistent

across

the

two

asset

universes.

The

Equal

Weighted

strategy

produced

the

highest

realized

returns,

while

the

Inverse

Variance

strategy

consistently

achieved

the

lowest

volatility,

the

smallest

drawdowns,

and

relatively

low

turnover.

In

contrast,

the

Maximum

Sharpe

Ratio

and

Markowitz

strategies

exhibited

higher

turnover,

higher

transac-

tion

costs,

and

weaker

realized

performance

despite

their

theoretical

opti-

mization

advantages.

The

Robust

Optimization

strategy

provided

a

com-

promise

between

return

generation

and

risk

control

but

did

not

consistently

outperform

the

Inverse

Variance

approach.

The

principal

conclusion

supported

by

the

results

is

that

stable

and

risk-aware

allocation

methods

can

be

more

eﬀective

in

practice

than

con-

ventional

strategies

that

depend

heavily

on

estimates

of

expected

returns.

Across

both

analyses,

simple

diversiﬁcation

and

variance-based

risk

control

delivered

more

consistent

portfolio

outcomes

than

more

complex

optimiza-

tion

procedures.

These

ﬁndings

suggest

that

portfolio

allocation

decisions

may

beneﬁt

more

from

robustness,

stability,

and

risk

management

than

from

aggressive

optimization

of

historical

return

estimates.

N.

Shahid,

J.

Guo,

H.

Liu

335

5

Limitations

of

the

study

The

analysis

is

based

on

a

single

year

of

historical

data

and

two

subsets

of

DJIA

constituents.

Consequently,

the

ﬁndings

may

be

inﬂuenced

by

the

speciﬁc

market

conditions

observed

during

the

study

period.

Although

the

consistency

of

results

across

both

asset

universes

strengthens

the

conclu-

sions,

the

performance

of

the

strategies

may

diﬀer

under

alternative

market

regimes

or

longer

investment

horizons.

In

addition,

the

two

asset

subsets

were

formed

by

ticker

order

rather

than

by

a

sector-stratiﬁed

procedure;

a

sector-balanced

split

of

the

DJIA

constituents

would

help

conﬁrm

whether

the

observed

consistency

in

strategy

rankings

is

robust

to

diﬀerences

in

sec-

tor

composition

across

the

two

universes.

6

Future

research

directions

Future

work

may

extend

the

analysis

to

multiple

years

of

data

and

ad-

ditional

equity

markets

to

examine

the

robustness

of

the

observed

results

across

diﬀerent

market

environments.

The

ﬁndings

of

this

study

may

also

be

combined

with

volatility-based

screening

and

Black-Litterman

portfolio

optimization

frameworks

to

investigate

whether

assets

consistently

favored

by

risk-aware

allocation

strategies

can

improve

portfolio

construction

and

asset

selection

decisions.

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