.


International Journal of Economics and Financial 
Issues

ISSN: 2146-4138

available at http: www.econjournals.com

International Journal of Economics and Financial Issues, 2018, 8(4), 256-264.

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018256

Investment Opportunities, Uncertain Implicit Transaction Costs 
and Maximum Downside Risk in Dynamic Stochastic Financial 
Optimization

Sabastine Mushori1*, Delson Chikobvu2

1Central University of Technology, P.O. Box 1881, Welkom, 9460, South Africa, 2University of the Free State, P.O. Box 339, 
Bloemfontein, 9300, South Africa. *Email: smushori@gmail.com

ABSTRACT

A dynamic stochastic methodology in optimal portfolio selection that maximizes investment opportunities and minimizes maximum downside risk 
while taking into account implicit transaction costs incurred in initial trading and in subsequent rebalancing of the portfolio is proposed. The famous 
mean-variance (MV) model (Markowitz, 1952) and the mean absolute deviation (MAD) model (Konno and Yamazaki, 1991) both penalize gains 
(upside deviations) and losses (downside deviations) in the same way. However, investors are concerned about downside deviations and are happy 
of upside deviations. Hence the proposed model penalizes only downside deviations and, instead, maximizes upside deviations. The methodology 
maintains transaction cost at the investor’s prescribed level. Dynamic stochastic programming is employed with stochastic data given in the form of 
a scenario tree. Consideration a set of discrete scenarios of asset returns and implicit transaction costs is given, taking deviation around each return 
scenario. Model validation is done by comparing its performance with those of the MV, MAD and minimax models. The results show that the proposed 
model generates optimal portfolios with least risk, highest portfolio wealth and minimum implicit transaction costs.

Keywords: Investment Opportunities, Downside Risk, Uncertain Implicit Transaction Costs 
JEL Classifications: C01, C58, D81, G11

1. INTRODUCTION

Individual investors, investment managers and fund managers 
are all concerned with achieving optimal portfolios of a set of 
investment assets. Models for portfolio selection have evolved 
over the years starting with Markowitz’s (1952) MV formulation 
to more recent stochastic optimization forms (Hiller and Eckstein, 
1993; Vladimirou and Zenios, 1997). Regardless of whether 
portfolios are selected for a bank’s derivative mix, an investor’s 
equity holdings or a firm’s asset and liability management, the 
common objective in all models is the minimization of some 
measure of risk while maximizing some reward measure. The MV 
model has enjoyed popularity over the years despite its criticisms. 
The MV portfolio analysis has the following simplifying 
assumptions:
i. The assets’ returns are multivariate normally distributed,
ii. The investor’s utility function is quadratic, and
iii. There are no transaction costs.

None of these is exactly true in actual markets. Many studies show 
that returns from hedge funds are not normally distributed (Brooks 
and Kat, 2002). Pratt (1964) concludes that a quadratic utility 
function is very unlikely because it implies increasing absolute 
risk aversion. Volatility treats risks and opportunities equally yet 
investors are concerned about downside deviations (losses) and are 
happy of upside deviations (gains). Hakansson (1971) explains that 
in the absence of transaction costs, myopic policies are sufficient 
to achieve optimality. The incorporation of transaction costs 
in any model provides essential “friction” which without it the 
optimization has complete freedom to reallocate the portfolio every 
time-period, which (if implemented) can result in significantly 
poorer realized performance than forecast, due to excessive 
transaction costs (Hakansson, 1971). These costs can turn high-
quality investments into moderately profitable investments or 
low-quality investments into unprofitable investments (D’ Hondt 
and Giraud, 2008). In this study, a stochastic multi-stage upside-
downside deviation (SMUDTC) model is proposed that takes into 



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018 257

account a risk-averse investor’s view of minimizing maximum 
downside risk while maximizing upside deviations (gains) in 
an uncertain environment. The model captures uncertainty in 
both portfolio risk and gain by way of scenarios, which is a 
representative and comprehensive set of possible realizations of 
the future. This is achieved by taking deviations around each return 
scenario. The SMUDTC model also takes into account uncertainty 
of implicit trading costs incurred by the investor during initial 
trading and in subsequent rebalancing of the portfolio.

2. LITERATURE REVIEW

As a way of overcoming the limitations of the MV model, 
alternative risk measures were developed. Konno and Yamazaki 
(1991) propose the MAD model in order to overcome the problem 
of computational difficulty inherent in the MV model. The MAD 
model does not require calculation of the variance-covariance 
matrix of asset returns and results in optimal portfolios with fewer 
assets (Simaan, 1997). Similar to the MV formulation, the MAD 
model penalizes both upside deviations and downside deviations. 
However, upside deviations are desirable to any investor while 
downside deviations are not. Thus the proposed model maximizes 
upside deviations and minimizes downside deviations in the 
presence of implicit transaction costs. The models, MV and MAD, 
are both deterministic.

A number of researchers in the literature have studied optimal 
portfolio selection in the presence of transaction costs. Gulpinar 
et al. (2004), incorporate proportional transaction costs in a 
multi-period MV formulation. Glen (2011) considers a MV 
portfolio rebalancing strategy with transaction costs comprising 
fixed charges and variable costs that include market impact. The 
variable transaction costs are assumed to be non-linear functions 
of the traded value. However, implicit transaction costs follow a 
random-walk process and hence, the use of a non-linear function to 
approximate such costs seems inappropriate. Xia and Tian (2012) 
estimate implicit transaction costs in Shenzhen A-stock market 
using the daily closing prices, and examine the variation of the 
cost of Shenzhen A-stock market from 1992 to 2010. The Bayesian 
Gibbs sampling method proposed by Hasbrouck (2009) is used to 
analyze implicit costs in the bull and bear markets. Kozmik (2012) 
discusses asset allocation with transaction costs formulated as 
multi-stage stochastic programming model. Transaction costs are 
regarded as proportional to the value of assets bought or sold, but 
no implicit trading costs are considered in the model. Conditional 
value-at-risk is employed as a risk measure. Brown and Smith 
(2011) study the problem of dynamic portfolio optimization in 
discrete-time finite-horizon setting and also consider proportional 
transaction costs. Lynch and Tan (2010) study portfolio selection 
problem with multiple risky assets. Analytic frameworks are 
developed for the case with many assets taking into account 
proportional transaction costs. While the study of optimal asset 
allocation has received fair consideration in the literature, it is 
important such studies to have accounted for implicit transaction 
costs for they are invisible and dependent on the chosen strategy. 
These costs are difficult to measure and can turn high-quality 
investments into moderately profitable investments or low-quality 
investments into unprofitable investments (D’Hondt and Giraud, 

2008). The model being proposed addresses the impact of implicit 
transaction costs by employing dynamic stochastic programming 
which takes into account scenarios and stages. Uncertainty of asset 
returns, implicit trading costs and risk is accounted for by using a 
set of scenarios. The model employs stochastic programming with 
recourse by rebalancing portfolio compositions at discrete time-
intervals as new information on asset returns become available. 
This study’s contributions include:
a. The development of a multi-stage stochastic model that 

maximizes portfolio gains and minimizes maximum downside 
risk in the presence of uncertain implicit transaction costs 
incurred during initial trading and in subsequent rebalancing 
of portfolios,

b. The development of a strategy that captures uncertainty of 
stock prices and corresponding implicit trading costs by way 
of scenarios in uncertain environments.

3. PROBLEM STATEMENT

A multi-period discrete-time optimal portfolio strategy is determined 
over an investment horizon [0,T]. The planning phase [0.t] where 
t<T, consists of non-overlapping time-intervals indexed by 
t = 1,2,…,τ, and (τ,T] is the period to maturity of the investment. 
Initial investment takes place at t = 0, and during the period (0,τ], 
the investor makes adjustments to his portfolio at each of the τ 
periods as new information on assets’ returns become available. 
This adjustment of the portfolio results in the investor incurring 
some transaction costs as he buys and sells shares of some securities. 
These transaction costs tend to erode the benefits of investment, 
hence the need to minimize them. Thus, the study considers an 
investor who is interested in maximizing portfolio gains (upside 
deviations) while keeping portfolio losses (downside deviations) to 
the minimum and implicit transaction costs at some prescribed level. 
This is achieved by employing dynamic stochastic programming 
in which uncertainty about future events is described by a discrete 
probability distribution of random parameters carried by a finite 
number of scenarios with prescribed probabilities. It is assumed that 
this discrete probability distribution is a reliable substitute of the true 
underlying probability distribution. Each complete realization of all 
uncertain parameters is a scenario along the multi-period horizon.

3.1. Scenario Generation
Let I={i:i=1,2,…,n} be a set of risky assets for an investment. The 
information available about the single uncertain parameter, the risky 
active yield, is a set of scenarios Rist, where s∈Ω={s: s=1,2,…,S} 
is a finite set of discrete scenarios. Each scenario has an associated 
probability ps such that 

1
1

S
ss

p
=

=∑ . Scenarios are arranged in 
the form of a tree spanning along the succession periods and being 
of length equal to the planning horizon τ. Each path represents 
a scenario, s, and each node of the tree represents a time when 
a decision is made and implemented. The decision process is 
non-anticipative, that is, a decision at a particular stage does not 
depend on the future realization of the random events. The recourse 
decision at period t is dependent on the outcome of period t−1. 
Given the event history up to time t, Rt,the uncertainty in period 
t + 1 is characterised by finitely many possible outcomes for the 
observations Rt+1. Below is an example of a scenario tree with 
a three-three branching structure in a 2-time period (Figure 1).



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018258

3.2. Model Constraints
An investor who has capital W0 to spend in his initial portfolio 
is considered. This capital is distributed among the n-securities 
of the initial portfolio. Let xt = [x1,1,t,x1,2,t...,x2,1,t, x2,2,t...,xi,s,t]

T, 
i = 1,…,n;s = 1,…,S;t = 1,…,τ, be the investor’s optimal strategy 
to be achieved at the end of the planning horizon. It is noted 
that xist is the proportion of wealth, Wt, of period t allocated 
to buy shares of asset i of scenario s in period t. Observe that 

1 1
1,   1, 2, , .

n S
it isti s

x x t τ
= =

= = = …∑ ∑  If aist and vist are the 
proportions of wealth, Wt, used to buy and sell, respectively, shares 
of security i of scenario s of period t, then

xist = xi,s,t−1+aist-vist,i = 1,2,…,n;s = 1,2,…,S;t = 1...,τ. It is 
assumed that the investor cannot buy and sell the same 
asset at each time when portfolio rebalancing takes place. 
That is, aist∙vist = 0. The expected return of asset i of period 
t  i s  g i v e n  b y  , 1, , ; 1, , ,it s ist ists Qr p R x i n t τ∈= ⋅ ⋅ = … = …∑  
where Q⊂Ω is a set of scenarios of asset i of period t. Thus, 
the gross expected portfolio return of period is given by 

1
,   1, 2, , ; 1, 2, , .

S
pt s ist ists

r p R x i n t τ
=

= ⋅ ⋅ = … = …∑  Suppose that 
the portfolio is self-financing, this results in the constraint.
0≤vit≤xit, (1)

W h e r e  it ist ss Qv v p∈= ⋅∑  a n d  .it ist ss Qx x p∈= ⋅∑  T h e 
constraint ensures that the volume of asset i of period t sold for 
portfolio rebalancing must not exceed the volume of the asset in 
the portfolio. In a self-financing portfolio being rebalanced, the 
amount of money got from selling asset i of period t should be at 
most the amount of money used to buy asset j (i≠j) of the same 
period. Hence the following results:

1,
0 , 1, , , 1, , ,

n
ist jsti A j j i

a v s S t τ
∈ = ≠

≤ ≤ = … = …∑ ∑  (2)

Where set A contains all assets for which volumes have been 
bought. The investor ensures that no short-selling takes place by 
having the constraint

0≤xist≤Uist, i=1,…, n;s=1,…, S; t=1,…,τ, (3)

Where Uist is the maximum proportion allowed for each asset i of 
scenario s in period t. Let the transaction cost rates for buying and 
selling shares of asset i in scenario s of period t during portfolio 
rebalancing be kist and list respectively. Thus either kist∙aist = 0 or 
list∙vist = 0 or both are zero. This results in the transaction cost for 
buying or selling shares of asset i of scenario s of period t being 
kist∙aist+list∙vist. The expected transaction cost of the portfolio of 
period becomes.

{ }
1

,   1, , ; 1, , .
S

s ist ist ist ist ists
p k a l v R i n t τ

=
⋅ ⋅ + ⋅ ⋅ = … = …∑  

Thus getting the net expected portfolio return, Npt, of period t as

{ }
1

, 1, , ; 1, , ,
S

pt pt s ist ist ist ist ists
N r p k a l v R i n t τ

=
= − ⋅ ⋅ + ⋅ ⋅ = … = …∑  

with the net portfolio wealth of period t given by

Wt = (1+Npt)∙Wt−1,t=1,…,τ (3)

3.3. Portfolio Risk
The study assumes that the investor intends to choose a 
portfolio with minimum of the maximum downside deviations 
of asset returns relative to expected portfolio return. Let the 
downside risk of asset i of scenario s in period t be defined by 
Mist = |min [0,Rist-rpt]|.

This gives the expected portfolio risk in period t as 1 .
S

s ist ists
p M x

=∑
The expected portfolio risk for the period [0,τ] becomes

1 1

1 S
s ist istt s

p M x
τ

τ = =∑ ∑

Letting 
1

S
t s ist ists

Z p M x
=

= ∑  results in the expected portfolio 
risk for the entire planning phase as

1

1
p tt

H Z
τ

τ =
= ∑

3.4. Portfolio Gain
Portfolio gain is defined as an upside deviation of an asset return 
from expected portfolio return at any period t. The study assumes 
that the investor wants to maximize upside deviations of asset 
returns resulting in better investment opportunities. Let the upside 
deviation of asset i of scenario s of period t be defined by

Nist = max [0,Rist-rpt]

This gives the expected upside deviation of the portfolio of period 
t as 

1

S
s ist ists

p N x
=∑ . Hence the expected upside deviation of the 

portfolio for all time-periods becomes

1 1

1 S
s ist istt s

p N x
τ

τ = =∑ ∑

Figure 1: A scenario tree



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018 259

If 
1

,
S

t s ist ists
Y p N x

=
= ∑  then the expected portfolio gain for the 

whole rebalancing period is given by

1

1
p tt

Y
τ

ϕ
τ =

= ∑

3.5. The Multi-stage Stochastic Optimization 
(SMUDTC) Model
There is a bi-criteria objective to be satisfied in the study by 
maximizing portfolio gain while minimizing maximum downside 
risk. Since the intention of the investor is to minimize portfolio 
risk, it is achieved by maximizing the negative of the downside 

risk, that is, maximizing 
1

1
.p ttH Z

τ

τ =
− = − ∑  This results in a 

single objective where the investor maximizes the sum of the 
negative loss and gain of the portfolio of any period t. Thus, the 
objective function becomes

( )
1 1

1
                  ( ).p p t tt tMaximize H Y Z

τ τ
ϕ

τ = =
− = −∑ ∑

Letting λ to be the minimum acceptable transaction cost and θ to 
be the minimum portfolio expected return, the following multi-
stage stochastic model with uncertain implicit transaction costs 
is obtained:

Maximize

1 1

1
t tt t

Y Z
τ τ

τ = =
 − ∑ ∑

Subject to Wt = (1+Npt) Wt-1, t=1,…,τ,
θ≤Npt, t=1,…,τ

1
0 , 1, , ; 1, , ,

S
t s ist ists

Z p M x i n t τ
=

= − = … = …∑

1
0 ,  1, , ; 1, , ,

S
t s ist ists

Y p N x i n t τ
=

= − − = … = …∑
{ }

1
  , 1, , ; 1, , ,

S
s ist ist ist ist ists

p k a l v R i n tλ τ
=

≥ + = … = …∑

1,
0 , 1, , ; 1, , ,

n
ist istj j i

i A

a v s S t τ
= ≠

∈

≤ ≤ = … = …∑∑

1
1 , 1, , ; 1, ,

S
ists

x i n t τ
=

= = … = …∑
0≤vit≤xit, i=1,…, n; t=1,…,τ,
0≤xist≤Uist, i=1,…, n; s=1,…, S; t=1,…,τ
0≤aist, i=1,…, n; s=1,…, S; t=1,…,τ

The model (4) has a non-linear objective function, and the third 
and fourth constraints are also non-linear. In order to transform the 
model into a stochastic linear programming model, the following 
changes are made.

For each scenario s, let Kist≥Mist = |min[0,Rist-rpt]|, s=1,2,…, S. This 
gives the expected portfolio risk of period t as 

1
,

S
s ist ists

p K x
=∑

with the expected portfolio risk for the period [0,τ] becoming 

1 1

1
.

S
s ist istt s

p K x
τ

τ = =∑ ∑  T h e n  l e t t i n g  1
S

t s ist ists
J p K x

=
= ∑  

gives the expected portfolio risk for the entire planning phase as

1

1
 ttE J

τ

τ =
= ∑

Similarly, for each scenario s, let Dist≥Nist=max[0,Rist-rpt], s=1,2,…, S. 
This results in 

1

S
s ist ists

p D x
=∑  as the expected upside deviation 

of the portfolio of period t. Therefore, the expected upside 
deviation (gain) of the portfolio for the period [0,τ] becomes 

1 1

1
 .

S
s ist istt s

p D x
τ

τ = =∑ ∑ . By letting 1 ,
S

t s ist ists
Q p D x

=
= ∑  the 

expected portfolio gain for the entire planning phase becomes

1

1
tt

F Q
τ

τ =
= ∑

The model (4) becomes equivalent to the following stochastic 
linear programming model

Maximize

1 1

1
  [  ]t tt tQ J

τ τ

τ = =
−∑ ∑

                                         (5)
Subject to Wt = (1+Npt) Wt-1, t=1,…,τ,

θ≤Npt, t=1,…,τ

1
0 , 1, , ; 1, , ,

S
t s ist ists

J p K x i n t τ
=

= − = … = …∑

1
 0 , 1, , ; 1, , ,

S
t s ist ists

Q p K x i n t τ
=

= − = … = …∑

1
0 , 1, , ; 1, , ,

S
t s ist ists

J p K x i n t τ
=

= − = … = …∑

1
0 ,  1, , ; 1, , ,

S
t s ist ists

Q p D x i n t τ
=

= − − = … = …∑
{ }

1
, 1, , ; 1, , ,

S
s ist ist ist ist ists

p k a l v R i n tλ τ
=

≥ + = … = …∑

1,
0 , 1, , ; 1, , ,

n
ist isti A j j i

a v s S t τ
∈ = ≠

≤ ≤ = … = …∑ ∑

1
1 , 1, , ; 1, , ,

S
ists

x i n t τ
=

= = … = …∑
0≤vit≤xit, i=1,…, n; t=1,…,τ,
0≤xist≤Uist, i=1,…, n; s=1,…, S; t=1,…,τ
0≤aist, i=1,…, n; s=1,…, S; t=1,…,τ

The following Theorem shows that models (4) and (5) are 
equivalent and yield the same optimal values.

3.6. Theorem 1
If x* is an optimal solution to (4), then (x*, E*, F*) is an optimal 
solution to (5). Conversely, if (x*, E*, F*) is an optimal solution 
to (5), then x* is an optimal solution to (4).

Proof

Without loss of generality, let * *istx x= . If x* is an optimal solution 
to (4), then (x*, E*, F*) is a feasible solution to (5), where



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018260

And

1

1
tt

E J
τ

τ =
= ∑ 1 1

1
.

S
s ist istt s

p K x
τ

τ = =
= ∑ ∑

1 1

1
 

S
s ist istt s

p M x
τ

τ = =
≥ ∑ ∑

1 1

1
min 0, 

S
s ist ist ptt s

p x R r
τ

τ = =
 = ⋅ ⋅ − ∑ ∑

1

1
tt

F Q
τ

τ =
= ∑ 1 1

1 S
s ist istt s

p D x
τ

τ = =
= ∑ ∑

1 1

1
 

S
s ist istt s

p N x
τ

τ = =
≥ ∑ ∑

1 1

1 
(max 0, )

S
s ist ist ptt s

p x R r
τ

τ = =
 = ⋅ ⋅ − ∑ ∑

If (x*, E*, F*) is not an optimal solution to (4), then there exists 
a feasible solution (x, E, F) such that E*<E and F*<F, where

E

1 1

1 S
s ist istt s

p K x
τ

τ = =
= ∑ ∑

1 1

1
 | min[0, ] |

S
s ist ist ptt s

p x R r
τ

τ = =
= ⋅ ⋅ −∑ ∑

And

F

1 1

1 S
s ist istt s

p D x
τ

τ = =
= ∑ ∑

1 1

1
 | mix[0, ] |

S
s ist ist ptt s

p x R r
τ

τ = =
= ⋅ ⋅ −∑ ∑

It is observed that

* *
1 1

1
  | min[0, ] |

S
s ist ist ptt s

p x R r E E
τ

τ = =
⋅ ⋅ − = <∑ ∑ .

And that *
1 1

1
  | min[0, ] |

S
s ist ist ptt s

E E p x R r
τ

τ = =
< = ⋅ ⋅ −∑ ∑ .

Also * *
1 1

1
max 0, 

S
s ist ist ptt s

p x R r F F
τ

τ = =
 ⋅ ⋅ − = < ∑ ∑  and that

*
1 1

1
max 0, 

S
s ist ist ptt s

F F p x R r
τ

τ = =
 < = ⋅ ⋅ − ∑ ∑ .

This is a contradiction since x* is an optimal solution of (4).

Conversely, if (x*,E*,F*) is an optimal solution of (5), then x* 
is an optimal solution of (4). Otherwise, there exists a feasible 
solution x to (4) such that

E*
*

1 1

1
  | min[0, ] |

S
s ist ist ptt s

p x R r
τ

τ = =
= ⋅ ⋅ −∑ ∑

1 1

1
 | min[0, ] | 

S
s ist ist ptt s

p x R r
τ

τ = =
< ⋅ ⋅ −∑ ∑

And
F*

*
1 1

1
(max 0, )

S
s ist ist ptt s

p x R r
τ

τ = =
 = ⋅ ⋅ − ∑ ∑

1 1

1
(max 0, )

S
s ist ist ptt s

p x R r
τ

τ = =
 < ⋅ ⋅ − ∑ ∑

=F

Which contradicts that (x*, E*, F*) is an optimal solution to (5). 
This completes the proof.

3.7. Transaction Cost Measurement
Transaction cost analysis has become increasingly important in 
helping firms and individual investors measure how effectively 
both perceived and actual orders are executed. Transaction costs 
are either implicit or explicit. Market fees, clearing and settlement 
costs, brokerage commissions, and taxes and stamp duties are all 
explicit costs. Implicit costs are invisible and are strongly related to 
the trading strategy. They can broadly be put into three categories 
which are market impact, opportunity costs and spread. When an 
investment is immediately executed without delay, implicit costs 
are largely a result of market impact or liquidity restrictions only. 
In such a case, market impact is defined as the deviation of the 
transaction price from the `unperturbed’ price that would have 
prevailed if the trade had not occurred. In the proposed model, 
immediate trade execution is assumed and market impact is taken 
to account for the total implicit transaction costs. The study applies 
the approach by Hau (2006) and considers the transaction price 
to be the asset’s last price of the month. The spread mid-point 
benchmark is used and the effective spread is evaluated as twice the 
distance from the mid-price measured in basis points. Taking PM as 
the mid-point of the bid-ask spread and PT as the transaction price, 
the effective spread (implicit transaction cost) is calculated as

200 | |
 

T M
Trade

M
P P

SPREAD
P

× −
=

4. DATA, MODEL APPLICATION AND 
RESULTS

The study uses historical monthly data of securities traded on 
the Johannesburg Stock Market from January 2008 to September 
2012. The following criteria are used to select securities available 
for portfolio selection:
a. Stocks with negative mean returns for the entire period of 

study are excluded from the sample,
b. Companies which were not on the list by January 2008 

and only entered the Johannesburg Stock Exchange (JSE) 
afterwards are excluded, and

c. Assets having the highest positive mean returns calculated for 
the entire period are taken to become our initial portfolio.



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018 261

The study uses historical simulation and takes empirical 
distributions computed from monthly returns as equi-probable 
scenarios. Taking Pi,t to be a monthly price of asset i considered in 
period t, a return scenario, Rist, for the asset i of period t is calculated 

as , , 1
, 1

i t i t
ist

i t

P P
R

P
−

−

−
= . Five scenarios for each asset in each period t 

are considered, giving a total of 5n scenarios in each period where 
n is the number of assets for selection. A scenario consists of an 
asset return and the corresponding implicit transaction cost. It is 
assumed that both asset return and the corresponding implicit cost 
are random since it is in the buying or selling of securities, whose 
prices are random, that implicit transaction costs are incurred by 
an investor. Implicit costs are calculated from the effective bid-
ask spread corresponding to each selected asset return. The initial 
portfolio is selected from 13 securities and empirical distributions 
of these 13 securities are considered. Each security has 54 historical 
monthly returns and random numbers are used to select an asset 
return and associated implicit transaction cost corresponding to 
a scenario of a security. This is done by numbering the months 
1–54. The transaction cost is given as a rate and each scenario 
is considered as equally likely to occur, giving a probability of 

1
5n

 for each scenario. It should be noted that scenarios are only 

considered for the proposed SMUDTC model and other models use 
mean returns, mean implicit transaction costs and risks calculated 
for the entire period in the study. The period under study is divided 
into two, giving in-sample and out-of-sample data. The in-sample 
period starts from January 2008 and ends March 2012. The second 
period starts from April 2010 to September 2012. In each period, 
the performance of each model is analysed. First-stage optimal 
portfolios generated by the SMUDTC model are compared with 
optimal portfolios from the MAD, MV and minimax models 
(Appendix). In a real-life environment, comparison of models is 
usually done by means of ex-post analysis. It is also noted that 
portfolio performances are usually affected by market trend, 
hence the need to subject the proposed SMUDTC model to these 
two periods and evaluate its performance. An investor who has 
R10,000 to spend on his initial portfolio is considered in the study.

4.1. In-sample Analysis
A GAMS software is used to solve the four models. In analysing 
the performance of each model, the gross portfolio mean return, 
portfolio risk, total implicit transaction costs incurred and the 
gross and net portfolio wealth are evaluated. Neither portfolio 
mean return nor portfolio risk is constrained, and diversification 
limits from 0.1 to 0.4 are used. The optimal portfolios generated 
by the four models are shown in Table 1. The phrase `Div. Lim’ 
stands for diversification limit.

It is observed that, for each given diversification limit, the 
SMUDTC optimal portfolios have the least gross mean portfolio 
return and the investor incurs the least implicit transaction costs. 
The MM optimal portfolios have the highest gross expected 
portfolio returns followed by the MV-generated optimal portfolios. 
However, the MM-investor incurs the greatest implicit trading 
cost followed by the MV-investor. It is clearly evident, that 
despite having the least expected portfolio returns, SMUDTC 
optimal portfolios have the greatest portfolio wealth for every 

diversification limit considered. The MAD-generated optimal 
portfolios have greater wealth than those of MV and MM optimal 
portfolios. The trend remains the same as portfolios become less 
diversified.

4.2. Out-of-sample Analysis
In a real-life environment, validation of models is usually done 
by ex-post analysis. This is achieved by making a comparative 
evaluation of model performance on in-sample and out-of-sample 
data sets. The period from April 2010 to September 2012 provides 
the out-of-sample data set for the SMUDTC model. Table 2 shows 
the performance of the SMUDTC model together with MV, MAD 
and minimax models (Appendix). The information in Table 3 shows 
that MM optimal portfolios have the highest gross expected portfolio 
returns followed by MV and MAD optimal portfolios respectively.

The information in the table shows that MM optimal portfolios 
have the highest gross expected portfolio returns followed by 

Table 1: Summary statistics of in-sample optimal 
portfolios
Div. Lim MAD MV MM SMUDTC
0.10

Mean return 0.027 0.030 0.034 0.003
Risk 0.024 0.001 0.034 0.001
Gross wealth 10265.76 10303.55 10339.30 10066.72
Implicit cost 710.10 835.42 901.70 40.22
Net wealth 9555.66 9468.12 9437.60 10026.50

0.15
Mean return 0.024 0.032 0.038 0.004
Risk 0.027 0.001 0.038 0.001
Gross wealth 10236.14 10316.92 10380.50 10065.00
Implicit cost 855.25 929.73 721.45 29.33
Net wealth 9380.89 9387.18 9659.05 10035.67

0.20
Mean return 0.023 0.033 0.041 0.003
Risk 0.028 0.001 0.041 0.001
Gross wealth 10225.12 10330.33 10409.00 10056.44
Implicit cost 1092.00 818.23 746.40 25.36
Net wealth 9133.12 9512.10 9662.60 10031.08

0.25
Mean return 0.022 0.033 0.043 0.003
Risk 0.029 0.001 0.043 0.001
Gross wealth 10216.15 10330.92 10428.00 10050.86
Implicit cost 396.25 760.18 883.25 23.20
Net wealth 9819.90 9570.74 9544.75 10027.66

0.30
Mean return 0.021 0.033 0.044 0.003
Risk 0.029 0.001 0.044 0.001
Gross wealth 10211.78 10331.02 10443.60 10045.83
Implicit cost 445.30 757.81 1023.30 53.84
Net wealth 9766.48 9573.21 9420.30 10028.63

0.35
Mean return 0.021 0.033 0.046 0.003
Risk 0.029 0.001 0.046 0.001
Gross wealth 10207.83 10331.02 10457.80 10051.63
Implicit cost 444.45 757.80 1083.60 18.79
Net wealth 9763.38 9573.22 9374.20 10032.84

0.40
Mean return 0.020 0.033 0.047 0.004
Risk 0.030 0.001 0.047 0.001
Gross wealth 10204.72 10331.02 10469.20 10047.63
Implicit cost 343.80 757.79 984.40 5.52
Net wealth 9860.92 9573.23 9484.80 10042.11



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018262

MV and MAD optimal portfolios respectively. It is also evident 
that the MM optimal portfolios have the highest portfolio risk 
with the SMUDTC optimal portfolios’ risks being the least for 
all diversification levels considered in the study. The SMUDTC 
optimal portfolios bear the least implicit transaction costs with the 
MM optimal portfolios having the highest transaction costs. It is 
also clear that the SMUDTC optimal portfolio has the greatest 
wealth for every diversification limit considered. All these features 
are evident in the in-sample analysis above. Thus, the SMUDTC 
investor has advantages of having high net expected portfolio 
returns and incurring the least implicit transaction costs during 
trading over his MM-, MV- and MAD- counterparts.

4.3. Sensitivity Analysis
Model sensitivity analysis is carried out using both in-sample and 
out-of-sample data. This is done by finding the sensitivity index 
(SI) for each parameter where the output percentage difference 
is calculated by varying one input parameter from its minimum 
value (zero in this case) to its maximum value (Hoffman and 
Gardner [1983]; Bauer and Hamby [1991]). The SI is calculated 

using the formula  max min
max

D D
Sensitivity index

D
−

= , where Dmin and 

Dmax are the minimum and maximum output values respectively 
resulting from varying the input parameter over its entire range. 
Zero is taken to be the minimum input value for each parameter 
with the maximum value constrained by the diversification limit 
considered and is given as the best weight of the chosen asset in 
the optimal portfolio. Diversification limits of 0.1, 0.2 and 0.3 
are considered. The SMUDTC model is stochastic and works 
by replacing one parameter (of the optimal portfolio) by a `less’ 
profitable one when the weight of the chosen asset is zero. The 
condition that the total weight of assets in the optimal portfolio 
must be unity provides for the inclusion of a less profitable asset 
in the ̀ new’ optimal portfolio. This results in a relative sensitivity 
value. Hence, the Dmin value of model output is a relative value. 
Therefore, by applying the method above we are finding relative 
sensitivity analysis of the model parameters.

Table 2 shows in-sample model sensitivity analysis where there 
are very small variations in portfolio wealth resulting from each 
asset being removed from the optimal portfolio. The proxy optimal 
portfolio has wealth increase of at most 0.05% and decrease of 
at most 0.049%. This shows that the SMUDTC model is not 
significantly influenced by individual parameter choices. The risk 
SI varies from 0 to 0.2, with zero being the modal SI. This again 
may imply that model output values are not strongly dependent 
on certain individual parameters. The cost SI range from −78.4% 
to 32.6%. These large values (numerically) are a result of very 
small implicit transaction costs incurred which have been used 
in the calculation. Table 1 shows the implicit transaction costs 
associated with SMUDTC optimal portfolios.

5. CONCLUSION

Portfolio selection that incorporates transaction costs in the 
literature is mostly devoted to proportional transaction costs. 
There is extensive use of the MV and MAD models in financial 
optimization yet both models penalise upside deviations (gains) 

Table 2: In-sample SMUDTC model sensitivity analysis
Div. Lim Parameter Max. value Cost SI Risk SI Wealth SI
0.1 X1 0.1 0.0000 0.2297 −0.00045

X2 0.1 0.1000 0.1228 −0.00024
X4 0.1 0.0000 0.0397 0.00002
X5 0.1 0.0000 0.1228 −0.00024
X6 0.1 0.0000 0.1561 −0.00017
X7 0.1 0.1000 0.1367 −0.00034
X8 0.1 0.0000 0.3262 −0.00033
X9 0.1 0.0000 0.0552 −0.00012
X10 0.1 0.0000 0.0776 −0.00032
X11 0.1 0.0000 0.0756 −0.00014

0.2 X1 0.2 0.0000 0.1356 −0.000073
X2 0.2 0.1000 −0.0410 −0.000047
X4 0.2 0.0000 −0.1798 0.000495
X6 0.2 0.0000 0.1893 0.00012
X9 0.2 0.0000 −01309 0.00022

0.3 X1 0.3 0.1000 −0.7836 −0.000509
X2 0.3 0.2000 −0.5172 −0.000867
X3 0.3 0.0000 0.0164 −0.000662
X4 0.3 0.0000 -0.4888 −0.000078

Table 3: Summary statistics of out-of-sample optimal 
portfolios
Div. Lim MAD MV MM SMUDTC
0.10

Mean return 0.027 0.029 0.031 0.005
Risk 0.002 0.002 0.1 0.0009
Gross wealth 10269.80 10289.02 10314.80 10064.57
Cost 432.00 426.34 502.10 15.36
Net wealth 9837.80 9862.68 9812.70 10049.21

0.15
Mean return 0.025 0.028 0.033 0.006
Risk 0.022 0.002 0.143 0.0009
Gross wealth 10251.75 10277.97 10332.30 10070.17
Cost 479.10 397.32 411.15 17.27
Net wealth 9772.65 9880.65 9921.15 10052.90

0.20
Mean return 0.024 0.027 0.035 0.007
Risk 0.023 0.002 0.2 0.0007
Gross wealth 10240.20 10268.25 10346.80 10073.55
Cost 223.20 378.64 503.60 8.00
Net wealth 10017.00 9889.61 9843.20 10065.55

0.25
Mean return 0.023 0.026 0.036 0.007
Risk 0.025 0.002 0.25 0.0006
Gross wealth 10229.25 10259.68 10356.25 10080.73
Cost 248.50 356.71 591.00 7.40
Net wealth 9980.75 9902.97 9765.25 10073.33

0.30
Mean return 0.023 0.026 0.036 0.008
Risk 0.025 0.002 0.25 0.0006
Gross wealth 10221.70 10255.02 10362.50 10081.56
Cost 272.80 340.06 550.00 6.64
Net wealth 9948.90 9914.96 9812.50 10074.92

0.35
Mean return 0.021 0.025 0.037 0.008
Risk 0.026 0.002 0.333 0.0006
Gross wealth 10214.90 10250.44 10368.30 10082.44
Cost 270.55 318.87 502.10 6.36
Net wealth 9944.35 9931.57 9866.20 10076.08

0.40
Mean return 0.021 0.025 0.037 0.007
Risk 0.027 0.002 0.333 0.0007
Gross wealth 10209.60 10247.81 10373.20 10075.40
Cost 215.20 306.70 440.40 6.80
Net wealth 9994.40 9941.12 9932.80 10068.60



Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018 263

and downside deviations (losses) in the same way. It is important 
to differentiate upside deviations from downside deviations since 
positive deviation is desirable to any investor while negative 
deviation is undesirable. Hence this paper proposes a new 
model that accounts for better investment opportunities (upside 
deviations), risk (downside deviations) and implicit transaction 
costs in a dynamic uncertain environment.

The study considers uncertainty in asset returns, portfolio risk 
and implicit trading costs incurred during initial trading and in 
subsequent rebalancing of the portfolio. The proposed model is 
tested using real data from an emerging market and its performance 
is compared with those of MV, MAD and minimax models. The 
results show that the proposed model generates optimal portfolios 
with least risk, highest portfolio wealth and minimum implicit 
transaction costs. The model is suitable for a risk-averse and 
conservative investor.

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Mushori and Chikobvu: Investment Opportunities, Uncertain Implicit Transaction Costs and Maximum Downside Risk in Dynamic Stochastic Financial Optimization

International Journal of Economics and Financial Issues | Vol 8 • Issue 4 • 2018264

APPENDIX

6. A REVIEW OF SOME VALIDATION 
MODELS

T h e r e  a r e  a  n u m b e r  o f  m o d e l s  o n  p o r t f o l i o  s e l e c t i o n 
presented in the literature to date. However, comparison in 
this study is restricted to the widely used mean-variance, 
MAD and minimax models. The following is a review of 
these models.

6.1. MV Model
The MV model is proposed by Markowitz (1952). This model 
minimizes portfolio variance (risk) subject to expected portfolio 
return achieving a prescribed level. The mathematical formulation 
of the model is as follows:

Maximize
1 1

n n
ij i ji j

x xϕ σ
= =

= ∑ ∑
Subject to

1
,

n
i ii

r xρ
=

≤ ∑

1
1 ,

n
ii

x
=

= ∑
0≤xi≤ui, i = 1,…,n

Where σij is the covariance between assets i and j, xi is the 
proportion of wealth invested in asset i, ri is the expected return 
of asset i in each period, ρ is the minimum rate of return desired 
by an investor, and ui is the maximum proportion of wealth which 
can be invested in asset i.

6.2. Minimax Model
The minimax model is proposed by Young (1998) in which 
minimum return is used as a measure of risk. It is a linear 
programming model which is formulated as follows:

Maximize Mp
Subject to

1
0 , 1, , ,

n
i it pi

w y M t T
=

≤ − = …∑

1
,

n
i ii

G w y
=

≤ ∑

1
,

n
ii

W w
=

≥ ∑
0≤wi, i=1,…,n.

Where yit is the return of one dollar invested in security i in period 
t, iy  is the mean return of security i, wi is the portfolio allocation 
to security i, Mp is the minimum return on the portfolio, G is the 
minimum level of return, and W is the total allocation.

6.3. MAD
Konno and Yamazaki (1991) propose the MAD model and show 
that it behaves in the same manner as the MV model when the 
assets’ returns are multi-variate normally distributed. The MAD 
model is formulated as given below:

Maximize

1

1 T
tt

y
T

β
=

= ∑
Subject to

1
0 , 1, , ,

n
t it ii

y a x t T
=

≤ + = …∑

1
0 , 1, , ,

n
t it ii

y a x t T
=

≤ − = …∑

1
,

n
i ii

r xρ
=

≤ ∑

1
1 ,

n
ii

x
=

= ∑
0≤xi≤ui, i = 1,…,n,

Where ait = rit-ri, 1
| |,

n
t it ii

y a x
=

= ∑  ri is the expected portfolio 
return in period t, rit is the return of security i of period t, ρ is 
the minimum rate of return desired by the investor, and ui is the 
maximum proportion of wealth that can be invested in asset i.