A Mixed R&D Projects and Securities PortfolioSelection Model1

Yong Fanga,b, Lihua Chenc,2, Masao Fukushimaa,3

aDepartment of Applied Mathematics and Physics, Graduate School of Informatics,

Kyoto University, Kyoto, 606-8501, Japan

bInstitute of Systems Science, Academy of Mathematics and Systems Sciences,

Chinese Academy of Sciences, Beijing, 100080, China

cDepartment of Management Science and Management Information Systems,

Guanghua School of Management, Peking University, Beijing, 100871, China

June 21, 2004; revised December 22, 2006

Abstract: The business environment is full of uncertainty. Allocating the wealth among var-

ious asset classes may lower the risk of overall portfolio and increase the potential for more

benefit over the long term. In this paper, we propose a mixed single-stage R&D projects

and multi-stage securities portfolio selection model. Specifically, we present a bi-objective

mixed-integer stochastic programming model. Moreover, we use semi-absolute deviation risk

functions to measure the risk of mixed asset portfolio. Based on the idea of moments approx-

imation method via linear programming, we propose a scenario generation approach for the

mixed single-stage R&D projects and multi-stage securities portfolio selection problem. The

bi-objective mixed-integer stochastic programming problem can be solved by transforming it

into a single objective mixed-integer stochastic programming problem. A numerical example

is given to illustrate the behavior of the proposed mixed single stage R&D projects and multi-

stage securities portfolio selection model.

1This work was partially supported by the Informatics Research Center for Development of Knowl-

edge Society Infrastructure, Graduate School of Informatics, Kyoto University, Japan and the National

Natural Science Foundation of China.2Corresponding author. Tel: +86-10-62765141.3Corresponding author. Tel: +81-75-753-5519; Fax: +81-75-753-4756.

E-mail addresses: yfang@mail.amss.ac.cn (Yong Fang), chenlh@gsm.pku.edu.cn (Lihua Chen),

fuku@i.kyoto-u.ac.jp (Masao Fukushima).

1

Key Words: Portfolio selection, R&D project portfolio selection, Semi-absolute deviation risk

function, Mixed-integer stochastic programming problem.

1 Introduction

During the past decade there has been a dramatic increase in the institutional invest-

ment. Although most of those investments remain focused on the traditional securities

investment, there is growing attention to various forms of alternative investment classes,

e.g., venture capital, private equity, private debt and real estate. With the extension of

investment asset classes, the overall portfolio risk can be lowered while the potential for

more benefit can be increased over the long term.

The mean variance methodology for portfolio selection proposed by Markowitz [22,

23] has been central to research activities in the traditional securities investment field.

Following Sharpe [28], some researchers proposed a series of linear risk functions [1, 20,

21], e.g., Konno and Yamazaki’s [16] mean absolute deviation risk function, Mansini and

Speranza’s [19] mean semi-absolute deviation risk function, Gini’s mean difference risk

function [31], Young’s [32] minimax risk function. Recently, Lai, Wang, Xu, Zhu and

Fang [18] formulated portfolio selection models with interval numbers. In their models,

the semi-absolute deviation risk function is extended to the interval case. Fang, Lai and

Wang [9] studied the portfolio selection problem based on the fuzzy decision theory.

For a long-term securities investment, an investor may adjust his/her portfolio posi-

tions timely with the varying environment to obtain more profit. Many researchers have

studied dynamic portfolio selection problems, e.g., Dantzig and Infranger [7], Dumas and

Luciano [8], Merton [24] and Zhu, Li and Wang [34]. Dynamic portfolio selection models

can be categorized into two classes, continuous time models and discrete time models.

Generally, the discrete time dynamic portfolio selection problem is more amenable to

mathematical programming methods. In recent years, with the development of com-

puter hardware and software, abundant work using stochastic programming methods

has been done in the study of multi-stage portfolio selection problems. Mulvey and

Vladimirous [25] proposed a stochastic network model for asset allocation. Klaassen [15]

2

gave a synthesis of financial asset-pricing theory and stochastic programming models for

asset-liability management. Zenios, Holmer, Raymond and Christiana [35] studied the

portfolio management for the fixed-income bond. Zhao and Ziemba [33] put forward a

multi-period stochastic programming asset allocation model with transaction costs. Ji,

Zhu, Wang and Zhang [12] proposed a stochastic goal programming model for multi-stage

portfolio management based on scenario analysis via moments approximation using lin-

ear programming. For a comprehensive treatment of portfolio selection and asset pricing,

see the monograph by Wang and Xia [30].

In today’s extremely competitive business environment, investors may consider in-

vesting their funds in other kinds of assets besides securities. Byrne and Lee [4] and

Keng [14] found that the mixed asset portfolio including listed property trusts, direct

property and financial assets always dominated the financial asset portfolio. Portfolio

selection of research and development (R&D) projects is one of the most important deci-

sion problems which corporations should face. In recent years, some researchers studied

R&D project portfolio selection problems by using mathematical programming methods,

e.g., Coffin and Taylor III [6], and Ringuest, Graves and Case [27]. Some securities and

R&D projects can be integrated into a mixed asset portfolio. Gustafsson, Reyck, De-

graeve and Salo [10] proposed a mixed asset portfolio selection model involving projects

and securities. In [10], apart from the fact that decision variables for projects are bi-

nary, the projects investment components are treated in the same way as the securities

investment components. However, the properties of projects and securities investments

are different. Generally, it will take much more time to execute R&D projects. Dur-

ing a long investment horizon, an investor may reallocate his/her securities investment

at any time while he/she cannot adjust the R&D projects. Therefore, we may consider

securities investment as multi-stage investment while R&D projects investment as single-

stage investment in the same investment horizon. In this paper, we construct a mixed

single-stage R&D projects and multi-stage securities portfolio selection model.

The paper is organized as follows. In Section 2, we present a bi-objective mixed-

integer stochastic programming model for the mixed single-stage R&D projects and

multi-stage securities portfolio selection problem. This model can be solved by trans-

forming it into a single objective mixed-integer stochastic linear programming model. In

3

Section 3, based on the idea of moments approximation [12], we give a scenario gener-

ation approach for the stochastic returns of single-stage R&D projects and multi-stage

securities. In Section 4, a numerical example is given to illustrate the behavior of the

proposed model and, in particular, its efficient frontier is constructed. In Section 5, we

test the proposed scenario generation method. Some concluding remarks are given in

Section 6.

2 Model Formulation

2.1 Problem Description

In this paper, we assume that an investor allocates his/her wealth among traditional

securities and R&D projects. Hence, in the mixed asset portfolio selection problem,

available investment asset classes are categorized into two types. The first class of assets

consists of traditional securities. The second class of assets consists of R&D projects.

An R&D project is a long term process. So it will take much time to obtain profit from

R&D projects. Once the investor decides to start a project, the budget of the project will

be paid. The R&D project cannot be adjusted during the whole investment horizon, since

otherwise it may result in the waste of the wealth. Hence the R&D projects investment

can be considered a single-stage investment in the whole investment horizon. However,

for securities investment components, the investor may reallocate them in any amount

at any time. So the securities investment can be considered dynamic investment in the

whole investment horizon. For a dynamic securities investment, the investor may obtain

more profit through re-balancing his portfolio positions timely according to his/her risk

preference instead of keeping it till the end of the investment horizon. Therefore, it is

suitable in practice to regard the securities investment as a dynamic process. The multi-

stage portfolio model allows the investor to adjust his portfolio positions dynamically at

discrete time points. There is a capital budget for each R&D project before investment.

The capital budget can be given by the investor or some experts. We assume that the

cost of carrying out a project will be the corresponding capital budget once the project

is started. That is, once the investor decides to invest in the project, the investment

4

amount of devoting to the project must be the capital budget of that project. The R&D

projects’ decision variables are binary, while those of securities are continuous.

2.2 Modelling

We assume that the securities component of mixed assets is composed of n risky securities

i = 1, 2, · · · , n offering random rates of returns, and a risk-free security f offering a

fixed rate of return. The projects component is composed of m R&D projects Pj, j =

1, 2, · · · ,m. Assume that the investor plans for a long-term investment. In the whole

investment horizon, the projects component investment is a single-stage investment,

while the securities component investment is a multi-stage investment. We divide the

investment horizon into T discrete time points. Then the securities investment is T -

stage investment. At time t = 0, the investor sets up his/her portfolio containing n + 1

securities and m projects. At the intermediate T − 1 discrete time points, the investor

may adjust his/her securities component of portfolio to obtain returns. However, at

those discrete time points, the investor cannot obtain additional returns from projects

investment, i.e., the wealth of the projects is still the capital budget of projects. At time

t = T , the investor will obtain the returns of projects since the projects are finished.

We assume that the investment policy is self-financing and the transaction costs are not

needed for the risk-free security. The stage t indicates the time period between time t−1

and t. We introduce some notations as follows.

Rj: the random variable representing the net return on the R&D project Pj, j =

1, 2, · · · ,m after deductng the cost (the budget);

αit: the amount of the risky security i, i = 1, 2, · · · , n bought at time t, t = 0, 1, · · · , T−1;

βit: the amount of the risky security i, i = 1, 2, · · · , n sold at time t, t = 0, 1, · · · , T − 1;

ξi: the unit transaction cost to buy the risky security i, i = 1, 2, · · · , n;

ζi: the unit transaction cost to sell the risky security i, i = 1, 2, · · · , n;

Bj: the capital budget of R&D project Pj, j = 1, 2, · · · ,m, i.e., the cost that the investor

pays once the project is decided to start;

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zj: the binary variable indicating whether project Pj, j = 1, 2, · · · ,m is started or not,

zj =

{1 if project Pj is selected for funding,

0 otherwise;

Yj: the amount of the total investment devoted to the R&D project Pj, j = 1, 2, · · · ,m,

i.e.,

Yj = Bjzj. (1)

The capital budget constraint at time t = 0 is given by

xf0 +n∑

i=1

(1 + ξi)xi0 +m∑

j=1

Yj = w0. (2)

Since the projects investment is single-stage and the securities investment is multi-

stage, the uncertain securities returns are dynamic processes and the uncertain projects

returns are static processes in the investment horizon. Stochastic process is usually

used to describe the uncertainty of a dynamic system. Although the varying process of

securities returns is continuous, the continuous process should be discretized to describe

the dynamic system when we deal with the mixed asset portfolio selection problem by

using a stochastic programming method. Let

rt = (r1t, r2t, · · · , rnt, R1t, R2t, · · · , Rmt), t = 0, 1, · · · , T − 1,

where Rjt = 0, j = 1, 2, · · · ,m, t = 0, 1, · · · , T − 2 and Rj,T−1 = Rj.

The vectors rt, t = 0, 1, · · · , T − 1 represent the discrete time stochastic process of

varying returns of n risky securities and m projects in T stages. We assume that the state

space of the stochastic process is discrete and finite. Any realization of the stochastic

process {rs0, r

s1, · · · , rs

T−1} is called a scenario of mixed asset portfolio selection problem.

The feature of the scenario tree is closely related to the reliability of a mixed single-

stage projects and multi-stage securities portfolio selection model. In the next section,

we will discuss the construction of the scenario tree in detail.

Given the scenario tree of the mixed single-stage projects and multi-stage securi-

ties portfolio selection problem, we assume that there are St scenarios in stage t, t =

1, 2, · · · , T and the probability of scenario s, s = 1, 2, · · · , St is pst , satisfying

St∑s=1

pst = 1.

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At time t, t = 1, 2, · · · , T − 1, under scenario s, s = 1, 2, · · · , St, the amount of the

risky security i is composed of the payoff at time t of the risky security i held at time

t− 1 and the amount balance between buying and selling at time t, i.e.,

xsit = xs

i,t−1(1+rsi,t−1)+αit−βit, i = 1, 2, · · · , n, s = 1, 2, · · · , St , t = 1, 2, · · · , T−1. (3)

At time T under scenario s, s = 1, 2, · · · , ST , the amount of the risky security i is given

by

xsiT = xs

i,T−1(1 + rsi,T−1), i = 1, 2, · · · , n, s = 1, 2, · · · , ST . (4)

Thus, the expected amount of the risky security i at each time t is given by

E(xit) =St∑

s=1

pstx

sit, i = 1, 2, · · · , n, t = 1, 2, · · · , T.

The amount invested in the risk-free security at time t, t = 1, 2, · · · , T − 1, under

scenario s, s = 1, 2, · · · , St is composed of the payoff at time t of the risky securities held

at time t− 1 and the cash balance between buying and selling the risky security, i.e.,

xsft = xs

f,t−1(1+rf,t−1)+n∑

i=1

(1−ζi)βit−n∑

i=1

(1+ξi)αit, s = 1, 2, · · · , St, t = 1, 2, · · · , T−1.

(5)

The amount invested in the risk-free securities at time T under scenario s, s = 1, 2, · · · , ST

is given by

xsfT = xs

f,T−1(1 + rf,T−1), s = 1, 2, · · · , ST . (6)

So the expected amount invested in the risk-free security at each time t is given by

E(xft) =St∑

s=1

pstx

sft, t = 1, 2, · · · , T.

The amount invested in the R&D project Pj, j = 1, 2, · · · ,m at time t, t = 0, 1, · · · , T−1 is Yj. The wealth of the R&D project Pj at time t, t = 1, 2, · · · , T − 1, under scenario

s, s = 1, 2, · · · , St is

Y sjt = Yj, j = 1, 2, · · · ,m, s = 1, 2, · · · , St, t = 1, 2, · · · , T − 1. (7)

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In the end of the investment horizon, i.e., at time T , the wealth of R&D project Pj, j =

1, 2, · · · ,m under scenario s, s = 1, 2, · · · , ST is given by

Y sjT = Yj(1 + Rs

j), j = 1, 2, · · · ,m, s = 1, 2, · · · , ST . (8)

So the expected wealth of the R&D project Pj at time t, t = 1, 2, · · · , T is given by

E(Yjt) =St∑

s=1

pstY

sjt.

Let x(t)s = (xsft, x

s1t, · · · , xs

nt) and Y (t)s = (Y s1t, Y

s2t, · · · , Y s

mt). Then the investor’s

overall wealth, i.e., the amount of mixed asset portfolio at time t under scenario s is

given by

wt(x(t)s, Y (t)s) = xsft +

n∑i=1

xsit +

m∑j=1

Y sjt, s = 1, 2, · · · , St, t = 1, 2, · · · , T,

and the investor’s overall expected wealth at time t is given by

St∑s=1

pstwt(x(t)s, Y (t)s) =

St∑s=1

pst(x

sft +

n∑i=1

xsit +

m∑j=1

Y sjt), t = 1, 2, · · · , T.

Generally, in the multi-stage portfolio selection problem, the investor tries to maxi-

mize the terminal wealth. So the expected wealth at time T ,

f(x(T )1, Y (T )1, · · · , x(T )ST , Y (T )ST ) =

ST∑s=1

psT (xs

fT +n∑

i=1

xsiT +

m∑j=1

Y sjT ), (9)

can be considered the objective function of the mixed asset portfolio selection problem.

We will use the semi-deviation risk function in our model. The expected semi-absolute

deviation of mixed asset portfolio wealth below the expected wealth at time t is defined

by

Dt(x(t)1, Y (t)1, · · · , x(t)St , Y (t)St)

=St∑

s=1

pst

∣∣ min{0, xsft +

n∑i=1

xsit +

m∑j=1

Y sjt −

St∑s=1

pst(x

sft +

n∑i=1

xsit +

m∑j=1

Y sjt)}

∣∣, (10)

8

and it can be used to measure the risk of the mixed asset portfolio at time t. In particular,

the terminal expected semi-absolute deviation of mixed asset portfolio wealth below the

expected wealth at time T , i.e., the terminal risk of mixed asset portfolio,

DT (x(T )1, Y (T )1, · · · , x(T )ST , Y (T )ST )

=ST∑s=1

psT

∣∣ min{0, xsfT +

n∑i=1

xsiT +

m∑j=1

Y sjT −

ST∑s=1

psT (xs

fT +n∑

i=1

xsiT +

m∑j=1

Y sjT )}

∣∣,

can be considered another objective function of the problem.

In the mixed single-stage projects and multi-stage securities portfolio selection prob-

lem, it is possible that the wealth in the intermediate stages fluctuates drastically. If

the risks in the intermediate stages were not considered, the investor might encounter

bankruptcy before the end of the horizon. So we add the risk controls of bankruptcy in

the intermediate stages, which are represented as the following constraints:

Dt(x(t)1, Y (t)1, · · · , x(t)St , Y (t)St) ≤ θt, t = 1, 2, · · · , T − 1, (11)

where θt are constants given by the investor.

In many financial markets, the securities are no short selling. So we add the following

constraints:

xsft, x

sit, α

sit, β

sit, xf0, xi0 ≥ 0, i = 1, 2, · · · , n, s = 1, 2, · · · , St, t = 1, 2, · · · , T.

zj = 0 or 1, j = 1, 2, · · · ,m.(12)

In addition, some other resource constraints such as human resources, material resources,

computer resources and equipment resources might be included in the problem.

2.3 Formulation

We assume that the investor wants to maximize the terminal wealth and minimize the

terminal risk under the constraints (1)–(8), (11) and (12) introduced in the previous

subsection. The mixed single-stage projects and multi-stage securities portfolio selection

problem can be formally stated as the following bi-objective mixed-integer stochastic

programming problem:

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(BSNLP) max f(x(T )1, Y (T )1, · · · , x(T )ST , Y (T )ST )

min DT (x(T )1, Y (T )1, · · · , x(T )ST , Y (T )ST )

s.t. (1)− (8), (11) and (12).

Problem (BSNLP) can be reformulated as a bi-objective mixed-integer stochastic

linear programming problem by using the following technique. Note that

∣∣ min{0, c}∣∣ =

1

2

∣∣c∣∣− 1

2c

for any real number c. Thus, by introducing auxiliary variables a+st, a

−st, s = 1, 2, · · · , St, t =

1, 2, · · · , T such that

a+st + a−st =

1

2

∣∣xsft +

n∑i=1

xsit +

m∑j=1

Y sjt −

St∑s=1

pst(x

sft +

n∑i=1

xsit +

m∑j=1

Y sjt)

∣∣, (13)

a+st − a−st =

1

2[xs

ft +n∑

i=1

xsit +

m∑j=1

Y sjt −

St∑s=1

pst(x

sft +

n∑i=1

xsit +

m∑j=1

Y sjt)], (14)

a+st ≥ 0, a−st ≥ 0, s = 1, 2, · · · , St, t = 1, 2, · · · , T, (15)

we may write

Dt(x(t)1, Y (t)1, · · · , x(t)St , Y (t)St) =St∑

s=1

2psta−st, t = 1, 2, · · · , T.

Hence, problem (BSNLP) can be rewritten as the following bi-objective mixed-integer

stochastic linear programming problem:

(BSILP) maxST∑s=1

psT (xs

fT +n∑

i=1

xsiT +

m∑j=1

Y sjT )

minST∑s=1

2psT a−sT

s.t.St∑

s=1

2psta−st ≤ θt, t = 1, 2, · · · , T − 1,

and (1)− (8), (12), (14) and (15).

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Thus the investor may determine his/her investment strategies by computing efficient

solutions of (BSILP).

An efficient solution of (BSILP) can be found by solving a single objective problem.

Specifically, for a given constant D0 that represents the tolerance level of risk accepted

by the investor, we may consider the following single objective problem instead of the

bi-objective problem (BSILP):

(MISLP-a) maxST∑s=1

psT (xs

fT +n∑

i=1

xsiT +

m∑j=1

Y sjT )

s.t.ST∑s=1

2psT a−sT ≤ D0,

and all constraints of (BSILP).

Any optimal solution of this problem constitutes an efficient portfolio of the mixed asset

portfolio selection problem.

Alternatively, for a given constant f0 that represents the minimum return level on

the portfolio required by the investor, we may consider another single objective problem

for (BSILP):

(MISLP-b) minST∑s=1

2psT a−sT

s.t.ST∑s=1

psT (xs

fT +n∑

i=1

xsiT +

m∑j=1

Y sjT ) ≥ f0,

and all constraints of (BSILP).

Any optimal solution of (MISLP-b) also constitutes an efficient portfolio of the mixed

asset portfolio selection problem.

Both (MISLP-a) and (MISLP-b) are mixed-integer stochastic linear programming

problems. Given a scenario tree of uncertain returns of securities and projects, these

two problems are standard mixed-integer linear programming problems, for which an

abundance of algorithms such as branch-and-bound algorithms and cutting-plane algo-

rithms [29, 26] is available. Because of the high computational complexity of the integer

programming problem, some heuristic algorithms are also effective to obtain an approx-

imate, but not necessarily optimal, solution quickly [2].

11

Problems (MISLP-a) and (MISLP-b) can be used interchangeably to generate the

efficient frontier of portfolios. In Section 4, we will give a numerical example to illustrate

the behavior of the proposed mixed single-stage R&D projects and multi-stage securities

portfolio selection model.

3 Generation of Scenario Tree

The properties of the scenario tree have an important impact on the mixed single-stage

R&D projects and multi-stage securities portfolio selection problem. The scenarios are

used to describe the market uncertainty. The complexity of the model is related to the

number of scenarios. A rational scenario tree can make the investor obtain better port-

folio by using the proposed model in practice. There have been proposed many methods

of generating scenario trees. The Monte Carlo simulation approach is commonly used to

generate scenarios of uncertain returns in the future. The vector autoregressive (VAR)

model is also used to generate multi-stage scenario trees [3, 17, 33]. Carino, Myers and

Ziemba [5] integrated the time series model and factor model to explore the generation

of scenarios. Mulvey [25] described an integrated scenario system to generate scenarios

for asset/liability management for pension fund of Tower Perrin company. Zenios and

McKendall [36] used massively parallel computing to give computing price scenarios of

mortgage-backed securities. Høyland and Wallace [11] proposed a nonlinear program-

ming model to generate scenarios that satisfy specified statistical properties. Recently, Ji,

Zhu, Wang and Zhang [12] put forward a linear programming model based on moments

approximation as well as descriptive statistical properties. Their linear programming

model overcomes the increasing complexity of Høyland and Wallace’s model due to non-

convexity and nonlinearity. Kaut and Wallace gave an evaluation of scenario-generation

methods for stochastic programming in [13].

In this section, based on the idea of Zhu’s moments approximation [12], we give a

method of generating a scenario tree for the mixed single-stage R&D projects and multi-

stage securities portfolio selection problem. Let the investment horizon be divided into

T discrete time points, i.e., the securities investment is T -stage investment, while the

projects investment is single-stage. Therefore at the intermediate discrete time points

12

t = 1, 2, · · · , T − 1, the investor cannot obtain an additional return from the projects

investment, i.e., in stages t = 1, 2, · · · , T − 1, the returns of projects are regarded as

zero. Thus, in stages t = 1, 2, · · · , T − 1, we only need to describe the uncertain returns

of securities by using scenarios analysis. In stage T , we need to describe the uncertain

returns of both securities and projects together by using scenarios analysis.

When the securities returns between abutting stages are not correlated or the auto-

correlations are very weak, we can assume that the securities returns are independent

and identically distributed. Thus, in this case, we can give scenarios directly in stages

t = 1, 2, · · · , T , regardless of the relationships between returns in different stages.

When the securities returns between abutting stages are correlated substantially, we

need to generate scenarios by using a time series model. The VAR model can describe

linear relationships between the current values and the lagged values as follows:

rt = C + H1rt−1 + H2rt−2 + · · ·+ Hprt−p + εt, εt ∼ N(0, Σ),

where the regressive coefficient vector C, matrices H1, H2, · · · , Hp and the covariance

matrix Σ can be estimated from the historical data. Given the initial return vector r0,

we can get the successive returns in turn if the disturbance vectors εt are determined.

Thus, the scenario generation is transferred from the returns series to the disturbance

vector series.

In the following we describe the scenario generation procedure in detail. At first, we

give a procedure of generating scenarios from stage 1 to stage T−1. Let St, t = 1, 2, · · · , T

denote the number of scenarios with non-zero probabilities in stage t.

Procedure 1

Step 1: Let t := 1 and S0 = 1. Let n10 denote the initial node. Test for the autocor-

relations of the securities returns. If the autocorrelations are very weak, go to Step 2.

Otherwise, go to Step 3.

Step 2: In this step, for each particular node nkt−1, k = 1, 2, · · · , St−1 at time t− 1, we

generate the succeeding scenarios in stage t. Since the probability distribution of the

securities returns ri,t−1, i = 1, 2, · · · , n is normal, an appropriate method is to bound

ri,t−1 between six times of the standard deviation around the expected value, i.e., the

13

intervals are [r̄i,t−1 − 3√

Σii, r̄i,t−1 + 3√

Σii], i = 1, 2, · · · , n, where r̄i,t−1 is the expected

value of ri,t−1, the covariance matrix Σ and higher order moments can be estimated

via historical data. Divide the interval [r̄i,t−1 − 3√

Σii, r̄i,t−1 + 3√

Σii] into mki,t−1 sub-

intervals of equal length, and then pick out the midpoints Ok,ji,t−1, j = 1, 2, · · · ,mk

i,t−1 of

these sub-intervals as a possible outcome of the security i. Then each n-dimensional

return vector (Ok,j11,t−1, O

k,j22,t−1, · · · , Ok,jn

n,t−1) can be considered a succeeding scenario for the

particular node nkt−1, k = 1, 2, · · · , St−1. There are sk

t−1 succeeding scenarios for node

nkt−1, where sk

t−1 = mk1,t−1m

k2,t−1 · · ·mk

n,t−1. Go to Step 4.

Step 3: In this step, for each particular node nkt−1, k = 1, 2, · · · , St−1 at time t− 1, we

generate the succeeding scenarios in stage t. Determine a VAR model as rt = C+H1rt−1+

H2rt−2 + · · ·+ Hprt−p + εt with εt ∼ N(0, Σ). Test for significance of the VAR model. If

the VAR model is not significant, we change lag orders to get a significant VAR model.

Estimate the regressive coefficient vector C, matrices H1, H2, · · · , Hp, the covariance

matrix Σ and higher order moments via historical data. Since the probability distribution

of εi,t−1, i = 1, 2, · · · , n is normal, an appropriate method is to bound εi,t−1 between six

times of the standard deviation around zero, i.e., the intervals [−3√

Σii, 3√

Σii], i =

1, 2, · · · , n. Divide the interval [−3√

Σii, 3√

Σii] into mki,t−1 sub-intervals of equal length,

and then pick out the midpoints εk,ji,t−1, j = 1, 2, · · · ,mk

i,t−1 of these sub-intervals as a

possible outcome of εi,t−1. Let

Ok,ji,t−1 = C+H1ri,t−2+H2ri,t−3+· · ·+Hpri,t−p−1+εk,j

i,t−1, i = 1, 2, · · · , n, j = 1, 2, · · · ,mki,t−1.

Then each n-dimensional return vector (Ok,j11,t−1, O

k,j22,t−1, · · · , Ok,jn

n,t−1) can be considered a

succeeding scenario for the particular node nkt−1, k = 1, 2, · · · , St−1. There are sk

t−1

succeeding scenarios for node nkt−1 where sk

t−1 = mk1,t−1m

k2,t−1 · · ·mk

n,t−1. Go to Step 4.

Step 4: In this step, we compute the probabilities of scenarios in stage t. First, taking

the expected value, covariance and partial higher moments as approximation goals, we

use the following optimization model to determine the probabilities of the succeeding

scenarios for each particular node nkt−1, k = 1, 2, · · · , St−1 at time t − 1. We give some

notations:

r̄t−1: the expected return vector of securities in stage t;

Ok,lt−1: the lth outcome vector of securities for node nk

t−1 at stage t, l = 1, 2, · · · , skt−1;

14

pk,lt−1: the probability of the lth outcome vector of securities for node nk

t−1, l =

1, 2, · · · , skt−1;

Σ: the covariance matrix of securities;

M3: the third-order center moment vector of securities;

M4: the fourth-order center moment vector of securities.

The optimization model is stated as follows:

(LP1) minn∑

i,j=1

µ1ij

(Σ−

ij + Σ+ij

)+

n∑i=1

µ2i

(M−

3i + M+3i

)+

n∑i=1

µ3i

(M−

4i + M+4i

)

s.t.skt−1∑l=1

Ok,lt−1p

k,lt−1 = r̄t−1,

skt−1∑l=1

(Ok,l

t−1 − r̄t−1

)(Ok,l

t−1 − r̄t−1

)′pk,l

t−1 + Σ− − Σ+ = Σ,

skt−1∑l=1

(Ok,l

t−1 − r̄t−1

)3

ipk,l

t−1 + M−3i −M+

3i = M3i, i = 1, · · · , n,

skt−1∑l=1

(Ok,l

t−1 − r̄t−1

)4

ipk,l

t−1 + M−4i −M+

4i = M4i, i = 1, · · · , n,

skt−1∑l=1

pk,lt−1 = 1, pk,l

t−1 ≥ 0, l = 1, · · · , skt−1,

Σ+ij, Σ

−ij,M

+3i ,M

−3i ,M

+4i ,M

−4i ≥ 0, i, j = 1, · · · , n,

where µ1ij, µ

2i , µ

3i > 0, i, j = 1, 2, · · · , n are given weights according to the investor’s

preference, Σ+, Σ− are positive and negative deviations of scenarios’ covariance ma-

trix, respectively, M+3 ,M−

3 are positive and negative deviations of the third-order cen-

ter moment vector of securities, respectively, M+4 ,M−

4 are positive and negative devia-

tions of the fourth-order center moment vector of securities, respectively, and M3i, M4i,(Ok,l

t−1 − r̄t−1

)i, etc. are the ith component of vectors M3, M4, Ok,l

t−1 − r̄t−1, etc.

By solving problem (LP1), we get the probabilities of the succeeding outcome vectors

of securities for the particular node nkt−1. Denote by pk

t−1 the probability of node nkt−1

at time t − 1. Thus, regarding the lth outcome vector of securities for node nkt−1 as

one of all scenarios in stage t, the probability of scenario plt, l = 1, 2, · · · , sk

t−1 can be

computed by plt = pk

t−1pk,lt−1. Note that pl

1 = p1,l0 since there is one initial node at time

0. Denote by vkt the number of succeeding scenarios with non-zero probabilities for node

15

nkt−1, k = 1, 2, · · · , St−1 and let St =

St−1∑k=1

vkt . Let t := t + 1. If t = T , then stop.

Otherwise, if the autocorrelations of securities returns are very weak, then go back to

Step 2, else go back to Step 3.

Through carrying out the above procedure, we obtain a set of scenarios for T − 1

stages. Due to the existence of uncertain returns of projects in stage T , the generation

method of scenarios in stage T is different from those in stages t = 1, 2, · · · , T − 1. In

the following, we give a procedure of generating scenarios in stage T .

Procedure 2

Step 1: For each node nkT−1, k = 1, 2, · · · , ST−1 at time T − 1, if the autocorrelations

of securities returns are very weak, then similarly to the case of stages t ≤ T − 1,

divide the interval [r̄i,T−1 − 3√

Σii, r̄i,T−1 + 3√

Σii] into mki,T−1 sub-intervals of equal

length, and pick out the midpoints Ok,ji,T−1, j = 1, 2, · · · ,mk

i,T−1 of these sub-intervals as a

possible outcome of the security i. If the securities returns between abutting stages are

correlated substantially, then similarly to the case of stages t ≤ T −1, divide the interval

[−3√

Σii, 3√

Σii] into mki,T−1 sub-intervals of equal length, and pick out the midpoints

εk,ji,T−1, j = 1, 2, · · · ,mk

i,T−1 of these sub-intervals as a possible outcome of εi,T−1. Let

Ok,ji,T−1 = C + H1ri,T−2 + H2ri,T−3 + · · ·+ Hpri,T−p−1 + εk,j

i,T−1,

i = 1, 2, · · · , n, j = 1, 2, · · · ,mki,T−1.

Step 2: Unlike securities, there are no historical return data of projects that can be

used to estimate parameters. However, the investor may use forecast data to estimate

parameters. Given the investment horizon, say three months, we can get some his-

torical returns of securities for three months at some time points. Considering the

market situation and state at some time points, projects management experts may

give possible returns of projects at these time points. Integrating the historical re-

turns of securities and possible returns of projects, we can estimate the expected value

r̂ = (r̂1, r̂2, · · · , r̂n, R̂1, R̂2, · · · , R̂m) and covariance matrix of the (m + n)-dimensional

random vector (r̄1, r̄2, · · · , r̄n, R1, R2, · · · , Rm), where r̄i represents the random return of

risky security i and Rj represents the random return of project Pj in the investment hori-

zon. Let δj denote the standard deviation of Rj. Divide the interval [R̂j − 3δj, R̂j + 3δj]

16

into hkj sub-intervals of equal length, and then pick out the midpoints ρk,i

j , i = 1, 2, · · · , hkj

of these sub-intervals as a possible outcome of Rj. Then each (m+n)-dimensional return

vector (Ok,j11,T−1, O

k,j22,T−1, · · · , Ok,jn

n,T−1, ρk,i11 , ρk,i2

2 , · · · , ρk,imm ) can be considered a succeeding

scenario for the particular node nkT−1, k = 1, 2, · · · , ST−1. There are sk

T−1 possible suc-

ceeding scenarios for node nkT−1, where sk

T−1 = mk1,T−1m

k2,T−1 · · ·mk

n,T−1hk1h

k2 · · ·hk

m.

Step 3: Similarly to the case of stages t ≤ T −1, we may use the following optimization

model to determine the probabilities of succeeding scenarios for each particular node

nkT−1, k = 1, 2, · · · , ST−1 at time T − 1. We give some notations:

r̄: the expected return vector of securities in stage T ;

Ok,lT−1: the lth outcome vector of securities for node nk

T−1 at stage T , l = 1, 2, · · · , skT−1;

ρk,l: the lth outcome vector of projects for node nkT−1 in the whole investment horizon,

l = 1, 2, · · · , skT−1;

Qk,l: the lth outcome vector of securities and projects in the whole investment hori-

zon, l = 1, 2, · · · , sT−1k , which is given by

Qk,l = (Uk,l1 , Uk,l

2 , · · · , Uk,ln , ρk,l

1 , ρk,l2 , · · · , ρk,l

m ),

where

Uk,li = (1 + Ok0,l0

i,0 )(1 + Ok1,l1i,1 ) · · · (1 + O

kT−2,lT−2

i,T−2 )(1 + Ok,li,T−1)− 1, i = 1, 2, · · · , n,

OkT−2,lT−2

i,T−2 is the preceding outcome of the security i for node nkT−1 at stage T − 1, node

nkT−2

T−2 is the predecessor of node nkT−1, O

kt−3,lt−3

i,t−3 is the preceding outcome of the security

i for node nkt−2

t−2 at stage t− 2, t = 3, 4, · · · , T , and node nkt−3

t−3 is the predecessor of node

nkt−2

t−2 , t = 3, 4, · · · , T and so on;

pk,lT−1: the probability of the lth outcome vector Qk,l of securities and projects for

node nkT−1, l = 1, 2, · · · , sk

T−1;

Σ: the covariance matrix of securities returns in one stage;

M3: the third-order center moment vector of securities returns in one stage;

M4: the fourth-order center moment vector of securities returns in one stage;

Φ: the covariance matrix of securities and projects;

17

N3: the third-order center moment vector of securities and projects;

N4: the fourth-order center moment vector of securities and projects.

The optimization model is stated as follows:

(LP2) minn∑

i=1

µ1i

(r̄−i + r̄+

i

)+

m+n∑i=1

µ2i

(r̂−i + r̂+

i

)+

n∑i,j=1

µ3ij

(Σ−

ij + Σ+ij

)

+n∑

i=1

µ4i

(M−

3i + M+3i

)+

n∑i=1

µ5i

(M−

4i + M+4i

)+

m+n∑i,j=1

µ6ij

(Φ−

ij + Φ+ij

)

+m+n∑i=1

µ7i

(N−

3i + N+3i

)+

m+n∑i=1

µ8i

(N−

4i + N+4i

)

s.t.skT−1∑l=1

Ok,lT−1p

k,lT−1 + r̄− − r̄+ = r̄,

skT−1∑l=1

Qk,lpk,lT−1 + r̂− − r̂+ = r̂,

skT−1∑l=1

(Ok,l

T−1 − r̄)(

Ok,lT−1 − r̄

)′pk,l

T−1 + Σ− − Σ+ = Σ,

slT−1∑l=1

(Ok,l

T−1 − r̄)3

ipk,l

T−1 + M−3i −M+

3i = M3i, i = 1, · · · , n,

slT−1∑l=1

(Ok,l

T−1 − r̄)4

ipk,l

T−1 + M−4i −M+

4i = M4i, i = 1, · · · , n,

slT−1∑l=1

(Qk,l − r̂

) (Qk,l − r̂

)′pk,l

T−1 + Φ− − Φ+ = Φ,

skT−1∑l=1

(Qk,l − r̂

)3

ipk,l

T−1 + N−3i −N+

3i = N3i, i = 1, · · · ,m + n,

skT−1∑l=1

(Qk,l − r̂

)4

ipk,l

T−1 + N−4i −N+

4i = N4i, i = 1, · · · ,m + n,

skT−1∑l=1

pk,lT−1 = 1, pk,l

T−1 ≥ 0, l = 1, 2, · · · , skT−1,

Σ+ij, Σ

−ij,M

+3i ,M

−3i ,M

+4i ,M

−4i ≥ 0, i, j = 1, · · · , n,

Φ+ij, Φ

−ij, N

+3i , N

−3i , N

+4i , N

−4i ≥ 0, i, j = 1, · · · ,m + n,

r̄−i , r̄+i ≥ 0, i = 1, · · · , n,

r̂−i , r̂+i ≥ 0, i = 1, · · · ,m + n,

where µ1i , µ

3ij, µ

4i , µ

5i > 0, i, j = 1, 2, · · · , n, and µ2

i , µ6ij, µ

7i , µ

8i > 0, i, j = 1, 2, · · · ,m+n

are given weights according to the investor’s preference, N3i, N4i,(Ok,l

T−1 − r̄)

i,(Qk,l − r̂

)i

are the ith component of vectors N3, N4, Ok,lT−1 − r̄t, Qk,l − r̂, respectively.

18

By solving problem (LP2), we get the probabilities of succeeding outcome vectors of

securities for the particular node nkT−1. Regarding the lth outcome vector of securities

for node nkT−1 as one of all scenarios in stage T , the probability of the lth scenario can be

computed by plT = pk

T−1pk,lT−1. Denote by vk

T−1 the number of succeeding scenarios with

non-zero probabilities for node nT−1k , k = 1, 2, · · · , ST−1 and let ST =

ST−1∑k=1

vkT . Thus, we

get ST scenarios with non-zero probabilities in stage T .

4 Numerical Example

In this section, we will give a numerical example to illustrate the proposed mixed single-

stage projects and multi-stage securities portfolio selection model. Assume that the

whole investment horizon is three months and the initial wealth w0 is U500000. The

securities investment is divided into three stages and each stage consists of one month.

We suppose that the investor chooses three stocks, 1 (Amoi Electronics), 2 (The first

Department Store) and 3 (National Electronic Power) from the Shanghai Stock Exchange

as the risky securities. We regard the bank deposit of 3 months as the risk-free security.

The monthly returns rf0, rf1, rf2 of the risk-free security in three stages are all 0.0017. All

the unit transaction costs ξi, ζi, i = 1, 2, 3 of risky securities are 0.0055 in the Shanghai

Stock Exchange. The projects investment is one stage. The projects component is

composed of three R&D projects P1, P2 and P3 which will be finished in three months.

The capital budgets of R&D projects are given by B1 = U80000, B2 = U100000 and

B3 = U150000.

We collect historical data of the three stocks from January, 1998 to May, 2004. The

data can be downloaded from the web-site www.stockstar.com. We use one month as a

period to get the historical rates of returns of seventy six periods. Through statistical

analysis for these data, we find that the autocorrelations of the three stocks returns series

are very weak. Based on the historical data, we estimate the mean of the monthly returns

of the three stocks as r̄ = (0.028442 0.00909 0.026909). We obtain the covariance matrix

19

of the monthly returns of the three stocks as

Σ =

0.021845 0.003748 0.004189

0.003748 0.007847 0.003394

0.004189 0.003394 0.009974

,

and the third-order and the fourth-order center moment vectors of the monthly returns

of the three stocks as

M3 = (0.002797 0.000648 0.000906) and M4 = (0.001914 0.000275 0.000397),

respectively. Vectors r̄,M3,M4 and matrix Σ are used in both Procedure 1 and 2.

By using the historical data of the three stocks from January, 1998 to May, 2004, we

consider three months as a period to get the historical rates of returns of twenty five pe-

riods. For the R&D projects component, there are no historical data to be used. In this

example, we give artificial data of three R&D projects by using random number genera-

tors. We assume that the random return of P1 follows the normal distribution with mean

0.10 and variance 0.025, the random return of P2 follows the normal distribution with

mean 0.12 and variance 0.030, the random return of P3 follows the normal distribution

with mean 0.18 and variance 0.045. We generate randomly twenty five periods data of

each R&D project according to the corresponding normal probability distribution. By

using these data, we estimate the mean of the quarterly returns of 1, 2, 3, P1, P2 and P3

as

r̂ = (0.104423 0.020881 0.088621 0.10158 0.112504 0.177704).

The covariance matrix of the quarterly returns of 1, 2, 3, P1, P2 and P3 is computed as

Φ =

0.088568 0.009269 0.011031 −0.00015 −0.0079 −0.00019

0.009269 0.017539 0.011235 −0.00143 −0.00382 −0.00184

0.011031 0.011235 0.035663 −0.01049 0.003947 −0.01347

−0.00015 −0.00143 −0.01049 0.027269 −0.00794 0.034994

−0.0079 −0.00382 0.003947 −0.00794 0.02931 −0.01018

−0.00019 −0.00184 −0.01347 0.034994 −0.01018 0.044906

.

20

The third-order and the fourth-order center moment vectors of the quarterly returns of

1, 2, 3, P1, P2 and P3 are obtained as

N3 = (0.0397 0.0015 0.0056 0.0008 − 0.0025 0.0018)

and

N4 = (0.0444 0.0008 0.0039 0.0021 0.0026 0.0057),

respectively. Vectors r̂, N3, N4 and matrix Φ are used in Step 2 of Procedure 2.

By using Procedures 1 and 2 described in Section 3, we generate a scenario tree for

the mixed single-stage projects and three-stage securities portfolio selection problem.

In stage 1, we divide the possible returns intervals of stocks 1, 2 and 3 into 5, 4 and

4 sub-intervals of equal length, respectively. So there are 80 variables in (LP1) for

the initial node. By solving (LP1), we have obtained an optimal solution of which 14

among 80 variables take non-zero values. Thus, we have got 14 scenarios with non-zero

probabilities. In stage 2, we divide the possible returns intervals of stocks 1, 2 and 3

into 4, 3 and 3 sub-intervals of equal length, respectively, so there are 36 variables in

(LP1) for each one of 14 nodes at time t = 1. By solving the corresponding (LP1) for

each one of 14 nodes at time t = 1, we have obtained an optimal solution of which 11

among 36 variables take non-zero values. So, for each node at time t = 1, we have got

11 scenarios with non-zero probabilities. Thus, we have got 154 scenarios with non-zero

probabilities in stage 2. In stage 3, we divide the possible returns intervals of stocks

1, 2 and 3 into 4, 3 and 3 sub-intervals of equal length, respectively. At the same time,

we divide the possible returns intervals of P1, P2 and P3 into 3, 3 and 3 sub-intervals of

equal length, respectively. So there are 972 variables in (LP2) for each one of 154 nodes

at time t = 2. We have solved problems (LP2) associated with those 154 nodes at time

t = 2, and obtained optimal solutions, in which the number of variables with non-zero

values ranges form 7 to 25, and the cumulative number of those variables amounts to

2507. Thus, we have got 2507 scenarios with non-zero probabilities in stage 3 altogether.

Using the scenario tree, we solve (MISLP-a) for a number of fixed values of the

terminal risk tolerance level D0, with the risk control parameters θt, t = 1, 2 both set

to be 10000. All computations were carried out on a WINDOWS PC using the LINDO

21

solver. Table 1 shows the terminal wealth of portfolios for fourteen possible risk tolerance

levels. Furthermore, the efficient frontier (EF-1) of portfolios is depicted in Figure 1.

If the available investment assets are only composed of the three stocks and the bank

deposit, i.e., the R&D projects are not included in (MISLP-a), then we can get some

other optimal portfolios for a number of fixed values of the risk tolerance level D0. Table

2 shows the terminal wealth of portfolios without R&D projects for fourteen possible

risk tolerance levels. The efficient frontier (EF-2) of portfolios without R&D projects is

depicted in Figure 1, too.

Table 1: Fourteen possible risk tolerance levels and the corresponding terminal wealth

of portfolios

D0 500 1000 2000 4000 6000 8000 10000

Terminal Wealth (U) 522449 528535 536713 550970 561713 577678 582117

D0 12000 14000 16000 18000 20000 25000 30000

Terminal Wealth (U) 585468 587879 589519 590676 590904 590943 590976

Table 2: Fourteen possible risk tolerance levels and the corresponding terminal wealth

of portfolios without R&D projects

D0 500 1000 2000 4000 6000 8000 10000

Terminal Wealth (U) 522449 528535 536713 550970 561713 567729 572750

D0 12000 14000 16000 18000 20000 25000 30000

Terminal Wealth (U) 577244 581274 584465 586522 587726 588952 589030

From Figure 1, we can find that the terminal wealth of EF-1 is generally larger than

that of EF-2 when the risk tolerance level exceeds 6000. The reason is that, under those

risk tolerance levels, the investor may invest his/her wealth in R&D projects that may

yield higher returns. The computational results show that investing some initial wealth

in R&D projects makes the investor obtain more benefit. Therefore, the expansion of

22

0 0.5 1 1.5 2 2.5 3

x 104

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6x 10

5

Risk

Ter

min

al W

ealth

EF−1EF−2

Figure 1: The efficient frontiers of the mixed asset portfolio selection problem

investment asset classes may lower the overall portfolio risk while increasing the potential

for more benefit over the long term.

5 Testing the Scenario Generation Method

There are at least two requirements to evaluate the scenario generation methods [13].

The first requirement is stability, i.e., those optimization problems which use scenario

trees generated with the same input should have approximately the same optimal objec-

tive values. The second requirement is that the scenario generation method should not

introduce any significant bias from the true solution.

Kaut and Wallace [13] define the in-sample stability and the out-of-sample stability.

We only need to solve the scenario-based optimization problems to test the in-sample

stability, while we need to evaluate the “true” objective function to test the out-of-

sample stability based on the knowledge of the distribution of the stochastic process.

Having in-sample stability means that the optimal solution obtained by solving the

optimization problem with each scenario tree is stable, i.e., we can get approximately

23

the same optimal value of the objective function. Having out-of-sample stability means

that the real performance of the solution is stable, i.e., the solution does not depend on

the choice of a scenario tree.

All computations were carried out on a WINDOWS PC using the LINDO solver. The

maximum size of an optimization problem solved by using the LINDO solver is limited.

Considering the computational difficulties of multi-stage optimization problems, we use a

simple two-period example to illustrate the proposed scenario generation method in this

section. A more efficient solver can deal with larger problems. To develop an efficient

algorithm for solving large-scale problems is one of our future subjects. We consider two-

month investments in three stocks, 1 (Amoi Electronics), 2 (The first Department Store)

and 3 (National Electronic Power) from the Shanghai Stock Exchange. The benchmark

tree has seventy six scenarios, and is based on the data in the period from January, 1998

to May, 2004. From the benchmark tree, we compute the moments and correlations

of the differentials of the return rates and estimate the statistical properties of the

three stocks. We use the benchmark tree as a representation of the true distribution.

Then we use Kaut and Wallace’s evaluation approach of scenario generation methods to

evaluate our method. For a given size of the tree, we generate 24 scenario trees, solve the

corresponding portfolio selection model for a given risk tolerance level, and then evaluate

the solutions on the benchmark tree.

Results of the test are presented in Table 3 and Table 4. We find that the proposed

scenario generation method gives a reasonable stability, both in-sample and out-of sam-

ple. We evaluate solutions based on different scenario trees in the in-sample tests while

we evaluate all solutions based on the same benchmark tree. As a result, the out-of-

sample values have a smaller variance than the in-sample values, as shown in Table 3

and Table 4. It is also observed that the true objective value of the solutions improves

as the number of scenarios increases.

Moreover, we solved the two-stage model on the benchmark tree and obtained the

“true” optimal objective values 513790 and 510845 for the given risk tolerance levels

D0 = 10000 and D0 = 6000, respectively. From Table 3 and Table 4, we may observe

that the scenario generation method does not introduce any significant bias, especially,

in the cases of more than 1600 scenarios.

24

Consequently we may conclude that the proposed method is stable and does not in-

troduce a significant bias. The results particularly show that we may get better solutions

when using more than 1600 scenarios. Hence, the tested scenario generation method is

suitable for the given mixed single stage R&D projects and multi-stage securities port-

folio selection model.

Table 3: Stability test for the given risk tolerance level D0 = 10000

# of scenarios

type of test objective value 900 1600 3600

in-sample Terminal Wealth average 512338 513314 513318

std. dev. 1073 3006 401

out-of-sample Terminal Wealth average 512576 512129 513309

std. dev. 971 508 391

Table 4: Stability test for the given risk tolerance level D0 = 6000

# of scenarios

type of test objective value 900 1600 3600

in-sample Terminal Wealth average 509864 510158 510280

std. dev. 685 1687 250

out-of-sample Terminal Wealth average 509696 509806 510268

std. dev. 533 328 240

6 Conclusion

Currently, investors invest in various asset classes to keep their competitive advantage.

Some securities and R&D projects can be integrated into a mixed asset portfolio. The

mixed asset portfolio increases the investors’ benefit opportunities. Considering the

different characteristics of securities investment and R&D projects investment, we have

25

proposed a mixed single-stage R&D projects and multi-stage securities portfolio selection

model. Specifically, a bi-objective mixed-integer stochastic programming model with the

constraints on the initial capital budget, the risk control in each intermediate stage and

no selling short of securities is presented. Moreover, the model uses the semi-absolute

deviation risk function to measure the risk of mixed asset portfolio. Based on the idea

of moments approximation, we give a scenario generation procedure for the stochastic

returns of single-stage R&D projects and multi-stage securities. Given a scenario tree,

efficient solutions of the bi-objective mixed-integer stochastic programming model can

be obtained by transforming it into a single objective mixed-integer linear programming

model. Thus the proposed model enables the investors to construct and manage their

mixed securities and R&D projects portfolio effectively.

Acknowledgements The authors are indebted to the referees for their helpful and

constructive comments.

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