The Errors Estimate of Optimal Combined Forecasting*


TheTheTheThe ErrorsErrorsErrorsErrors EstimateEstimateEstimateEstimate ofofofof thethethethe MultistageMultistageMultistageMultistage CombinedCombinedCombinedCombined
IIIInvestmentnvestmentnvestmentnvestment RiskRiskRiskRisk AssessmentAssessmentAssessmentAssessment∗∗∗∗

YYYYuuuu JikeJikeJikeJike ZhouZhouZhouZhou ZongfangZongfangZongfangZongfang1

School of Management and Economics, University of Electronic Science & Technology
Chengdu Sichuan, 610054, P.R.China

AbstractAbstractAbstractAbstract

Investment risk is economic development faced serious risk. The multistage combination investment
risk assessment (MCIRA) can reduce the assessment error, but how to survey the error which produces
by the MCIRA models, has the important significance. From theoretical side, the errors upper-bound of
the MCIRA models is determined in this paper. We also give the relationships between the errors of the
general MCIRA models, the simple average models and the errors of each investment risk assessment
model in the combination.

Keywords: investment risk; errors estimate; simple average models; MCIRA models.

∗This research has been supported by National Natural Science Foundation of China (No. 70971015).
1Zhou Zongfang, professor, zhouzf@uestc.edu.cn

Journal of Risk Analysis and Crisis Response, Vol. 1, No. 2 (November 2011), 106-109

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                   106

mailto:zhouzf@uestc.edu.cn


Zhou Zongfang and Yu Jike

1.1.1.1. IntroductionIntroductionIntroductionIntroduction

Since the combined investment risk assessment
methods were brought forward, a number of scholar have
studied the manifold methods of determining the
combined weighted coefficients, such as: the
equal-weight or the simple average method, the recursive
equal-weight method, the superiority-matrix method,
novel particle swarm, real options and so on[1]-[8]. To our
great regret, there were few research results available on
the errors bounds of multistage combined investment risk
assessment problems [9]-[10].

In this paper, we first determine the errors maximum
of the simple average combined investment risk
assessment model, then the errors maximum of the
general multistage combined investment risk assessment
model will be discussed, at last we will give the
relationships between the errors of the general multistage
combined investment risk assessment (MCIRA) model
and the simple average models and the errors of each
investment risk assessment model in the combined
assessment.

We utilize n investment risk assessment models for
the same investment risk assessment problem within T
given stages. To determining the combined risk
assessment weights, the multistage combined investment
risk assessment (MCIRA) problem is described as below:

On the assumption that yt (t=1,2, …, T) are actual
observations of investment risk in the t-th stage; fit
(t=1,2, …, T; i=1,2, …, n) are the risk assessment values
in advance of the i-th investment risk assessment model;
eit = ytttt ---- fit are errors between the observations and risk
assessment values of the i-th model in the t-th stage. The
combined investment risk assessment values are:

∑
=

=
n

i
itit ff

1
µ , t=1,2, …, T (1)

where µ i (i=1,2, …, n) are called the combined risk

assessment weights, which satisfies∑
=

n

i
i

1

µ =1, (i =1,2, …,

n); et =∑
=

n

i
ite

1

= ytttt ---- ft (t =1,2, …, T) are called errors of

the combined risk assessment.

Let ∑∑∑
===

==
n

i
iti

T

t

T

t
t eeJ

1

2

11

2 )( µ

µµµµ )(
1 1 1

)]([ n
n

i

n

j

T

t
jtitji Eee ′== ∑∑ ∑

= = =

(2)

be the errors square sum of the MCIRA model, where
)( 21 ′= n,μ,,μμμ ⋯ is called weight

vector; )( 21 ′= iTiii ,e,,eeE ⋯ (i=1,2,…, n) are error
vectors of the i-th model, jijiij EEEE ′== ,

nnijn EE ×= )()( is a n × n symmetric

matrix, i
T

t
itii JeE == ∑

=1

2 (i=1,2, …, n) is just the

investment risk assessment errors square sum of the i-th
investment risk assessment model.

The matrix )(nE provides the error messages of
each investment risk assessment model. We call )(nE the
risk assessment errors information matrix of the MCIRA
model and assume the matrix )(nE is invertible (or
replace by the errors vectors Ei (i=1,2, …, n) are linear
independence).

DefinitionDefinitionDefinitionDefinition 1:1:1:1: If *(n)
** μEμJ )( ′= , ( *iμ ≥ 0,

i=1,2, …, n) is minimal about µ , the µ * is called the
optimal combined assessment weight vector, the J* is
called the minimal errors square sum of the optimal
MCIRA model.

2.2.2.2. TheTheTheThe errorserrorserrorserrors boundsboundsboundsbounds ofofofof simplesimplesimplesimple averageaverageaverageaverage modelmodelmodelmodel

DefinitionDefinitionDefinitionDefinition 2:2:2:2: In equation (1), if
n

μi
1

= , i=1,2, …,

n , then the corresponding MCIRA model is called the
simple average model.

Let errors square sum of simple average model is:

)0()0( )( μEμJ (n)A ′= (3)

where )
1
,,

1
,

1
()0( ′=

nnn
μ ⋯ .

If the errors square sum of the i-th investment risk
assessment model is Ji, let their maximum and minimum

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The Errors Estimate of the Multistage Combined Investment Risk Assessment

be respectively:

{ }ini JJ ≤≤= 1max max and { }ini JJ ≤≤= 1min min .

TheoremTheoremTheoremTheorem 1:1:1:1: If the MCIRA model is the simple
average model, then

maxJJA ≤ . (4)

CorollaryCorollaryCorollaryCorollary 1:1:1:1: If the MCIRA model is the simple
average model, then minA JJ < if and only if

∑∑
= =

<
n

i

n

j
ij JnE

1
min

2

1

(5)

TheoremTheoremTheoremTheorem 2:2:2:2: Supposing (n)E is an positive definite
matrix, if the errors square sum of each investment risk
assessment model in the combination is the same constant
C ( i.e. Ji = C, i=1,2,…, n), then

AJ < C (6)

Above theorem indicates that the simple average
model can reduce the assessment errors, if Ji = C
(constant), i=1,2, …, n. For example, under the
hypothesis of theorem 2, if ),,,( 21 ′= iTiii eeeE ⋯ ,
i=1,2,…,n are n (T ≥ n) orthogonal vectors in
T-dimension space (i.e. nnjinnij(n) EEEE ×× ′== )()( is a
scalar matrix, and its diagonal elements are constant C),
then the errors square sum of simple average model is

C
n
1

.

3.3.3.3. TheTheTheThe errorserrorserrorserrors estimateestimateestimateestimate ofofofof thethethethe generalgeneralgeneralgeneral MCIRAMCIRAMCIRAMCIRAmodelsmodelsmodelsmodels

Let the errors square sum of the general MCIRA
model (In order to distinguish other peculiar
circumstances, said that it is the general MCIRA model）
be:

μEμμJJ (n)′== )( (7)

where µ is a weight vector.

Lemma:Lemma:Lemma:Lemma:[11] If nξξξ ,,, 21 ⋯ are n different
(n-dimension) weight vectors, and constants

nααα ,,, 21 ⋯ satisfy 01
1

>=∑
=

i

n

i
i ,αα , then

∑∑
==

<
n

i
ii

n

i
ii ξJαξαJ

11

)()( (8)

or ∑∑∑
===

<′
n

i
i(n)ii

n

i
ii(n)

n

i
ii ξEξαξαEξα

111

)()()(

TheoremTheoremTheoremTheorem 3:3:3:3: If µ = ),,( 21 ′nμ,μμ ⋯ is a weight

vector, and 01
1

>=∑
=

i

n

i
i ,μμ , then

∑
=

<′==
n

i
ii(n) JμμEμμJJ

1

）（ . (9)

In other words, the errors square sums of the general
MCIRA model don’t exceed the weight sum of errors
square sums of each investment risk assessment models
in the combination.

CorollaryCorollaryCorollaryCorollary 2:2:2:2: If µ is any nonnegative weight
vector, then

i) max(n) JμEμJ <′= (10)

ii) max
n

i
iA JJn

J ≤< ∑
=1

1
(11)

Form the above results, we know that the errors
square sum of the general MCIRA model doesn’t exceed
the maximum of errors square sum of each model in the
combination. We can reduce the investment risk
assessment errors by combining assessment model. The
general MCIRA model's errors that showed by errors
square sum are bounded above, and the superbound
is maxJ .

4.4.4.4.AnAnAnAn ExampleExampleExampleExample

Considering a general MCIRA model: the reciprocal
variance weight multistage combined investment risk
assessment model, its weight vector is:

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Zhou Zongfang and Yu Jike

)
1

,,
1

,
1

(
1

),,,(
21

1

21 ′=′=

∑
=

n
n

i
i

n JJJJ
μμμμ ⋯⋯ ,

From theorem 3, its errors square sum J* satisfies
inequation as below:

)
111

(
1

1
2

2
1

1

1

1
n

n
n

i i

n

i
ii

* J
J

J
J

J
J

J

JμJ +++=<
∑

∑
=

=

⋯

cn

i i

M

J

n
==

∑
=1

1

Where Mc is called the harmonic mean of J1, J2 , …,Jn.
Consequently the errors square sum J* of the reciprocal
variance weight model is smaller then the harmonic mean
Mc of each investment risk assessment model in the
combination. In particular, if Ji >0, i=1,2,…,n, since the
harmonic mean don’t exceed the arithmetic average value,
we express:

max

n

i
iC

* JJ
n

ΜJ ≤≤< ∑
=1

1
(12)

the above inequation is just inequation (11).

5.5.5.5. ConclusionsConclusionsConclusionsConclusions

In this paper, we have estimated the errors boundary
of the simple average method and the general MCIRA
models, and indicated that the errors is the bounded to the
above theoretically. We have also applied the
mathematics analysis techniques to determine the existent
maximum of the errors square sum of the general MCIRA
models. This article research to carries on the appraisal
accurately to the investment risk, dodges the investment
risk effectively, has the very important theory and the
practical significance. The research conclusion is also
suitable for the general combination risk assessment
domain, such as combination credit risk assessment[12],
insurance risk assessment, disaster risk assessment and so
on. We certainly believe that these studies will be of a
great significant theoretical value and potential practical
significance in the risk management domain.

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	1.Introduction
	2.Theerrorsboundsofsimpleaveragemodel
	3.TheerrorsestimateofthegeneralMCIRAmodels