Sci. Rep. Kanazawa Univ.
Vol. 56, pp. 15–34, 2012

A method for portfolio evaluation based on linear
expansion and its application to an empirical
portfolio
Naila-Al-MAHMUDA1, SEIRO OMATA2, KENTA KOBAYASHI3
1 Graduate School of Natural Science and Technology, Kanazawa University,

Kanazawa, Ishikawa, Japan.
2 Institute of Science and Engeneering, Kanazawa University, Kanazawa,

Ishikawa, Japan.
3 Graduate School of Commerce and Management, Hitotsubashi University,

Kunitachi, Tokyo, Japan.
(Received December 20, 2012 and accepted in revised form January 23, 2013)

Abstract This paper introduces a new scheme for portfolio evaluation founded ona linear expansion in a basis of economic indicators. The core idea is to express theprice movement of a complex set of financial instruments by several carefully selectedindicators whose behaviour is by far more understandable than the complicated relationsof financial derivatives contained in the portfolio. The principles of selection of theeconomic indicators for a given portfolio are discussed in detail and the method is tested
on a basic portfolio containing one stock.
Key words: Portfolio analysis, Economic indicator, Expansion method.

1

Introduction

The property of financial institutions, such as banks, should be evaluated based on international standard law, Basel II accord. According to this agreement, financial institutions
are required to quantify the operational risks. The property, or financial portfolio, of
the institution is a collection of financial instruments, such as stocks, bonds, derivatives,
etc., and is often very complex. Therefore, it is necessary to find a suitable model forevaluation of such complex portfolios. Moreover, the decision on which assets should
be included in the portfolio, when and how many of these assets should be purchased orsold, always involves the expected return on the portfolio, and the risk associated withthis return [8].
In 1933, Cowles examined return but ignored any consideration of risk [9], while

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An expansion method for portfolio evaluation and its application

Markowitz developed techniques [10] for evaluating portfolio performance based on risk
and return. He showed that the risk on a portfolio return depends not only on the riskinessof the individual components associated with the portfolio but also on the relationshipamong the components. On the basis of this context, Sharpe [4] developed the capital
asset price model. This model assumes that the risk-return profile of a portfolio can be
optimized, that the asset return is normally distributed and that there is no transaction
cost.
In the early 1990s, Value at Risk (VaR) was the only common risk measure that could
be defined for all types of business. In 1994, JPMorgan Chase & Co., an American multinational banking corporation of securities, first published the methodology of VaR. VaR
describes the losses that can occur over a given period of time at a given confidence level
due to market risks [5]. Having the ability to express the market risk of an entire portfolio
in one number, it is a well established and accepted risk measure in the risk management
practice. In recent years, it has become not only a risk measurement method but also a
process that helps decision-making and investment guiding.
According to Hendricks [6] there are several methods that can be used for portfolio’s
VaR calculation. Those methods which are based on historical information can be classified into three groups: Variance-covariance method, Monte-Carlo simulation method,and Historical simulation method.
According to the variance-covariance approach, returns on the portfolio are normally
distributed and the portfolio is a linear function of its risk factors. Its main problem isthat the returns distribution may have fat tails and thus the VaR will underestimate theexpected losses. The Monte Carlo method generates large number of random market scenarios. The fundamental assumption in this method is that the portfolio price changes arenormally distributed and the distribution is generated from the market scenarios. Thenfor each scenario, the profits and the losses of the portfolio are computed and the corresponding volatility is calculated. In case of complex portfolios including derivatives
like options, Monte Carlo simulation can be used to evaluate the VaR, even though it
requires a longer time to be performed. Historical simulation method uses the historical
distribution of returns of a portfolio to simulate volatility which measures the VaR. On
the other hand, the historical simulation method does not assume that the asset returns
are normally distributed. Also, it is relatively easy to implement. It assumes that theasset returns are independent and identically-distributed and it applies equal weight to all
returns over the whole period.
The main purpose of this study is to evaluate a given financial portfolio through thehistorical simulation approach based on certain important economic indicators, which
improves the volatility (risk) estimations provided by the standard historical simulation
approach. Based on a detailed analysis of the relationship between the portfolio andeconomic indicators, economic indicators relevant to the portfolio are chosen from thecurrent market. The value of the portfolio is then approximated by a linear combina-

Naila-Al-Mahmuda et al.

17tion of the values of the selected indicators. The coefficients of the linear expansion arecalculated by a fitting method using known historical data for both the indicators and theportfolio. By extrapolation we can then construct an approximate distribution of expected
portfolio price changes from which the VaR and other properties can be deduced.
The main difficulty of portfolio evaluation up to now has been the fact that the portfolio consists of a large number of instruments that are correlated to each other. In manycases, especially for financial institutions, it is usual to have more than 10,000 instruments in a portfolio. This implies the necessity for a huge number of calculations, evenjust for the initial calculation of the correlations between the instruments. Such calculations are time-consuming even when modern supercomputers are employed. Moreover,considering a large number of instruments separately entails the growth of error and possibility of VaR overestimation.
On the other hand, the new method uses only a few representative economic indicators, usually of the order of tens, which is a very small number comparing to the numberof financial instruments associated with the portfolio. Subsequently, the computation canbe significantly accelerated and the time constraint overcome. The method can also provide a clearer view of the otherwise complex system because it reduces the study of thesystem to the analysis of a few representative indicators. These, if chosen correctly, canserve as efficient gauges for expected reactions of the whole portfolio to specific changes
in the market.
The paper is organized as follows: Section 2 introduces the new method using economic indicators and section 3 explains the method of choosing the economic indicators.
In section 4 we continue with an empirical analysis of the proposed method on the example of a portfolio consisting of one stock, Toyota Motor Corporation, using a basis of 18
indicators.

2

Methodology

The main idea of the proposed method is to approximate given financial instrument(s)by a linear combination of time-series data of selected economic indicators. There arethree equally important steps in building the method. The first step is to give general
guidelines for the selection of economic indicators, while the second step is to develop an
asset pricing model using the selected economic indicators. The third step is to calibrate
the method for a given portfolio.
Let a be the vector of time-series data of the considered instrument’s closing prices
from the first day up to the m-th day of the considered period :a = (a1 , a2 , · · · , am ).

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An expansion method for portfolio evaluation and its application

Further, let ri denote the vector of time-series data of i-th indicator values from the firstday up to m-th day of the considered period:iri = (r1i , r2i , · · · , rm).

Then we consider the following linear approximationna ' ∑ αi ri ,

(2.1)

i=1

where αi ∈ R denotes the coefficients of the linear expansion, and n is the total number ofindicators. The values of the coefficients αi are obtained from the least-squares approachas the minimizers of the following functional:nI(α1 , α2 , · · · , αn ) = |a − ∑ αi ri |2 → min,i=1

where |.| is an m-dimensional Euclidean norm.

2.1

Data management

According to the Dow theory, at any time the stock market has three kinds of trends. The
primary trends last from less than one year to several years, the secondary trends last fromten days to three months and the minor trends last from one day to three weeks or more.
The thirdly classified trends are subject to some degree of manipulation and thus thesetrends are not important for our analysis, or even can be misleading. Therefore, instead of
the raw data we consider the means of the daily closing prices over certain short periodsof several days. This smoothes out local fluctuations while preserving the global tendencyof the data. We calculate the averaged value xJ0 at J-th day by the following expression:
xJ0 =

J+d
1
∑ x j,
2d + 1 j=J−d(2.2)

where d is a suitably chosen natural number expressing the time-span over which averages are taken.
In order to be able to observe the influence of individual indicators on the value ofthe portfolio without any bias, the time-series data of both the portfolio and the economic
indicators are scaled from the actual time-series data by the following expression:x=x0 − x̄0
,
x̄0

(2.3)

Naila-Al-Mahmuda et al.

19

where x represents the scaled time-series data, x0 is the smoothed time-series data from
(2.2) and x̄0 is the mean of x0 .
In what follows, we will use the notion of correlation coefficient between two timeseries data a and b defined by

ρ (a, b) =cov(a, b),âb̂

where â, b̂ are the standard deviations of a, b, respectively, and the covariance of a and b
is given by the formulacov(a, b) =1 m
∑ (a j − ā)(b j − b̄).
m j=1

The value of the correlation coefficient determines the degree of relationship between
two data sets.
As a measure of quality of approximation of data set a by another set b, we will use
the following cummulative relative errore(a, b) =3

1 m |a j − b j |∑ |a j | .
m j=1

(2.4)

Principles of selection of economic indicators

Concepts for evaluation of the risk associated with a portfolio are based on models describing changes in the factors that influence the value of individual assets and the correlation between these factors. In this section, we discuss the basic guidelines for theselection of a representative set of such factors, called economic indicators in the sequel.
There are various factors that may influence the portfolio prices such as basic market
factors, market indices that can judge the health of economy, and currency exchange, afactor related to trading.

3.1

Basic market factors

Market factors are all the market participants selling or taking delivery of the productsthat include any type of goods, services and information. As an example, in crude oilmarket, the market factors are the oil companies, refiners and the physical trading companies. Financial derivatives, such as forwards, futures and options are also important
market factors in the financial industry. Derivatives are settled against the spot pricebased on the daily transaction of company’s stocks. From the viewpoint of indicator
selection, our method considers derivatives only in the sense of futures (see section 3.2).

20

An expansion method for portfolio evaluation and its application

Indicators falling into the category of basic market factors are selected in the following
steps:
1. Selection of region and market: regions are considered based on where the components of a given portfolio get their earnings such as revenues, return on investment,
etc., and how they pay their costs such as production cost, operating cost, laborcost, overhead cost, etc. On the basis of the important regions the financial markets are specified depending on where the portfolio is traded.
2. Selection of candidates for basic market factors: a relatively large number of market factors holding top positions in the financial markets obtained in step 1.
3. Selection of related industry categories: In order to select the market factors weneed to assess the performance of individual factor not only by region or country
but also by industry. We adopt the Industry Classification Benchmark (ICB) systemdeveloped by Dow Jones. This system divides the market into increasingly specificcategories according to similar functions and related products. According to thecomposition of the portfolio, suitable categories from ICB system are considered
in order to select the market factors.
4. Computation of correlation coefficients: determine the degree of correlation ofmarket factors from step 2 and the portfolio.
5. Final selection: in general, a market factor is selected as an indicator if it is amember of the candidates from step 2, falls under an industry category selected instep 3 and the correlation coefficient is comparatively high.
The selection of market factors outlined in the above five steps can be modified according to the focused product and business area of the portfolio components. For a specificportfolio, a detailed analysis of its structure becomes an important ingredient of the selection of indicators from the group of basic market factors. In this case, the correlation
coefficients may not be high but there exists strong relation with some of the portfolio
components. On the other hand, a highly correlated market factor can be excluded, if
there is no reasonable relation with any of the portfolio components or, more probably, if
its connection with the portfolio is too strong.

3.2

Market indices

A market index is a value computed from a group of data to measure a certain sectorof the market. These data may be derived from any sources, including company performance, prices, productivity, and employment. We select this area in order to identify the
impact of country’s economic health on the portfolio. Moreover, market indices are capable of involving other factors, for example, domestic and foreign political events such as

Naila-Al-Mahmuda et al.

21

war and terrorism, as well as natural disasters, while the physical market factors may not
be able to reflect these events. The financial indices measure the performance of selectedcompanies in order to evaluate the economic trends.
To give a few examples, the Dow Jones Industrial Average detects the USA market
trend, the consumer price index is an economic index detecting the variation in prices
for different consumer goods and services over time, the real GDP measures the levelof prices of all new, domestically produced, final goods and services in an economy andthe commodity index tracks the overall performance of commodity markets. Commodity indices often use the prices of future contracts for commodities that are listed oncommodity exchanges, and can be grouped into subcategories such as energy, metals,
agriculture, livestock, etc.

3.3

Currency exchange

Currency exchange rates can affect the returns on portfolio, especially when the portfolio
consists of components from foreign markets. A mutual fund is an example of such kindof portfolio that holds international stocks. Fluctuations in the value of the base currency
relatively to any foreign currency can affect the value of the portfolio. For example, when
the base currency decreases relative to a foreign currency, it results in a higher valuationof the portfolio as measured in base currency.
Moreover, there may exist a nonlinear relationship between the currency exchange
rates and the gaining of a portfolio component. For example, a portfolio component
can use foreign currency to accumulate its input while the revenue is calculated in local
currency. In order to maintain the stability and consistency of the approximation method,the exchange rates should not be considered only as components of the linear expansion
but should also be used, e.g., as weights for certain indicatros, such as commodity indices.

4

Empirical analysis

According to the methodology discussed in section 2, we now provide a detailed empirical analysis and model testing. We discuss the optimal choice of economic indicators,
approximate the given portfolio by a linear expansion and examine the properties of the
expansion on the example of a portfolio consisting of one component, Toyota Motor
Corporation (TMC) stock. Although our method is primarily designed for the analysis ofcomplex multi-component portfolios, the analysis of a simple one-component portfolio
is the first indispensable step in the validation of the model. Toyota Motor was chosenbecause it is one of the Japan’s biggest employers and the backbone of the country’s
manufacturing sector. Moreover, the changes in the entire stock market are related to thestock price of TMC.

22

4.1

An expansion method for portfolio evaluation and its application

Data Description

We use daily closing prices of TMC stock. The data were gained from Google financeand Forex database, ranging from January, 2006 to December, 2010. Collected data arerefined according to the formulae (2.2) and (2.3) with d=2 (see Fig. 1).

Figure 1: (a) closing prices of TMC stock from January, 2006 to December, 2010 and (b) the corresponding
averaged and scaled data.

4.2

Economic indicators

In this subsection, we perform the selection of specific indicators for our empirical portfolio according to the procedure discussed in section 3. First, we study the activities of
TMC from the geographical viewpoint. We would like to observe the activities of TMC

23

Naila-Al-Mahmuda et al.

related to its earnings and expenditures throughout the world and accumulate relevant
information, as suggested in step 1 of section 3.1.
As of end of March, 2012, Toyota conducts its business worldwide with 50 overseas
manufacturing companies in 27 countries. Toyota’s vehicles are sold in more than 160
countries.
Export from Japan (%)

RegionProduction (%)

Sales (%)

Japan

48

23

North America

18

36

45

Europe

8

15

16

Asia

20

22

7

Others6

432

From the above table (obtained from the Japan Automobile Manufacturers Association, Inc.), it is found that the most important regions for TMC are the domestic region,
Japan, and the overseas region, North America. Therefore, we have selected two financial markets: Tokyo Stock Exchange (TSE) and the New York Stock Exchange (NYSE).
It is also found from the databook of TMC that the highest percentage of Toyota’s revenue is sales revenue and the sales revenue mostly consists of the income from export tooverseas from Japan. On the other hand, the expenses associated with administering thebusiness on a day to day basis include the fixed cost, such as overhead cost that remains
the same regardless of the number of products produced, and variable costs, such as materials, which can vary according to the quantity to be produced. Moreover, an expense
incurred by activities not related to the core operations of the business can be denoted byindirect costs, such as advertising the products locally or internationally.
As in step 2 in subsection 3.1, we have considered the top 100 companies that are
most actively traded on the Tokyo Stock Exchange as the candidates to select indicators
under the basic market factors group. Since TMC is keeping more manufacturing companies in order to maintain the level of quality, we choose the basic market factors fromthe domestic market only.
According to the step 3 of subsection 3.1, we adopt the Nikkei sectors and components for the selection of related industry categories. Nikkei is the most widely quoted
average of Japanese equities, similar to the Dow Jones Industrial Average. The Nikkei
represents the performance of the market and components have been rebalanced so thatit can represent the Japan’s industrial structure. Nikkei component stocks are balanced
based on six sectors. For each sector we have analyzed the structure of the portfolio
and its relationship to the corresponding market factors. Brief discussion on indicator
selection based on six sectors of Nikkei is given below:• Financial sector: This sector is one of the important sectors for TMC. Toyota’s
expertise extends beyond its automobile manufacturing business into a variety of

24

An expansion method for portfolio evaluation and its application

other fields, including housing, financial services, communications, marine technology, biotechnology and afforestation. On the basis of company analysis, thefinancial services section is much correlated with automobile manufacturing business while the others are not much important to it. We have selected the Mizuho
Financial Group because it is highly correlated with TMC stock comparing to othermarket factors in financial sector. One of the operating subsidiaries, Mizuho Bank,is the only bank, other than Japan Post Bank, to have branches in every prefecture in Japan. Moreover, the Mizuho Financial Group offers financial services
including banking, securities, trust and asset management services in 30 countries
throughout the world.
• Capital goods: JTEKT is selected from the subsector machinery under capital
goods sector. The main activities of JTEKT are manufacturing and sale of machine tools, such as metal cutting applications that are used in electric devices or
automotive engines and chassis. It also manufactures and sales auto parts, such as
steering systems, driveline components etc. Moreover, it is a member of Toyotagroup and the correlation is high. Since TMC needs different types of machinery
for its business operations, it is assumed that it collects most of its machineries
from the group member, JTEKT.
• Technology: Electric machinery is the most important subsector related to our empirical portfolio under the technology sector. Hoya and Aisin are two importantfactors under this sector from the viewpoint of TMC stock. Hoya is a global technology company and the leading supplier of innovative and necessary high-tech
products and services. The Aisin Group offers a wide lineup of products covering
almost every automobile-related field and it is also a member of Toyota Group.
• Materials: Raw materials in automobile industry may differ from a manufacturer
to another, direct or by-products. Steel and petroleum-based products represent
an increasingly large percentage of automobile components. Besides, aluminium,
glass, rubber and plastics are the main materials used in automobile manufacturingsector.
In cars, steel is used to create the underlying chassis or cage beneath the body that
forms the skeleton of the vehicle. It is also used in a variety of areas throughout the
body to accommodate the engine or other parts, such as door beams, roofs, etc.,
and exhausts are often made from stainless steel. Steel manufacturing has evolved
greatly, so carmakers can make different types of steel for different areas of thevehicle. In January, 1934, the Aichi Steel Corporation was founded by the TMC as”Steel Production Department” for the research and creation of specialty steel for
car manufacture.
Nippon Electric Glass Co. Ltd manufactures and sales a wide range of glass related

Naila-Al-Mahmuda et al.

25

products, such as glass for electronic devices, heat-resistant glass etc. On the other
hand, Toyota door glasses aren’t just an ordinary part of the vehicle, but rather
considered as one of the most essential auto part. These two companies are highly
correlated to each other.
In 1949, Toyota Motor founded Toyota Gosei Co. Ltd, a rubber research operationand continues to own nearly 45% of the company. 95% of sales revenue of thiscompany is composed from manufacture of auto components and Toyota Motoris one of the major customers of its products. This company is key partner in the
globalization activities of Toyota Motor Corporation in North America, Europe
and Southeast Asia.
• Consumer goods: Toyota Tsusho is the trading unit for the Toyota Group, which
includes Toyota Motor and auto parts maker Denso. The company deals with awide range of goods, including metals such as steel and aluminium, machinery andelectronics, energy and chemicals, and various consumer products. Toyota Tsusho
exports Toyota vehicles around the world, but plenty of non-automotive consumer
products also contribute to its business. Moreover, Toyota Motor Corporation owns
22% of the company.
• Transportation & Utilities: Toyota started research and development of fuel cellvehicles in 1992 with the aim of developing zero-emission ecology cars so thatthese cars will reduce the amount of CO2 emitted by the transportation sectors.
For example, as a result of hybrid technology advancement, it is developing thefuel cell hybrid vehicle (FCHV) based on its own proprietary technology. Japanese
automakers are working to reduce the cost of manufacturing a fuel-cell system forsuccessful adoption of this vehicle. In this regard, more than 20 Japanese companies including TMC and Tokyo Gas Co. Ltd-Japan’s number one gas company,
are working together for the development of hydrogen supply infrastructure. In
1980’s the Tokyo Gas Co. Ltd started to develop fuel cells, which constitute potential applications for natural gas. In the field of transportation fuel, this company
is trying to spread the use of natural gas vehicles. Also it is trying to develop thetechnology for organizing a hydrogen supply infrastructure based on natural gas.
This technology is intended to disseminate the use of fuel cell vehicles.
Next, we deliberate on the selection of the remaining indicators related to market
indices and currency exchange. The economic impacts on automobile industry associated with movements in crude oil prices is one of the important points. Crude oil pricemovements can be driven by several factors, including fundamental changes in supply
and demand, speculation and geopolitical concerns [1]. Japan imports about 80% of its
oil and natural gas since its natural resources are extremely low. The overall demand
for automobiles declined during 2008-2010 due to a global financial crisis. The financial

26

An expansion method for portfolio evaluation and its application

downturn also affected auto manufacturers all around the globe. Despite the decline in demand Toyota had a competitive edge enhanced. It had also created hybrid engine designs
that consumed less fuel and produced less environmental pollution and thus maintaining
their overall sales at a higher level even the recession period was worsened by the rapid
increase in the prices of fuel.
The West Texas Intermediate (WTI) crude oil price, quoted in U.S. dollar (USD) andacquired from the U.S. Energy Information Administration, is chosen as a representativeof the world oil price. Oil price increased in the first six months of 2008 and then collapsed. Peak oil theory, originated by M. King Hubbert in 1956, would imply that the
sharp fall in oil prices in 2008 was due to the sudden discovery of new oil reserves [7].
On the other hand, our portfolio prices also experienced a downward trend in the same
period (see Fig. 2).
10000
TMC
WTI

140

8000

120

80

4000

60

Stock price (WTI)

Stock price (TMC)

100
6000

40
2000
20

0

0
Oct’06

Aug’07

Jun’08time

April’09

Feb’10

Dec’10Figure 2: Data of the TMC stock price and the WTI oil price.

In addition, Speir (2004) concludes that oil price volatility alone explains not morethan a half of gasoline price movements. This result is supported by Ashton and Upton (2004). The U.S. Gulf coast area is one of the most important regions for energy
resources and infrastructure. Over 40 percent of total U.S. petroleum refining capacity
is located along the Gulf coast. The Gulf coast has plentiful access to gasoline from its
own refineries and it supplies a large proportion of the gasoline. Therefore, we have also
considered U.S. Gulf coast conventional gasoline spot price as one of the indicators.
In principle, building linear expansion only through commodity producing companies is inadequate for getting exposure to the given portfolio. Portfolio returns are not
only linked to company specific risk but also closely related to the prices of commodities. Thus, the diversified portfolio holders carefully watch the benchmark of commodity
indices. Protection against inflation is also one of the main considerations to include
commodity indices as indicators. Moreover, the commodities are an important part of the

27

Naila-Al-Mahmuda et al.

Currency

Market Indices

Basic Market Factors

production process of many goods. We have selected the commodity indices Gasoline
index, Rubber index, and Crude oil index from the Tokyo Commodity Exchange Inc. In
this section, we have also chosen the Dow Jones Industrial Average (DJIA) as an indicator in order to gauge the performance of industrial sector within the American economy.
Dow is one of the most closely watched U.S. benchmark indices, tracking targeted stock
market activity.
The majority of Toyota vehicles are sold in the U.S., Japan, Europe and Australia andhence, it is exposed to fluctuating economic and political conditions of those markets.
The Japanese currency, one of the major currencies in the world, strengthened 45 percent
over five years. The stronger yen makes it harder for companies, such as TMC to makevehicles in Japan and sell them profitably in the overseas, since Toyota is keeping moremanufacturing companies in Japan in order to maintain the level of quality. Therefore, we
consider three of major exchange rates: U.S. dollar, Euro and Australian dollar (respective to the Japanese yen), which have significant impact on the earnings and expenditureof our empirical portfolio. We also convert the WTI crude oil prices and gasoline spot
prices from USD to Japanese yen to reflect the effect of the exchange rates.
According to the above discussion, we have selected the following 18 economic indicators (see Table 1).
Correlation
coefficient

Nikkei Sectors

Subsectors

Market Factors

Financial

BankingMizuho Financial Group

0.91

Capital Goods

Machinery

JTEKT Corporation

0.95

Hoya Corporation

0.91

Technology

Materials

Electric Mechinery

Aisin Seiki Co. Ltd.

0.88

Glass & Ceramics

Nippon Electric Glass Co. Ltd.

0.87

RubberToyota Gosei Co. Ltd.

0.63

Steel Products

Aichi Steel Corporation

0.78

Crude OilWest Texas Intermediate Crude Oil

0.25

Gasoline

U.S. Gulf Coast Conventional Gasoline

0.46

Consumer Goods

RetailToyota Tsusho Corporation

0.96

Transportation &Utilities

Gas

Tokyo Gas Company Limited

0.91

Gasoline index

0.71

Rubber index

0.53

Crude Oil index

0.43

Dow Jones Industrial Average

0.78

USD/JPY

0.93

EUR/JPY

0.85

AUD/JPY

0.78

Table 1: Selected 18 economic indicators

28

4.3

An expansion method for portfolio evaluation and its application

Approximation of the portfolio

In order to approximate the value of the portfolio, we have divided the total period (January, 2006 to December, 2010) into two smaller subperiods [2]. The first period is used
to determine the coefficients of expansion (2.1), and will be called “historical period”.
In practice, the data of economic indicators and portfolio prices are known for this period, which makes the calculation of the least-squares approximation possible. The second period, called “expansion period” is where the portfolio is actually evaluated and inpractice corresponds to the future. For experimentation and analysis purposes, we canchoose the starting point of expansion period arbitrarily. Considering the financial crisisof 2007-2008, a downturn in global economic activity leading to the “Great Recession”,
we decided to start the expansion from August, 2007.
First, we need to determine a suitable length for the historical period, so that it yields
a sufficiently precise approximation in the expansion period. The dependence of the relative error of the approximation on the length of historical period was investigated forseveral lengths of expansion period.
The results are shown in Fig. 3 and suggest a monotone decrease in the error withincreasing length of the historical period and with decreasing length of the expansion
period. A suitable length of the historical period can then be set based on prescribedvalues for the error tolerance. In our case, for example, based on the requirement thatthe relative errors should be below 4% for expansion periods up to 50 days, the historicalperiod was set to 400 days. The behaviour of the expansion may vary according to thestarting point - we address this topic in the following subsection.
10
EP7
EP15
EP30
EP50

9

relative error for expansion period(EP)

8
7
6
5
4
3
2
1
0
50

100

150

200
historical period

250

300

350

400

Figure 3: Dependence of relative error on the length of historical period for several choices of expansion
period.

Naila-Al-Mahmuda et al.

4.4

Analysis of the approximation

4.4.1

Error analysis

29

The resulting approximation of the portfolio with starting point August, 2007 and historical period 400 days is depicted in Fig. 4 (a). Here we used the linear approximation for
the TMC stock price obtained from the historical period and extended it by substituting
the known data for economic indicators during the expansion period. The figure thus
gives the comparison of evolution of the extrapolated linear expansion with the motionof the real portfolio prices. Fig. 4 (b) shows the relative error evolution from the starting
point up to the end of available data in December, 2010. The portfolio prices are scaled
according to the equation (2.3) but relative errors are computed by (2.4) using the actual
prices. The obtained values of expansion coefficients αi from (2.1) for each indicator are
shown in Table 2.

Figure 4: (a) real portfolio value (red line) and its approximation (green line), and (b) the corresponding
relative error.

30

An expansion method for portfolio evaluation and its application

Economic indicator

Coefficient

Aisin Seiki Co. Ltd.

0.76

Tokyo Gas Company Limited

0.39

JTEKT Corporation

0.38

Dow Jones Industrial Average

0.32

EUR/JPY

0.31

U.S. Gulf Coast Conventional Gasoline

0.21

Mizuho Financial Group

0.17

Nippon Electric Glass Co. Ltd.

0.11

Crude Oil index

0.009

Rubber index-0.02

USD/JPY-0.07

Toyota Tsusho Corporation-0.08

AUD/JPY-0.12

West Texas Intermediate Crude Oil-0.14

Hoya Corporation-0.18

Aichi Steel Corporation-0.25

Toyota Gosei Co. Ltd.

-0.27

Gasoline index-0.48

Table 2: Indicator coefficients of TMC stock for historical period January 4, 2006 to August 17, 2007

One can observe that the error is kept below 5% for almost one year. Moreover, thecorrelation coefficient of the actual prices and the approximated prices within the period
from starting point to December, 2010, is as high as 0.916272. These facts support theapplicability of the approximation method.
Focusing on the economic event of 2008 and 2009, known as “Great Recession 20082009” we examine the behaviour of our expansion from January, 2008 to December,2009. The purpose is to test the robustness of the expansion method in situations with
rapid and significant changes in the economy. The recession began by the end of 2007
and the recovery phase tentatively began in mid-2009 [3]. Japan went collectively into
the recession at the middle of 2008.
It is found that the error of the expansion during the whole recession period is lessthan 10% (see Fig. 5), which suggests that the expansion has a relatively stable behaviour
even in extreme situations, such as economic recession.

31

Naila-Al-Mahmuda et al.
0.8
actual Toyota stock priceour approximation
recession period
relative error=4.8%

0.6

stock price (scaled)

0.4

0.2

0

-0.2

-0.4

-0.6
Oct’06

Aug’07

Jun’08time

April’09

Feb’10

Dec’10Mar’09Jun’09

0.1

0.09

0.08

relative error

0.07

0.06

0.05

0.04

0.03

0.02
Dec’07May’08

Oct’08time

Figure 5: Graphical representation of different periods and the relative error for recession period.

4.4.2

Feasible expansion period

In order to find the expansion period, for which the approximation gives reasonable results, we have calculated the evolution of relative errors for a large number of different
starting points. The results for five selected starting points are shown in Fig. 6.
We observe that the error for starting points falling into the recession period cross the
5% borderline within approximately one month. Since the error continues to rise after 4
weeks for all starting points, we conclude that the safe expansion period (in the sense ofallowed error of 5%) is not more than 3-4 weeks. If we allow errors only up to 3%, theexpansion method works for circa 2 weeks.

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An expansion method for portfolio evaluation and its application

0.07
Sp Aug’07
SP Jan’08
SP June’08
SP Nov’08
SP Feb’10

0.06

relative error

0.05

0.04

0.03

0.02

0.01

0
0

5

10

15
expansion period(days)

20

25

30

Figure 6: Evolution of relative error for different starting points.

4.4.3

Ability to follow changes in indicators

In this section, we verify the robustness of the expansion method by investigating its ability to accurately pursue rapid changes in certain aspects of the market. If the method isrobust in this sense, it can become a convenient tool for the evaluation of impacts on theportfolio of various expected or forseen changes in the market, such as abrupt changes inoil prices and exchange rates, decline or growth of economy in some countries, etc.
The robustness is tested on past data by picking an indicator and a test period during
which the indicator displays a significant monotone change. The value of this referenceindicator within this period is replaced by a simple linear interpolation and the remaining indicators are generated with random changes correlated to the reference indicator.
Values of indicators obtained in this way are substituted into the approximate expansion
for the portfolio, the predicted value at the end of the testing period is calculated and thisexperiment is repeated for a large number of times, giving a probability distribution for
the value of the portfolio at the end of the test period.
In particular, we chose the DJIA since its value decreases almost monotonely from
August, 2007 to August, 2008 (see Fig. 7). The setting for the approximate expansion isas in section 4.4.1. The value of DJIA is prescribed as a linear function interpolating between its values on 19th of August, 2007 and 22nd of August, 2008. The historical period
for the approximation is then from 4th of January, 2006 to 18th of August, 2007. The
remaining 17 indicators are random values correlated with the above prescribed DJIA.
The results of 10,000 experiments yielded a probability distribution for TMC stock
price on 22nd of August, 2008 (see Fig. 8). The mean of this distribution was foundto be 0.0063, which corresponds the value of 5031.50, and is close to the actual value

33

Naila-Al-Mahmuda et al.
4

1.6

x 10

stock price

1.4
1.2
1
0.8
0.6

0

Oct’06

Aug’07

Jun’08time

April’09

Feb’10

Dec’10Figure 7: Original and interpolated DJIA data.

of 4896.50. The stock prices move within the interval (4770.00, 4920.00) (red mark in
the figure) from 19th to 27th of August, 2008. Several different test periods within themonotone decrease of the DJIA were examined with similar agreement of the predicted
and actual value of TMC stock price. Therefore, we conclude that the proposed method
has a good potential to accurately respond to changes in market factors.
700
600

Mean = 0.0063
Actual = −0.0207

500
400
300
200
100
0
−1.5

−1

−0.5

M

0.5

1

1.5

Figure 8: Probability distribution for TMC stock on Aug 22, 2008 obtained from 10,000 simulations.

5

Conclusion

In this paper, we introduced a method for portfolio evaluation based on the approximationof the portfolio by a linear combination of a relatively small number of market indicators.

34

An expansion method for portfolio evaluation and its application

In this way, it is expected that the analysis of a huge number of financial instruments(often including complex derivatives) and their complicated correlations can be reduced
to the investigation of several carefully picked simple gauges, which will provide a clear
view of the tendencies and properties of the portfolio.
The method was tested on a one-component portfolios, and favorable results wereobtained. Namely, it was found that the suggested approximation gives a good agreementwith real data for certain feasible extrapolation periods, whose lengths can be estimated.
Moreover, the approximation is robust with respect to large-scale changes in the market
situation and therefore can be used as a decision tool for investment policies.
It is assumed that the full strength of the developed technique can be brought into playby its application to complex portfolio. Thus, the next step of the research will include
the extension of the ideas contained in the present work for such complicated sets offinancial instruments, and refinement of the statistical methods used for the analysis of
the method.

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