# Model Based On Efficiency Measurement Of Investment Corporations

This research presents the results of a survey done on the stock portfolio performance of 25 Iranian Investment companies admitted to Tehran Stock Exchange Corporation through 2006-2007[45]. To assess the portfolio return, the weighted average of returns of each segment of portfolio is calculated and the standard deviation of the portfolio return covering this period is taken as the risk criterion of the portfolio as a whole. Then a suitable model according to the econometric criteria is selected by Eviews software. One of the techniques of configuring the deterministic frontier function is COLS (Corrected Ordinary Least Squares) that is adjusted to the presented model. With the use of it, the portfolio deterministic frontier function of the investment companies admitted to Tehran Stock Exchange Corporation is obtained. Upon this deterministic frontier function, the efficiency ratio for each company is attained and finally graded. The resulted ranking is compared to the resulted ranking according to the Sharp model for validation of the model.

## 1 Introduction

Today, estimating the efficiency ratio and ranking of companies has attracted policy makers' attention. Increasing or at least protection of financial capital is the most important purpose of investment. So it seems that evaluation of portfolio performance is important for investors.

Investment firms' efficiency is one of the important matters that fascinate the directors of these companies as well as supervising part of the financial market of a country. Very close competition in open-economy society, forces directors of firms and financial institutions to use their greatest effort for attaining a higher level of efficiency by getting closer to the production frontier and also selecting a suitable scale for their economic activities.

Investment firms in Iran have a background of 35 years. Investigating the activity experience of these firms up to 1995 shows that they have never had an important role in the transactions of Tehran Stock Exchange Corporation Market, and only since 1990 that Stock Exchange Corporation has begun its activity again, they have had a considerable role in the exchange market.

Since no independent research related to the application of deterministic frontier function by using COLS method for ranking investment firms has been done so far and most researches--as will be mentioned in the literature review--have directly evaluated investment firms, this research could be a new outlook towards investment attraction approaches.

The statistical society of this research is 25 investment firms admitted to TSE for years 2006-2007.

To assess the portfolio return, the weighted average of returns of each segment of portfolio is calculated and the standard deviation of the portfolio return covering this period is taken as a whole. Then the best model according to the econometric criteria is selected by the Eviews software. Using one of the techniques of estimation of the deterministic frontier function namely COLS to this model, the portfolio deterministic frontier function of the investment firms admitted to Tehran Stock Exchange Corporation is assessed. Upon this deterministic frontier function, the efficiency ratio for each firm is attained and finally ranked.

Mainly, Methods of estimating frontier functions are divided into two groups: Estimating frontier cost function and frontier production function. Two techniques are used for assessing frontier functions:

A- The econometrical technique that is estimated by using COLS or MLE

B- Non-parametric methods and definitely linear programming methods

An investment firm is evaluated according to its portfolio. The portfolio of Iranian firms usually has two parts, stock exchange and non-stock exchange. Investment firms announce their stock exchange and non-stock exchange portfolio by presenting announcement to the stock exchange periodically. The non-stock exchange part of the investment firms portfolio is eliminated for simplification reasons (ibid [45]).

## 2 Literature review

In this section, the most important researches done on the evaluation of investment firms are studied.

In 1963 William Sharp defined the expected performance of a portfolio by two criteria: expected return rate (Ei) and variations in return (risk) that is measured by standard deviation of return [50]. He examined the performance of 34 investment companies in the 1954-1963 spans. Treynor, in 1965, used interest of practicing of portfolio oscillation as a criterion for risk, instead of using the general risk [56]. He ranked the same 34 investment companies that Sharp has investigated.

In 1968, Jensen developed a model to evaluate the performance of investment companies and viewed the issue of evaluation of investment portfolio performance at two dimensions [31]. Jensen used the theory of pricing of investment assets that had been stated by "Sharp" and "Treynor" to design a model for evaluating portfolio performance. Sharp and Treynor emphasized firmly on absolute scales of performance, while Jensen sought relative scale. He studied 115 investment companies. In his Ph.D. thesis in 1967, he presented evidence showing that the examined investment companies have performed well in reducing risk (Hedging) for their stockholders [32]. Another study was done by Malkiel in 1995. He examined investment companies in the span of 1971-1991[36].

The other different researches have been done on evaluation of the investment companies, summarized in Table 1.

A few researches have been done in Iran about assessing investment companies. Heibati (1999), in a research entitled "Evaluation of Heavy Investment Companies on the Basis of Analytical Hierarchy Process"; used the analytical hierarchy process. Researcher first selected 13 criteria for evaluating the management performance of investment companies. He considered six investment companies. By examining the initial 13 criteria, it was evident that AHP model has six basic variables fitted in four independent groups. Then investment companies relative to selected criteria were evaluated and ranked [30].

Eslami et al. (1996) examined the single indexical model, for the companies admitted in TSE Corporation during 1992-1996[17]. Haji Bozorgi, in 1995, worked on examining the performance coordination of investment companies and their purposes in the investment market as a Master of Science thesis. Researcher tended to examination of performance of 4 investment companies during the years 1991-1995[29].

A research under the title of "extra return of active management in the investment companies" was done for Ph.D. thesis in accounting with the purpose of comprehensive investigation into performance of Iranian investment companies admitted to TSE and on their portfolio management. The researcher has used Jensen α criterion as an indicator for showing the probable extra return of the investment companies [37].

In another research under the title of "Utilization of AHP Technique in Ranking Important Factors in Evaluating Investment Companies", it was attempted to take an important step in this relation by using AHP and prioritizing important factors in evaluating the investment companies [52]. ArabMazar and Mashayekh evaluated the performance of 14 investment companies admitted in TSE for the period of years 1995-2001. The researcher used "Sharp" model for calculating the portfolio risk and return and "Jensen" model for performance evaluation [4].

Most of researches, as mentioned before, have evaluated investment companies by using Sharp, Treynor and Jensen Criteria.

## 3 Efficiency and efficiency measuring methods

The technical efficiency of a given firm is defined to be the factor by which the level of production for the firm is less than its frontier output. Suppose that for an ith firm, frontier output is; therefore, technical efficiency for ith firm is specified by . Technical efficiency for individual firms is predicted by obtaining the ratio of the observed values to the corresponding estimated frontier values, , in which is estimated by MLE or COLS method.

The amounts by which a firm lies below its production and profit frontiers, and the amounts by which it lies above its cost frontier, can be regarded as measures of inefficiency.

There are two methods for measuring efficiency. The first is linear programming that is divided into two groups:

1) Data Envelopment Analysis (DEA) that is much extended and has many operational instances today. Its interesting property is that it does not need to introduce the function form.

2) Deterministic Frontier Analysis in which a special functional form is considered for estimating efficiency frontier.

The second efficiency measuring method is Stochastic Frontier Analysis that uses econometrics models.

## 4 Estimation Techniques of Deterministic Frontier

As an econometric view, the estimation of deterministic frontier is interesting because the concept of maximality (or minimality) puts a bound on the dependent variable. In the literature review of econometrics, several techniques for estimating deterministic frontier function are developed.

There are three common methods for determining Deterministic Frontier Function, such as: Mathematical Programming, MLE (Maximum Likelihood Estimation) and COLS (Corrected Ordinary Least Squares). In mathematical programming, we can use the common methods for determining desired function; also if measuring relative efficiency is the main purpose of the study, we can use the DEA method (Data Envelopment Analysis).

DEA is a non-parametric model used for estimating efficiency ratio and ranking companies. DEA models can be input-oriented or output-oriented and can also be specified as CRS (Constant Return to Scale) or VRS (Variable Return to Scale) models. Output-oriented models with regard to input factors amounts, makes output maximum. Conversely, input-oriented models with regard to given output level minimize input factors [7].

In deterministic frontier model , supposing that distribution function for the error term i.e. is given, we can use the MLE method for determining vector and consequently deterministic frontier production function: by estimating parameters of function (distribution function ) and by having relation by means of the MLE method, estimation of will be specified.

COLS method will be introduced in section 4-2.

## 4.1 Literature on Estimation Techniques of Deterministic Frontier

Theories related to efficiency were first proposed by Farrell. He categorized economic efficiency into two parts: technical and allocative efficiency and used the concept of maximum production or production frontier to measure them [20].

Several methods of calculating technical efficiency were developed by other economists such as Timmer [58], Upton [57], Greene [27], Forsund, Lovell and Schmidt [22], Kopp [33] by estimating production frontier function and using Linear Programming (LP) and Corrected Ordinary Least Squares (COLS).

Generally, two decades after Farrell, most economic researchers have been attracted towards estimating Deterministic Production Frontier Functions (DPFF) that were mainly estimated by LP and COLS methods. In the past years economists have been attracted towards Stochastic Frontier Production Function.

SFA (SFA is acronyms for stochastic frontier Analysis) methodology was originally proposed independently by Aigner et al [2] and Meeusen and Van Den Broeck [38], and has since generated an extensive Literature, both methodological and empirical (e.g. Forsund et al.(ibid [22]); Schmidt [46], Bauer [10]; Battese [9]; Greene [28]). However, in practice, we cannot be confident about the validity of these parametric assumptions that are used to estimate the model. The parametric form of the frontier function might be wrong due to several reasons. An alternative to the parametric SF is the deterministic nonparametric approach where no specific parametric assumptions are made on the model. In these nonparametric approaches, the statistical properties of envelopment estimators like DEA and FDH (FDH is acronyms for Free Disposal Hull, respectively) (Farrell (ibid [20]); Charnes et al [13]; Deprins et al [15]), rely on the so-called "deterministic" assumption. This assumption implies that no noise is allowed in these deterministic frontier (DF) models.

Recently, Cazals et al [12], Aragon et al [6] and Daouia and Simar [14] have proposed robust versions of the FDH estimator, robust to extreme values and/or outliers since they do not envelop all the data. But these approaches still rely heavily on the deterministic assumption, where no noise is allowed. In the presence of panel data, in a series of papers, Park and Simar [40], Park et al. ([41],[42],[43]) consider the semi parametric estimation of SF panel models under various assumptions on the joint distribution of the random firm effects and the regressors and on various dynamic specifications. Fan et al [19] propose a two-step pseudo-likelihood estimator in a semi parametric model where the production frontier is not specified, but distributional assumptions are imposed on the stochastic components as in Aigner et al (ibid [2]).

An average production frontier is then estimated through standard kernel methods, the shift for the frontier is obtained through a moment condition, as in the MOLS approach (see Kumbhakar and Lovell [34]) and the remaining parameters of the stochastic components are estimated by maximizing a pseudo likelihood function.

In some paper, a new approach is proposed to handle nonparametric SF models [51]. The method is based on the local maximum likelihood principle (see Tibshirani and Hastie [53], or Fan and Gijbels [18]), which is nonparametric. Their approach extends and generalizes Fan et al, Note that their estimator is obtained through a one step maximization procedure. As pointed out by Gozalo and Linton [26], the idea to use the Local Likelihood method for SF models was first suggested by Kumbhakar and Tsionas [35] for a particular case of the model proposed here.

For long time before, econometricians have been estimating average production functions. It has only been since the pioneering work of Farrell (1957) that serious consideration has been given to the possibility of estimating so-called frontier production functions, in an effort to bridge the gap between theory and empirical work (ibid [20]). The beginning point for any discussion of frontier and efficiency measurement is the work of Farrell (1957), who provided definitions and a computational framework for both technical and allocative inefficiency. Farrell's approach is non-parametric. This approach has been extended and applied by Farrell and Fieldhouse [21], Seitz [47, 48], Todd [55], Afriat [1], Dugger [16] and Meller [39].

Aigner and Chu [3] were the first to follow Farrell's suggestion. They specified a homogeneous Cobb-Douglas production frontier, and required all observations to be on or beneath it. The restrictive homogeneous Cobb-Douglas specification has been relaxed by Førsund and Jansen [24] and Førsund and Hjalmarsson [23], among others.

In this paper, (eliminating) COLS is used for estimating deterministic frontier. One of the most important properties of this method is that COLS solves the model parametrically and this is an advantage. And if a new firm enters the data set, it does not need further running the model like DEA, but we can determine the firm's ranking by computation of efficiency ratio and comparison with efficiency ratio of other firms.

## 4.2 Corrected Ordinary Least Squares (COLS)

In this method, for obtaining a functional model, first a suitable statistical model based on OLS was fitted upon the existing data. As an example, if it is supposed that the obtained model is, this model should justify itself existing variables and their adequacy statistically. To obtain deterministic frontier function from this model, we can consider 3 methods [8].

In transferential method, we shift the obtained model upward at the size of α≥0 to such an extent the function envelope all the existent data. For this purpose, we can use the trial & error method. If, in this phase, has a normal distribution, by placing α=1.96σ in which σ is standard deviation of error term relative to the obtained model, we can envelope all data with 98% Probability. The advantage of this method is the speed in obtaining the answer, and the disadvantage is weak precision in determining coefficients of deterministic frontier function model.

In the replacing method, dependent variable amount of existent data below the obtained model curve i.e. data that for them ; is replaced by their amount in the model i.e. and this set of new data were fitted by OLS, variables of new model take no change. Only their coefficients will be changed in this method. Then the new data below this new model is replaced again by the above method and this phase is repeated so that obtained model ultimately became near to deterministic frontier function. After that by repeating this method, changes of model coefficients toward previous period show no noticeable change, we can obtain deterministic frontier function by the transferential transferring method.

In this method, precision in determining coefficients of deterministic frontier function model is more, but the speed of getting the answer is less.

In the eliminating method, below existent data obtained model curve eliminated and a model with the same previous variables but with different coefficients fitted by OLS. This method continued on remaining data until the final model is obtained.

Then by using transferential method in this phase, deterministic frontier function is obtained. In this method, precision in determining model coefficients is more than precision in transferential method and less than replacing method and the speed of obtaining the answer is less than transferential method and more than replacing.

A difficulty with the COLS technique is that even after correcting the constant term, some of the residuals may still have the wrong sign, so that these observations end up above the estimated production frontier. This makes the COLS frontier a somewhat awkward basis for computing the technical efficiency of individual observations. One response to this problem is provided by the stochastic frontier approach. Another way of resolving the problem is to estimate by OLS, and then to correct the constant term by shifting it up until no residual is positive, and one is zero. Gabrielson [25] and Greene (ibid [27]) have both shown that this correction provides a consistent estimate of .

Another difficulty with the COLS technique is that the correction to the constant term is not independent of the distribution assumed for u. Thus the one-parameter gamma distribution and the exponential distribution yield systematically different estimates of technical efficiency.

## 5. The Model

In this section, we tend to provide a suitable model for deterministic frontier function of investment firms' portfolio.

## 5.1 Variables

Return of a set of stock portfolio equals weighted average return of each stock return in portfolio that is calculated by equation (1).

(5.1)

In this equation, is market value of the desired asset divided by market value of the whole organizer stock exchange of portfolio.

Where,

: Portfolio return

: Return of individual stock

: Individual stock share in portfolio

: Number of various stock in portfolio.

So for assessing and , we needed some information about combination of monthly portfolio of the under-investigation investment firms and also common stocks return of different companies which exist in the portfolio composition of every under-investigation investment firm.

Monthly portfolio compositions of investment companies were provided from Tadbirpardaz Company by investing the existent data banks.

Suppose that there is T historical period and expresses portfolio return of the th period, then portfolio return average is defined as follows:

Also standard deviation can be calculated as follows:

This estimation of portfolio standard deviation was used as an indicator for measuring general risk of portfolio, and we can compare it directly with the standard deviation of other portfolios [44].

Since suitable criterion for investment return average during time is geometric return average, we have used this method for calculating average return (ibid [45]). The formula is shown in Equation (5.2):

(5.2)

In the research, time series data and segmental data are used together. The application of the combinatorial data in econometrics has many priorities toward use of time series data or segmental data. The Combinatorial data considers information of different segments and their dynamics concurrently. The combinatorial data conclude past procedures of variables and create certainty in viewpoint spotting variables dynamics [54].

In Table 2, measured risk and return of 25 under-study companies from 2006 to 2007 is presented.

## 5.2 Dependant and Explicative variables

The model presented in this research has two variables: risk of portfolio as explicative and return of portfolio as independent variable

## 5.3 Model formulation

Capital assets pricing model (CAPM) [49], is a simple but fine expression about necessary risk & return for every individual stock or portfolio. This model formulates the base of investments, i.e. the higher the supposed risk, the higher the necessary return must be and vice versa. And upon this model, there is a direct relation between risk and return of every share or portfolio.

According to this model, There is a direct relation between risk and return by Equation (5.3):

(5.3)

The return of which is return vector of the investment firms' portfolio and risk is risk vector of these firms' portfolio.

## 6 Model implementation and results analysis

In this section, we try to estimate parameters by using ordinary least squares (OLS).

The two best criteria to select the suitable model are (multiple coefficient of determination) and t-statistics or P-value [5, 11]

In this research, by using portfolio risk and return of the investment companies admitted to TSE Corporation and using Eviews software system, a model as the below form was estimated:

Return: portfolio return of the investment companies

Risk: portfolio risk of the investment companies

This model with regard to estimating several different models such as Double-Log,Semilog,hyperbola, exponential models, by trial and error, and the form of efficient frontier obtained from Markowitz theory and regression criteria (R2 ant t-statistics) was selected as a suitable model. Also, theoretically, this model is suitable, because there is a positive relation between risk and return and has the same shape as efficient frontier shape in Markowitz model (ibid [49]). By obtaining this model, the first phase for using deterministic frontier function method to determine efficiency is provided. So, the suitable estimated model upon data is as follows:

By using the model, we can estimate portfolio return, by comparing to the real values of portfolio return we can introduce efficiency ratio.

In this section, we have obtained a curve that some company’s return are below and some are above. As all companies should lie below frontier function we used Eliminating COLS. In the eliminating method, we eliminate those companies the estimated return of which is more than real value of their return, i.e. Efficiency Ratio <1.

On the remaining data, a model is estimated with the same previous variables but with different coefficients by OLS. This method is repeated (for 4 times) until the final model obtained on the remaining data the R2 of which is near one and coefficients of model variables are significant (with regard to t-statistic and P-value). In Table 3, estimated parameters by eliminating COLS are presented.

As it is shown in Table 3, the definition coefficient of the model is near 1 and the coefficients are significant. And the model (6.1) is as fellows:

(6.1)

At this step, we need to shift estimated function, so that all companies return lay below the estimated curve. In the transferring method, If ε (residual or error term) has a normal distribution, by placing α=1.96δ in which δ is standard deviation of errors in model, we can envelope all data with probability of 98%. For the model (6.1), we check the test of normality on error values.

Since the residuals(errors) have a normal distribution in this step(as demonstrated in the Fig.1), we shift the estimated function in a measure α=1.96δ (α=1.96*0.023662= 0.046), so the constant coefficient in COLS model is equal with -0.127(0.046+ (-0.17352)).

The transferring method results in Deterministic frontier function (Fig.2) with the under formula:

## Return = -0.127+0.54*risk^ (1/3)

Now, for all companies, efficiency ratio is equal to. And According to these efficiency ratios, we rank these companies in Table 4.

## 7 Validation of model

We also rank 25 investment companies during 2 years in research by the Sharpe ratio. And we want to compare hereby ranking results of R/V ratio in Sharpe's work with the ranking results of eliminatory COLS. If results are the same nearly, our model has necessary validation. We calculate the Spearman's rank correlation coefficient between the results of two kinds of ranking namely ranking on the basis of the Sharpe Ratio and the eliminating COLS for comparing the results of these two kinds of rankings. If the Spearman's rank correlation coefficient is near 1, we understand that the results of these two kinds of ranking are nearly the same and so, our model has necessary validation. An alternative interpretation of the ratio gives rise to the name-reward-to- variability ratio (R/V). The numerator shows the difference between the fund's average return and the pure interest rate; it is thus the reward provided by/to the investor for bearing risk. The denominator measures the standard deviation of the rate of return; it shows the amount of risk actually borne. The ratio is thus the reward per unit of variability. Table 5 shows the values of the R/ V ratio for the 25 funds and ranks of them.

The Spearman's rank correlation coefficient is calculated as follows:

=the difference between the assigned ranks to the ith phenomenon.

= the number of the ranked phenomena

As can be understood from the above table and by calculating the Spearman's rank correlation coefficient between the results of two methods by the 0.96 value, the COLS method in this research obtain the same ranking with the ranking of R/V in Sharpe' work and this is the reason for the validity of our model.

## 8 Conclusion

By estimating the best model on data and applying the eliminating COLS , deterministic frontier function of investment firms portfolios obtained and efficiency ratios of investment firms measured by the use of this deterministic frontier. Then according to these ratios, investment firms ranked. This deterministic frontier is ascending that accord with existent theories.

Whereas many investors do not have financial capability and necessary expertise in construction of suitable portfolio, they commend the exercise of this important task to investment firms. But that the investors select which investment firm, the obtained rankings help them.

It is suggested that other estimations techniques of deterministic frontier function be used for the estimation of efficiency ratio and ranking of investment firms. Also by the use of DEA, SFA, COLS models estimate efficiency ratios of investment firms and compare these methods. Moreover it is suggested that various models of performance evaluation combine with each other for performance evaluation and ranking of investment firms.

For an example, output results of DEA and COLS and PLP models combine with each other and final ranks of investment firms in TSE obtain.

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