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NITHYA.J

Assistant Professor, Department of Commerce and

International Business, Dr.G.R.D. College of

Science, Coimbatore-641004, Tamil Nadu

Optimal Portfolio

Construction With

Markowitz Model

Among Large Cap’s In

India

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Abstract

The aim of this project is to construct effective portfolio for the large cap companies. This study enables to

know the performance of some Nifty Fifty companies having larger market capitalization. The analysis given

on this project is on the basis of risk & return and on Sharpe index model. This project suggests best

investment decision in selected large cap industries. By closely watching the daily price changes the trend and

time of entry and exit in the market can be predicted. In short the decisions as which script has to make sell

decisions and the composition of funds of asset allocation for each script can be analyzed by the way of

modern approach using Single Index Model (Sharpe model).

1. Introduction

The risk of a portfolio comprises systematic risk, also known as un- diversifiable risk, and unsystematic risk

which is also known as idiosyncratic risk or diversifiable risk. Systematic risk refers to the risk common to all

securities, i.e. market risk. Unsystematic risk is the risk associated with individual assets. Unsystematic risk

can be diversified away to smaller levels by including a greater number of assets in the portfolio (specific

risks "average out"). The same is not possible for systematic risk within one market. Depending on the

market, a portfolio of approximately 30-40 securities in developed markets such as UK or US will render the

portfolio sufficiently diversified such that risk exposure is limited to systematic risk only. In developing

markets a larger number is required, due to the higher asset volatilities.

A rational investor should not take on any diversifiable risk, as only non-diversifiable risks are rewarded

within the scope of this model. Therefore, the required return on an asset, that is, the return that compensates

for risk taken, must be linked to its riskiness in a portfolio context—i.e. its contribution to overall portfolio

riskiness—as opposed to its "stand alone risk." In the CAPM context, portfolio risk is represented by higher

variance i.e. less predictability. In other words the beta of the portfolio is the defining factor in rewarding the

systematic exposure taken by an investor.

A rational investor always attempts to minimize risk and maximize return on his investment. Investing in

more than one security is a strategy to attain this often-conflicting goal. In 1952, Harry M. Markowitz

developed a model that could be used to systematically operationalise the old adage – don’t put all eggs in

one basket. Markowitz's portfolio model is concerned with selecting optimal portfolio by risk adverse

investors. According to the model risk adverse investors should select efficient portfolios, the portfolio that

maximizes return at a given level of risk or minimize risk at a given level of return, which can be formed by

combining securities having less than perfect positive correlations in their returns. Markowitz model was

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theoretically elegant and conceptually sound. However, its serious limitation was the volume of work well

beyond the capacity of all except a few analysts.

To resolve the problem William F. Sharpe developed a simplified variant of the Markowitz model that

reduces substantially its data and computational requirements (Sharpe 1963). As per Sharpe’s model, the

construction of an optimal portfolio is simplified if a single number of measures the desirability of including a

stock in the optimal portfolio. If we accept his model, such a number exists. In this case, the desirability of

any stock is directly related to its excess return-to-beta ratio. If the stocks are ranked from highest to lowest

order by excess return to beta that represents, the desirability of any stock's inclusion in a portfolio. The

number of stocks selected depends on a unique cut off rate such that all stocks with higher ratios will be

included and all stocks with lower ratios excluded.

Markowitz’s original work still defines the main analytical tool for choosing ―optimal‖ portfolios. In practice,

however, most of the work of the portfolio manager is done in preparing the inputs for the Markowitz model

(the forecasts for portfolio return and portfolio variance), and in interpreting the outputs of the model. The

most recent advances in portfolio management have focused on ways to analyze in more detail the different

sources contributing to the total risk in a portfolio.

A major consequence of our understanding of a number of alternative investment strategies as being

inherently short event risk is to reassess their role as diversifiers for conventional stock-and-bond portfolios.

The nature of correlation changes dramatically during eventful times. In the fall of 1998, for example, the

correlation of market-neutral hedge funds to broad markets approached 1.0. Therefore, one should probably

not rely on diversification arguments in advocating hedge-fund investments. Instead, one should probably

note whether an investor is particularly well paid for assuming a fund’s risks and also decide whether an

investor is in a unique position to assume risks that others wish to lay off or not taken on.

Assuming that the rational investor seeks to maximize the expected return for a given level of volatility, or

equivalently seeks to minimize portfolios ex- ante risk for any given expected return, Markowitz [1952, 1959]

triggered the development of modern portfolio theory with the introduction of the mean-variance framework.

The concept of portfolio efficiency quantifies existing link between risk and return of a portfolio and the

complete set of optimal (or efficient) portfolios consequently forms the mean-variance frontier. Built upon the

mean-variance framework, the Capital Asset Pricing Model (CAPM) as developed by Sharpe [1964], Lintner

[1965] and Mossin [1966] states that under some certain conditions and taking different levels of investors'

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risk tolerance into account, the portfolio that provides the highest reward per unit of risk, better known as

Maximum Sharpe Ratio (MSR) portfolio should be held by all market participants.

Although mean-variance analysis and the CAPM are two pillars of modern finance, these models have been

scrutinized since their introduction. Especially simplified assumptions, namely the aim of rational and risk

averse investors to maximize economic utilities without influencing prices, having homogeneous investment

views based on all information to be available at the same time to all investors, trading without any costs and

holding a well diversified portfolio, have been strongly criticized. While Roll [1977] passed criticism on the

observability of the tangential portfolio, Merton [1980] found that already small changes in return estimates

can lead to completely different optimal weights in a portfolio construction process. In a nutshell, due to the

CAPM general assumptions and the question about availability of risk and return estimates, the mean-

variance framework is known to have difficulties in its practical implementation.

Extending the work from Merton [1987], Malkiel and Xu [2006] found that if investors do not hold the

market portfolio, unsystematic risk is positively related to stock returns. According to Martellini [2008], taken

together with the fact that asset pricing theory implies a positive premium for systematic risk, these findings

suggest a positive relationship between total volatility and expected returns. However, Haugen and Baker

[1991] were the first to provide empirical evidence for the inefficiency of market capitalization-weighted

indices and exclusively focused on risk in their alternative portfolio construction process. Repeatedly

investing into a stock portfolio constructed to expose investors to minimum risk as measured by variance

provided a higher Sharpe Ratio (SR) than the Wilshire 5000 index in the period 1972-1989 and therewith

inspired further risk-based investing approaches. This paper adopts the question of how to construct an

optimal portfolio regarding different risk objectives and contributes to existing literature by examining using

Sharpe’s single index method.

2. Statement Of Problem

The report has been prepared for the basis of studying the various avenues or sectors where investors can

invest their savings in a group of securities or portfolios. Creation of portfolios helps to reduce risk, without

sacrificing returns. Portfolio management deals with the analysis of individual securities as well as with the

theory and practice of optimally combining securities into good portfolios. Portfolio is constructed to

diversify the risks and maintain perfect negative correlation between the securities held together. Framing

portfolio by selecting securities based on brand identity and recent performance does not turn up anticipated

return in long term. This paper is built around cooking up the portfolio by balancing the positive and negative

correlation existing between the securities and in turn getting returns closer to the anticipated results.

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2.1 Objectives

The main objective of this study is to create an optimum portfolio using Sharpe’s single index model.

In this study secondary data sources are used.

We also consider the movement of share prices, expected returns, standard deviation and beta values.

The stock price movements, closing index points of the companies and beta values for the past four years

are collected for analysis.

The risk free rate of return is known.

All the values obtained above are interpreted and analysed using Sharpe’s single index model.

2.2 Scope Of The Study

Selection of companies is restricted to Nifty fifty only.

Of the fifty company of the index, the companies are chosen and analysed based on their performance in

the past four fiscal years.

No other factors other than the Share price movements, index movements, rate of return on government

securities and beta values for the securities for the past four years are taken for analysis.

2.3 Limitations

1. Only four years data has been considered for the construction of optimal portfolio.

2. Due to time constraint we are doing for only randomly selected fifteen scrip’s.

3. The portfolio is constructed purely on the basis of Sharpe’s model which basically considers the stock

price movements and does not take into consideration company specific factors, industry specific factors

and economic specific factors.

3. Review Of Literature

TANJA MAGO_C, M.S.(2009), the techniques that are used to solve the problem of optimal portfolio

selection all have their drawbacks. To solve these problems, we propose a new approach driven by utility-

based multi-criteria decision making setting, which utilizes fuzzy measures and integration over intervals. The

novel approach for an optimal portfolio selection has shown significant impact on the improvements of the

existing techniques both theoretically and experimentally. A common criterion for this assessment is the

expected return-to-risk trade-off as measured by the Sharpe ratio. Given that the ideal, maximized Sharpe

ratio must be estimated, we develop, in this paper, an approach that enables us to assess ex ante how close a

given portfolio is to this ideal. For this purpose, we derive the large-sample distribution of the maximized

Sharpe ratio, as obtained from sample estimates, under very general assumptions. This distribution then

represents the spectrum of possible optimal return-risk trade-offs that can be constructed from data.

Seegopaul, Hamish; Gupta, Francis; Prestbo, John (2005), For many plan sponsors, the greatest allocation to a

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single asset class is generally to domesticlarge-cap equities, so even a marginal difference in the performance

of this asset class over a long period can result in significant differences in the value of overall plan.Gotoh,

Jun-ya; Takeda, Akiko(2011), the reformulations of the robust counterparts of the value-at-risk (VaR) and

conditional value-at-risk (CVaR) minimizations contain norm terms and are shown to be highly related to the

ν-support vector machine (ν-SVM), a powerful statistical learning method. For the norm-constrained VaR and

CVaR minimizations, a nonparametric theoretical validation is posed on the basis of the generalization error

bound for the ν-SVM.Dale, Garry (2009), The adviser's job, often with the help of a computerized modeling

tool, is to balance the asset weighted portfolio against the client's assessed risk profile. Admittedly, this

sounds complex, but, in simple terms, a client's risk profile is a measure of how much capital they are

prepared to lose over a given period of time. Not, as many understand it, where the client fits on a scale of one

to ten. Jacobson, Brian J (2006), one of the challenges of using downside risk measures as an alternative

constructor of portfolios and diagnostic device is in their computational complexity, intensity, and

opaqueness. The question investors, especially high-net-worth investors who are concerned about tax

efficiency, must ask is whether downside risk measures offer enough benefits to offset their implementation

costs in use. A final insight is an outline of how to forecast risk using distributional scaling. Marketing

business weekly (2008), Calibration procedure proved superior to existing methods of averaging two or more

separate optimizations made with one or more risk models. First, optimality is guaranteed since there is no ad

hoc averaging done after optimization. Second, overly conservative solutions are explicitly avoided as the

primary tracking error constraint is always binding. Third, there often is a substantial synergistic benefit when

both risk models affect the solution. Bilbao, A; Arenas, M; M Jiménez; B Perez Gladish; M V Rodríguez

(2006), Expert estimations about future Betas of each financial asset have been included in the portfolio

selection model denoted as `Expert Betas' and modeled as trapezoidal fuzzy numbers. Value, ambiguity and

fuzziness are three basic concepts involved in the model which provide enough information about fuzzy

numbers representing `Expert Betas' and that are simple to handle. Clarke, Roger (2002),The ex-ante

relationship is a generalized version of a previously developed "fundamental law of active management" and

provides an important strategic perspective on the potential for active management to add value. The ex post

correlation relationship represents a practical decomposition of performance into the success of the return-

prediction process and the "noise" associated with portfolio constraints.Rudin, Alexander M; Morgan,

Jonathan S,(2006), Despite the importance of diversification in portfolio construction, our current methods of

measuring it are inefficient. Implementation in hedge fund strategies reveals that various hedge funds offer

less diversification than may have been thought, and that there has been reduced diversification in the past

several years,Kangari, R, Riggs, L S,(1988)Two major obstacles are risk evaluation associated with each

project and the correlation coefficient between projects, which describes the efficiency of the diversification.

The probabilistic approach that is suggested is a more realistic approach to the evaluation of correlation. It is

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not possible for a contractor to completely diversify a portfolio, so industry risk cannot be

eliminated.Borkovec, Milan; Domowitz, Ian(2010)Accounting for trading costs ex ante delivers superior net

returns, broader diversification, lower turnover, and a portfolio robust to noisy alpha signals, relative to

standard mean-variance stock selection and portfolio construction. Mitigation of transaction costs, leading to

improvement in realized returns and better alignment of return with risk, begins at the portfolio construction

stage and therefore should not be controlled only at the level of trading desks.

4. Research Methodology

This is a descriptive study on the construction of portfolio of stocks. The data taken for the study is Secondary

in nature. The data has been collected from various websites like National Stock Exchange (NSE) and also

from the databases of Ebsco and Proquest. The study is conducted with the financial data for the past four

years from 1 April 2008 to 31 March 2012. The sample size of the study is limited to 15 large cap companies.

They are a combination of stocks from various sectors namely Banking, Information Technology, Energy,

FMCG, Infra, Pharma, etc.. The sampling technique adopted is Random Sampling.

4.1 Tools Used For Data Analysis

Beta Coefficient

Beta coefficient is the relative measure of non-diversifiable risk. It is an index of the degree of movement of

an asset’s return in response to a change in the market’s return.

β=Correlation*σ(Y)/σ(X)

Where, σ(Y) = Standard Deviation of Individual Stock,

σ(X) = Standard Deviation of Market

Return

The total gain or loss experienced on an investment over a given period of time, calculated by dividing the

asset’s cash distributions during the period, plus change in value, by its beginning-of-period investment value

is termed as return.

Return = ((Today’s market price – Yesterday’s market price)/Yesterday’s market price)*100

Efficient Portfolio

A portfolio that maximizes return for a given level of risk or minimizes risk for a given level of return is

termed as an efficient portfolio.

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Correlation

A statistical measure of the relationship between any two series of numbers representing data of any kind is

known as correlation.

Risk-Free Rate Of Return (Rf)

Risk-free rate of return is the required return on a risk free asset, typically a three month treasury bill.

Excess Return-Beta Ratio

Excess Return- Beta Ratio = Ri-Rf/βi

Where, Ri= the expected return on stock,

Rf = the return on a riskless asset,

βi= the expected change in the rate of return on stock associated with one unit change in the market return.

Cut-Off Point

Ci = (MV*Σ(Ri-Rf) β/NRV)/ (1+MV*Σβ^2/NRV)

Where MV=variance of the market index

NRV = variance of a stock’s movement that is not associated with the

movement of market index that is stock’s unsystematic risk.

Investment To Be Made In Each Security

N

Xi=Zi/∑Zi

i=1

Where, Xi= the proportion of investment of each stock.

And Zi=βi/NRV (Ri-Rf/βi-Ci)

Where, Ci = the cut-off point.

5. Analysis And Discussions

The best model to measure the risk is standard deviation and beta and using this stock return is calculated.

Table 1: Return, Standard Deviation, Beta and Residual Variance of Individual Stock

Portfolio Scrip’s Individual Return α β Unsystematic Risk

ITC 45.15630781 2.48941059 0.561263549 6.197165084

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HDFC -8.034497417 3.792903111 1.148901333 14.38611401

INFOSYS 2.956756271 1.923710021 0.692962931 3.700660244

ICICI 72.82353214 3.39112593 1.549568331 11.49973507

ONGC -39.13308354 3.322595538 0.825152399 11.03964111

SBI 62.29244616 2.734717596 1.143667091 7.478680331

TATA MOTORS 60.63047122 4.300381038 1.30728804 18.49327707

M&M 69.60074429 3.386513204 1.074686915 11.46847168

NTPC 0.866573602 1.964663737 0.722779541 3.859903601

BHEL -85.35497731 3.593234935 1.032742164 12.9113373

CIPLA 51.13273521 1.938771035 0.498426256 3.758833127

WIPRO 49.03034737 2.803540103 0.886436543 7.85983711

DLF -36.97339488 3.937281052 1.572556532 15.50218208

RANBAXY 49.53198033 2.989680009 0.745920128 8.938186558

SAIL -10.1237203 3.231801617 1.296612145 10.44454169

Table 2: Excess return to beta ratio

Portfolio Scrip’s (Ri-Rf)/β Rank New Ranking

ITC 65.77357 2 CIPLA

HDFC -14.1653 12 ITC

INFOSYS -7.62414 9 M&M

ICICI 41.6784 7 RANBAXY

ONGC -57.4113 14 SBI

SBI 47.2624 5 WIPRO

TATA MOTORS 40.07569 8 ICICI

M&M 57.09639 3 TATA MOTORS

NTPC -10.2015 10 INFOSYS

BHEL -90.6276 15 NTPC

CIPLA 86.05633 1 SAIL

WIPRO 46.01609 6 HDFC

DLF -28.7515 13 DLF

RANBAXY 55.35711 4 ONGC

SAIL -14.1628 11 BHEL

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CIPLA yielded the maximum return among the companies selected and Bharat Heavy Electricals Limited

yielded lower return following that ONGC and DLF yielded lower return. Pharma and FMCG have shown a

higher return in all the companies chosen for the analysis. It shows that Pharma is the growing sector and it is

most preferred investable securities in India. Beta is greater than 1 in Mahindra and Mahindra, State Bank of

India, ICICI, Tata Motors, SAIL, HDFC, DLF and BHEL, which shows that these securities have more risk

and at the same time the reward per unit of risks is also more. But in case of other companies with regards to

beta it is less than 1 which shows it is less risky when compared to market risk.

Sharpe has provided a model for the selection of appropriate securities in a portfolio. The excess return of any

stock is directly related to its excess return to beta ratio. It measures the additional return on a security (excess

of the risk less asset return) per unit of systematic risk. The ratio provides a relationship between potential

risk and reward. Ranking of the stocks are done on the basis of their excess return to beta. Based on the excess

return to beta ratio the scrip’s are ranked from 1 to 15, with Cipla being in the first rank and BHEL being in

the last. The excess return to beta ratio was calculated using 8.24% as risk free rate of return.

Table 3: Cut-off point calculation for 15 companies

Scrip's New(Ri-Rf)/β (Ri-Rf)β/New

USR

Σ(Ri-Rf)β/New

USR

MarketVar*Σ(Ri-

Rf)β/New USR

CIPLA 86.0533 5.687379398 5.687379398 19.32693481

ITC 65.7735 3.341892869 9.029272267 30.68340342

M&M 57.096 5.749556017 14.77882828 50.22162771

RANBAXY 55.357 3.445805809 18.22463409 61.93121477

SBI 47.262 8.265778228 26.49041232 90.02010172

WIPRO 46.016 4.600165897 31.09057822 105.6524519

ICICI 41.678 8.699877204 39.79045542 135.2165003

TATA MOTORS 40.075 3.703284478 43.4937399 147.8010551

INFOSYS -7.624 -0.989405019 42.50433488 144.4388446

NTPC -10.201 -1.380681883 41.123653 139.7469915

SAIL -14.162 -2.279726714 38.84392628 131.9999913

HDFC -14.165 -1.318923544 37.52500274 127.5180062

DLF -28.7515 -4.586181493 32.93882125 111.9331781

ONGC -57.411 -3.541066306 29.39775494 99.8998754

BHEL -90.627 -7.48621515 21.91153979 74.46011096

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Scrip's β^2/New

USR

Σβ^2/New USR 1+MV*Σβ^2/New USR Cut Off Point

CIPLA 0.066085602 0.066085602 1.224573046 15.7825904

ITC 0.050785206 0.116870808 1.397152067 21.96139142

M&M 0.100691043 0.217561851 1.739321824 28.87425835

RANBAXY 0.062246653 0.279808504 1.9508493 31.74577082

SBI 0.174891585 0.454700089 2.545168408 35.36901583

WIPRO 0.099964625 0.554664714 2.884869641 36.6229553

ICICI 0.20866171 0.763326424 3.593946876 37.62339982

TATA MOTORS 0.092387723 0.855714147 3.907900171 37.82109281

INFOSYS 0.12975957 0.985473717 4.348851017 33.21310481

NTPC 0.13533938 1.120813097 4.808763253 29.06090073

SAIL 0.160970084 1.281783181 5.355774118 24.64629545

HDFC 0.093039463 1.374822645 5.671942166 22.48224726

DLF 0.159522223 1.534344868 6.214032888 18.0129684

ONGC 0.061678823 1.59602369 6.423630754 15.55193305

BHEL 0.082601603 1.678625293 6.704328713 11.10627389

Cut Off 37.82109281

Table 4: Selection of stocks among 15 companies

Scrip’s Cut Off

CIPLA 15.78259

ITC 21.96139

M&M 28.87426

RANBAXY 31.74577

SBI 35.36902

WIPRO 36.62296

ICICI 37.6234

TATA MOTORS 37.82109

Table 5: Proportion of funds invested

Scrip’s z x

CIPLA 6.395374073 41.96%

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ITC 2.530425592 16.60%

M&M 1.806077157 11.85%

RANBAXY 1.463401973 9.60%

SBI 1.443729097 9.47%

WIPRO 0.924188657 6.06%

ICICI 0.519553808 3.41%

TATA MOTORS 0.159321617 1.05%

Table 6: Proportion of Investment in each Stock

Scrip’s Proportion Of Investment

CIPLA 41.96%

ITC 16.60%

M&M 11.85%

RANBAXY 9.60%

SBI 9.47%

WIPRO 6.06%

ICICI 3.41%

TATA MOTORS 1.05%

5.1 Cutoff Point

The selection of the stocks depends on a unique cut-off rate such that all stocks with higher ratios of excess

return to beta are included and stocks with lower ratios are left out. The cumulated values of Ci start declining

after a particular Ci and that point is taken as the cut-off point and that stock ratio is the cut-off ratio C. In

Table 3 the highest value of Ci is taken as the cut-off point that is C*. Here Corporation Bank has the highest

the cut-off rate of C*=37.82109. All the stocks having Ci greater than C* can be included in the portfolio.

5.2 Construction Of Optimum Portfolio

After determining the securities to be selected, one should find out how much should be invested in each

security. The percentage of funds to be invested in each security can be estimated. As already mentioned all

the stock with Ci greater than cut off point can be included in the portfolio. Here the top five companies

according to excess return to beta ratio is taken for calculating the proportion of investment.

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5.3 Portfolio Investment

Table 6 shows the proportion of investment in each stock. And it indicates the weights on each security and

they sum up to 100 percentage. By using Sharpe index model thus we are able to find out the proportion of

investments to be made for an optimal portfolio. The maximum investment should be made in CIPLA with a

proportion of 42%. Following that ITC, Mahindra and Mahindra and RANBAXY are the next three

companies where major percentage of investment can be made. Evidently, the companies chosen for the

investments are growing at a steady rate in the recent years.

6. Findings

The Pharma and FMCG sectors are the major contributors for the portfolio. Among the selected 15 stocks,

pharmaceutical sector is performing well. The beta values for those stocks are lesser than 1 which indicates,

minimal risk is involved in those stocks. So we consider 2 stocks from Pharma, two stocks from Banking

sector and two from manufacturing according to the cut-off value calculated. The CIPLA forms the major

contribution with 42% followed by ITC with 17%.

7. Recommendations

The recommended proportion investments to the companies according to constructed portfolio are Cipla

42%, ITC 17%, M&M 12%, Ranbaxy 10%, SBI 9%, Wipro 6%, ICICI 3%, and Tata Motors 1%.

Investors expecting high return for the minimal risk can go for Cipla (β<1).

To investors who are risk lovers can go for M&M.

Investors better not to think about BHEL, ONGC, DLF, and HDFC.

Figure 1: Proportion Investment In Portfolio

41.96%

16.60%

11.85%

9.60%

9.47%

6.06%

CIPLA

ITC

M&M

RANBAXY

SBI

WIPRO

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8. Conclusion

Though there are 15 stocks that meet the criteria for being included in the Portfolio, the portfolio is

constructed with the top 8 stocks that meet the criteria to be included in the portfolio according to the Sharpe

Index Model. Those stocks are: CIPLA, ITC, Mahindra & Mahindra, Ranbaxy, State Bank of India, Wipro,

ICICI and Tata Motors. The portfolio predominantly consists of stocks from the pharmaceutical sector. The

share market is more challenging, fulfilling and rewarding to resourceful investors willing to learn the trade

for having effective returns with minimum risk involved. The optimal portfolio analysis and risk, return trade

off are determined by the challenging attitudes of investors towards a variety of economic, monetary, political

and psychological forces prevailing in the stock market. Thus the portfolio construction table would help an

investor in investment decisions. And the investor would select any company among the fifteen companies

from the above portfolio table. I also hope this dissertation will help the investors as a guiding record in future

and help them to make appropriate investment decisions.

9. References

AJ Du Plessis, M Ward (2009), A note on applying the Markowitz portfolio selection model as a passive investment

strategy on the JSE, Investment Analysts Journal, No. 69

Axioma, Inc.; Research Shows Use of Multiple Risk Models Improves Portfolio-Construction Results

Anonymous. Marketing Business Weekly (Oct 19, 2008): 325.

Axioma, Inc.; Research Shows Use of Multiple Risk Models Improves Portfolio-Construction Results

Anonymous. Marketing Business Weekly (Oct 19, 2008): 325.

Bilbao, A; Arenas, M; M Jiménez; B Perez Gladish; Rodriguez, An extension of Sharpe's single-index model:

portfolio selection with expert betas, The 57. 12 (Dec 2006): 1442-1451.

Borkovec, Milan; Domowitz, Ian; Kiernan, Brian; Serbin, Vitaly, Portfolio Optimization and the Cost of

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