Construction of Optimal Portfolio-Treynor-Black Model: A ...

Construction of Optimal Portfolio-Treynor-Black Model: A study

With Reference to Selected Exchange Traded Funds

Srinivas K. R1. and Dr. B. Shivaraj

2

Research Scholar1, Department of Studies in Business Administration, B.N Bahadur Institute of

Management Sciences, Manasa gangothri, University of Mysore,Mysuru-570 006.

Professor (Retd.)2, Department of Studies in Business Administration, B.N Bahadur Institute of

Management Sciences, Manasa gangothri, University of Mysore,Mysuru-570 006.

Abstract

This study empirically examines the construction of the optimal portfolio by using

Treynor-Black model of selected Exchange Traded Funds listed on national stock exchange in

India. A sample of twelve ETFs consisting of equity, gold and indices exchange traded funds

listed on the National Stock Exchange in India. The study has been analyzed on construction of

optimal portfolio (active portfolio) of selected ETFs with the help of excess return, standard

deviation, beta, alpha and unsystematic risk, over a period of six years from January 2014 to

December 2019.

Keywords: Exchange Traded Funds, Optimal portfolio, Active portfolio, Treynor-Black Model,

Alpha, Abnormal returns and Unsystematic risk.

Introduction

Exchange Traded Funds (ETFs) were first created in the year 1993. After nearly a decade

in 2001, India saw its first ETF, the Nifty ETF Fund or Nifty BeEs, launched by benchmark

mutual fund. Benchmark mutual fund continued to be the pioneer of ETFs in India and launched

the first debt exchange-traded fund in 2004 called as Liquid BeEs. This was a Fixed Income

ETF. In fact, it was the first money market ETF that gave India the privilege of being the first

GORTERIA JOURNAL ISSN: 0017-2294

VOLUME 34, ISSUE 2 - 2021 Page No:90

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country to launch an ETF catering to the needs of risk-averse investors. Subsequently, in 2007,

the fund house launched the first gold exchange-traded fund called Gold Be Es. While Goldman

Sachs bought Benchmark Mutual Fund in 2011 and Reliance Mutual Fund bought Goldman

Sachs in 2015, these ETFs are still operational. At the end of 2019, there were 71 ETFs that

could track various debt and equity indices with total assets under management of around INR

1.47 trillion. Since its launch in the early nineties, the global ETF markets have experienced

phenomenal growth. At the end of 2019, there were around 7,797 ETFs or Exchange Traded

Product in the world with total assets under management crossing USD 5.78 trillion. Globally,

the US market dominates the ETF segment with a 70% share with the Asia Pacific market

holding around 11%share. Also, equity ETFs accounts for nearly 76% of the global ETF markets

while fixed income ETFs for 20%.

An important part of any investment decision making process should be the

determination of capital allocation between the risky portfolio and risk-free assets. The optimal

capital allocation risk aversion and as well as risk-seeking for the risk return trade off of the

optimal risky portfolio. Capital allocation and security selection are technically identical with the

aim of maximizing the return on investment with the construction of optimal portfolio. Treynor-

Black model is a model for security selection published by Jack Treynor and Fischer Black in

1973.Active portfolio model that seeks to maximize portfolios Sharpe ratio (SR) through a

combination of an actively managed portfolio component built with selected mispriced securities

and passively managed market index portfolio component.

Objective of the Study

To construct an optimal portfolio of selected ETFs with the help of Treynor-Black Model

To identify the proportion of money to be invested by investors in active portfolio and

passive portfolio

Review literature

Alex Kane and Robert R. (2004) aimed to identify and reduce the threshold of

profitable forecasting ability for portfolio management, for that database of monthly forecasts of

abnormal returns for 105 stocks over 37 months, provided by an investment firm that used these

forecasts to construct its portfolios. Using a database of realized returns on these stocks and the



https://kristal.ai/products/?refCode=etfhistoryblog&tag=GLOBAL_ETF&type=WW

market index, author estimated forecast discount functions and applied them to the forecasts

prior to the construction of the risky portfolios. These findings lend a real meaning to the concept

of nearly-efficient capital markets. If a selected stock was representative of competitive

investment firms, this suggests that competition leads to a degree of information efficiency that

reduces the forecast precision of super-marginal firms to a level as low as what the author

observe in the experiment.

Alexander D. Brown (2015) examined the added benefits of actively managing a

portfolio of securities from an individual investor’s perspective. More specifically, managing a

market portfolio with the combination of a selected few actively managed securities can, in some

instances, create an excess return. The author formed 20 separate region-based portfolios and 20

separate industry-based portfolios for each year from 2000 to 2014 with the original notion of

finding patterns in creating excess return by actively managing a combination of passive and

active portfolios. Illustrating this model is the easy part as the financial and statistical concepts

are consistent with the vast amount of portfolio optimization model in the world, however, after

researching this method of active management, the analysis firmly state that this method is not

easy for an individual investor to implement. To actively and successfully implement this

method, it would take the work of a team of security and economic analysts to come up with the

inputs (forecasts) for the mispriced securities and the market as a whole.

Research Methodology

Data: The study is based on secondary data. Secondary data sources include websites,

newspapers, SEBI manuals and textbooks. There are 12 ETFs consisting of equity, gold and

indices ETFs traded on NSE in India for period of six years from January 2014 to December

2019 for the purpose of constructing optimal portfolio of selected funds by using Treynor-Black

Model.

Tools Used

Construction optimal portfolio of selected funds by using Treynor-Black Model with the help of

following formulas:



1. Initial position of sample funds in the active portfolio

𝑤𝑖𝑜 =

𝛼𝑖

𝜎2 (𝑒𝑖)

Construction of optimal portfolio under TB model initially

needed to calculate appraisal ratio i.e., the ratio of alpha to

unsystematic risk.

2. Scaled initial position

𝑤𝑖 =𝑤𝑖

∑𝑖=1

𝛼𝑖

𝜎2(𝑒𝑖)

𝑛

After calculating, the next step is to find out initial position of

active portfolio, the higher the weight assigned to fund had

higher alpha. In negative alpha, a negative weight in the

active portfolio was expected.

3. Alpha of the active portfolio

𝛼𝐴 = 𝑤𝑖𝛼𝑖

𝑛

𝑖=1

The portfolio alpha is the weighted average of the alpha for

each fund, using the share in the portfolio as the weight(𝑤𝑖),

and the portfolio specific risk (𝛼𝑖), where the portfolio

variance is the weighted sum of the asset-specific risk.

4. Residual variance of the active portfolio

𝜎2 𝑒𝐴 = ∑𝑖=1𝑛 𝑤𝑖

2𝜎2(𝑒𝑖)

The portfolio residual variance or unsystematic risk is the

weighted average of the residual variance for each fund, using

the share in the portfolio as the weight.

5. Initial position in the active portfolio

𝑤𝐴𝑂 =

𝛼𝐴

𝜎2(𝑒𝐴 )

𝐸(𝑅𝑀 )

𝜎𝑀2

Initial position of active portfolio denote by wAO the appraisal

ratio of active portfolio excess of market return to market

variance.

6.Beta of the active portfolio

𝛽𝐴 = ∑𝑖=1 𝑛 𝑤𝑖𝛽𝑖 The portfolio beta is the weighted average of the beta for each



fund, using the share in the portfolio as the weight, and the

portfolio systematic risk (beta).

7. Adjusted (for beta) position in the active portfolio

𝑤𝐴. =

𝑤𝐴𝑜

1 + 1 − 𝛽𝐴 𝑤𝐴0

The beta of the active portfolio does not change the beta of

the overall portfolio; an active portfolio with a large beta will

be reduced to a smaller weight in the overall portfolio in order

to have the original beta of the passive portfolio remain

unchanged upon mixing the active and passive portfolios.

8.Final weights in passive portfolio and in security

𝑤𝑀. = 1 − 𝑤𝐴

. ; 𝑤𝑖. =

𝑤𝐴. 𝑤𝑖

Once the weight of the adjusted active portfolio is calculated,

the weight of the passive portfolio can be found by

subtracting the adjusted weight of the active portfolio from

one.

Table No-1: Table represents excess of return, standard deviation, Beta, Residual SD and

Alpha of selected Exchange Traded Funds

Funds Excess of

Return

Standard

deviation

Beta Residual

S.D

Alpha

ICICI Prudential Nifty ETF 0.41 4.929 1.094 7.190 0.068

Goldman Sache CNX Nifty

-0.388 3.867 0.58 10.145 -0.396

Nippon India ETF Nifty BeES

0.224 3.896 1.104 0.605 0.61

Kotak Nifty ETF

0.377 3.939 0.984 1.792 0.168

SBI ETF Gold

-0.183 3.733 -0.333 12.348 0.67

ICICI Gold

-1.312 11.480 -0.462 128.737 -0.598

HDFC Gold ETF -0.204 3.704 -0.36 11.852 0.066



Kotak Gold ETF

-0.118 3.854 -0.328 13.313 0.54

ICICI Prudential Nifty 100

0.1 5.635 1.011 17.154 0.213

MotilalOswal Nifty 100

0.819 6.834 0.154 46.358 1.266

Nippon India PSU Bank

-0.55 9.661 1.644 54.154 -2.248

Reliance Nifty 100 ETF

0.324 4.611 0.923 9.092 0.874

Nifty 50 0.265 3.78 1 0.000 0

From the above table 1,it is found that Motilal Oswal Nifty 100 outperformed its

benchmark index by securing the highest alpha of 1.266 because of the highest positive excess

return with lowest beta. Similarly, Goldman Sache CNX Nifty, ICICI Gold ETF and Nippon

India PSU Bank underperformed its benchmark index by securing the negative alpha value

because of negative of excess return and beta. The rest of the sample funds were an average

performance to its benchmark index.

Table No-2: Table showing weight of selected Exchange Traded Funds

Funds

σ(e)2

α/σ2(e)

=Wo(i) W(i)

W in

%

ICICI Prudential Nifty ETF 51.696 0.0013 0.00076 0.076

Goldman Sache CNX Nifty

102.921 -0.0038 -0.00222 -0.222


0.366 1.6666 0.96037 96.037

Kotak Nifty ETF

3.211 0.0523 0.03015 3.015

SBI ETF Gold

152.473 0.0044 0.00253 0.253

ICICI Gold ETF

16573.215 0.0000 -0.00002 -0.002

HDFC Gold ETF

140.470 0.0005 0.00027 0.027

Kotak Gold ETF

177.236 0.0030 0.00176 0.176


294.260 0.0007 0.00042 0.042

MotilalOswal Nifty 100 2149.064 0.0006 0.00034 0.034




2932.656 -0.0008 -0.00044 -0.044


82.664 0.0106 0.00609 0.609

Total 1.735 1.00 100.00

From the above table 2, it is found thatGoldman Sache CNX Nifty, ICICI Gold ETF and

Nippon India PSU Bank had negative weight because of negative alpha. Nippon India ETF Nifty

BeES secured highest weight of 96.037% because of less residual standard deviation. Rest of the

sample fund had small positive weight with highest residual standard deviation.

Table No-3: Table showing weight of active portfolio and passive portfolio of selected funds

Funds

α A

σ 2 (e

A) β A W0 (A)

W A*

total

WM*

W* A

each

fund

ICICI Prudential Nifty

ETF

0.00005 0.00003 0.00083 0.93566 0.84835

0.15165 0.06430

Goldman Sache CNX

Nifty

0.00088 0.00051 -0.00129 --- ---

--- -0.18810

Nippon India ETF Nifty

BeES

0.58582 0.33759 1.06024 --- ---

--- 81.47272

Kotak Nifty Exchange

Traded Fund

0.00506 0.00292 0.02967 --- ---

--- 2.55756

SBI ETF GOLD

0.00170 0.00098 -0.00084 --- ---

--- 0.21482

ICICI Gold ETF 0.00001 0.00001 0.00001 --- --- -0.00176



---

HDFC Gold ETF

0.00002 0.00001 -0.00010 --- ---

--- 0.02297

Kotak Gold ETF

0.00095 0.00055 -0.00058 --- ---

--- 0.14895


0.00009 0.00005 0.00042 --- ---

--- 0.03539


0.00043 0.00025 0.00005 --- ---

--- 0.02880


0.00099 0.00057

-0.00073 --- ---

--- -0.03747


0.00533 0.00307 0.00562 --- ---

--- 0.51688

Total 0.60133 0.34652 1.093 0.93566 0.84835 0.15165 84.835

From the above table 3, it is found that the weight of active portfolio was 84.83% rest of

15.17%secured by passive (market) portfolio. Among a selected sample of active portfolio

Nippon India ETF Nifty BeES had highest weight of 81.47% with highest alpha, residual

standard deviation squared and a beta of active portfolio of 0.58582, 0.33759 and1.06024,

respectively. Goldman Sache CNX Nifty, ICICI Gold ETF and Nippon India PSU Bank had the

negative weight. W0 of 0.93566 is greater than W* so it is better to move the proportion of

investment on passive i.e., market index portfolio.



Table No-4: Table showing Beta, Risk Premium, Standard deviation, Sharpe ratio, M2 and

Benchmark risk of market portfolio, Active portfolio and complete portfolio

Particulars

Beta (β)

Risk

Premium

S.D (σ)

Sharpe

Ratio

M 2 Bench

mark

Risk

Nifty 50 (Market) 1 0.265 3.780 0.070 ------- --------

Active Portfolio 1.079 0.891 4.175 0.214 0.542 --------

Funds --- --- --- --- --- ---

ICICI Prudential Nifty ETF 0.00083 0.00031 0.0037 --- --- ---

Goldman Sache CNX nifty

-0.00129 0.00086 -0.0086

--- --- ---


1.06024 0.21512 3.7416

--- --- ---

Kotak Nifty Exchange Traded

Fund

0.02967 0.01137 0.1188

--- --- ---

SBI ETF Gold

-0.00084 -0.00046 0.0095

--- --- ---

ICICI Gold

0.00001 0.00003 -0.0002

--- --- ---

HDFC Gold ETF -0.00010 -0.00006 0.0010 --- --- ---

Kotak Gold Exchange Traded

Fund

-0.00058 -0.00021 0.0068

--- --- ---


0.00042 0.00004 0.0024

--- --- ---


0.00005 0.00028 0.0023

--- --- ---


-0.00073 0.00024 -0.0043

--- --- ---




0.00562 0.00197 0.0281

--- --- ---

Complete Portfolio 1.079 0.796 4.115 0.194 0.466 0.076

Optimal portfolio Sharpe

ratio

--- --- ---

1.344

--- ---

From the above table 4, it is found that the SR measures the amount of a portfolio’s

excess return per one unit of total risk measured by the standard deviation. The SR of the active

portfolio is superior to that passive portfolio due to the increased risk premium. M2 is a measure

of risk-adjusted return relative to the market. M2 return of active portfolio comparison with an

average passive portfolio (market index). In the present study, in terms of M2, active portfolio

and complete portfolio are outperformed to the passive portfolio by improving 0.542 and 0.466,

respectively. Optimal portfolio is a combination of market portfolio and a portfolio of selected

fund with superior expected returns, the overall SR must exceed one of the markets in this study

optimized portfolio SR increases to1.344 compared to SR of market 0.070, active portfolio 0.214

and complete portfolio 0.194 by maximizing information ratio of the active portfolio.



Findings

Table No-5: Table showing final result of Treynor-Black model active portfolio

management

Particulars

Beta (β)

Risk

Premium

S.D (σ)

Sharpe

Ratio

M 2

Bench

Market

risk

Nifty 50

(Market) 1 0.265 3.780 0.070 ------- --------

Active

Portfolio 1.079 0.891 4.175 0.214 0.542 --------

Complete

Portfolio 1.079 0.796 4.115 0.194 0.466 0.076

Optimal

portfolio

S.R ---- ------- ----- 1.344 ---- -----

The final results are presented in above table 5, as mentioned; the end goal of TB

optimization is maximization of the optimal portfolio’s SR. Therefore, as an optimal portfolio is

a combination of market portfolio and a portfolio of selected ETFs with superior expected

returns, the overall SR must exceed one of the markets, active portfolio and complete portfolio.

The study indicates that SR of the optimized portfolio was 1.344 indicating exceed in the SR of

the market, active portfolio and complete portfolio 0.070,0.214 and 0.194, respectively.

Conclusion

In this study, twelve ETFs and Nifty 50 as a benchmark were selected to analyze the

construction of optimal portfolio with the help of the Treynor-Black model for which the

secondary data were extracted from various sources. Analysis revealed that the total weight age

of active portfolio was 84.83%, rest of 15.17%secured by passive (market) portfolio. In terms of



M2, active portfolio and complete portfolio are outperformed to passive portfolio, whereas, SR

of optimized portfolio was exceeding SR of the market, active portfolio and complete portfolio.

References

Articles

1. Alex Kane and Robert.R (2004). Active portfolio management the power of the Treynor

Black model, https://www.researchgate.net/publication/228637827.

2. Alexander D. Brown (2015), The power of an actively managed portfolio: an Empirical

example using the Treynor-Black Model, Oxford May 2015.

Text books

1. Sharpe, W. F., Alexander G. J., & Bailey J. V. 2008. Investments (6th ed.). Englewood

Hills: PHI Learning Private Limited.

2. ZviBodie, Alexa Kane, Marcus A. & Mohanty P. 2016. Investments (10th ed.). New

York: McGraw-Hill.

3. Reilly and Brown, 2011, Investment analysis and portfolio management (10th

ed.),

Cengage Learning.

Websites

1. www.nseindia.com

2. www.moneycontrol.com

3. www.rbi.com

4. www.amfiindia.com.



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