Applying Cauchy-Schwarz Maximization to Optimal Portfolio Decision with Short Sales Allowed

Hsien-Tang Tsai 1(蔡憲唐)National Sun Yat-Sen University

Duan Wei 2(韋伯韜)Finance & Monetary Division of National Policy Foundation

Hsin-Hung Chen3*(陳信宏)Cheng Shiu University(正修科技大學)

Hwai-Hui Fu4(傅懷慧)

Fortune Institute of Technology

Abstract

Since Sharpe introduced the Sharpe Ratio, many financial institutions have used

this measure to select portfolios and to evaluate the performance of mutual funds.

Conventionally, fund managers must establish an efficient frontier of portfolios and

select a portfolio with the highest expected Sharpe Ratio. Otherwise, they can revise

the objective function of the Markowitz mean-variance portfolio model and resolve

non-linear programming to obtain the maximum Sharpe Ratio portfolio directly. This

paper proposes an algorithm with closed-form solution for an optimal Sharpe Ratio

portfolio with short sales allowed by applying the Cauchy-Schwarz maximization.

Without solving a non-linear programming problem, this algorithm is simple, time-

saving and cost-saving. Moreover, two empirical cases are discussed to confirm the

efficiency and effectiveness of the proposed algorithm.

Key words: Sharpe ratio, mean-variance portfolio model, efficient frontier,

Cauchy-Schwarz maximization

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1. Introduction

Since Sharpe (1966) introduced the Sharpe Ratio, many financial institutions

have used this measure to evaluate the performance of mutual funds and to select

portfolios. As Sharpe (1994) mentioned, the Sharpe Ratio is built on Markowitz’s

mean-variance paradigm, which assumes that the mean and standard deviation of the

distribution of one-period returns are sufficient statistics for evaluating the prospects

of an investment portfolio. The hypothetical efficient frontier generated by Markowitz

mean-variance portfolio model comprises the set of portfolios with the highest

achievable expected returns for given standard deviations or the lowest achievable

standard deviations for given expected returns. It normally exhibits the higher-risk-

higher-return characteristic, which often causes dilemmas for portfolio management

decision makers. The Sharpe Ratio solves this problem because it summarizes the two

measures (mean and variance) as one (the Sharpe Ratio).

In recent years, several studies have presented various measures for evaluating

the performance of portfolios. For example, Murthi, Choi and Desai (1997)

introduced the DEA portfolio efficiency index (DPEI), calculated by data

envelopment analysis. Dowd (2000) developed the generalized Sharpe Ratio, which

he thought to be superior to the standard Sharpe ratio. Campbell, Huisman and

Koedijk (2001) constructed a performance index similar to the Sharpe ratio based on

Value-at-Risk (VaR). Besides, there are articles proposing several other approaches to

select optimal portfolio, such as Chunhachinda et al. (1997), Lucas and Klaassen

(1998), and Prakash, Chang and Pactwa (2003). However, these measures and

approaches are new and have not been broadly applied in the real world.

Markowitz (1959) suggested the use of the Expected Utility Maximum Method

to solve this dilemma. This theory uses the maximum of a utility function and an

indifference curve to determine optimal proportions of investments in securities.

Many studies addressed this issue in the 1960s and 1970s. For example, Borch (1969),

Feldstein (1969), Hanoch and Levy (1970), Chipman (1973) all discussed the

procedures and conditions for applying the method of maximizing the utility function.

Specifically, Hanoch and Levy (1970) found that the portfolio with the maximum

utility function is not necessarily on the efficient frontier under certain conditions.

Other extensive studies such as those by Tsiang (1972), Levy (1974), Levy and

Markowitz (1979), Kroll, Levy and Markowitz (1984) concerned whether an astute

selection from the mean-variance efficient portfolios and using approximation yield a

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portfolio with precisely maximum expected utility.

Although utility functions have been proposed to help investors to optimize their

portfolios, they are not appropriate for financial institutions and fund managers. The

most important problems for financial institutions in using utility functions pertain to

the choice of such functions and the associated parameters. Utility functions include

too many patterns and parameters. Consequently, the Sharpe Ratio is still the most

important index used by financial institutions and fund management units to measure

the performance of mutual funds. Moreover, this ratio can be used to select the

optimal portfolio on the efficient frontier generated by the Markowitz mean-variance

model because it considers both the mean and the standard deviation of the portfolio

returns.

Since many financial institutions use the Sharpe Ratio to evaluate the

performance of mutual funds, fund managers tend to select a portfolio that can

generate the highest Sharpe Ratio. Traditionally, fund managers need to establish the

efficient frontier of portfolios and select a portfolio with the highest Sharpe Ratio.

Otherwise, they can revise the objective function of Markowitz mean-variance model

and use non-linear programming to obtain the maximum Sharpe Ratio portfolio

directly. These models will be described below. However, the number of company

stocks is increasing in stock exchange markets. For example, the New York Stock

Exchange (NYSE) market already has more than 2,800 company stocks. These

companies include a cross-section of leading companies, midsize and small

capitalization companies. The National Association of Securities Dealers Automated

Quotations (NASDAQ) stock market lists approximately 3,600 electric companies. In

a stock market with large number of securities and trade volume such as NYSE or

NASDAQ, the computing time will be long and the process will be complicate by

using traditional non-linear programming method to obtain the optimal portfolio.

As a matter of fact, the Cauchy-Schwarz maximization in multivariate statistics

can be applied to generate a closed-form solution to decide the highest Sharpe Ratio

portfolio of the mean-variance model efficient portfolios. Therefore, the purpose of

this paper is to propose a simple algorithm with closed-form solution to find the

optimal Sharpe Ratio portfolio with short sales allowed by applying the Cauchy-

Schwarz maximization. Without solving a non-linear programming problem, this

algorithm is simple, time-saving and cost-saving. Real world data will be used to

confirm the proposed algorithm.

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2. Markowitz Mean-Variance Model with Short Sales Allowed

The Markowitz mean-variance model states that if the portfolio consists of n

securities, then its efficient frontier will be,

Min (1)

s.t. ;

;

The following definitions are used.

: expected rate of return of portfolio;

: variance of return rates of portfolio;

: mean rate of return of security i ; ;

: investment proportion of security i ; ;

: covariance of returns of securities i and j;

S = : covariance matrix of n securities;

Some constraints can be added to the model if the investment statement restricts

the investment weights of specific securities. For example, government funds

regulations usually limit the weights of stocks investments and restrict the short

selling of securities. In this paper, the investment weights of specific securities or

asset classes are unrestricted, and short sales are allowed.

3. Sharpe Ratio

Sharpe (1966) presented the Sharpe Ratio to evaluate the performance of funds.

This ratio is one of the measures first utilized in evaluating portfolios. It has been

considered to be a reward-to-variability ratio since it measures the fund’s return above

the risk-free rate (excess return) divided by the standard deviation of the return. The

measure followed closely his earlier work on the capital asset pricing model (CAPM),

dealing specifically with the capital market line (CML). Sharpe’s portfolio

performance measure is stated as follows.

Sharpe Ratio = ( - Rf) / σ (2)

4

where,

is the average rate of return of the portfolio over a specified period

Rf is the average rate of return of risk-free assets over the same period

is the standard deviation of the rate of return of the portfolio over the period

This composite measure of portfolio performance is clearly similar to the

Treynor measure, but it seeks to measure the total risk of the portfolio by including

the standard deviation of returns rather than considering only the systematic risk by

using beta. Because the numerator is the portfolio’s risk premium, this measure

indicates the risk premium return earned per unit of total risk.

4. Optimal Portfolio Selection by Maximizing the Sharpe Ratio

4.1 Mean-variance model with maximum Sharpe Ratio

The efficient frontier generated by Markowitz mean-variance model comprises

the set of portfolios with the highest achievable expected returns for given standard

deviations or the lowest achievable standard deviations for given expected returns;

therefore, it does not determine the optimal portfolio. If fund managers wish to select

the portfolio with the highest Sharpe Ratio, they need to establish the efficient frontier

of portfolios and select a portfolio with the highest expected Sharpe Ratio. Otherwise,

they can revise the objective function of the Markowitz mean-variance model as

follows, and obtain the precise maximum Sharpe Ratio portfolio directly. This model

was mentioned by Elton and Gruber (1995).

Max ( -Rf)/σ (3)

s.t. ;

In equation (3):

= ;

σ = ;

Rf = risk-free return rate;

Let e represent the vector of excess return rates (return rates of the securities

above the risk-free rate) of the n securities. Equation (3) becomes,

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Max e / (4)

s.t. ;

Traditionally, non-linear programming problem as Eq. (3) or Eq. (4) have had to

be solved to obtain the optimal solution. This process is time-consuming. Therefore,

this paper proposes a new simple algorithm with closed-form solution of finding the

optimal Sharpe Ratio portfolio by applying the Cauchy-Schwarz maximization.

4.2 Cauchy-Schwarz Maximization and Application

According to Johnson and Wichern (1992), the extended Cauchy-Schwarz

inequality in multivariate statistics gives rise to the following Maximization Lemma.

Let B be p×p positive definite matrix and d be a given p×1 vector. Then for an

arbitrary nonzero p×1 vector x,

Max = B-1d (5)

with the maximum attained when x = c B-1d for any constant c≠0.

Let the covariance matrix of securities S, the vector of excess return rates of

securities e, and the investment weights matrix w in Eq. (4) replace B, d and x,

respectively, in Eq. (5). Equation (5) becomes,

Max ( e / )2 = S-1e (6)

with the maximum attained when w = c S-1e for any constant c≠0.

In fact, the left side of Eq. (6) equals the square of the Sharpe Ratio. Normally,

the portfolio with the maximum Sharpe Ratio has a positive expected return rate.

Hence, Eq. (6) should yield the same optimal portfolio as Eq. (4) if a suitable constant

c is selected to ensure . Restated, if the covariance matrix and the excess

return rates vector of securities are known, the optimal Sharpe Ratio portfolio can be

obtained by the following algorithm using the Cauchy-Schwarz maximization (CSM-

Algorithm):

Step 1：compute w = S-1e to obtain the primary solution of the investment weights

matrix.

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Step 2：normalized the optimal solution to w*= cS-1e, where c = 1/ and wi are

the investment weights of the primary solution.

5. Empirical Analysis

Two empirical cases will be considered to confirm the CSM-Algorithm. The

first example concerns a mutual fund, which invests in Dow Jones Industrial Index

stocks; all positive investment weights for the portfolio with the maximum Sharpe

Ratio are obtained. The second case involves an asset allocation decision with

investment in five asset classes, and the portfolio with maximum Sharpe Ratio is

found to include a negative investment weight, implying that fund managers or

investors need to short sell the asset class to obtain the optimal Sharpe Ratio portfolio.

5.1 Optimal Portfolio with all positive investment weights

A mutual fund manager is assumed to decide to invest in five stocks in the Dow

Jones Industrial Index, which are 3M (MMM), Johnson&Johnson (JNJ), Coca-Cola

(KO), Wal-Mart (WMT), and Home Depot (HD). Since most financial institutions use

the Sharpe Ratio to evaluate the performance of mutual funds, the manager should

determine a portfolio that can generate the highest Sharpe Ratio. Annual returns over

the period 1983-2002 are considered to estimate the mean return rates and covariance

matrix of the five securities. Table 1 presents the means and standard deviations of

these five stock returns. Table 2 presents the covariance matrix of the return rates of

five securities. In fact, past history alone is not recommended as a predictor of future

returns. These data are used as examples of real world security and portfolio

moments.

Table 1 The Means and Standard Deviations ofthe Five Stock Returns

Securities MMM JNJ KO WMT HD

Mean (%) 11.04 17.38 18.82 28.96 37.39Standard

Deviation (%) 13.94 22.38 25.55 39.43 56.98Resource: Merrill Lynch International Bank Limited

Table 2 The Covariance Matrix of the Five Stock Returns

Covariance MMM JNJ KO WMT HD

MMM 184.66 66.66 115.47 -3.81 -37.38

JNJ 66.66 475.80 372.77 387.95 363.65

7

KO 115.47 372.77 620.05 263.75 386.62

WMT -3.81 387.95 263.75

1477.0

0

1609.8

6

HD -37.38 363.65 386.62

1609.8

6

3084.7

3

The data in Tables 1 and 2 were used to find the efficient frontier portfolios of

the Markowitz mean-variance model as in Eq. (1). Table 3 and Fig. 1 present the

means, standard deviations and investment proportions of portfolios on the efficient

frontier. Kroll, Levy and Markowitz (1984) called these portfolios “E-V efficient

portfolios” because they are obtained by minimizing the variance (V) for a given

expected rate of return (E). A risk-free return rate of 5% was assumed and the Sharpe

Ratios of the efficient portfolios were calculated. The right column in Table 3 and Fig.

2 show the results. Table 3 states that the portfolio with the maximum Sharpe Ratio

portfolio should be close to the portfolio with ( ,) = (18%, 15.62%).

Table 3 The Efficient Frontier Portfolios of the Five Stocks

MMM JNJ KO WMT HD (%) Sharpe Ratio

0.8589 0.2243 -0.0979 0.0103 0.0044 12 12.56 0.55748

0.7411 0.1914 -0.0232 0.0550 0.0357 14 12.45 0.72272

0.6232 0.1584 0.0516 0.0998 0.0671 16 13.56 0.81123

0.5054 0.1254 0.1263 0.1445 0.0984 18 15.62 0.83219*

0.3876 0.0924 0.2011 0.1892 0.1298 20 18.32 0.81884

0.2698 0.0594 0.2758 0.2339 0.1611 22 21.41 0.79394

0.1519 0.0264 0.3506 0.2787 0.1925 24 24.75 0.76754

0.0341 -0.0066 0.4253 0.3234 0.2238 26 28.26 0.74318

-0.0837 -0.0396 0.5000 0.3681 0.2552 28 31.87 0.72175

-0.2016 -0.0726 0.5748 0.4128 0.2865 30 35.55 0.70320

-0.3194 -0.1056 0.6495 0.4576 0.3179 32 39.29 0.68719

-0.4372 -0.1386 0.7243 0.5023 0.3492 34 43.07 0.67334

-0.5551 -0.1716 0.7990 0.5470 0.3806 36 46.88 0.66130

*The numbers of the left five columns are the proportions of stocks on the efficient frontier.

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If the fund manager wants to obtain the portfolio with the precise maximum

Sharpe Ratio, he (she) can apply Eq. (3) and solve the non-linear programming

problem. The data in Tables 1 and Table 2 are also used to determine the portfolio

with the precise optimal Sharpe Ratio portfolio and the results in the first row of Table

4 are obtained.

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Figuire 1 The Efficient Frontier of the Five Stocks

Figuire 2 The Sharpe Ratios of the Five Stocks Efficient Portfolios

As mentioned above, the CSM-Algorithm is a good method to acquiring the

same result as that of Eq. (3) without solving the non-linear programming problem. To

obtain the excess return rates vector of securities, we have to use the data of Table 1

and deduct risk-free return rate 5%. Then, the inverse of the covariance matrix in

Table 2 was multiplied by the excess return rates vector of securities and the optimal

investment weights vector of five stocks was obtained - (MMM, JNJ, KO, WMT, HD)

= (0.02733, 0.00678, 0.00662, 0.00766, 0.00520). A constant to make the sum of the

investment weights equal to one can be easily found to normalize these weights. The

inverse of the sum of these weights, 0.0536, is 18.6584. The primary weights were

therefore multiplied by 18.6584 and the results in the second row of Table 4 were

obtained. These weights are the same as those generated by traditional non-linear

programming. In other words, if 18.6584 is the constant c in Eq. (6), then the CSM-

Algorithm will find the portfolio with the optimal Sharpe Ratio.

Table 4 The Maximum Sharpe Ratio Portfolio of the Five Stocks

Solutions MMM JNJ KO WMT HD (%) Sharpe Ratio

Non-linearprogramming 0.50989 0.12656 0.12353 0.14298 0.09704 17.92 15.53 0.8322

CSM-Algorithm 0.50989 0.12656 0.12353 0.14298 0.09704 17.92 15.53 0.8322

5.2 Optimal Portfolio with a negative investment weight

The second example presents an empirical illustration based on an asset

allocation problem involving five U.S. and international asset classes: cash, bonds and

three stock indexes. For cash, we use the London InterBank Offered Rate (LIBOR) to

estimate returns. The three stock returns are based on the S&P500, the Morgan

Stanley Capital International (MSCI) and the Taiwan Stock Exchange (TWSE)

indexes. The TWSE index is used to represent and estimate the rate of return of a

developing country’s stock market, which has higher return and risk than those of a

developed country market. Bonds’ returns are computed using the Salomon Smith

Barney U.S. government bond index. Annual returns over the period 1984-2001 are

considered. Table 5 illustrates the means and standard deviations of the returns of

these five asset classes. Table 6 presents the covariance matrix of the returns of the

five asset classes. Table 5 indicates that an asset class with a higher mean return rates

usually has a higher standard deviation of returns. The only exception is MSCI, which

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has a lower mean rate of return but a higher standard deviation than the S&P500.

Table 5 The Means and Standard Deviations of the Five Asset Classes Returns

Assets Cash Bonds S&P500 MSCI TWSE

Mean (%) 6.40 8.71 12.34 11.08 24.21Standard

Deviation (%) 1.77 4.95 14.48 16.32 48.31Resource: Merrill Lynch International Bank Limited

Table 6 The Covariance Matrix of the Five Asset Classes

Covariance Cash Bonds S&P500 MSCI TWSE

Cash 3.12 4.21 3.62 4.76 2.28

Bonds 　4.21 24.49 23.25 18.91 -12.54

S&P500 　3.62 23.25 209.78 175.73 22.91

MSCI 　4.76　18.9

1

　175.73 266.18 361.36

TWSE 　2.28　-

12.54

　22.91

　361.36

2333.7

8

The data in Tables 5 and 6 were used to determine the efficient frontier

portfolios of the Markowitz mean-variance model. Table 7 and Fig. 3 present the

means, standard deviations and investment proportions of efficient frontier portfolios.

Additionally, the right column of Table 7 and Fig. 4 provide the Sharpe Ratios of the

efficient portfolios. Table 7 indicates that the portfolio with the maximum Sharpe

Ratio should be close to the portfolio with ( ,) = (8%, 2.74%).

The data in Tables 5 and 6 and Eq. (3) are also used and applied to solve the

non-linear programming problem and find the portfolio with the optimal Sharpe

Ratio. Table 8 presents the optimal investment weights of asset classes, the mean

return rate and the standard deviation of the portfolio with the maximum Sharpe

Ratio.

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Table 7 The Efficient Frontier Portfolios of the Five Asset Classes

Cash Bonds S&P500 MSCI TWSE (%) Sharpe Ratio

1.0210 -0.0297 0.0326 -0.0304 0.0060 6.50 1.75 0.85535

0.9123 0.0590 0.0739 -0.0632 0.0180 7.00 1.94 1.03327

0.6950 0.2364 0.1564 -0.1287 0.0408 8.00 2.74 1.09527*

0.4777 0.4139 0.2389 -0.1942 0.0636 9.00 3.80 1.05132

0.2605 0.5913 0.3214 -0.2596 0.0865 10.00 4.97 1.00671

0.0432 0.7687 0.4040 -0.3251 0.1093 11.00 6.17 0.97234

-0.1741 0.9462 0.4865 -0.3906 0.1321 12.00 7.40 0.94643

-0.3914 1.1236 0.5690 -0.4561 0.1549 13.00 8.63 0.92655

-0.6087 1.3010 0.6515 -0.5216 0.1777 14.00 9.88 0.91094

-0.8260 1.4785 0.7340 -0.5871 0.2006 15.00 11.13 0.89841

-1.0433 1.6559 0.8166 -0.6525 0.2234 16.00 12.39 0.88816

-1.4779 2.0108 0.9816 -0.7835 0.2690 18.00 14.90 0.87242

-1.9125 2.3656 1.1466 -0.9145 0.3147 20.00 17.42 0.86092

-2.3470 2.7205 1.3117 -1.0454 0.3603 22.00 19.95 0.85218

-2.7816 3.0754 1.4767 -1.1764 0.4059 24.00 22.48 0.84530

*The numbers of the left five columns are the proportions of the asset classes on the efficient frontier.

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Figuire 3 The Efficient Frontier of the Five Asset Classes

The data in Table 5 are used to obtain the vector of excess return rates of

securities and thus apply the CSM-Algorithm. The inverse of the covariance matrix in

Table 6 is multiplied by the excess return rates vector of securities and the optimal

investment weights vector of five asset classes, (Cash, Bonds, S&P500, MSCI,

TWSE) = (0.33158, 0.08392, 0.05963, -0.04935, 0.01541), is obtained. For this

example, the constant should be 2.2666, which is the inverse of the total primary

weights, to make the sum of the investment weights equal to one. The primary

weights were therefore multiplied by 2.2666 and the results in the second row of

Table 8 were obtained.

As in the first example, this solution verifies that the CSM-Algorithm yields the

same result as traditional non-linear programming. The Cauchy-Schwarz

maximization can also determine the portfolio with the optimal Sharpe Ratio in this

example. Notably, the optimal investment proportions of the maximum Sharpe Ratio

portfolio include a negative number. The optimal investment weight of MSCI is –

11.186%, implying that investors should short sell MSCI index securities to achieve

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Figuire 4 The Sharpe Ratios of the Efficient Portfolios with Five Asset Classes

the highest Sharpe Ratio. This example illustrates that when the maximum Sharpe

Ratio portfolio includes negative investment weights, the Cauchy-Schwarz

maximization remains effective in finding the portfolio with the optimal Sharpe Ratio.

Table 8 The Maximum Sharpe Ratio Portfolio of the Five Asset Classes

Solutions Cash Bonds S&P500 MSCI TWSE μ(%) Sharpe Ratio

Non-linearprogramming 0.75155 0.19021 0.13517 -0.11186 0.03494 7.74 2.49 1.1003

CSM-Algorithm 0.75155 0.19021 0.13517 -0.11186 0.03494 7.74 2.49 1.1003

6. Conclusion

The objective of most fund managers is to select the portfolio that can generate

the highest Sharpe Ratio. Theoretically, the Cauchy-Schwarz maximization is

appropriate for obtaining the optimal portfolio. This paper proposed a simple

algorithm with closed-form solution by applying the Cauchy-Schwarz maximization

to find the highest Sharpe Ratio portfolio. Moreover, two empirical examples

demonstrated that the CSM-Algorithm can correctly generate the optimal portfolio,

regardless of whether the optimal investment weights of the securities include

negative numbers.

As in Kroll, Levy and Markowitz (1984) and Campbell, Huisman and Koedijk

(2001), these two empirical examples used historical data to estimate the means

vector and covariance matrix of the return rates of securities and asset classes. In fact,

we do not recommend past history alone as a predictor of future returns. Forecasting

technology, such as time-series or regression analytic tools, can be applied to obtain a

more precise means vector and covariance matrix. Moreover, real-world mutual funds

are usually invested in more than five corporate stocks, and even though only five

stocks were used to create the portfolio in the first example. However, the CSM-

Algorithm remains useful when the number of securities in portfolio is increased.

In a stock market with large number of securities and trade volume such as

NYSE or NASDAQ, the computing time will be long and the process will be

complicate by using traditional non-linear programming method to obtain the optimal

portfolio. Therefore, the proposed CSM-Algorithm is a very important tool for

investment decision makers who are seeking to determine the optimal portfolio.

The CSM-Algorithm provides a closed-form solution for optimal portfolio. This

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algorithm, which can reduce computing time and costs, is better and easier than the

traditional non-linear programming method. We believe the proposed algorithm with

closed-form solution is meaningful and will complete portfolio model theory.

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