1.
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
1
3.
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
3
4.
Markowitz (1979), Kroll, Levy and Markowitz (1984) concerned whether an astute
selection from the mean-variance efficient portfolios and using approximation yield a
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-
4
5.
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.
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,
n n
Min σ 2 = ∑∑ w i w js ij = w T Sw
i =1 j=1
(1)
n
s.t. ∑w r
i =1
i i = wT r = μ ;
n
∑w
i =1
i =1;
The following definitions are used.
µ : expected rate of return of portfolio;
σ2 : variance of return rates of portfolio;
r i : mean rate of return of security i ; r = (r1 , r 2 ,..., r n ) ;
T
w i : investment proportion of security i ; w = (w 1 , w 2 ,..., w n ) T ;
s ij : covariance of returns of securities i and j;
S = (s ij ) nxn : 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
5
6.
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 = ( µ - Rf) / σ (2)
where,
µ 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
σ 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 ( µ -Rf)/σ (3)
n
s.t. ∑w
i =1
i =1;
6
7.
In equation (3):
µ = wT r ;
σ = w T Sw ;
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,
Max w T e / w T Sw (4)
n
s.t. ∑w
i =1
i =1;
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,
( x T d) 2 T
Max T = d B-1d (5)
x Bx
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 ( w T e / w T Sw )2 = e T 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.
7
8.
Hence, Eq. (6) should yield the same optimal portfolio as Eq. (4) if a suitable constant
n
c is selected to ensure ∑w
i =1
i = 1 . 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.
n
Step 2：normalized the optimal solution to w*= cS e, where c = 1/ ∑ w i and wi are
-1
i =1
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.
5.1Optimal 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.
8
9.
Table 1 The Means and Standard Deviations of
the Five Stock Returns
Securities MMM JNJ KO WMT HD
Mean (%)
11.04 17.38 18.82 28.96 37.39
Standard
Deviation (%) 13.94 22.38 25.55 39.43 56.98
Resource: 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
KO
115.47 372.77 620.05 263.75 386.62
WMT
-3.81 387.95 263.75 1477.0 1609.86
0
HD
-37.38 363.65 386.62 1609.8 3084.73
6
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 ( µ , σ) = (18%, 15.62%).
Table 3 The Efficient Frontier Portfolios of the Five Stocks
MMM JNJ KO WMT HD µ (%) σ(%) 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
9
10.
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 360.66130 46.88
*The numbers of the left five columns are the proportions of stocks on the efficient frontier.
40
35
30
Expected Return(%)
25
20
15
10
5
0
10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00
Standard Deviation(%)
Figuire 1 The Efficient Frontier of the Five Stocks
10
11.
0.90
0.85
0.80
Sharpe Ratio
0.75
0.70
0.65
0.60
0.55
0.50
10 15 20 25 30 35 40
Expected Return(%)
Figuire 2 The Sharpe Ratios of the Five Stocks Efficient Portfolios
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.
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.
11
12.
Table 4 The Maximum Sharpe Ratio Portfolio of the Five Stocks
MMM JNJ KO WMT HD µ (%) σ(%) Sharpe
Solutions Ratio
Non-linear 0.50989 0.12656 0.12353 0.14298 0.09704 17.92 15.53
programming 0.8322
CSM-Algorithm 0.50989 0.12656 0.12353 0.14298 0.09704 17.92 15.53 0.8322
5.2Optimal 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
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 24.21
Standard
Deviation (%) 1.77 4.95 14.48 16.32 48.31
Resource: 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
12
13.
S&P500
3.62 23.25 209.78 175.73 22.91
MSCI
4.76 18.91 175.7 266.18 361.36
3
TWSE
2.28 -12.5 22.91 361.3 2333.78
4 6
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 ( µ , σ) = (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.
Table 7 The Efficient Frontier Portfolios of the Five Asset Classes
Cash Bonds S&P500 MSCI TWSE µ (%) σ(%) 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
13
14.
-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.
30.00
25.00
Expected Return(%)
20.00
15.00
10.00
5.00
0.00
0.00 5.00 10.00 15.00 20.00 25.00
Standard Deviation(%)
Figuire 3 The Efficient Frontier of the Five Asset Classes
14
15.
1.15
1.10
1.05
Sharpe Ratio
1.00
0.95
0.90
0.85
0.80
5.00 10.00 15.00 20.00 25.00
Expected Return(%)
Figuire 4 The Sharpe Ratios of the Efficient Portfolios with 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
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.
15
16.
Table 8 The Maximum Sharpe Ratio Portfolio of the Five Asset Classes
Cash Bonds S&P500 MSCI TWSE μ(%) σ(%) Sharpe
Solutions Ratio
Non-linear
programming 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
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.
16
17.
References
Borch, K. 1969. A note on uncertainty and indifference curves. Review of Economic Study,
36: 1-4.
Campbell, R., R. Huisman and K. Koedijk. 2001. Optimal portfolio selection in a Value-at-
Risk framework. Journal of Banking & Finance, 25: 1789-1804.
Chipman, J. 1973. The ordering of portfolios in terms of mean and variance. Review of
Economic Study, 40: 167-190.
Chunhachinda, P., K. Dandapani, S. Hamid, A.J. Prakash. 1997. Portfolio Selection and
skewness: Evidence from international stock markets. Journal of Banking &
Finance, 21: 143-167.
Dowd, K. 2000. Adjusting for risk: An improved Sharpe Ratio. International Review of
Economics and Finance, 9: 209-222.
Elton, E.J. and M.J. Gruber. 1995. Modern Portfolio Theory and Investment Analysis. John
Wiley & Sons.
Feldstein, M.S. 1969. Mean variance analysis in the theory of liquidity preference and
portfolio selection. Review of Economic Study, 36: 5-12.
Hanoch, G. and H. Levy. 1970. Efficient portfolio selection with quadratic and cubic utility.
The Journal of Business, 43: 181-189.
Johnson, R.A. and D.W. Wichern. 1992. Applied Multivariate Statistical Analysis, Prentice
Hall, New Jersey, Sixth Edition.
Kroll, Y., H. Levy. and H.M. Markowitz. 1984. Mean-variance versus direct utility
maximization. The Journal of Finance, 39: 47-61.
Levy, H. 1974. The rationale of the mean standard deviation analysis: comment. American
Economic Review, 64: 434-441.
Levy, H. and H.M. Markowitz. 1979. Approximating expected utility by a function of mean
and variance, American Economic Review, 69: 308-317.
Lucas, A. and P. Klaassen. 1998. Extreme Returns, Downside Risk, and Optimal Asset
Allocation, Journal of Portfolio Management, 25(1) : 71-79.
Markowitz, H.M. 1952. Portfolio selection. The Journal of Finance, 7: 71-91.
Markowitz, H.M. 1959. Portfolio selection. John Wiley & Sons.
Murthi, B.P., Y.K. Choi and P. Desai. 1997. Efficiency of mutual funds and portfolio
performance measurement: A non-parametric approach. European Journal of
Operational Research, 98: 408-418.
Prakash, A.J., C.H. Chang and T.E. Pactwa. 2003. Selecting a portfolio with skewness :
Recent evidence from US, European, and Latin American equity markets. Journal
of Banking & Finance, 27: 1375-1390.
Sharpe, W.F. 1966. Mutual fund performance. The Journal of Business, 34: 119-138.
Sharpe, W.F. 1994. The Sharpe Ratio. The Journal of Portfolio Management, 21: 49-58.
Tsiang, C. 1972. The rationale of the mean standard deviation analysis, skewness preference,
and demand for money. American Economic Review, 62: 354-371.
17
Be the first to comment