17 декабря 2015г.
Abstract
Securities evaluation criteria are presented to select the assets for portfolio diversification. Modifications of the Sharpe coefficient are proposed, based on the new introduced risk measures. The proposed selection criteria are evaluated according to the efficiency of a composed portfolio, measured by the Sharpe coefficient. Comparative analysis of the introduced securities evaluation criteria is held.
Risk.
Keywords: assets selection, diversification, reward-to-variability ratio, risk measure, ‘beta’ coefficient, Value-at-
Introduction
Investment portfolio analysis recently has become widely used due to the securities market development as is demonstrated by many publications on this subject. One of the most discussed problems is the assets selection for portfolio diversification. There are different criteria developed to evaluate the securities to be included in a portfolio. From the point of view of portfolio diversification, the risk of a portfolio is reduced when non-correlated assets are added. However in practice it can be difficult to select perfectly non-correlated assets. The Sharpe coefficient (reward-to- variability ratio) and the Treynor coefficient (reward-to-volatility ratio) can be used to evaluate the securities to be included into a portfolio. One of the discussed problems is the way to measure risk or volatility of a security.
The Treynor coefficient uses stock betas from the CAPM to evaluate systematic risk, i.e. the return risk associated with market movements. Though being widely criticized, the beta-method is often used in financial analysis, and stood numerous empirical tests. When returns and factors are jointly normally distributed and independent over time, the classical method provides the most efficient unbiased estimator of factor risk premiums in linear models. However many empirical studies report important beta variation over time. A standard approach to modeling and estimating time-varying betas has not yet emerged. Especially betas are biased, inconsistent, and inefficient in emerging markets, as has been shown in. Given the fact that unstable betas might have serious consequences on the efficiency of beta based risk evaluations, there is a strong need for a better understanding of stock betas.
The Sharpe coefficient is a fundamental performance measure. Nevertheless, there have been some improvements of this ratio. The classical Sharpe ratio is based on normal distribution mean-variance analysis. When distributions are nonnormal or have fat tails, the performance rankings are not accurate. Two principal approaches to generalize the Sharpe ratio can be distinguished.
One strand of the literature is based on the use of utility functions. Hodges generalizes Sharpe ratio applying the exponential utility function with Arrow-Pratt risk-aversion index, which is constant for exponential utility independent of wealth. Horges’ reason for choosing exponential utility is the assumption of its equivalency to quadratic utility and mean- variance analysis.
Another generalized Sharpe measure is based on the family of negative power utility functions, also called constant relative risk aversion. ‘Gama’-generalized Sharpe ratio depends on risk-aversion parameter and the initial wealth, so it does not have a unique value, as does the ordinary Sharpe ratio or Horges’ generalized Sharpe ratio.
One of the recent approaches is the use of utility functions with hyperbolic absolute risk aversion (HARA). One of the strong points of this approach is that such utility functions allow the derivation of a generalized two-funds separation theorem thus leading to sample capital market evaluation formulas, and to the generalization of the Sharpe ratio and the Traynor ratio as well.
Nevertheless, utility function approaches, though important, are rather subjective. The degree of investor’s risk- aversion and the selection of utility function remain discussed questions.
Another strand of literature aims at applying risk measures which are based on downside risk considerations. Ziemba and Schwartz propose to find the downside standard deviation, and the total variance is twice the downside variance. Thus, a superior investor is not penalized for good performance. Ziemba calls the obtained performance ratio ‘the symmetric downside risk Sharpe measure’.
Another downside risk measure is Value-at-Risk (VaR), a widely used concept for quantifying the risk of portfolios. VaR has received an official recognition after having been recommended by various regulating financial institutions as a portfolio risk-measurement tool. Thus, The Bank for International Settlements (BIS) recommends VaR method for defining the Market Risk Capital of a bank. Moreover since the publication of the market risk measurement system RiskMetricsTM of J.P. Morgan in 1994 VaR has gained increasing acceptance and can now be considered as the industry’s standard tool to measure market risks.
There are two main groups of models to calculate VaR. Parametric models such as delta-normal are based on statistical parameters such as the mean and the standard deviation of the risk factor distribution. Non-parametric models are simulations or historical models.
The aim of this paper is to propose new securities selection criteria based on the Sharpe coefficient and to evaluate their efficiency. We modify the Sharpe coefficient using the new introduced ( R - VaR )- and ( R - Rlow )-risk measures. The new measures are based on VaR- and Rlow -values, which refer to the downside measures.
Diversification is more efficient when non-correlated assets are added. However, for a given asset being positively or negatively, strongly or weakly correlated with other assets at the same time, it is difficult to select perfectly non- correlated assets to diversify the portfolio. In such case the Sharpe and Treynor coefficients can be used to range the assets according to the reward-to-variability and reward-to-volatility ratio, respectively. The Sharpe coefficient, known as the ‘reward-to-variability ratio’, is defined to be:
where Rp is the mean-return of the portfolio p,R f is the risk-free asset return, and s pis the standard deviation of the portfolio p.
The Treynor coefficient (the ‘reward-to-volatility ratio’) is assumed to be:
where ß p is the ‘beta’-coefficient of the portfolio p, that is defined in the Market Model.
The Sharpe coefficient (1) and Treynor coefficient (2) can be equally used for assets evaluation [11]:
where Ri is the mean return of the asset i,i = 1, n , s iis the standard
deviation and biis the coefficient of sensitivity of the asset i to
market movement. The choice of a security
i* to be added
into a portfolio can be based on the maximization of the Sharpe coefficient (3) or the Treynor coefficient (4),
i.e
It means a preference is given to the asset having the largest market prime per one risk-unit, measured by the standard
deviation (the Sharpe coefficient) or by the ‘beta’-value (the Treynor
coefficient).
The choice of the coefficient depends
on the set of the financial
assets in the investor’s
portfolio. The risk for an investor, possessing other assets that are not included in the portfolio, should be measured
by the ‘beta’-coefficient since this coefficient evaluates risk relatively to the market. When all instruments are included
in the portfolio
under consideration, the standard deviation can be seen as a suitable risk-measure, and the Sharpe coefficient can be used as asset evaluation criterion.
We introduce a new parameter, termed ‘low-mean’ return of the asset i, defined as:
where Z -
is the set of indices
t such that R < Ri , and p is the probability of the return R . Rilow is the mean-return of a left (‘bad’) part of the return distribution of the asset i,
i.e. the mean-value for the returns, which are less than the
mean return of the asset Ri
In a similar way we obtain the
mean-return of a right (‘desirable’) part of a return distribution of the asset i:
where Z +
is the set of indices
t such that R > Ri , and p is the probability of the return R .
In terms of ‘low-mean’ and ‘upper-mean’ returns the full variability of the return of the asset can be described by the difference of the ‘upper-mean’ and the
‘low-mean’ returns ( Riupper - Rilow ).
We define a new risk-measure, namely the difference between the asset mean return and the ‘low-mean’ return ( Ri - Rilow ). This risk-measure is especially suitable for assets with asymmetric distributions.
For the case of a symmetric distribution the following
equality holds:
The value-at-risk (VaR) is a measure
widely used in financial
analysis. For a known asset return distribution, VaR defines the return
that can be achieved with some probability level [2]:
where a is the confidence level, which is usually
set equal to
0.01, 0.05, or 0.1.
While the basic concept of VaR is simple, many complications can arise in practical
use. A major drawback
of VaR approach
is that optimization problems, aiming at computing
optimal portfolios with respect to VaR are typically hard to solve numerically. The reason is that VaR is in general not a convex function. In this respect a related concept, conditional Value-at-Risk (CVaR)
has recently been suggested as an alternative downside risk measure,
which is determined as the expected mean loss after the VaR. This risk-measure is more consistent than VaR, due to some important
properties such as subadditivity and convexity. CVaR is proved
to be a coherent risk measure in the sense introduced by Artzner,
Delbaen, Eber and Health. A more detailed study of CVaR and its application for assets selection can be a subject for further research.
We propose another new risk measure, namely the difference between
the asset mean return and the VaR-value for a -confidence level ( Ri - VaRi ). The choice of the confidence level depends
on the investor’s attitude
to risk. Risk preference allows setting
high confidence-level, that increases
VaR-value and decreases
investor subjective evaluation of risk, measured by ( Ri - VaRi )-value. And on the contrary,
risk aversion implies
low confidence-level.
On the base of the risk-measures introduced in 3.1, we propose the following
modifications of the Sharpe coefficient for the asset i :
The coefficient (9) describes the amount of excessive return (market
prime) referred to one unit of risk, measured as a deviation of asset mean return from its ‘low’-
mean return. This coefficient may be recommended to evaluate especially the assets characterized by asymmetric distribution.
The coefficient (10) describes the amount of excessive return per one unit of risk, measured
as a deviation of asset mean return from its VaR-value. VaR-value can be estimated for different a -confidence levels,
which are set regarding
the investor’s risk preferences.
Conclusion In this paper new Silow - and SiVaR - securities selection criteria have been proposed, based on the introduced (Ri - Rilow )- and ( Ri - VaRi )- risk-measures.
( Ri - Rilow )-value can be a suitable
risk-measure especially
for the asymmetric distribution of the return of an asset. ( Ri - VaRi )-value allows the investor to set acceptable deviation of the return from the VaR-value for different confidence levels, considering his risk preferences. For further
research measures based on conditional VaR can be
considered, since CVaR possesses such important properties as subadditivity and convexity.
References
1. Luenberger D.G., Investment science, Oxford University Press, 1998, New York
2. Sharpe
W.F., Alexander G.J., Beiley
J.V., Investments, Prentice Hall International, Inc., 1995, New Jersey
3.
Ammann M., Reich C. VaR for Nonlinear
Financial Instruments – Linear Approximation Or Full Monte Carlo? Swiss Society for Financial Market Research, Financial Markets and Portfolio
Management, 2001, Vol. 15, No.3, p.363-378
4.
Breuer W.,
Gürtler M. Performance Evaluation and
Preferences Beyond Mean-Variance, Swiss Society for Financial
Market Research,
Financial Markets
and Portfolio Management, 2003, Vol. 17, No.2, p.213-233
5.
Ebner M., Neumann
T., Time-Varying Betas of German Stock Returns, Swiss Society for Financial Market Research, Financial
Markets and Portfolio Management, 2005, Vol. 19, No.1,
p.29-46
6.
Gatev G., Malakhova A.,
Securities Selection
Criteria For Investment Portfolio Diversification,
Proceedings of the Fifth International School on Optimization of Production and Energy Management, 2005, Sofia,
p.73-80 (in French)
7.
Gatev G., Malakhova
A., The Assets Selection Problem for Investment Portfolio Diversification, Proceedings of the EESD Conference (Engineering Education in Sustainable Development), 2006, INSA de Lyon, p.30
8.
Iqbal J., Brooks R. Alternative Beta Risk Estimators And Asset Pricing Tests In Emerging
Markets: The Case of Pakistan,
Journal of Multinational Financial Management, 17, 2007, p.75-93
9.
Jagannathan R., Wang Z. Emperical Evaluation of Asset-Pricing Models: A Comparison of the SDF and Beta Methods,
The Journal of Finance, 2002,
October, Vol. 57, No.5, p.2337-2367
10.
Künzi-Bay A., Mayer J. Computational Aspects of Minimizing Conditional Value-At-Risk, Computational Management Science, 3, Spring-Verlag, 2006, p.3-27
11.
Malakhova A.A. Innovation methods
of risk measurement on Russian securities market,
Proceedings of the international conference on innovative development of Russian economics: Moscow, Moscow Institute
of Economics, Statistics and Informatics, 2009, p. 458-462.
(in Russian)
12.
Malakhova A.A., Zyryanova I.I. Financial
Assets Risk Measurement Problem, Proceedings of the international conference on Innovative Development of the Modern
Science, Ufa, 2014, p.159-165
(in Russian)
13.
Rockafellar R.T., Uryasev S., Zabarankin
M. Generalized Deviations in Risk Analysis, Finance and Stochastics, 10, Spring-Verlag, 2006, p.51-74
14.
Ziemba W.T. The Stochastic Programming Approach
to Asset, Liability, and Wealth Management, The Research Foundations of The Association for Investment Management and Research, USA, 2003
15.
Zorin A. Value-at-Risk As a Risk Measure for Trading
Strategy, The Securities Market Journal, No. 12, 2005, p.75-77 (in Russian)
16.
Zyryanova I.I., Malakhova
A.A. Counteraction to unfair financial practices via creating a single mega-regular of the market:
Proceedings of the international conference on Economics
and management nowadays:
Krasnoyarsk, Siberian
Institute of Business
and Administration, 2013,
p.114-117 (in Russian)
