A suitable financial risk measure is especially important. Distinct risk measures have different impacts on asset pricing, portfolio hedging, capital allocation, and investment performance evaluation. Standard deviation、VaR and ES are commonly used risk measurement tools,but there are lots of defects. For example, the standard deviation makes the upside and downside movements of returns be same punishment; VaR depends only on the probability of more extreme realizations but not on their values, and is not subadditivity risk measure; Expected Shortfall(ES) is a conditional expected value below the quantile, which although overcomes VaR's drawbacks, but only depends on catastrophic loss, thus being more conservative. Newey and Powell (1987) define the "asymmetric least squares" (ALS) and put forward the concept of expectile. It has some excellent properties, such as:strictly monotone increasing; positive homogeneity; translation invariance; superadditivity; prudentiality and sensitiveness to tail events. What's more, it's an average that balances between conditional upside mean and conditional downside mean.In this paper, the expectile is extended to asset allocation, and put forward to Adjexpectile's concept based on Levy theorem. Its economic and financial meaning can be understood as follows:when the portfolio return is lower than its τ quantile, after trading off conditional downside mean punishment, the size of the losses below the expected return. The coherent risk measure of Adjexpectile is discussed. Generally Adjexpectile satisfies subadditivity and positive homogeneity, further meeting the convexity, providing a theoretical basis for Adjexpectile portfolio optimization. Also, the relationship between Adjexpectile and shortfall、VaR、standard deviation, as well as Adjexpectile risk contribution and Euler decomposition is discussed. To Adjexpectile portfolio optimization, the nonparametric method is used to convert Adjexpectile into a linear programming problem. In the empirical analysis part, six asset indexes are enployed to the optimization allocation. Because the characteristics of six indexes data are more complex, having left skewness, also having right skewness and leptokurtosis, so for a given τ value, the average of four CARE modeles is taken to estimate different index α value into portfolio optimization calculation. The conclusions are summarized as follows:(1) Portfolio efficient frontier areas:to standard deviation as a risk measure, 99% Adjexpectile portfolio frontier is better than 99% shortfall frontier, overlaps the mean standard deviation portfolio frontier; to Shortfall as risk measurement, 99% Adjexpectile portfolio frontier is better than 99% shortfall portfolio frontier. The charm of Adjexpectile as risk measure is shown. (2) Portfolio risk diversification areas:mean Adjexpectile asset allocation is more dispersed than the mean standard deviation and mean Shortfall. This further embodies its advantage.
ZHOU Jing, LUO Le
. Adjexpectile as A Risk Measure: Properties、Optimization and Asset Allocation Applications[J]. Chinese Journal of Management Science, 2018
, 26(5)
: 51
-61
.
DOI: 10.16381/j.cnki.issn1003-207x.2018.05.006
[1] Markowitz H. Portfolio election[J]. Journal of Finance, 1952,7(1):77-91.
[2] Fishburn P. Mean risk analysis with risk associated with below target returns[J]. American Economic Review,1977,67(2):116-126.
[3] Kuan C, Yeh J, Hsu Y. Assessing value at risk with CATE,the conditional autoregressive expectile models[J]. Journal of Econometrics, 2009,150(2):261-270.
[4] Artzner P, Delbain F, Eber J M, et al. Coherent measures of risk[J]. Mathematical Finance, 1999,9(3):203-228.
[5] Acerbi C, Tasche D. On the coherence of expected shortfall[J]. Journal of Banking and Finance, 2002,26(7):1487-1503.
[6] Yamai Y, Yoshiba T. On the validity of Value-at-Risk:Comparative analyses with expected shortfall[J].Monetary and Economic Studies, 2002,20(1):57-85.
[7] Koenker R, Bassett G S. Regression quantiles[J]. Econometrica,1978,46(1):33-50.
[8] Aigner D J, Amemiya T, Poirier D J. On the estimation of production frontiers:Maximum likelihood estimation of the parameters of a discontinuous density function[J]. International Economic Review, 1976,17(2):377-396.
[9] Newey W K, Powell J L. Asymmetric least squares estimation and testing[J].Econometrica, 1987,55(4):819-847.
[10] Lam K, Sin C Y, Leung R. A theoretical framework to evaluate different marginsetting methodologies[J]. Journal of Futures Markets,2004,24(2):117-145.
[11] Yao Q, Tong H. Asymmetric least squares regression estimation:A nonparametric approach[J]. Nonparametric Statistics, 1996,6(2):273-292.
[12] Efron B. Regression percentiles using asymmetric squared error loss[J]. Statistica Sinica, 1991,1(1):93-125.
[13] Huang C F, Litzenberger R H. Foundations for financial economics[M]. New Jersey:Prentice Hall, 1988.
[14] Hamao Y, Musulis R, Ng V. Correlations in price changes and volatility across international stockmarkets[J]. The Review of Financial Studies,1990,3(3):281-307.
[15] Manganelli S. Asset allocation by penalized least squares[R]. Working Paper, European Central Bank, 2007.
[16] Dowd K. Beyond value at risk:The new science of risk management[M]. New York:Wiley, 1999.
[17] Duffie D, Pan Jun. An overview of value at risk[J]. Journal of Derivatives, 1997,4(3):7-49.
[18] Basak S, Shapiro A. Value-at-Risk based risk management:Optimal policies and asset prices[J]. The Review of Financial Studies, 2001,14(2):371-405.
[19] Uryasev S, Rockafellar R T. Optimization of conditional Value-at-Risk[J].Journal of Risk,1999,29(1):1071-1074.
[20] Bertsimas D, Lauprete G, Samarov A. Shortfall as a risk measure:Propertyes,optimization and applications[J]. Journal of Economic Dynamics & Control, 2004,28(7):1353-1381.
[21] Bassett G W, Koenker R, Kordas G. Pessimistic portfolio allocation and Choquet expected utility[J]. Journal of Financial Econometrics, 2004,2(4):477-491.
[22] Levy H. Stochastic dominance and expected utility:Survey and analysis[J].Management Science,1992,38(4):555-593.
[23] Delbaen F. Coherent risk measures on general probability spaces[M].Berlin Heidelberg:Springer,2002.
[24] Engle R F, Manganelli S. CAViaR:Conditional autoregressive Value-at-Riskby regression quantiles[J].Journal of Business & Economic Statistics, 2004,22(4):367-381.
[25] Taylor J W. Estimating Value-at-Risk and expected shortfall using expectiles[J]. Journal of Financial Econometrics, 2008,6(2):231-252.
[26] Kuester K, Mittnik S, Paolella M S. Value-at-Risk prediction:A comparison of alternative strategies[J]. Journal of Financial Econometrics, 2006,4(1):53-89.
[27] 苏幸, 周勇. 条件自回归expectile模型及其在基金业绩评价中的应用[J].中国管理科学,2013,21(6):22-19.
[28] 谢尚宇, 姚宏伟, 周勇. 基于ARCH-Exp-ectile方法的VaR和ES尾部风险测量[J].中国管理科学, 2014,22(9):1-9.
[29] 简志宏, 曾裕峰, 刘曦腾. 基于CAViaR模型的沪深300指数期货隔夜风险研究[J]. 中国管理科学, 2016,24(9):1-10.
[30] Litterman R. Hot spotsTM and hedges[J].Journal of Portfolio Management,1996,Special Issue:52-75.
