포트폴리오 최적화

Portfolio optimization

포트폴리오 최적화는 어떤 목적에 따라 고려되는 모든 포트폴리오 집합 중에서 최고의 포트폴리오(자산분포)를 선정하는 과정이다. 목표는 일반적으로 기대수익률과 같은 요소를 최대화하고 재무위험과 같은 비용을 최소화한다. 고려되는 요인은 유형(예: 자산, 부채, 수익 또는 기타 기초)에서 무형(예: 선택적 분할)에 이르기까지 다양할 수 있다.

현대 포트폴리오 이론

현대 포트폴리오 이론은 1952년 해리 마코위츠 박사 논문에서 소개되었다.[1][2] 마코위츠 모델을 보라. 그것은 투자자가 주어진 위험의 양에 따라 포트폴리오의 예상 수익을 최대화하기를 원한다고 가정한다. 효율적인 포트폴리오로 알려진 이 기준을 충족하는 포트폴리오의 경우, 더 높은 기대 수익을 달성하려면 더 많은 위험을 감수해야 하기 때문에 투자자들은 위험과 예상 수익 사이의 트레이드오프를 직면하게 된다. 효율적 포트폴리오의 이러한 리스크 예상 수익 관계는 효율적 프런티어라고 알려진 곡선으로 그래픽으로 표현된다. 각각 효율적인 개척지의 한 점으로 대표되는 모든 효율적인 포트폴리오가 잘 전달된다. 높은 순간을 무시하면 위험증권에 유의미한 과잉투자로 이어질 수 있지만, 특히 변동성이 높은 경우 수익분배가 가우스적이지 않을 때 포트폴리오를 최적화하는 것은 수학적으로 어려운 일이다.[3][4]

최적화 방법

포트폴리오 최적화 문제는 제한된 효용 최대화 문제로 지정된다. 포트폴리오 효용 함수의 공통 공식은 이를 리스크 비용을 뺀 예상 포트폴리오 수익률(거래 및 금융 비용의 순액)으로 정의한다. 후자의 구성요소인 위험원가는 포트폴리오 위험과 위험 회피 매개변수(또는 위험 단가)를 곱한 것으로 정의된다. 실무자들은 다변화를 개선하고 위험을 더 제한하기 위해 추가 제약조건을 추가하는 경우가 많다. 그러한 제약조건의 예로는 자산, 부문 및 지역 포트폴리오 가중치 한도가 있다.

Specific approaches

Portfolio optimization often takes place in two stages: optimizing weights of asset classes to hold, and optimizing weights of assets within the same asset class. An example of the former would be choosing the proportions placed in equities versus bonds, while an example of the latter would be choosing the proportions of the stock sub-portfolio placed in stocks X, Y, and Z. Equities and bonds have fundamentally different financial characteristics and have different systematic risk and hence can be viewed as separate asset classes; holding some of the portfolio in each class provides some diversification, and holding various specific assets within each class affords further diversification. By using such a two-step procedure one eliminates non-systematic risks both on the individual asset and the asset class level. For the specific formulas for efficient portfolios,[5] see Portfolio separation in mean-variance analysis.

One approach to portfolio optimization is to specify a von Neumann–Morgenstern utility function defined over final portfolio wealth; the expected value of utility is to be maximized. To reflect a preference for higher rather than lower returns, this objective function is increasing in wealth, and to reflect risk aversion it is concave. For realistic utility functions in the presence of many assets that can be held, this approach, while theoretically the most defensible, can be computationally intensive.

Harry Markowitz[6] developed the "critical line method", a general procedure for quadratic programming that can handle additional linear constraints and upper and lower bounds on holdings. Moreover, in this context, the approach provides a method for determining the entire set of efficient portfolios. Its application here was later explicated by William Sharpe.[7]

Mathematical tools

The complexity and scale of optimizing portfolios over many assets means that the work is generally done by computer. Central to this optimization is the construction of the covariance matrix for the rates of return on the assets in the portfolio.

Techniques include:

Optimization constraints

Portfolio optimization is usually done subject to constraints, such as regulatory constraints, or illiquidity. These constraints can lead to portfolio weights that focus on a small sub-sample of assets within the portfolio. When the portfolio optimization process is subject to other constraints such as taxes, transaction costs, and management fees, the optimization process may result in an under-diversified portfolio.[14]

Regulation and taxes

Investors may be forbidden by law to hold some assets. In some cases, unconstrained portfolio optimization would lead to short-selling of some assets. However short-selling can be forbidden. Sometimes it is impractical to hold an asset because the associated tax cost is too high. In such cases appropriate constraints must be imposed on the optimization process.

Transaction costs

Transaction costs are the costs of trading in order to change the portfolio weights. Since the optimal portfolio changes with time, there is an incentive to re-optimize frequently. However, too frequent trading would incur too-frequent transactions costs; so the optimal strategy is to find the frequency of re-optimization and trading that appropriately trades off the avoidance of transaction costs with the avoidance of sticking with an out-of-date set of portfolio proportions. This is related to the topic of tracking error, by which stock proportions deviate over time from some benchmark in the absence of re-balancing.

Improving portfolio optimization

Correlations and risk evaluation

Different approaches to portfolio optimization measure risk differently. In addition to the traditional measure, standard deviation, or its square (variance), which are not robust risk measures, other measures include the Sortino ratio, CVaR (Conditional Value at Risk), and statistical dispersion.

Investment is a forward-looking activity, and thus the covariances of returns must be forecast rather than observed.

Portfolio optimization assumes the investor may have some risk aversion and the stock prices may exhibit significant differences between their historical or forecast values and what is experienced. In particular, financial crises are characterized by a significant increase in correlation of stock price movements which may seriously degrade the benefits of diversification.[15]

In a mean-variance optimization framework, accurate estimation of the variance-covariance matrix is paramount. Quantitative techniques that use Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.[16] Allowing the modeling process to allow for empirical characteristics in stock returns such as autoregression, asymmetric volatility, skewness, and kurtosis is important. Not accounting for these attributes can lead to severe estimation error in the correlations, variances and covariances that have negative biases (as much as 70% of the true values).[17]

Other optimization strategies that focus on minimizing tail-risk (e.g., value at risk, conditional value at risk) in investment portfolios are popular amongst risk averse investors. To minimize exposure to tail risk, forecasts of asset returns using Monte-Carlo simulation with vine copulas to allow for lower (left) tail dependence (e.g., Clayton, Rotated Gumbel) across large portfolios of assets are most suitable.[18]

More recently, hedge fund managers have been applying "full-scale optimization" whereby any investor utility function can be used to optimize a portfolio.[19] It is purported that such a methodology is more practical and suitable for modern investors whose risk preferences involve reducing tail risk, minimizing negative skewness and fat tails in the returns distribution of the investment portfolio.[20] Where such methodologies involve the use of higher-moment utility functions, it is necessary to use a methodology that allows for forecasting of a joint distribution that accounts for asymmetric dependence. A suitable methodology that allows for the joint distribution to incorporate asymmetric dependence is the Clayton Canonical Vine Copula. See Copula (probability theory)#Quantitative finance.

Cooperation in portfolio optimization

A group of investors, instead of investing individually, may choose to invest their total capital into the joint portfolio, and then divide the (uncertain) investment profit in a way which suits best their utility/risk preferences. It turns out that, at least in the expected utility model,[21] and mean-deviation model,[22] each investor can usually get a share which he/she values strictly more than his/her optimal portfolio from the individual investment.

See also

References

  1. ^ Markowitz, H.M. (March 1952). "Portfolio Selection". The Journal of Finance. 7 (1): 77–91. doi:10.2307/2975974. JSTOR 2975974.
  2. ^ Markowitz, H.M. (1959). Portfolio Selection: Efficient Diversification of Investments. New York: John Wiley & Sons. (reprinted by Yale University Press, 1970, ISBN 978-0-300-01372-6; 2nd ed. Basil Blackwell, 1991, ISBN 978-1-55786-108-5)
  3. ^ Cvitanić, Jakša; Polimenis, Vassilis; Zapatero, Fernando (2008-01-01). "Optimal portfolio allocation with higher moments". Annals of Finance. 4 (1): 1–28. doi:10.1007/s10436-007-0071-5. ISSN 1614-2446. S2CID 16514619.
  4. ^ Kim, Young Shin; Giacometti, Rosella; Rachev, Svetlozar; Fabozzi, Frank J.; Mignacca, Domenico (2012-11-21). "Measuring financial risk and portfolio optimization with a non-Gaussian multivariate model". Annals of Operations Research. 201 (1): 325–343. doi:10.1007/s10479-012-1229-8. S2CID 45585936.
  5. ^ Merton, Robert. September 1972. "An analytic derivation of the efficient portfolio frontier," Journal of Financial and Quantitative Analysis 7, 1851–1872.
  6. ^ Markowitz, Harry (1956). "The optimization of a quadratic function subject to linear constraints". Naval Research Logistics Quarterly. 3 (1–2): 111–133. doi:10.1002/nav.3800030110.
  7. ^ The Critical Line Method in William Sharpe, Macro-Investment Analysis (online text)
  8. ^ Rockafellar, R. Tyrrell; Uryasev, Stanislav (2000). "Optimization of conditional value-at-risk" (PDF). Journal of Risk. 2 (3): 21–42. doi:10.21314/JOR.2000.038.
  9. ^ Kapsos, Michalis; Zymler, Steve; Christofides, Nicos; Rustem, Berç (Summer 2014). "Optimizing the Omega Ratio using Linear Programming" (PDF). Journal of Computational Finance. 17 (4): 49–57. doi:10.21314/JCF.2014.283.
  10. ^ Talebi, Arash; Molaei, Sheikh (17 September 2010). M.A., M.J. Proceeding of 2010 2nd IEEE International Conference on Information and Financial Engineering. p. 430. doi:10.1109/icife.2010.5609394. ISBN 978-1-4244-6927-7. S2CID 17386345.
  11. ^ Shapiro, Alexander; Dentcheva, Darinka; Ruszczyński, Andrzej (2009). Lectures on stochastic programming: Modeling and theory (PDF). MPS/SIAM Series on Optimization. 9. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). Mathematical Programming Society (MPS). pp. xvi+436. ISBN 978-0-89871-687-0. MR 2562798.
  12. ^ Zhu, Zhe; Welsch, Roy E. (2018). "Robust dependence modeling for high-dimensional covariance matrices with financial applications". Ann. Appl. Stat. 12 (2): 1228–1249. doi:10.1214/17-AOAS1087. S2CID 23490041.
  13. ^ Sefiane, Slimane and Benbouziane, Mohamed (2012). Portfolio Selection Using Genetic Algorithm Archived 2016-04-29 at the Wayback Machine, Journal of Applied Finance & Banking , Vol. 2, No. 4 (2012): pp. 143-154.
  14. ^ Humphrey, J.; Benson, K.; Low, R.K.Y.; Lee, W.L. (2015). "Is diversification always optimal?" (PDF). Pacific Basin Finance Journal. 35 (B): B. doi:10.1016/j.pacfin.2015.09.003.
  15. ^ Chua, D.; Krizman, M.; Page, S. (2009). "The Myth of Diversification". Journal of Portfolio Management. 36 (1): 26–35. doi:10.3905/JPM.2009.36.1.026. S2CID 154921810.
  16. ^ Low, R.K.Y.; Faff, R.; Aas, K. (2016). "Enhancing mean–variance portfolio selection by modeling distributional asymmetries" (PDF). Journal of Economics and Business. 85: 49–72. doi:10.1016/j.jeconbus.2016.01.003.
  17. ^ Fantazzinni, D. (2009). "The effects of misspecified marginals and copulas on computing the value at risk: A Monte Carlo study". Computational Statistics & Data Analysis. 53 (6): 2168–2188. doi:10.1016/j.csda.2008.02.002.
  18. ^ Low, R.K.Y.; Alcock, J.; Faff, R.; Brailsford, T. (2013). "Canonical vine copulas in the context of modern portfolio management: Are they worth it?" (PDF). Journal of Banking & Finance. 37 (8): 3085. doi:10.1016/j.jbankfin.2013.02.036. S2CID 154138333.
  19. ^ Chua, David; Kritzman, Mark; Page, Sebastien (2009). "The Myth of Diversification". Journal of Portfolio Management. 36 (1): 26–35. doi:10.3905/JPM.2009.36.1.026. S2CID 154921810.
  20. ^ Adler, Tim; Kritzman, Mark (2007). "Mean-Variance versus Full-Scale Optimization: In and Out of Sample". Journal of Asset Management. 7 (5): 71–73. doi:10.2469/dig.v37.n3.4799.
  21. ^ Xia, Jianming (2004). "Multi-agent investment in incomplete markets". Finance and Stochastics. 8 (2): 241–259. doi:10.1007/s00780-003-0115-2. S2CID 7162635.
  22. ^ Grechuk, B., Molyboha, A., Zabarankin, M. (2013). "Cooperative games with general deviation measures", Mathematical Finance, 23(2), 339–365.