If the market is inefficient and there is friction, it will lead to the existence of transaction costs, and the liquidity of open-end funds is directly related to transaction costs. The portfolio problem of market friction was first proposed by Magil and Constantinides, and then further studied by Davis and Norman. Davis( 1990) and others used the stochastic control method to analyze the transaction cost related to the liquidity of securities in the presence of market friction, and found that it is reasonable to keep within a certain risk interval and make the minimum transaction when it is close to the boundary of this interval. Shreve, Akian( 1995) and others studied the multidimensional portfolio problem with transaction costs by using the viscosity theory, and solved a problem of maximizing the ultimate wealth of three assets by using the finite difference method. However, the method proposed by Davis, Shreve, Akian, etc. Ignoring the huge transaction cost brought by the fixed cost, Eastham and Hastings solved this problem effectively with the pulse control method. Morton and Pliska( 1995) also studied the optimal portfolio management with fixed transaction costs. Although the transaction cost in their model is not the real transaction cost, their method has a certain guiding role in solving the corresponding portfolio problems.
Recent studies believe that the liquidity of securities is the decisive factor of securities value. Compared with liquid securities, the pricing of illiquid securities always has a certain discount. For example, Amihud and Mendelssohn (199 1) and Kamara (1994) have confirmed that there is an income gap of more than 35 basis points between illiquid medium-term bonds and liquid government bonds; Whitelaw( 199 1) and others also confirmed a similar phenomenon. Brito (1977), Subrahanyam (1979), Amihud and Mendelssohn (1986), Duma, Luciano (199 1), BuDoukhi and whitelaw (65433). The research results of portfolio liquidity function by Constantinides and Mehra( 1998) mainly focused on exogenous transaction costs and restrictions on borrowing or selling, while Longstaff (2001) later focused on the endogenous illiquidity function of trading strategy and securities value. Longstaf solves the intertemporal portfolio problem that investors are restricted by liquidity.
(B) Portfolio theory based on style investment
Style investment began in 1992 with william sharpe's paper Asset Allocation: Style Management and Performance Evaluation. Foreign research on style investment mainly focuses on the following aspects:
First, the investment style analysis. At present, the recognized style analysis methods mainly include style analysis based on and combination. The former is the income-based style analysis put forward by Sharp, who thinks that the investment style of fund managers in the past period can be judged by comparing the relationship between the income of funds and the income of selected style index; The latter mainly divides the investment style of the fund according to the characteristics of the stock actually held by the fund. Kahn( 1996) found that for small sample funds, risk prediction based on portfolio analysis is more relevant than income analysis. Kaplan(2003) found that for large-cap stocks, the results of the two style analysis methods are similar, but for small and medium-sized market value portfolios and growth portfolios, the results of the two methods are significantly different.
Second, the research on the performance and causes of style investment. Style investment often shows small market value effect (the return from investing in small-scale company stocks is higher than that from investing in large-scale company stocks) and BV/MV effect (net assets/market value). Banz( 198 1) first found that the average return on shares of the smallest company was higher than that of the largest company 19.8%. Reinganum (198 1) also found a similar phenomenon. For BV/MV effect, Stattman (1980) found that the average return of American companies' stocks was positively correlated with BV/MV. Fama and French( 1992) also proved that BV/MV effect is obvious in American market. There are several explanations for this: First, Fama and French (1993, 1995), Johnson (1997) and others think that the excess return of style investment is compensation for risk, which is ignored by the orthodox capital asset pricing model; Secondly, Lakonishok, Shleifer and Vishny( 1994) think that the excess return is caused by investors' overreaction to the past performance of a stock; Thirdly, Daniel and Tietmann (1997) think that because companies with certain attributes have some common characteristics, there may be some business problems at the same time, which leads to the above two effects; Fourthly, it is considered that the choice of calculation method and data processing are caused by human factors.
Thirdly, the periodicity of style investment and the study of style conversion strategy. From the perspective of value/growth or large-cap stocks and stocks, style investment has different performances in different periods and is cyclical. Frank et al. (2002) research shows that small-cap stocks/large-cap stocks in American and Japanese stock markets always perform poorly or well in the range. David, Robert and Christopher (1997) found that the return rate of value/growth portfolio has obvious periodicity by analyzing the data of the United States, Canada and other countries. Because style investment is cyclical, investors can get better returns through style conversion. Levi, and Liodakis( 1999) think that when there is no significant difference between the returns of the two relative styles, investors have the opportunity to improve portfolio performance through style conversion; Other scholars such as KevinQ Q. W ang(2003) and Georgi(2003) also studied this phenomenon respectively.
Fourthly, the research on the influence of style investment on the securities market. Lee and Andrei (199 1) use the style investment theory to explain why funds listed on the same stock market rise and fall together, although they hold completely different stocks. Fruit et al. (1999) also used the concept of style investment to explain why the same kind of stocks performed differently on different exchanges. Sorensen, Lazzara (1995), Andersson (1997) and Fochtman( 1995) have also studied the relationship between a certain style and a certain influencing factor (such as macroeconomic factors, price trends, etc.). ).
(C) Long-term portfolio theory based on continuous time
For a long time, markowitz's mean-variance theory has played an important role in guiding people's short-term investment. But in fact, the optimal portfolio of long-term investment and short-term investment is different.
Samuelson (1963, 1969) and others first described the restrictive conditions for long-term investors and short-term investors to make the same decision. Merton (1969, 197 1, 1973) also made a long-term and in-depth study on this issue. Their research tells people that investment opportunities will change with time, and long-term investors always pay attention to the impact on long-term investment opportunities and hope to arbitrage from them. Jin, Homberg (1996); Balduzzi; Lynch (1999); Barberis(2000) and others have established empirical models for portfolio selection of long-term investors, which are based on Samuelson( 1963,1969); Mosin (1968); Merton (1969, 197 1,1973); Stiglitz (1979); Rubinstein (1976a, b); Breeden (1979), etc., finally completed the empirical test of the early theoretical literature. They assume that an investor with limited life has the Hara (hyperbolic absolute risk aversion) effect of ending wealth, and it turns out that approximation is not used, and the optimal portfolio weight is linear. Balduzzi and Lynch come to the conclusion that ignoring the real transaction cost will increase the utility cost by 0.8%, reaching16.9%; Barberis found that even after the uncertainty of many parameters is included in the model, there are still enough income expectations, so that long-term investors can always allocate more assets in stocks.
Morton (1973) proposed the hedging effect of the long-term impact of interest rates. When the risk aversion coefficient of investors is greater than 1, the demand for risky assets is not only affected by the risk premium of assets, but also by the covariance of the adjustment of expected rate of return and expected forward interest rate. For the intertemporal budget constraint in intertemporal theory, Campbel( 1993) thinks that the intertemporal budget constraint of investors is approximately linear when the consumption-wealth ratio remains unchanged or changes little; Tepla(2000) extended the selection criteria of static portfolio to a dynamic intertemporal model under the constraints of lending and short selling. Campbell and Viceira(200 1) also expounded this part of the conclusion.
Jeremy Siegel (1994) thinks that the risk of stocks in long-term investment is lower than that of bonds or even national debt, and stocks are the safest investment assets in long-term investment. Viceira (1999,2000) proved that opportunities lead to greater utility loss when ignoring the market in the optimal investment strategy. Campbell, Chan, Viceira (200 1) and others use VaR (first-order vector autoregressive) model to analyze the consumption and portfolio selection of long-term investors. The research shows that the predictability of stock returns increases investors' demand for stock investment, and long-term inflation bonds can increase the effectiveness of stabilizing investors; The research of John Y.Campbell, George Chacko and Jorge Rodriguez(2004) also shows that conservative long-term investors have a positive demand for intertemporal arbitrage. These studies have made outstanding contributions to the establishment of long-term portfolio framework.
For the asset allocation problem of long-term investment, using continuous-time mathematics to analyze dynamic portfolio selection can at least be traced back to the research work of Robert Merton (1969-1973). Duffle (1996); Karatezas, Shref (1998); Morton( 1990) gives a general method of portfolio selection in continuous time. Chacko and Viceira( 1999) discussed the influence of time-varying risks on investment. Cox, Huang (1989); Cox, Leland (1982); Pliska( 1996) and others put forward the "saddle method" of intertemporal consumption and portfolio selection, which uses the SDF (random discount factor) attribute in the complete market to transform the dynamic problem into a static one, making the result easier to solve. Campbell and Viceira(2002) systematically discussed the long-term portfolio selection for the first time in their book Strategic Asset Allocation: Portfolio Selection for Long-term Investors. They created an intertemporal empirical analysis method, which can be compared with mean variance analysis; Facts have proved that long-term inflation index bonds are risk-free assets for long-term investors. It reveals the condition that stocks are safer assets for long-term investors than for short-term investors. It proves how labor income affects portfolio selection.
(D) Portfolio theory based on VaR
VaR method was only paid attention to by scholars who studied the theory of securities portfolio in 1950s. It was originally used to measure the market risk of some financial companies trading securities. The introduction of VaR method makes up for the deficiency of the original portfolio theory in measuring portfolio risk to some extent.
Foreign scholars have defined VaR from different angles.
Joroin( 1996) is considered as the worst-case loss within a given probability confidence level; Sironi and Resti( 1997) think that it is the potential maximum loss within a defined period under certain probability conditions.
Luciano (1998) thinks that it is the possible loss of a single position or the whole portfolio under certain probability conditions; Given the distribution of asset (portfolio) value change, risk is defined according to the possibility that the value change exceeds a critical point.
Mauser, Rosen and Jorion(200 1) estimated the optimization problem of portfolio selection under the condition of VaR by historical simulation or Monte Carlo simulation respectively. However, VaR still has many defects.
Artzner et al. (1999) put forward the concept of risk consistency measure, in which consistency is judged by four axiomatic assumptions. Because VaR does not satisfy the subadditivity of the four conditions, it means that the principle of portfolio risk diversification is rejected under some conditions, and VaR is not a coherent risk measure.
Based on this, Pflug, Rockafellar, Uryasev (2000, 2002); Acerbi and Tasche(2002) put forward Conditional Value (CVaR) as a measure of risk to modify VaR. CvaR is defined as the loss that exceeds the expectation of some conditions of vaR, and only the downside risks are considered. If the confidence interval corresponding to VaR is (1-α), then α-CVaR is the average loss exceeding α-VAR; In view of the fact that VaR can't compare the risk exposures from different markets, Giuseppe Tardivo(2002) put forward the concept of benchmark -VaR, that is, the maximum deviation of a fund or portfolio from the benchmark within a certain period of time and a certain confidence interval; Emmer et al. (200 1) introduced the concept of venture capital (CaR) to measure risk rather than variance. In view of the fact that VaR only measures the risk of portfolio under normal market conditions, Embrechts et al. (1997) combined the extreme theory of measuring extreme situations with VaR, and put forward a method to measure the extreme risk of market. McNeil and Frey(2000) used extreme value theory to study the tail characteristics of time series in Swiss financial market, and concluded that extreme value method is more robust and accurate than VaR.
After defining the risk measurement indicators such as VaR and CVaR, the research on portfolio selection based on them is carried out accordingly.
Rockafellar et al. (2000) and Anderson et al. (200 1) considered the problem of portfolio optimization when CVaR was used as a risk measure, and proved that CVaR is a convex function and can be used to construct an effective optimization method. Rockafellar and others also proposed a linear programming method, which can minimize both VaR and CVaR. After introducing the concept of venture capital (CaR), Emmer et al. established a "mean -CaR model" for portfolio selection, and deduced the optimal solution and efficient boundary in analytical form. Young( 1998) put forward a portfolio model of maximum and minimum return (MMR): under the constraint that the average return of the portfolio exceeds a certain minimum return level, the minimum return in any period is maximized, and the decision-making goal is to consider the optimal return among the most unfavorable returns. The risk measure is the smallest possible return rather than variance.
In addition, Bogentoft et al. (2001); Topaloglou et al. (2002); Castellacci and Siclari(2003) also studied the problem of portfolio selection based on VaR and CVaR.
(E) Portfolio theory based on non-utility maximization
Cover is one of the early scholars of non-utility maximization portfolio theory. He proposed a pan-portfolio model under discrete-time conditions. The outstanding advantage of this model is that it doesn't need to know the market parameters and related statistical information, such as interest rate and price fluctuation, or even describe the dynamic mechanism of price change in discrete time, and only needs to track the performance-weighted changes of different securities weights to realize the optimal constant portfolio. Cover also describes the asymptotic behavior of universal combination, and illustrates that universal combination has good explanatory power with examples.
Hellwing put forward a universally applicable pricing method of economic resources-the principle of preserving value, that is, the intrinsic value (future income value) of resources does not change with time. Helwing used this method to investigate the portfolio optimization of the securities market under the conditions of discrete time and finite state space, and showed good explanatory power.
Buckley and Korn think that for those investors who passively follow the index, the ideal portfolio always consists of all the securities that enter the index from the perspective of investigating the index tracking error under random cash flow. This will inevitably lead to the deviation between the performance of cash accounts held by capital asset investors and the performance of the index (that is, lead to tracking errors). Based on this, Buckley and Korn give the relevant model in this case (that is, the general continuous-time model based on half-saddle), analyze the impulse control problem caused by investors, and give the general conditions for the existence of the optimal control strategy. In addition, they also discussed the existence and uniqueness of value maintenance strategies in some diffusion markets, solved the unique value maintenance measurement problem of option hedging theory in incomplete markets (that is, the minimum saddle measurement problem), and investigated the influence of additional constraints on portfolio strategies.
(vi) Behavioral finance and behavioral portfolio theory
In the past 20 years, empirical financial research has continuously found evidence of predictable stock returns, and the theoretical basis and empirical test of EMH have been strongly challenged. The empirical study of the securities market has found many abnormal phenomena that EMH and capital asset pricing model can't reasonably explain. Faced with a series of financial anomalies, people began to question the traditional financial theory with the efficient market hypothesis as the core. Because behavioral finance can better explain these phenomena, behavioral finance, which was previously ignored, has attracted more and more scholars' attention.