# Performance Evaluation Model of Short-Term Mutual Funds Based on Return-Variance-Liquidity

Authors: Ye Tian; Guanjiang Wan; Jianing Shi; Yufei Liu; Wenbin Liu; Wanfang Shen
Journal: Security and Communication Networks
 (2022)

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## Abstract

In the past, research on evaluating the portfolio efficiency of mutual funds was mostly based on the mean-variance framework proposed by Markowitz. However, for Chinese short-term mutual funds (CSTMF), short-term mutual funds with different lock-up periods will automatically roll over after the lock up period to restart the funds unless instructed to terminate. By reducing the liquidity of their investments, the lock-up period generates an opportunity cost for investors. The liquidity impacts of different CSTMFs will depend on their lock-up periods. In this paper, we evaluate the portfolio efficiency of the funds by justifying using the lock-up period as a measurement to measure the liquidity risk faced by the investors from the perspective of investors. Therefore, here, we propose to use the return-variance-liquidity framework to evaluate the performance of the funds. Furthermore, we select 28 CSTMF funds to conduct empirical analysis with appropriate DEA models. Firstly, we carry out some comparisons with the traditional mean-variance framework. The results show that the length of the lock-up periods has a significant impact on the fund’s performance, which has important guidance for the investment strategy. Finally, we conduct the benchmarking analysis for the 28 funds and introduce a new DEA model, present how to reduce liquidity risk under the premise of given return and variance, and provide valuable investment advices by combining these with sensitivity analysis. Overall, there are three conclusions: (1) most inefficient funds can be improved using portfolios consisting of funds with shorter lock-up period. (2) For some inefficient funds with shorter lock-up period, the current return and variance levels can be achieved by a portfolio. (3) Among the funds with higher efficiency scores, funds with shorter lock-up period have higher stability.


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## Body

## 1. Introduction

At present, the financial market in China has diversified investments and financial methods, such as stocks, futures, options, foreign exchange, and funds. Many of those investment methods, such as stocks, options, and futures, are accompanied by relatively high risks. At this stage, the options of relatively low risk investments mainly include the Chinese bank financial product (CBFP), money market fund (MMF), and Chinese short-term mutual fund (CSTMF) [1, 2].Both CSTMFs and CBFPs have lock-up periods, during which investors cannot redeem them. Their main differences are as follows:(1)
The minimum purchase amount of the CBFPs is generally high, while CSTMFs A-class shares only have a lower minimum purchase amount.(2)
Most of the CBFPs are repaid with the principal and interest after the expiration. However, once the CSTMFs have expired, as long as the investor does not apply for redemption, the share held by the investor will be automatically transferred to the next operation period.(3)
The CBFPs are often invested in by agents and have a strong ability to cope with the asset price fluctuation with more investment portfolio targets, while the CSTMFs have relatively fewer investment targets and shorter duration of portfolio target. CSTMFs, specifically, can only invest in short-term, high-security varieties, such as agreement saving, short-term financing bills, and central bank bill.(4)
The operation of the asset pool of CBFPs is not transparent in terms of information disclosure, while the CSTMFs’ information disclosure is more detailed and transparent [3]. It means that the investors could clearly know the daily return of CSTMFs, however, CBFPs always show the return on maturity date.Intuitively, differences between CSTMFs and CBFPs are briefly described as Table1.Table 1
Differences between CSTMFs and CBFPs.
TypesMinimum purchaseRedemption rulesAmount of portfolio targetsInformation disclosureCSTMFsLowDepend on investorsRelatively smallTransparentCBFPsHighAfter expirationLargeNontransparentTable1 shows that there are many differences between CSTMFs and CBFPs. However, CSTMFs share much more common points with MMFs’ [3]. The investment targets of the two are similar, the unit net value of them is always 1(RMB), and the return of them is calculated daily. The fundamental difference between the two is that MMFs can be redeemed on a daily basis, while CSTMFs can be purchase at any time and redeemed only after the specified lock-up period (usually 7 days, 14 days, 30 days, and 60 days). Hence, we can deem CSTMFs as MMFs with lock-up period.By the above comparisons of three relatively safe and relatively stable investment methods, it can be seen that the CSTMFs have lock-up period, however, the daily changes in return are transparent, and thereby it is an improvement based on the original CBFP. At the same time, CSTMFs have the following characteristics: low investment threshold, many types of lock-up periods can be selected, they can be purchased at any time, and investors have the right not to redeem them at maturity. They are valued by the amortized cost method, and the returns are calculated daily and carried over according to the lock-up period [2]. At that time, the China Banking Regulatory Commission officially suspended short-term CBFPs within 30 days. The supply of large-scale short-term financial market was in short, and CSTMFs then had the space. The ability to purchase CSTMFs through the internet is also one of the reasons for its substantial growth.Up to June 30, 2018, according to the data from the funds’ quarterly report, a total of 54 CSTMFs (calculated with the consolidation of A and B share) totaled RMB 712.493 billion (the current exchange rate of RMB and US Dollar is 6.8 : 1), increasing by RMB 334.155 billion compared with the end of 2017, with an increase of about 88%. On average, the size of a single fund was as high as RMB 13.194 billion, increasing by 84.83% compared with the end of 2017. Of these, 25 funds doubled in size, and as many as 29 funds had a scale of over RMB 10 billion, an increase of 13 from the end of 2017. The change in the market share of funds from December 31, 2016, to June 30, 2018, is shown in Table2.Table 2
The changes in the CSTMFs’ market share.
Date2016/12/312017/6/302017/12/312018/6/30Market share (RMB 100 million)1741273037837125Data sources: data of the funds’ quarterly report.Although the emergence of MMF has had a certain impact on CSTMF in China, the CSTMFs generally have higher returns than MMFs. In China, financial products similar to CSTMF include financial products issued by large securities companies or their asset management companies, and insurance products issued, underwritten, or managed by insurance companies. Because the historical data from both types of products are difficult to obtain, in this paper, we evaluate the portfolio efficiency of CSTMF to provide investment strategies and reference for fund investors.Besides, the key point of this paper is to consider lock-up period and liquidity risk when evaluating the funds from the perspective of investors, which has not actually been done in case study. The specific method will be introduced in the subsequent part.


## 2. Evaluation of Efficiency of Funds

There have been many studies on the evaluation of efficiency of funds, such as [4–6]. In terms of evaluation methods, there are mainly index methods and frontier methods.The literature evaluating funds by index methods includes the following: Liu et al. [5] adjusted the traditional Sharpe ratio model using VaR and ES with empirical nonparametric estimating method and nonparametric smooth kernel estimating method to replace the variance of return of investment. Besides, they also consider the time-varying VaR and ES of return volatility and simulated the volatility by the GARCH model. Yang and Meng [7] used the index method to rank the fund performance using the Sharpe ratio, and they proposed a generalized Sharpe ratio considering the higher moment, while using the parametric and the nonparametric method to calculate the generalized Sharpe ratio. Xue and Hu [8] applied the asymmetric generalized autoregressive conditional heteroskedasticity (EGARCH) model to estimate VaR values to replace the standard deviation in the Sharpe ratio, and they combined the Sharpe ratio with the liquidity risk and showed that the modified Sharpe ratio is valid. Shi [9] used the generalized autoregressive conditional heteroskedasticity model based on the generalized error distribution to calculate the VaR and ES. This paper uses them instead of standard deviation in the standard Sharpe ratio, and then, it gets the modified Sharpe ratio as a fund performance evaluation. It can be seen from the above literature that the index method could rank the funds’ profitability, however, it cannot take into account the investor's preference for the two dimensions of risk and return. For example, the risk averse are likely to evaluate the lowest-risk funds within the given return range (risk-oriented), and they will not choose the Sharp ratio to evaluate the funds.In 1952, the Mean-Variance model proposed by the American economist Markowitz [10] made the frontier method developed into the core method of modern portfolio theory. The frontier method not only allows investors to evaluate funds based on their risk-return preference but also provides a variety of possible improvements for funds that are not on the frontier. For example, Morey and Morey [11] proposed a quadratic DEA model with the variance as the input and the expected return as the output over the same period. Briec and Kerstens [12] further extended the analysis of multilevel mean-variance portfolio analysis in several ways and introduced a more general efficiency measure to reduce risk and expand return over all time periods. Lamb and Tee [13] discussed what risk and return measures could justifiably be combined and constructed in a diversification-consistent model. However, in practical applications, various constraints of market friction make the real frontier difficult to calculate.The data envelopment analysis (DEA) method proposed by Charnes et al. [14] is a powerful tool for studying the relative efficiency of the same type of production decision making units (DMU). It approximates the real frontier using piecewise linear facets. When the sample size is relatively large, we can obtain a very similar result to the nonlinear frontier evaluation method with very small number of calculations [15]. For example, Yang and Luo [4] introduced the super efficiency DEA into the performance evaluation of Chinese investment funds, and they established a model based on super efficiency DEA to compute funds’ efficiency for their performance assessments. Liu [6] applied the four-stage DEA method and the bootstrapped DEA method to control the effects of exogenous environmental variables and random shocks and empirically analyzed the performance of 78 QDII funds in China in 2013. Zhou et al. [16] proposed a multiperiod portfolio optimization model by taking transaction costs into consideration and constructed the definition of multiperiod portfolio efficiency and the corresponding nonlinear model based on the real frontier. Then, they used the DEA model to approximate the frontier and estimate the efficiencies of the multiperiod portfolio with transaction costs. Gu and Ning [17] used the DEA evaluation method, combined with China's national conditions, to build a set of evaluation system for the performance of mixed open-end funds, and they proposed suggestions for improving the performance of domestic mixed open-end funds. Hu [18] introduced the time dimension into the performance evaluation of funds from a dynamic perspective, established the Malmquist-DEA index evaluation model of panel data, and empirically examined the performance characteristics of China's stock open-end funds. Basso and Funari [19] used the DEA model to evaluate fund performance and introduced more DEA fund performance indicators and conducted empirical tests. Galagadera and Silvapulle [20] used the DEA method to evaluate the performance of 257 Australian mutual funds from 1995 to 1999, and they conducted logistic regression analysis to examine the dependence of efficiency on several influence factors of mutual funds. Basso and Funari [21] studied the application of fund size in performance evaluation of mutual funds, and they analyzed the relationship between performance scores and fund size, and they constructed the corresponding DEA model to empirically analyze a set of European equity mutual funds. Premachandra et al. [22] summarized the existing research on the performance evaluation of funds and proposed a two-stage DEA model. Then, they decomposed the overall efficiency into two parts and evaluated the relative performance of 66 large mutual fund in the U.S. from 1993 to 2008. Besides, they provided improvement benchmarks for inefficient funds. Liu et al. [15] demonstrated that with adequate data size, the classic DEA models can effectively sample portfolios to take into consideration sufficient diversification, and thus, they can be used as an effective tool in computing the portfolio efficiency for their performance assessments.However, the above-mentioned evaluation methods based on the mean-variance model proposed by Markowitz and its extensions may not be suitable here. When selecting funds, besides the return and risk, investors may also pay attention to other aspects, such as lock-up period [23, 24]. Because of different investor’s preferences, the selection criteria for other aspects will be different. Specifically, for the CSTMFs, in this paper, we consider the impact of the lock-up period on investors. The funds have lock-up period, which will lead to opportunity cost. The economic definition of opportunity cost refers to the benefits an individual, investor, or business misses out on when choosing one alternative over another. The opportunity cost caused by the lock-up period of the funds here refers to the market opportunity that investors lose during the lock-up period. Specifically, because of the lock-up period, investors' capital liquidity is reduced, and investment funds cannot be redeemed in time, which may result in investors losing opportunities for other investment options.Opportunity cost is represented by liquidity risk in finance. Taking commercial bank as an example, Yao [25] defined liquidity risk as a possibility of loss to the profit of commercial banks and threat to their existence due to the lack of sufficient cash to pay the debt, guarantee customers to withdraw deposits, and meet loan demand. Liquidity risk includes the financing risk and market liquidity risk [26]. Financing liquidity risk refers to the risk when a commercial bank is affected in its daily operations or financial situation if it needs to meet the funding needs. Market liquidity risk refers to the risk of a commercial bank losing its market value when it sells its assets because of market fluctuations or insufficient depth.A lot of studies have been done on the causes and countermeasures of the liquidity risk of commercial banks. For example, Diamond's [27] model framework believes that the bank converts depositors' current deposits into long-term loans of borrowers so that banks play a role of maturity transformation in the above process of liquidity transformation. Therefore, the banks’ creation of liquidity and risk is mainly because of the mismatch between deposit and loan maturity. Allen and Douglas [28] proposed that commercial banks provide liquidity mainly by obtaining short-term deposits and providing long-term loans. It makes them face the risk of deposit being withdrawn at any time. Therefore, they believed that the commercial bank's own operating system led to the emergence of liquidity risk. Lev [29] analyzed the liquidity risk problem based on the information asymmetry model, confirming that there were the differences of interests and their own conditions that resulted in the problem of information asymmetry between the investors and banks and brought liquidity risks to the banks. There are also related pieces of literature, such as Xu and Wang [30] and Zhong [31], who studied the relationship between the term structure of loan and the liquidity risk of commercial banks. There are many pieces of research on the liquidity risk measurement of commercial banks. The common measurement indicators of liquidity risk include: loan-to-deposit ratio, cash-to-assets ratio, and liquidity ratio [32]. For example, the National People's Congress enacted the Law of the People's Republic of China on Commercial Banks in 1995, which clearly states, “The ratio of loan balance to deposit balance shall not exceed 75%.”Since the investors of CSTMFs only consider their own liquidity risk after investment, investors investing in the funds can be analogized as commercial bank loans. The following parts of this paper focus on the measurement of liquidity risk of commercial bank loans in recent years.The 2008 financial crisis brought more stringent regulation to liquidity. Basel III, released on December 16, 2010, introduced liquidity coverage ratio (LCR) and net stable funding ratio (NSFR) as the liquidity indicators. The liquidity coverage ratio (LCR), as a measure indicator of asset liquidity, refers to the ratio of the stocks of high-quality liquid assets owned by commercial banks to the total net cash outflows of 30 working days under the conditions of significant liquidity pressure. The net stable funding ratio (NSFR) is a measure of the stability of funds and is defined as the ratio between the available stable funding (ASF) and the required stable funding (RSF). The NSFR aims to make commercial banks obtain funds with a more stable financing method. The LCR standard is not less than 100%, and the NSFR standard is greater than 100%. In May 2018, the China Banking and Insurance Regulatory Commission issued the “Measures for Liquidity Risk Management of Commercial Banks,” which was revised twice, in 2015 and 2017, and it introduced three new liquidity risk supervision indicators (net stable fund ratio, liquidity matching rate, and high-quality liquidity asset adequacy ratio), which were implemented from July 1, 2018.Few pieces of research currently evaluate such CSTMFs that have lock-up period leading to liquidity risks. From the analysis of the above-mentioned related commercial banks' liquidity problems, it can be known that the loan liquidity is mainly controlled by the ratio of ASF, and the screening of stable funds is mainly determined by the term and reputation of the loans (see definition of stable funds [33]). Because of the development of digital technology and information technology, effective management is feasible [34–38].It indicates that the focus of the loan liquidity risk measurement is the length of money recovery and the creditworthiness of the borrower. Hence, the term of the loan and reputation have a crucial impact on the control of liquidity risk. The money that investors invest in CSTMFs can be analogized to commercial bank loans, and no defaults of the funds have been found to break the contract. This paper considers the length of the lock-up period of the funds as the liquidity risk indicator in the DEA model. Although some scholars have suggested to evaluate hedge fund performance using the lock-up period and minimum investment amounts as inputs [23, 24], no further empirical work has been found.To make the later evaluation work clear and easy to understand, in this paper, we define the following closed-end fund chain: in the N-day investment period, the former (N-S + 1) trading days are the start date of a fund with a lock-up period of S days. The entire fund is formed by a closed-end fund of each period in N days, which is called a closed-end fund chain. Therefore, the purpose of this paper is to evaluate the entire closed-end fund chain.One of the key issues in evaluating the efficiency of CSTMF with the characteristics of closed-end fund chains is how to deal with the lock-up period. When investors consider whether to invest in a CSTMF, they must not only consider their return but also consider their lock-up period to rationally arrange the proportion of fund allocation, so that the liquidity risk can be reduced under the guarantee of return. At present, the lock-up period and liquidity risk have not been considered when evaluating the funds using the DEA method. In the following sections, we will apply the DEA method to evaluate the funds based on the lock-up period.In this work, we try to find the DEA models with reasonable discrimination power, which are consistent with the evaluation of CSTMF and universally applicable through empirical analysis, by comparison of various DEA models. We conduct empirical analysis on the DEA models using the data of 28 funds to provide useful reference for investors to invest in the funds.


## 3. DEA Model and Preliminary Evaluation Results of CSTMFs

As for the selection of indicators for the efficiency evaluation of CSTMFs, this paper refers to the literature of fund evaluation [11], taking return as a desirable output indicator and variance as an undesirable output indicator (as an input indicator in model calculation). According to the characteristics of the closed-end fund chain, based on the lock-up periods of sample funds, for each CSTMF, the paper regards each fund of the same period (a period per lock-up period) as different funds. Hence, we calculate the mean of daily return and variance of each fund with the moving average method in the sample time period. Then, by calculating the mean of the return and variance of each fund, which is included in each CSTMF during the time period, we use them as the return indicator and risk indicator of each CSTMF. In addition, we introduce the lock-up period of the funds as an input indicator, which is used to measure liquidity risk.Firstly, we apply the BCC model, which was initially proposed by Banker, R. D., Charnes, A. and Cooper, W. W., to evaluate the efficiency of the funds [15]. Since the funds’ investors can only control the input (lock-up period), the BCC input-oriented model is selected. The classic BCC input-oriented model is as follows [39]:(1)Minθ−εS1−+S2−+S+,Subject to:∑j=128vjλj+S1−=θ​v0,,∑j=128tjλj+S2−=θ​t0,∑j=128ejλj−S+=e0,∑j=128λj=1,θ≤1,λj,S1−,S2−,S+≥0,j=1,⋯,28,where ε represents the non-Archimedean infinitesimal, θ represents the score of efficiency, vj,tj is the input vector, vj and tj represent variance and lock-up period of jth DMU, respectively, ej is the output that represents the mean of the return of jth DMU, S1−, S2−, and S+ are the slack variables of inputs and output, respectively, λ1,⋯,λ28 are unknown variables, v0, t0, and e0 represent the variance, lock-up period, and return of the DMU0 being evaluated, respectively.It is worth noting that the lock-up period of the funds is in the majority of 7 days, 14 days, 30 days, and 60 days, and the sample size of other types of funds is too small. To make the evaluation results comparable, we mainly take the funds with these four kinds of lock-up periods as the research objects and select 28 funds with such four kinds of lock-up periods for evaluation. Among them, there were 8 funds with 7 days of lock-up period, 6 funds with 14 days of lock-up period, 8 funds with 30 days of lock-up period, and 6 funds with 60 days of lock-up period.We could observe the following from the calculation result (Table3) with the application of the BCC model (1): the classical BCC input-oriented model has insufficient discriminability. Hence, we further select the BCC input-oriented model with the Russell measure.(2)Minθ1+θ22−εS+,Subject to:∑j=128vjλj=θ1v0,∑j=128tjλj=θ2t0,∑j=128ejλj−S+=e0,∑j=128λj=1,0≤θ1≤1,0≤θ2≤1,λj,S+≥0,j=1,⋯,28,where θ1 and θ2 represents the ratio of shrink or expansion along the direction of variance and lock-up period, respectively. Therefore, when compared with model (1), model (2) is more flexible and contains more information.Table 3
Calculation results of BCC models (1) and (2).
Fund codeModel (1)Model (1) slacksModel (1)Model (2)Model (2) slackModel (2) Ranking(Lock-up period)EfficiencyRankingEfficiency380007 (7)0.9990.02070.644—9550012 (7)0.9990.15990.5320.04313471007 (7)1.000—11.000—1660016 (7)0.9990.215120.5030.00115217025 (7)0.9990.11980.5600.00811485118 (7)0.9990.180100.5210.05914690012 (7)0.9990.183110.5370.06912000037 (7)1.000—11.000—1470014 (14)0.576—150.567—10380001 (14)0.5000.125180.3040.01718530014 (14)1.000—11.000—1202303 (14)0.944—130.910—8000322 (14)0.5000.133160.2700.01920485120 (14)0.5000.118170.260—21000089 (30)1.000—11.000—1110050 (30)0.933—140.916—7001057 (30)0.2860.031220.177—23270046 (30)1.000—11.000—1090023 (30)1.000—11.000—1470030 (30)0.2580.033230.2520.02122380010 (30)0.2330.013240.149—25530028 (30)0.4710.142200.4690.00416530029 (60)0.1170.121270.0700.01226000791 (60)0.214—250.162—24485122 (60)0.1170.095260.066—27202305 (60)0.1170.027280.060—28110052 (60)0.350—210.284—19470060 (60)0.47314.385190.3530.02617(Data from Choice database, which is developed by East Money Information Co., Ltd.).From the calculation results of model (2), it can be seen that among all the funds, 6 funds are DEA efficient, of which 2 funds have a 7-day lock-up period, 1 has a 14-day lock-up period, and 3 have 30-day lock-up period. The difference of efficiency scores between efficient units and inefficient units is large. The funds with lower rankings mostly have a longer lock-up period, for example, 4 funds with 60-day lock-up period ranked after 20. By comparing the original data, it is found that although the return gap between the funds is not large, the variance and the length of the lock-up period are large. The difference in efficiency scores is mainly affected by the variance and the length of the lock-up period.At the same time, we apply the traditional mean-variance model to calculate the efficiency of the above-mentioned funds to present the impact on the efficiency scores of the funds before and after considering the liquidity risk. The results are shown in Table4.Table 4
Comparison of calculation results of traditional mean-variance model and model (2).
Fund codeEfficiency of mean-variance modelRanking of mean-variance modelEfficiency of model (2)Ranking of model (2) (Lock-up period)(Lock-up period)(Lock-up period)380007 (7)0.140120.6449550012 (7)0.035180.53213471007 (7)0.55861.0001660016 (7)0.003260.50315217025 (7)0.067140.56011485118 (7)0.023200.52114690012 (7)0.041170.53712000037 (7)0.47381.0001470014 (14)0.340100.56710380001 (14)0.061160.30418530014 (14)1.00011.0001202303 (14)0.066150.9108000322 (14)0.023210.27020485120 (14)0.011240.26021000089 (30)1.00011.0001110050 (30)0.83350.9167001057 (30)0.002270.17723270046 (30)1.00011.0001090023 (30)1.00011.0001470030 (30)0.151110.25222380010 (30)0.031190.14925530028 (30)0.47190.46916530029 (60)0.013220.07026000791 (60)0.012230.16224485122 (60)0.008250.06627202305 (60)0.002280.06028110052 (60)0.140130.28419470060 (60)0.47370.35317It can be seen from Table4 that after considering the lock-up period as the input variable, the efficiency rankings of the funds have changed significantly, especially the funds with a 7-day lock-up period and 60-day lock-up period. After considering the lock-up periods, the rankings of the funds in the 7-day lock-up period rose by an average of nearly 6, while the rankings of the funds in the 60-day lock-up period fell by an average of nearly 4.


## 4. Improvement Strategy of Investment

This section aims to help investors improve their investment strategies. For this purpose, the concept of DEA projection is first introduced: if the DMU is not efficient, then the original input vector and output vector can be adjusted by solving the corresponding DEA model to make them efficient. The adjusted point is called the “projection” of the DMU on the production frontier [40].The projection formula of model (2) used in this paper is as follows:(3)v^0=θ10v0,t^0=θ20t0,e^0=e0+s+0,where θ10, θ20, and s+0 represent the optimal solutions of the model (2), X^0=v^0,t^0T, Y^0=e^0, and X^0,Y^0 be the projection vectors of DMU being evaluated on the DEA efficient frontier. The projection results are shown in Table 5.Table 5
The projection results of model (2).
Fund codeEfficiency of model (2)SlacksRankingBenchmarks(Fund code (lock-up period))(Lock-up period)471007 (7)000037 (7)530014 (14)000089 (30)270046 (30)090023 (30)380007 (7)0.644—90.1660.834————550012 (7)0.5320.043131.000—————471007 (7)1.000—11.000—————660016 (7)0.5030.001151.000—————217025 (7)0.5600.008111.000—————485118 (7)0.5210.059141.000—————690012 (7)0.5370.069121.000—————000037 (7)1.000—1—1.000————470014 (14)0.567—100.7690.231————380001 (14)0.3040.017181.000—————530014 (14)1.000—1——1.000———202303 (14)0.910—8—0.696—0.180—0.124000322 (14)0.2700.019201.000—————485120 (14)0.260—210.9680.032————000089 (30)1.000—1———1.000——110050 (30)0.916—7———0.586—0.414001057 (30)0.177—23—0.932———0.068270046 (30)1.000—1————1.000—090023 (30)1.000—1—————1.000470030 (30)0.2520.021221.000—————380010 (30)0.149—250.1050.895————530028 (30)0.4690.00416——1.000———530029 (60)0.0700.012261.000—————000791 (60)0.162—24——0.8170.183——485122 (60)0.066—270.8460.154————202305 (60)0.060—280.0290.971————110052 (60)0.284—19——0.3780.622——470060 (60)0.3530.02617——1.000———As can be seen from Table5, the benchmarks of most funds are composed of the funds with a lock-up period of 7 days and 14 days. For example, the lock-up period of the fund 485122 is 60 days, and its benchmarks consist of funds 471007 and 000037 with a lock-up period of 7 days. In other words, the portfolio return level consisting of funds with shorter lock-up period is not lower than the longer lock-up period. It will help reduce liquidity risk. Since the investors of the funds mainly focus on liquidity risk, they can formulate specific plans to improve investment, which is based on the benchmarks. Consider fund 485122, for example. The projection point is equivalent to investing 84.6% in 471007 and the remaining 15.4% in 000037, and the value of corresponding slacks is zero. Hence, the projection and 485122 have the same mean return. This portfolio has a lower variance, which clearly largely reduces the liquidity risk of investors. Specifically, compared to the investor's total investment on the fund 485122, the portfolio of 84.6% of fund 471007 and 15.4% of fund 000037 can achieve the same level of return with lower variance and lower liquidity risk. Obviously, investors should give priority to investing in this portfolio. Hence, the improvement plan of investment derived from the benchmarking analysis of model (2) in Table 5 is reasonable.Below, we will further discuss whether it is possible to further reduce the liquidity risk under the premise of given returns and variances. Referring to the DEA models that can keep the material balance conditions proposed by Rødseth [41] in the measurement of environmental efficiency, the following model is introduced at first:(4)Min t,Subject to:∑j=128vjλj+S−=v0,∑j=128ejλj−S+=e0,∑j=128tjλj+S∗−=t,∑j=1nλj=1,λj,S−,S+,S∗−≥0,j=1,⋯,28,where vj represents the variance of jth DMU, ej represents the return of jth DMU, tj represents the length of the lock-up period of jth DMU, t represents the minimum length of the lock-up period that can be achieved, S−, S+, and S∗− are the slack variables of variance, return, and length of the lock-up period, respectively, v0 and e0 represent the variance and return of the DMU0 being evaluated, respectively, and model (4) is applied to the fund evaluation. Specifically, this model is used to study how much liquidity risk can be further reduced under the given current return and variance levels.As Table6 shows, at the given return and variance level, the liquidity risk of the funds with a relative longer lock-up period can be further reduced. For example, the lock-up period of fund 110050 is 30 days. We apply model (2) to calculate and obtain its benchmarks: two funds with the same lock-up period of 30 days, which are fund 000089 and fund 090023, respectively. According to Table 6, using model (4) to calculate, there is a special fund with a lock-up period of 7 days in the benchmarking combination. In other words, according to the calculation result of model (2), compared with the investor's total investment on fund 110050, the portfolio with 58.6% of the fund 000089 and 41.4% of the fund 090023 has the same level of return and smaller variance, however, it does not play a role in reducing liquidity risk. If we take the benchmarks of fund 110050 calculated by model (4), investors can choose the portfolio consisting of 31.8% of the fund 000089, 53.7% of the fund 090023, and 14.5% of the fund 000037. Compared with totally investing in fund 110050, this portfolio has the same level of return and variance, and since this portfolio contains one fund with a lock-up period of 7 days (000037), it has a lower liquidity risk than the former portfolio. Hence, the investment improvement plan obtained from the benchmarking analysis results of model (4) in Table 6 can reasonably further reduce liquidity risk for the investor.Table 6
Calculation and projection results of model (4).
Fund codebBenchmarks (fund code (lock-up period))(Lock-up period)380007 (7)550012 (7)471007 (7)660016 (7)217025 (7)485118 (7)690012 (7)000037 (7)530014 (14)000089 (30)270046 (30)090023 (30)380007 (7)7.0000.0970.0050.0160.0110.0120.0060.0080.846----550012 (7)7.0000.1310.1190.1430.0420.1220.1130.1340.196----471007 (7)7.000--1.000---------660016 (7)7.0000.1060.1150.1060.1420.1080.1210.1210.181----217025 (7)7.0000.0930.0890.173-0.1080.0740.1050.359----485118 (7)7.0000.1300.1250.1330.0710.1230.1240.1370.155----690012 (7)7.0000.1450.1150.1540.0280.1250.1030.1310.199----000037 (7)7.000-------1.000----470014 (14)7.0000.0270.0060.152-0.0210.004-0.789----380001 (14)7.0000.0840.1020.155-0.1130.0890.1200.336----530014 (14)14.000--------1.000---202303 (14)12.868-------0.745-0.089-0.166000322 (14)7.0000.1090.1220.1240.0750.1180.1230.1330.196----485120 (14)7.0000.1090.1170.1100.1190.1110.1210.1230.190----000089 (30)30.000---------1.000--110050 (30)26.658-------0.145-0.318-0.537001057 (30)8.569-------0.932---0.068270046 (30)30.000----------1.000-090023 (30)30.000-----------1.000470030 (30)7.0000.0810.0260.2310.0010.0640.0110.0370.550----380010 (30)7.0000.0610.0030.0090.0090.0080.0040.0050.901----530028 (30)11.682-------0.3720.6090.018--530029 (60)7.0000.1060.1200.1140.1100.1130.1250.1280.184----000791 (60)10.808-------0.834---0.166485122 (60)7.0000.1030.1050.0990.1200.1000.1090.1090.255----202305 (60)7.0000.0150.0010.0030.0030.0020.0010.0020.973----110052 (60)14.428-------0.677---0.323470060 (60)7.000--0.124----0.875----


## 5. Sensitivity Analysis of Decision

To further discuss the rationality of investment strategy with shorter lock-up period under given return and variance level, in the following, we will analyze the sensitivity of the DMU with efficiency scores greater than 0.9 in model (2), and the specific method is to reduce 5% of the respective return and observe the change in efficiency scores. The results are shown in Table7.Table 7
Sensitivity analysis results of model (2).
Fund code (lock-up period)Efficiency of model (2)Efficiency ranking of model (2)Original efficiency of model (2)Original ranking of model (2)(After decreasing 5% of return)(After decreasing 5% of return)471007 (7)1.00011.0001000037 (7)1.00011.0001530014 (14)1.00011.0001202303 (14)0.265200.9108000089 (30)0.143251.0001110050 (30)0.296180.9167270046 (30)0.398161.0001090023 (30)0.567101.0001It can be seen from Table7 that the efficiency scores of the two funds with a 7-day lock-up period and one fund with 14-day lock-up period is still 1, whose returns are reduced by 5%. However, the efficiency scores of the three funds with a 30-day lock-up period is significantly reduced. Therefore, the funds with a shorter lock-up period have more stable efficiency scores and are not easily affected by the changes of return.Meanwhile, according to the calculation results of model (2) in Table5, 6 funds with the efficiency scores greater than 0.9 are selected with the 14-day lock-up period and the 30-day lock-up period, and the respective outputs are reduced by 5%. Then, we calculate its benchmarks using model (4). The results of the benchmarks are shown in Table 8.Table 8
Sensitivity analysis results of model (4).
Fund code (lock-up period)Benchmarks (fund code (lock-up period))380007(7)550012(7)471007(7)660016(7)217025(7)485118(7)690012(7)000037(7)530014(14)110050(30)090023(30)530014(14)--------1.000--202303(14)0.0460.0380.0400.0440.0390.0380.0370.718---000089(30)0.0140.0030.0060.0060.0050.0020.0020.962---110050(30)-------0.830--0.170270046(30)0.0130.0050.077-0.0040.0050.0040.893---090023(30)-------0.308-0.692-It can be seen from Table8 that if the returns are reduced by 5%, most funds with longer lock-up period can be replaced by several funds with shorter lock-up period. For example, the fund 000089 with a lock-up period of 30 days can be seen from Table 6 that its lock-up period cannot be further reduced under original return. When the return is reduced by 5%, it can be completely replaced by the funds’ portfolio with a 7 day-lock-up period.From the above analysis, it can be seen that compared with the funds with longer lock-up periods, the funds with shorter lock-up periods have more stable efficiency scores, and the portfolio consisting of the funds can significantly reduce the liquidity risk of investors.


## 6. Conclusion

CSTMFs have different lock-up periods, and investment in the funds with different lock-up periods leads investors to face different liquidity risks. In this paper, we use the lock-up period as a method to measure the liquidity risk for the funds with different lock-up periods. 28 funds with four different kinds of lock-up periods are selected for empirical analysis, and the mean of the return, variance, and lock-up periods are used as the evaluation indicators. The BCC-DEA input-oriented model with the Russell measure is used to evaluate the funds. At the same time, the results obtained by applying this model are compared with the results of the traditional mean-variance model. The results show that the lock-up periods have a greater impact on ranking, which have important reference value for investors to choose the investment strategy. To further study the investment strategy, we present a benchmark analysis of the evaluation results and use the relevant models to give suitable portfolios to reduce the liquidity risk with the given return and variance level. At the same time, combining with the sensitivity analysis, we present valuable investment suggestions.Through the studies above, the following conclusions are obtained: (1) the performance of most of the inefficient funds with longer lock-up period can be improved through portfolios consisting of some funds with shorter lock-up period. (2) For some inefficient funds with shorter lock-up period that cannot be improved through a portfolio consisting of the funds with the same or shorter lock-up periods, the current return and variance levels can be achieved by a portfolio consisting of some funds with shorter and longer lock-up periods. (3) Among the funds with higher efficiency scores, funds with shorter lock-up period have higher stability. In addition, the funds with longer lock-up periods that slightly reduced the return are easily replaced by the portfolio with a shorter lock-up period, which can significantly reduce liquidity risk without affecting the levels of return and variance. Consequently, given the conclusions above, this research suggests that including more funds with a shorter lock-up period in the portfolio of the investor will help reduce liquidity risk. Meanwhile, the standard of expected return and variance will not be impacted significantly for the portfolio of the investor.Since some markets with high liquidity, such as stock market, future market, and so on, do not possess explicit liquidity structure, the further related studies will focus on the investment ways with known liquidity structure. It is possible that the framework and method in this paper could be utilized to evaluate other types of funds with more complex structure of lock-up period, and the measure of liquidity risk might be adjusted. It is beyond the scope of this paper and will be addressed elsewhere.



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*Source: 5106350-2022-05-14.xml*