In financial markets, short sellers will be required to post margin to cover possible losses in case the prices of the risky assets go up. Only a few studies focus on the optimization and performance evaluation of portfolios in the presence of margin requirements. In this paper, we investigate the theoretical foundation of DEA (data envelopment analysis) approach to evaluate the performance of portfolios with margin requirements from a different perspective. Under the mean-variance framework, we construct the optimization model and portfolio possibility set on considering margin requirements. The convexity of the portfolio possibility set is proved and the concept of efficiency in classical economics is extended to the portfolio case. The DEA models are then developed to evaluate the performance of portfolios with margin requirements. Through the simulations carried out in the end, we show that, with adequate portfolios, DEA can be used as an effective tool in computing the efficiencies of portfolios with margin requirements for the performance evaluation purpose. This study can be viewed as a justification of DEA into performance evaluation of portfolios with margin requirements.
In financial markets, the potential losses on short sales can be huge when the prices of the risky assets go up; therefore, in practice, the short sellers will be required to post margin or collateral to cover possible losses. In practice, margin requirements were firstly developed in the Securities Act of 1933, the Banking Act of 1933, and the Securities Exchange Act of 1934.
Most studies on margin requirements focus on margin calculation, relationship among margin requirements and stock prices, market volatility, investor behaviors, and so forth. For example, Lintner [
Portfolio optimization and performance evaluation, on the other hand, are also an important topic in financial studies. There are numerous researches on this point, such as Markowitz [
In this work, we consider the situation where short selling is allowed and deposit of certain percentage margin according to the shorting is needed to avoid loss by default of short seller. In this situation, we examine the DEA (data envelopment analysis) approach, a nonparametric technique proposed by Charnes et al. [
This paper is organized as follows. In Section
Consider the problem of selecting a portfolio from
In real investment, short selling is sometimes relevant for mutual funds, and even more for hedge funds. At the same time, in order to reduce the potential losses in case of default, short sellers are obliged to deposit and maintain a minimum amount of cash or discounted risky assets (margin requirements) with their broker-dealer. In particular, initial margin requirements set the minimum margin deposit, with which a position can be opened, and maintenance requirements set a floor, below which the margin deposit is not allowed to fall as long as the position remains open. Under Regulation
For simplicity, we do not distinguish the initial margin and maintenance margin in this work. We denote by
If all margins are required to be cash, no discounted risky assets are allowed; then, we can set
We assume that the investors are allowed to lend and to borrow the cash at the same interest rate
Thus, we can construct the following optimization model under the mean-variance framework proposed by Markowitz [
Accordingly, we can develop the following optimization model, which maximize the expected value subject to the constraints that the variance of the portfolio is no greater than the given level and the margin requirements are satisfied. Consider
If we can set
For the purpose of using DEA to evaluate the performance of portfolios with margin requirements, we must construct the theoretical foundation, that is, the convexity of the following portfolio possibility set with the strongly free disposability principle proposed by Liu et al. [
Note that model (
The portfolio possibility set in (
Let
For any
Since
For any
Thus,
Note that
Since
Thus,
Therefore,
It is evident that the frontier functions determined by models (
The frontier
The frontier
By using the data in Section
Exact frontiers of portfolios with different margin requirements.
In classical economics, performance is often evaluated by efficiency. Farrell [
Efficiencies with different orientations.
input-oriented radial measure:
output-oriented radial measure:
In order to define the efficiency of portfolio under the mean-variance framework, the portfolio frontier can be regarded as the production frontier and the portfolio possibility set is in nature the production possibility set in classical economics. And the performance of a portfolio can be measured via its closeness to the frontier. According to the extended strongly free disposability discipline of Liu et al. [
Since we have proved that the portfolio possibility set in (
DEA is a nonparametric approach based on the inputs-outputs analysis. It has been widely used to evaluate the performance of profit and nonprofit organizations [
Suppose that there are
Similarly, we can construct the following output-oriented DEA model:
The
Basically, the DEA frontiers will approximate the exact frontier of portfolios with margin requirements. When there exist sufficient portfolios, the efficiency scores of DEA models will be close to the real efficiencies of portfolios. Thus, there is a quite reasonable conclusion: to make DEA work, there have to be enough portfolios. The advantage of this approach is that there is no need to know the analytical expression of the exact frontier, so that one can handle variety of complex situations. Moreover, since DEA models are linear programming models, they will reduce the computation dramatically.
In this section, we will verify the validity of the above models with simulation. We select 150 stocks from the Chinese stock market. These monthly data are from January 2005 to January 2008. The descriptive statistics is listed in Table
Descriptive statistics of selected stocks.
Maximum | Minimum | Median | Mean | |
---|---|---|---|---|
Expected return | 0.0186 | 0.0007 | 0.0086 | 0.0087 |
Variance | 0.0113 | 0.0028 | 0.0072 | 0.0072 |
Covariance | 0.0113 | 0.0005 | 0.0040 | 0.0038 |
The exact frontiers with cash margin, cash, and discounted risky asset margin are shown in Figure
We then randomly generate proportions of risky assets to construct portfolio samples with margin requirements. The sample sizes are set to be 100, 500, 1000, and 2000, respectively. The top curve in Figure
Comparison of DEA frontiers with cash margin.
The top curve in Figure
Comparison of DEA frontiers with cash and discounted risky asset margin.
Table
Correlation of efficiencies with different sample sizes and margin requirements.
Sample size | 100 | 500 | 1000 | 2000 |
---|---|---|---|---|
Cash margin | 0.8778 | 0.9816 | 0.9818 | 0.9994 |
Cash and discounted risky asset margin | 0.7190 | 0.9761 | 0.9833 | 0.9967 |
In this work, we consider the situation that short selling is allowed and margin is required to avoid losses by default of short seller. Only a few studies focus on the optimization and performance evaluation of portfolios in this situation. This paper constructs the optimization model for portfolios with margin requirements and the corresponding portfolio possibility set. The convexity of the set is then proved, to ensure that the DEA frontier can approximate the exact frontier. The concept of efficiency in classical economics is extended to represent the performance of portfolios with margin requirements. The input-oriented and output-oriented DEA models are constructed, which greatly simplify the calculation. Finally, the simulation example shows that, with the increase in sample size, DEA frontier gradually approximates the exact frontier of portfolios with margin requirements, which illustrates the feasibility and effectiveness of the proposed approach.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research is supported by the Chinese National Natural Science Foundation (no. 71371067 and no. 70901024).