Interfaces with Other DisciplinesA minimax portfolio selection strategy with equilibrium
Introduction
One of the key issues facing an individual is how to allocate wealth among alternative assets. Almost all financial institutions have the same problem. The portfolio theory was build to solve this problem––portfolio selection problem.
The pioneering work on portfolio theory is, without doubt, the literature published by Markowitz [21], [22] in 1950s. He formulated the portfolio selection problem as a choice of the mean and variance of a portfolio of assets. He proved the fundamental theorem of mean–variance theory, namely for a given level of variance one maximizes expected return, and for a given level of expected return one minimizes variance. These two principles led to the formulation of an efficient frontier from which the investor could choose his/her preferred portfolio, depending on individual risk return preferences.
In the forty years since then, portfolio theory was then improved, enlarged, and completed in several directions. Let us mention two of these lines of research. One dealt with alternative portfolio selection models, for example, mean–semivariance model [20], [22], mean–absolute deviation model [18], [31], mean–variance–skewness model [8], [17], and minimax type models [5], [9], [11], [29], [33]. Another group of subsequent publications concerned the equilibrium of the mean–variance capital market, for instance, the most spectacular breakthrough works by Sharp [30], Lintner [19], and Mossin [23] under the name of CAPM, the famous review articles by Roll [27] and Ross [28], and some later works by Konno and Shirakawa [15], [16], Nielsen [24], Pardalos et al. [25], and Pardalos and Tsitsiringos [26].
The classic mean–variance model depends on inputs of the expectation and the variance of random returns of the underlying assets. In the real world, however, we do not know the exact values of the expectation and the variance and we can only estimate them. Recent research reveals that the investment characteristics of an optimal portfolio in Markowitz's mean–variance model can be very sensitive to estimation errors in problem inputs. In particular, it has been found that errors in the asset means can be much more damaging than errors in other parameters; see Best and Grauer [2], [3], [4], Chopra et al. [6], Chopra and Ziemba [7], and Hensel and Turner [12]. This stimulates us to consider such an optimal portfolio selection problem where the expected return of each underlying asset varies in an estimated interval while the covariance between any two asset returns is given and fixed. In this paper we propose a new minimax model to choose optimal portfolios for this problem. An optimal portfolio is defined as that one that maximizes the worst (minimally) possible expected rates of returns on portfolios. As showed by our results, this leads to an efficient frontier similar to that obtained by the mean–variance model of Markowitz. In addition, we establish a sufficient condition for the existence of a unique nonnegative equilibrium price vector of risky assets in the capital market under the minimax framework. Further, we derive an explicit formula for the equilibrium prices and some properties for the equilibrium.
It should be pointed out that our minimax portfolio selection model is different from other minimax models in existing literature, such as the representative ones mentioned above. Firstly, among them, some dealt with immunization problems [11], some minimized the maximum loss over all past observation periods for a given level of return [33], some are based on game theoretical approach [29], some used scenario analysis method [9], and some minimized the maximum risk of the individual assets [5]. To the best of our knowledge, none of them has a set up similar to our minimax type. Secondly, we are able to provide an analytic equilibrium price system while others are not.
The paper is organized as follows. In Section 2, the optimal portfolio selection problem is stated and a minimax model is established. The optimal strategy is found in Section 3. The existence of a nonnegative equilibrium price system is established in Section 4. Some properties of equilibrium are discussed in Section 5. Finally in Section 6, our concluding comments and discussions are presented.
Section snippets
Problem statement and modelling
We consider in this paper a capital market with n risky assets offering random rates of returns and a risk-less asset offering a fixed rate of return. An investor allocates his/her wealth among the n risky assets and the risk-less asset. As usual, we assume that no costs and no taxes are associated with transactions, all assets are infinitely divisible and have no limitation on short-sales, and the total value of the assets as well as the total number of shares of each asset remain constant
Optimal strategy
We begin this section with a known result from [10]. Lemma 3.1 Let X be a nonempty set and Y be a nonempty compact topological space. Let be lower semicontinuous on Y. Suppose that F is concavelike on X and convexlike on Y, that is to say:andThen
It is clear that, in our problem (Pω), is compact, f is
Equilibrium price system
Along the lines of Refs. [15], [16], in this section we discuss the existence and uniqueness and the explicit expression of an equilibrium price system (p1,p2,…,pn)T under which the market is cleared, i.e., the total demand matches the total supply of each asset.
Let us further assume that in the capital market
- (1)
the total number of shares of asset j is xj0 (j=1,2,…,n+1).
- (2)
there are m investors i=1,2,…,m with the following characteristics:
- •
Investors i=1,2,…,m have the same estimated intervals for the
- •
Some properties of equilibrium
In this section we use the results proved in the previous sections to establish some properties of equilibrium.
For the sake of convenience, letThen,Let Theorem 5.1 Separation property Let p be the price vector defined by (4.9). Then, after the transaction, each investor i holds a portfolio which is composed of the (n+1)th (riskless) asset and a nonnegative multiple λi of the market endowment x0=(x01,x02,…,x0n) of risky assets, where ∑i=1mλi=1; a
Conclusions and discussion
This paper has dealt with the problem of optimal portfolio selection and equilibrium when the target is to maximize the weighted criteria under the worst possible evolution of the rates of returns on risky assets. The resulting problem taken the form of minimax problem. The optimal “minimax portfolio” was analytically presented, which, in fact, can be obtained using linear programming technique. Furthermore, a sufficient condition for the existence and uniqueness of a nonnegative equilibrium
Acknowledgements
This research was supported in part by a Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 200267), a grant of the National Natural Science Foundation of China (No. 10171115), a “Tenth Five-Year Plan” project of Ministry of Education of China (No. 01JA630009), a grant of the Natural Science Foundation of Guangdong Province (No. 011193), a CERG grant (CityU1081/02E) and a research grant of City University of Hong Kong.
The authors would like to thank three
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