Bounds for Goal Achieving Probabilities of Mean-Variance Strategies with a No Bankruptcy Constraint

Scott, Alexandre; Watier, François

doi:10.4236/am.2012.312a278

Cited by 1 publication

(1 citation statement)

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“…Unfortunately, it does not seem possible to express the investor's stopping time rule as a simple first passage time for Brownian motion, and so it appears that one has to make do with Monte Carlo approximations, or bounds on the goal achieving probabilities. See Scott and Watier for more details on a numerical scheme suitable to deal with that case.…”

Section: Resultsmentioning

confidence: 99%

Goal achieving probabilities of cone‐constrained mean‐variance portfolios

Labbé

Watier

2013

Appl Stoch Models Bus & Ind

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In this paper, we establish closed-form formulas for key probabilistic properties of the cone-constrained optimal mean-variance strategy, in a continuous market model driven by a multidimensional Brownian motion and deterministic coefficients. In particular, we compute the probability to obtain to a point, during the investment horizon, where the accumulated wealth is large enough to be fully reinvested in the money market, and safely grow there to meet the investor's financial goal at terminal time. We conclude that the result of Li and Zhou [Ann. Appl. Prob., v.16, pp.1751-1763, (2006] in the unconstrained case carries over when conic constraints are present: the former probability is lower bounded by 80% no matter the market coefficients, trading constraints, and investment goal. We also compute the expected terminal wealth given that the investor's goal is underachieved, for both the mean-variance strategy and the aforementioned hybrid strategy where transfer to the money market occurs if it allows to safely achieve the goal. The former probabilities and expectations are also provided in the case where all risky assets held are liquidated if financial distress is encountered. These results provide investors with novel practical tools to support portfolio decision-making and analysis.Appl. Stochastic Models Bus. Ind. 2014, 30 544-572 C. LABBÉ AND F. WATIER Li and Zhou [7] compute this probability and establish the following astonishing result that they call the 80% rule: the chances of achieving the goal when using the switch-when-safe strategy are at least 80% (whereas they might be as low as 50% with the optimal mean-variance strategy). The 80% lower bound is universal, in the sense that it holds no matter the market coefficients, the length of the investment horizon, or the target.A very natural question is whether the remarkable 80% rule carries over, for example, when there are trading constraints and/or within the frame of an incomplete market. Scott and Watier [8] provide an affirmative answer when short-selling of stocks is prohibited (in a complete market). In this paper, we establish that it actually extends to more general portfolio constraints that encompass short-selling prohibition and incomplete markets. Precisely, the vector whose components are the amounts invested in each of the n stocks must remain within a given closed convex conic subset of R n . As it is of interest for an investor to know what to expect if the goal were unachieved, we take one further step and study the expected value of terminal wealth in that event.Another drawback of the mean-variance strategy (in its classical formulation) is that it does not preclude the wealth from becoming negative, or even arbitrarily small. Therefore, unless an investor is willing (or can afford) to go deep into debt, then the strategy, with or without the switching feature, may be impossible to implement in practice. In order to account for this kind of risk, let us say that the investor falls in financial distress if at some point his wealth ...

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Section: Resultsmentioning

confidence: 99%