3,731 research outputs found
Continuous-time mean-variance efficiency: the 80% rule
This paper studies a continuous-time market where an agent, having specified
an investment horizon and a targeted terminal mean return, seeks to minimize
the variance of the return. The optimal portfolio of such a problem is called
mean-variance efficient \`{a} la Markowitz. It is shown that, when the market
coefficients are deterministic functions of time, a mean-variance efficient
portfolio realizes the (discounted) targeted return on or before the terminal
date with a probability greater than 0.8072. This number is universal
irrespective of the market parameters, the targeted return and the length of
the investment horizon.Comment: Published at http://dx.doi.org/10.1214/105051606000000349 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Continuous-Time Markowitz's Model with Transaction Costs
A continuous-time Markowitz's mean-variance portfolio selection problem is
studied in a market with one stock, one bond, and proportional transaction
costs. This is a singular stochastic control problem,inherently in a finite
time horizon. With a series of transformations, the problem is turned into a
so-called double obstacle problem, a well studied problem in physics and
partial differential equation literature, featuring two time-varying free
boundaries. The two boundaries, which define the buy, sell, and no-trade
regions, are proved to be smooth in time. This in turn characterizes the
optimal strategy, via a Skorokhod problem, as one that tries to keep a certain
adjusted bond-stock position within the no-trade region. Several features of
the optimal strategy are revealed that are remarkably different from its
no-transaction-cost counterpart. It is shown that there exists a critical
length in time, which is dependent on the stock excess return as well as the
transaction fees but independent of the investment target and the stock
volatility, so that an expected terminal return may not be achievable if the
planning horizon is shorter than that critical length (while in the absence of
transaction costs any expected return can be reached in an arbitrary period of
time). It is further demonstrated that anyone following the optimal strategy
should not buy the stock beyond the point when the time to maturity is shorter
than the aforementioned critical length. Moreover, the investor would be less
likely to buy the stock and more likely to sell the stock when the maturity
date is getting closer. These features, while consistent with the widely
accepted investment wisdom, suggest that the planning horizon is an integral
part of the investment opportunities.Comment: 30 pages, 1 figur
Continuous time mean-variance portfolio selection with nonlinear wealth equations and random coefficients
This paper concerns the continuous time mean-variance portfolio selection
problem with a special nonlinear wealth equation. This nonlinear wealth
equation has nonsmooth random coefficients and the dual method developed in [7]
does not work. To apply the completion of squares technique, we introduce two
Riccati equations to cope with the positive and negative part of the wealth
process separately. We obtain the efficient portfolio strategy and efficient
frontier for this problem. Finally, we find the appropriate sub-derivative
claimed in [7] using convex duality method.Comment: arXiv admin note: text overlap with arXiv:1606.0548
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