  In this report we derive the strategic (deterministic) allocation to bonds
and stocks resulting in the optimal mean-variance trade-off on a given
investment horizon. The underlying capital market features a mean-reverting
process for equity returns, and the primary question of interest is how
mean-reversion effects the optimal strategy and the resulting portfolio value
at the horizon. In particular, we are interested in knowing under which
assumptions and on which horizons, the risk-reward trade-off is so favourable
that the value of the portfolio is effectively bounded from below on the
horizon. In this case, we might think of the portfolio as providing a
stochastic excess return on top of a "guarantee" (the lower bound).
  Deriving optimal strategies is a well-known discipline in mathematical
finance. The modern approach is to derive and solve the Hamilton-Jacobi-Bellman
(HJB) differential equation characterizing the strategy leading to highest
expected utility, for given utility function. However, for two reasons we
approach the problem differently in this work. First, we wish to find the
optimal strategy depending on time only, i.e., we do not allow for dependencies
on capital market state variables, nor the value of the portfolio itself. This
constraint characterizes the strategic allocation of long-term investors.
Second, to gain insights on the role of mean-reversion, we wish to identify the
entire family of extremal strategies, not only the optimal strategies. To
derive the strategies we employ methods from calculus of variations, rather
than the usual HJB approach.
