Thou shalt buy and hold

An investor holding a stock needs to decide when to sell it over a given investment horizon. It is tempting to think that she should sell at the maximum price over the entire horizon, which is however impossible to achieve. A close yet realistic goal is to sell the stock at a time when the expected relative error between the selling price and the aforementioned maximum price is minimized. This problem is investigated for a Black-Scholes market. A stock 'goodness index' α, defined to be the ratio between the excess return rate and the squared volatility rate, is employed to measure the quality of the stock. It is shown that when the stock is good enough, or to be precise when α≥1/2, the optimal strategy is to hold on to the stock, selling only at the end of the horizon. Moreover, the resulting expected relative error diminishes to zero when goes to infinity. On the other hand, one should sell the stock immediately if α≤0. These results justify the widely accepted financial wisdom that one should buy and hold a stock - if it is good, that is.