We propose a new approach in the estimation of the optimal hedge ratio that allows the hedge ratio to vary over time but without the necessity of frequently rebalancing the portfolio. We apply this in the context of the US and UK equity markets using weekly spot share prices and future share prices during the period January 5, 1999 to September 29, 2009. Our method is to test for cointegration in the presence of two potentially unknown structural breaks by determining the timing of each via the underlying data. The empirical findings reveal that the spot and future prices are strongly cointegrated in each market. The estimated parameters disclose that the optimal hedge ratio is not constant in case of the US and the UK. We find one negative and one positive shift in the optimal hedge ratio in the US. However, we find only one significant and positive shift in the optimal hedge ratio in the UK. The implication of these findings from the perspective of both investors as well as policy makers is elaborated on in the main text.