A currency investment strategy is used to maximize the return on a portfolio of foreign currencies relative to any appreciation of the corresponding foreign exchange rates. Given the uncertainty in the estimation of the future currency values, we employ robust optimization techniques in order to maximize the return on the portfolio for the worst-case foreign exchange rate scenario. Currency portfolios differ from stock-only portfolios in that a triangular relationship exists among foreign exchange rates to avoid arbitrage. Although the inclusion of such a constraint in the model would lead to a nonconvex problem, by choosing appropriate uncertainty sets for the exchange and the cross exchange rates we obtain a convex model that can be solved efficiently. Alongside robust optimization, an additional guarantee is explored by investing in currency options in order to cover the possibility that foreign exchange rates will materialize outside the specified uncertainty sets. We present numerical results that show the relationship between the size of the uncertainty sets and the distribution of the investment among currencies and options, and the overall performance of the model in a series of backtesting experiments.