Conic financial theory is based on the existence of a two price economy (bid and ask prices) for market valuation and risk measures. We study the portfolio selection problem within the framework of conic finance. With this approach we obtain a risk level from the market and select the weights of each asset to maximise the bid price. We compare different distortion measures to obtain the expected future cashflows, and we implement several algorithms to calibrate the model and solve the problem under different constraints on size and weights.