Consider the return generating process of three assets X, Y, and Z. Returns are explained by two uncorrelated factors, f1,t, the consumer confidence and f2,t, inflation:

RX,t = 0.05 - f1,t 2f2,t; RY,t = 0.10 - 4f1,t; RZ,t = -0.008 - f1,t + 2f2,t

As a portfolio manager, you would like to replicate the performance of the well-diversified TVX small cap index, which has a factor loading of two with respect to the first factor and a factor loading of three with respect to the second factor. Assume also that the standard deviation of the first factor is 4% and of the second factor is 9%. Finally, the risk-free rate is 2%.

1. Find the expected returns and the factor betas of the three assets. (6 marks)

2. Determine the tracking portfolio of the TVX small cap index using the three assets above. Find its expected return and standard deviation. (5 marks)

3. Find the composition of the first pure factor portfolio, which is by construction a well-diversified portfolio. Write its factor equation and estimate its expected return, risk premium, and risk. Interpret your answer. (4 marks)

4. Assume that the alpha of the TVX small cap index is 2%. Determine if there is any arbitrage opportunity. If yes, provide full details of the arbitrage strategy (using all available assets) and calculate the arbitrage profit. (5 marks)