Today I discussed the notions ¡­

Today I discussed the notions of return, mean return, and risk-premium ("mean return - interest rate") of individual assets and portfolios. We showed that the Q-expectation of the return equals the interest rate r for any risk-netural probability measure Q. To quantify risk in a portfolio the concept of its beta was introduced. I showed that the ratio of the risk premium of an arbitrary portfolio and the risk premium of the "benchmark" portfolio (the state price density) is equal to the beta. Finally, I introduced the basic portfolio optimization problem of maximizing expected utility of terminal wealth, and proved that the existence of an optimal solution (trading strategy) to this portfolio problem implies the existence of a risk-netural probability measure (i.e., the absence of arbitrage). Moreover, such a measure comes out explicitly (in terms of the marginal utility of terminal wealth) thanks to the first order conditions at a maximum.