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Guys I'm stuck with a problem... Consider the portfolio choice problem of a risk-averse individual with a strictly increasing utility function. There is a single risky asset, and a risk free asset. Formulate an investor's choice problem and comment on the first-order conditions. What is the minimum risk premium required to induce the individual to invest all his wealth in the risky asset?

Since we know that the choice problem an investor must solve can be expressed as:

$\max_{a} \mathcal{E}[U(Y_1)] = \max \mathcal{E}[U(Y_0(1+r_f)+a(r_i-r_f))]$

Where $U( )$ is the utility of money function and $\mathcal{E}$ the expectation operator. Moreover, $Y_1$ is the wealth at time 1 whereas $Y_0$ the wealth at time 0, whereas $a$ is the portion that should be invested in the risky asset.

By differentiating into the expectation we can solve the maximization problem and we have: $\mathcal{E}[(U'(Y_0(1+r_f)+a(r_i-r_f))(r_i-r_f)]=0$

The FOC that solves the problem, that is written on the solution of the exercise, is

$\mathcal{E}[U'(Y_0(1+r_i)(r_i-r_f))]\geq0$

Since $a=1$. I don't get why the FOC is this... Can anybody explain me better?

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Try to formulate the problem as a constrained optimization problem, and examine the KKT (Karush-Kuhn-Tucker) complementary slackness conditions.

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