Investor Choice with Risky and Risk-Free Assets

Summary

The document frames a portfolio choice problem for a risk-averse investor choosing how much initial wealth to place in one risky asset, with the remainder earning a risk-free return. It writes expected utility as a function of terminal wealth and gives the interior first-order condition: expected marginal utility of wealth times the risky asset’s excess return equals zero.

The question concerns why a stated condition at the all-risky allocation is an inequality rather than an equality, and what risk premium would induce that allocation. The response points toward formulating the allocation as a constrained optimization problem and applying the Karush-Kuhn-Tucker complementary slackness conditions. It does not work through those conditions or derive a minimum premium. The setup also leaves assumptions about return distributions and feasible allocations unspecified, so it does not establish a numerical threshold or a general formula for one.

Key ideas

• An interior portfolio choice sets the expected marginal-utility-weighted excess return to zero.
• At the all-risky boundary, the optimality condition can differ from the interior equality.
• The allocation constraint makes Karush-Kuhn-Tucker conditions relevant at the boundary.
• The document does not derive the premium threshold needed for full investment in the risky asset.

Tags

# Investor choice problem


# Investor choice problem












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?

## Answer by Dimitris (score 3)

https://quant.stackexchange.com/a/19419

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

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