Optimal Allocation

With the utility function in hand, we can now discuss optimal decision-making. Utility can be linked to various choices, such as asset allocation or investment strategies. The goal is to maximize the expected utility of the decision. Clearly, decision-making affects wealth (wealth is a function of the decision process), and wealth, in turn, determines utility (utility is a function of wealth). Thus:

\[u(W) = u(W(x)),\]

where:

• $u(W)$ is the utility function
• $W(x)$ is the wealth function, which depends on the decision variable $x$

For a risk-averse individual, the utility function is increasing and concave. This means the highest utility is achieved at the first-order condition (FOC), where the first derivative of the utility function with respect to the decision variable is zero.

Optimal Portfolio Allocation Problem

Consider a specific wealth function:

\[\tilde W = (W_0 - \sum_{j=1}^{n} a_j) (1 + r_f) + \sum_{j=1}^{n} a_j (1 + \tilde r_j)\]

where:

• $W_0$ is initial wealth
• $a_j$ is the amount invested in asset $j$
• $W_0 - \sum_{j=1}^{n} a_j$ is the amount invested in the risk-free asset
• $r_f$ is the risk-free rate
• $\tilde r_j$ is the random return of asset $j$
• $n$ is the number of risky assets
• $\tilde W$ is the random wealth after investment

We can rewrite the wealth function as:

\[\tilde W = W_0 (1 + r_f) + \sum_{j=1}^{n} a_j (\tilde r_j - r_f)\]

The individual’s objective is to maximize the expected utility of wealth (not just maximize wealth itself!):

\[\max_{\{a_j\}} E[u(W)] = E[u(W_0 (1 + r_f) + \sum_{j=1}^{n} a_j (\tilde r_j - r_f))]\]

The first-order condition (FOC) is:

\[\frac{\partial E[u(W)]}{\partial a_j} = 0 \\ E[u'(W) \cdot (\tilde r_j - r_f)] = 0, \quad \text{for } j = 1, 2, ..., n\]

This condition determines the optimal allocation for each asset $j$.

Intuition 1: Risk Premium $\tilde r_j - r_f$

The term $\tilde r_j - r_f$ is called the risk premium of asset $j$. It is the excess return an investor expects for taking on the additional risk of a risky asset compared to the risk-free asset.

From the FOC, $E[u’(W) \cdot (\tilde r_j - r_f)] = 0$. Since $u’(W)$ is always positive, $\tilde r_j - r_f$ must sometimes be positive and sometimes negative for the expectation to be zero.

If an asset always has a strictly positive risk premium, the FOC cannot be zero, and the optimal allocation would be infinite (subject to borrowing constraints, the investor would allocate all wealth to that asset).

Intuition 2: Risk Aversion

Claim:

A risk-averse individual (with an increasing, concave utility function) will invest in risky assets if and only if the expected risk premium is positive for at least one asset.

An individual will invest in risky assets if and only if at least one $j$ has $E[\tilde r_j - r_f] > 0$.

Proof:

For an individual to invest zero (or even short sell) in risky asset $j$, it must be that:

1. The optimal allocation for asset $j$ is zero or negative, $a_j \leq 0$.
2. The maximum utility is achieved at $a_j = 0$ or less.
3. Therefore, when $a_j = 0$, the slope should be less than or equal to zero:

\[E[u'(W) \cdot (\tilde r_j - r_f)] \leq 0\]

Since $u’(W) > 0$, this implies:

\[E[\tilde r_j - r_f] \leq 0, \quad \forall j = 1, 2, ..., n\]

Therefore, an individual will not invest in risky assets if and only if all expected risk premia are less than or equal to zero. If at least one asset has a positive expected risk premium, the individual will invest in risky assets.