Optimal Capital Allocation

Investment Theory

Lorenzo Naranjo

Fall 2024

Introduction

The Problem

• If passive investors possess the same information, they should agree on which combination of risky assets provides the best trade off between risk and return.
  • Some investors may think this optimal portfolio of risky assets carries too much risk.
  • Others may think the opposite.
• It is possible to reduce or increase the risk of a portfolio by investing or borrowing a risk-free asset.
• The capital allocation decision is then about how much to invest in this well-diversified portfolio of risky investments and how much to allocate to a risk-free asset.

The Solution

• The solution to the capital allocation problem has two dimensions:
  • Model what is available to invest.
  • Understand what investors want.
• The first point requires us to determine the investment opportunity set.
  • Combining a risky asset with the risk-free rate generates an investment set called the capital allocation line (CAL).
• The second point has to do with how investors feel when taking risks.
  • Investors dislike risk but like returns.
  • In finance and economics, we capture these two opposite effects using a utility function.

The Capital Allocation Line

The Economic Setup

• We analyze the investment opportunity set generated by a risky asset Q and a risk-free asset.
• We denote by r_{f} the risk-free rate of return.
• The expected return of the risky asset is denoted by \mu_{Q} whereas the standard deviation or volatility of fund returns is denoted by \sigma_{Q}.

The Investment Opportunity Set

• A portfolio P that invest w in Q and 1 - w in the risk-free asset has the following expected return and volatility: \begin{aligned} \mu_{P} & = (1 - w) r_{f} + w \mu_{Q}, \\ \sigma_{P} & = |w| \sigma_{Q}. \\ \end{aligned} \tag{1}
• If we only consider portfolios in which we invest in the risky asset, i.e. w \geq 0, we can combine both equations to get \mu_{P} = r_{f} + \left( \frac{\mu_{Q} - r_{f}}{\sigma_{Q}} \right) \sigma_{P}.

The Sharpe Ratio

• The Sharpe ratio of the risky-asset is defined as: \mathit{SR} = \frac{\mu_{Q} - r_{f}}{\sigma_{Q}}. \tag{2}
• The expected return of any portfolio formed by combining the risk-free and the risky asset is given by a line with intercept r_{f} and slope coefficient \mathit{SR}: \mu = r_{f} + \mathit{SR} \times \sigma. \tag{3}
• This line is called the Capital Allocation Line of Q or just CAL(Q).

The Capital Allocation Line

Figure 1: The figure shows the capital allocation line of the risky portfolio Q.

The CAL as a Production Function

• The CAL transforms risk into expected returns.
• The CAL of a risky asset can then be seen as a production function where the input is risk and the output is expected return.
• The Sharpe ratio of a risky asset is the marginal rate of transformation (MRT) of risk into expected return.
• Investors should then try to maximize this trade-off by maximizing the Sharpe ratio of their portfolios.

Example 1 Suppose that r_{f} = 5\%, \mu_{Q} = 12\% and \sigma_{Q} = 20\%. The Sharpe ratio of Q is \mathit{SR} = \frac{0.12 - 0.05}{0.20} = 0.35.

Suppose that you want a portfolio P on the CML but with 5% volatility. If w denotes the weight in the market, this means that 0.05 = w \times 0.20, or w = 25\%. Thus, a portfolio that invest 25% in Q and 75% in the risk-free asset has 5% volatility. The expected return of this portfolio is: \mu_{P} = 0.75 \times 0.05 + 0.25 \times 0.12 = 7.8\%. The position of P in the CML is illustrated in Figure 1.

Investor’s Utility

Expected Utility

• Investors seek to get the maximum return for the minimum risk.
• A standard way in economics to capture this trade-off is by using a utility function.
  • A utility function allows us to rank different combinations of risk and expected return.
• A simple way to do this in finance is to define: U(\mu, \sigma) = \mu - \frac{1}{2} A \sigma^{2}.
• The coefficient A denotes how sensitive is a particular investor to risk measured here by \sigma^{2}.
  • We call A the coefficient of risk-aversion.
  • It is common in applications to use values for A between 1 and 4.

Indifference Curves

• There are several pairs (\mu, \sigma) that provide the same utility, i.e. for a given U and \sigma, we can always find a \mu such that: \mu = U + \frac{1}{2} A \sigma^{2}.
• These functions are called indifference curves since an investor with risk-aversion coefficient A is indifferent among any of these combinations of \mu and \sigma.
• Let’s fix A = 3 and U to be either 2%, 6% or 10%. We can now plot the corresponding indifference curves.

Figure 2: The figure shows indifference curves for different levels of utility.

Certainty Equivalent

• The curves in the graph represent all combinations of (\mu, \sigma) that provide the same utility, i.e. the investor is indifferent among these choices of risk and return.
• Indifference curves that provide higher utility are always above indifference curves that provide lower utility.
• Each indifference curve can be characterized by its certainty equivalent, which represents the expected return that would provide the same level of utility with no risk, that is when \sigma = 0.
• The utility level can therefore be interpreted as the certainty equivalent of a particular portfolio.

Maximizing Utility

The Investment Opportunity Set

• Optimal portfolio choice is about maximizing utility given the constraints imposed by the investment opportunity set.
• Remember that for a given w that determines the weight in the risk asset Q, the investment opportunity set is characterized by: \begin{align*} \mu & = (1 - w) r_{F} + w \mu_{Q}, \\ \sigma^{2} & = w^{2} \sigma_{Q}^{2}. \end{align*}

The Optimal Portfolio

• The utility of investing w in Q and the rest in the risk-free asset: \begin{align*} U & = \mu - \frac{1}{2} A \sigma^{2} \\ & = (1 - w) r_{F} + w \mu_{Q} - \frac{1}{2} A w^{2} \sigma_{Q}^{2}. \end{align*}
• The first-order condition (FOC) is: \frac{dU}{dw} = (\mu_{Q} - r_{F}) - A w \sigma_{Q}^{2} = 0.
• The optimal w^{*} is given by: w^{*} = \frac{\mu_{Q} - r_{F}}{A \sigma_{Q}^{2}}.

The Characteristics of the Optimal Portfolio

• The amount allocated to the risky asset is smaller if the risk aversion or if its variance are larger.
  • The investor’s risk aversion and the variance of the asset play the same role and are indistinguishable.
• The mount allocated to the risky asset increases with its expected return, but decreases with the risk-free rate.
• The resulting expected return and standard deviation of the optimal portfolio are given by: \begin{align*} \mu^{*} & = (1 - w^{*}) r_{F} + w^{*} \mu_{Q}, \\ \sigma^{*} & = w^{*} \sigma_{Q}. \end{align*}

Example 2 Consider an agent with a risk-aversion coefficient equal to 3. If r_{f} = 5\%, \mu_{Q} = 12\% and \sigma_{Q} = 20\% as in Example 1, we have that w^{*} = \frac{0.12 - 0.05}{3 \times 0.20^{2}} = 58.33\%. Therefore, the portfolio that maximizes the utility for the investor consists investing 41.67% in the risk-free asset and 58.33\% in the risky asset. The expected return and volatility of this portfolio are \begin{aligned} \mu^{*} & = (1 - w^{*}) \times 0.05 + w^{*} \times 0.12 = 9.08\%, \\ \sigma^{*} & = w^{*} \times 0.20 = 11.67\%. \end{aligned}

The Optimality Condition

• The investor wants to maximize utility, i.e., to achieve the highest level of utility possible.
• The point where the indifference curve is tangent to the capital market line determines the optimal portfolio.
• At this point, the marginal rate of substitution (MRS) between risk and return equals the marginal rate of transformation (MRT) between risk and return.

Figure 3: The figure shows that optimal portfolio choice occurs where the marginal rate of substitution equals the marginal rate of transformation between risk and return.

Computing the MRS

• The MRS characterizes the trade-off between risk and return for the same level of utility.
• In an indifference curve, we must have that 0 = dU = \frac{\partial U}{\partial \mu} d\mu + \frac{\partial U}{\partial \sigma} d\sigma = d\mu - A \sigma d\sigma. Thus, \text{MRS} = \frac{d\mu}{d\sigma} = A \sigma.

Computing the MRT

• In our setup, the MRT characterizes the ability to transform risk into expected return.
• The MRT is therefore equal to the Sharpe ratio of any risky portfolio in the investment opportunity set, i.e. \text{MRT} = \frac{\mu - r_{f}}{\sigma}.

Finding the Optimum Again

• The optimal portfolio with mean \mu^{*} and standard deviation \sigma^{*} satisfies \text{MRS} = A \sigma^{*} = \frac{\mu^{*} - r_{f}}{\sigma^{*}} = \text{MRT}.
• We could have used this equation to find the optimal portfolio weights.
  • Just replace \mu^{*} = r_{f} + w (\mu_{Q} - r_{f}) and \sigma = w \sigma_{0} in the expression above, and solve for w.

Is There Another Way?

• Yes, we could try maximizing utility subject to the constraint of the capital allocation line given by (3), i.e., forming a Lagrangian: \mathcal{L} = \mu - \frac{1}{2} A \sigma^{2} - \lambda (\mu - r_{f} - \mathit{SR} \sigma).
• The FOC are \begin{aligned} \frac{\partial \mathcal{L}}{\partial \mu} & = 1 - \lambda = 0, \\ \frac{\partial \mathcal{L}}{\partial \sigma} & = -A \sigma + \lambda \mathit{SR} = 0,\\ \frac{\partial \mathcal{L}}{\partial \lambda} & = \mu - r_{f} - \mathit{SR} \sigma = 0. \\ \end{aligned}
• The solution is the same as before, i.e., \text{MRS} = \text{MRT}.