Question

In Example 7.3 we showed that a person with a CARA utility function who faces a Normally distributed risk will have expected utility of the form \(E[U(W)]=\mu_{W}-(A / 2) \sigma_{W}^{2},\) where \(\mu_{W}\) is the expected value of wealth and \(\sigma_{W}^{2}\) is its variance. Use this fact to solve for the optimal portfolio allocation for a person with a CARA utility function who must invest \(k\) of his or her wealth in a Normally distributed risky asset whose expected return is \(\mu_{r}\) and variance in return is \(\sigma_{r}^{2}\) (your answer should depend on \(A\) ). Explain your results intuitively.

Short answer

Answer: The optimal amount to invest in the risky asset (k) can be calculated using the formula \(k = \frac{\mu_{r}}{A\sigma_{r}^{2}}\), where \(\mu_{r}\) is the expected return of the risky asset, \(A\) is the person's risk aversion constant, and \(\sigma_{r}^{2}\) is the variance in the return of the risky asset.

Step-by-step solution

Write down the wealth equation

The person's wealth (\(W\)) can be expressed as the sum of their initial wealth minus the amount invested in the risky asset and the returns from the risky investment. This can be written as: \(W = W_{0} - k + kR\), where \(W_{0}\) is the initial wealth, \(k\) is the amount invested in the risky asset, and \(R\) is the returns from the risky investment.

Calculate expected value and variance of wealth

We need to calculate the expected value (\(\mu_{W}\)) and variance (\(\sigma_{W}^{2}\)) of wealth. The expected value of wealth is given by: \(\mu_{W}=E[W]=W_{0}-k+k\mu_{r}\), where \(\mu_{r}\) is the expected return from the risky asset. The variance of wealth is given by: \(\sigma_{W}^{2}=Var[W]=(k\sigma_{r})^{2}=\sigma_{r}^{2}k^{2}\), where \(\sigma_{r}^{2}\) is the variance in return of the risky asset.

Calculate expected utility

Plug in the expected value and variance of wealth into the equation for expected utility: \(E[U(W)] = \mu_{W} - (A / 2) \sigma_{W}^{2} = (W_{0} - k + k\mu_{r}) - (A / 2)(\sigma_{r}^{2}k^{2})\)

Maximize expected utility with respect to k

Now, we need to find the optimal investment amount \(k\) that maximizes expected utility. To do this, differentiate \(E[U(W)]\) with respect to \(k\) and set the derivative equal to zero: \(\frac{\partial E[U(W)]}{\partial k} = \mu_{r} - Ak\sigma_{r}^{2} = 0 \)

Solve for optimal k

Solve the equation for optimal investment amount \(k\): \(k = \frac{\mu_{r}}{A\sigma_{r}^{2}}\)

Interpret the result

The optimal amount to invest in the risky asset (\(k\)) depends on the expected return of the risky asset (\(\mu_{r}\)), the person's risk aversion constant (\(A\)), and the variance in return of the risky asset (\(\sigma_{r}^{2}\)). This result intuitively makes sense because as the expected return increases, the person would want to invest more in the risky asset. However, as the risk aversion increases or the variance in return increases, the person would want to invest less in the risky asset to protect their wealth from the potential downside.

Key concepts

CARA Utility Function

The CARA (Constant Absolute Risk Aversion) utility function is a tool used to measure an investor’s risk tolerance. It expresses the idea that an individual's dislike for losing a dollar is constant, regardless of how wealthy they are. In mathematical terms, the utility of wealth is given by a negative exponential function:
\( U(W) = -\frac{e^{-AW}}{A} \)
where \( A \) is a parameter that represents the degree of risk aversion: a higher \( A \) implies more risk aversion. The beauty of the CARA utility function lies in its simplicity and in the fact that it leads to decision-making that is independent of wealth levels. This characteristic makes it a useful tool for analyzing investment choices where risk and return are balanced.

Normal Distribution

In finance and statistics, the normal distribution is a continuous probability distribution that is symmetrical around the mean, meaning that it depicts that the occurrence rates of values near the mean are higher as compared to those far from the mean.

It is defined by two parameters: the mean \( \mu \) which indicates the center of the distribution, and the variance \( \sigma^2 \) which measures the spread or dispersion of the distribution. When an investment’s returns are normally distributed, it simplifies the risk analysis, allowing for straightforward calculations of expected values and variance. In the context of investment decisions, assuming normal distribution for asset returns is a common practice, although real-life returns can sometimes deviate from this model.

Expected Utility

Expected utility is a key concept in economics and finance that represents the average utility an investor expects to achieve from a portfolio. It’s calculated by taking into account the utility of all possible outcomes, each weighted by its probability. The formula for expected utility given a CARA utility function and normally distributed returns simplifies to:

\( E[U(W)] = \mu_W - \frac{A}{2} \sigma_W^2 \)

where \( \mu_W \) is the expected value of wealth and \( \sigma_W^2 \) is the variance of wealth. Expected utility is used to evaluate the desirability of portfolios with different risk-return profiles. Investors aim to maximize this quantity within their risk tolerance constraints, leading to different optimal portfolio allocations.

Investment Risk Analysis

Investment risk analysis is the process of identifying and assessing the risks associated with investing. It involves analyzing the probability of different investment outcomes and their potential impact on an investor’s wealth.
Key tools in risk analysis include the expected return and the variance or standard deviation of returns, which measure the average outcome and the uncertainty or risk of the investment, respectively.

By using the CARA utility function, investors can quantify their risk aversion and determine the optimal investment that maximizes expected utility given the trade-off between risk and return. The solution to the problem highlights that with a CARA utility function, the optimal investment level \( k \) in a risky asset is inversely proportional to the product of risk aversion \( A \) and the variance of the asset's return \( \sigma_r^2 \). This illustrates the balance investors seek between their desire for higher returns and their tolerance for risk in their portfolio allocation decisions.