Question

Investment in risky assets can be examined in the state-preference framework by assuming that \(W^{*}\) dollars invested in an asset with a certain return, \(r,\) will yield \(\mathrm{W}^{*}(\mathrm{l}+r)\) in both states of the world, whereas investment in a risky asset will yield \(\mathrm{W}^{*}\left(1+r_{g}\right)\) in good times and \(\left.\mathrm{W}^{*}\left(\mathrm{l}+r_{b}\right) \text { in bad times (where } r_{g}>r>r_{b}\right)\) a. Graph the outcomes from the two investments. b. Show how a "mxied portfolio" containing both risk-free and risky assets could be illustrated in your graph. How would you show the fraction of wealth invested in the risky asset? c. Show how individuals' attitudes toward risk will determine the mix of risk- free and risky assets they will hold. In what case would a person hold no risky assets? d. If an individual's utility takes the constant relative risk aversion form (Equation 8.62 ), ex plain why this person will not change the fraction of risky asset held as his or her wealth increases.

Short answer

Answer: An individual's risk preferences play a significant role in determining the mix of risk-free and risky assets in their investment portfolio. Risk-averse individuals will prefer to invest more in risk-free assets to minimize potential losses, while risk-seeking individuals will allocate a higher proportion of their portfolio to risky assets in the hope of higher returns. The investment decisions of someone with constant relative risk aversion (CRRA) utility remain unchanged as their wealth increases, as their risk preferences remain constant.

Step-by-step solution

a. Graph the outcomes of investing in risky and risk-free assets

To graph the outcomes of investing in risky and risk-free assets, let's first represent the return of each investment on the y-axis and the amount invested in each asset on the x-axis. For the risk-free asset, its return will be a straight line with a slope of \((1+r)\), while the risky asset will have two possible returns \((1+r_g)\) and \((1+r_b)\). We can plot these lines on a graph with appropriate labels for clarity.

b. Mixed portfolio illustration on the graph

To illustrate a mixed portfolio on the graph, we can show a combination of both risk-free and risky assets by creating an additional line that represents the returns of the mixed portfolio. This line will lie between the risk-free and risky asset lines as it takes on characteristics of both investments. We can represent the fraction of wealth invested in the risky asset by the slopes of the lines emanating from the point of intersection of the risk-free and risky asset lines. The lines with greater slope indicate a higher proportion of risky assets, whereas lines with lower slopes have a higher proportion of risk-free assets.

c. Individual attitudes toward risk and asset mix

Individuals' attitudes towards risk will determine the mix of risk-free and risky assets they hold. Risk-averse individuals will prefer to invest more in risk-free assets to minimize their exposure to potential losses. Conversely, risk-seeking individuals will allocate a higher proportion of their portfolio to risky assets in the hope of higher returns. In the extreme case, a person would hold no risky assets if they are entirely risk-averse, meaning they would only invest in the risk-free asset to avoid any significant losses.

d. Constant relative risk aversion and investment decisions

If an individual's utility takes the constant relative risk aversion (CRRA) form, as given in Equation 8.62, their investment behavior is invariant to an increase in wealth. This is because the risk preferences of a person with constant relative risk aversion remain constant as their wealth increases. As a result, someone with CRRA utility will not change their investment decisions (i.e., the fraction of risky assets held in their portfolio) as their wealth increases. This is because their risk preferences remain the same despite an increase in wealth, and their preference for a balance between risk-free and risky assets remains unchanged.

Key concepts

Investment in Risky Assets

Understanding how to invest in risky assets is essential for any financial portfolio management. A risky asset is one that has variability in returns; it can yield higher gains in certain scenarios (referred to as 'good times') and losses in others ('bad times'). In contrast, a risk-free asset offers a certain return with no variability. When plotting these on a graph, as seen in step 1 of our exercise solution, the risk-free asset appears as a straight line, reflecting its stable return, while the risky asset shows two different outcomes, depicting potential fluctuations in returns.

Investors must consider their risk tolerance when deciding how much to invest in such assets. While some individuals may accept the higher risk for a chance at better returns, others may opt for a smaller allocation towards risky assets to preserve their capital. This exercise allows us to visualize the outcomes of different investment decisions and better understand the implications of adding risk to our portfolios.

Mixed Portfolio

A mixed portfolio, as explained in step 2 of our solution, involves diversification through combining both risk-free and risky assets. By balancing between the two, investors can tailor their portfolios to meet their risk preferences and investment objectives. On a graph, a mixed portfolio is represented by a line that demonstrates a blend of the characteristics of risk-free and risky assets. The slope of this line helps to indicate the fraction of wealth invested in the risky asset.

Determining the right mix depends on an investor's risk tolerance. An ideal mixed portfolio strikes a balance that maximizes returns while maintaining an acceptable level of risk exposure. This concept reflects the very essence of portfolio theory, illustrating how diversification can optimize investment outcomes.

Constant Relative Risk Aversion

Constant Relative Risk Aversion (CRRA) is a key measurement of how an investor's willingness to take risks remains unchanged, despite variations in wealth. According to the step 4 explanation, an individual with CRRA maintains a constant attitude towards risk, which implies that they do not alter the proportion of their investment in risky assets as their wealth grows.

For individuals conforming to CRRA, their risk tolerance is scaled to their overall wealth. As such, a proportional investment strategy is employed—using the same percentage of wealth to invest in risky assets irrespective of the total capital they hold. This utility function helps explain why some investors maintain a steady investment strategy, regardless of how much money they have or make, highlighting a fundamental principle of behavioral finance.