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 \(W^{*}(1+r)\) in both states of the world, whereas investment in a risky asset will yield \(W^{+}\left(1+r_{g}\right)\) in good times and \(W^{*}\left(1+r_{b}\right)\) in bad times (where \(r_{g}>r>r_{b}\) ). a. Graph the outcomes from the two investments. b. Show how a "mixed 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 7.42 ), explain why this person will not change the fraction of risky assets held as his or her wealth increases. \(^{25}\)

Short answer

Answer: An individual's attitude towards risk influences the mix of risk-free and risky assets they hold in their investment portfolio. The more risk-tolerant they are, the larger the fraction of wealth they allocate towards risky assets. Conversely, the less risk-tolerant they are, the larger the fraction of wealth they allocate towards risk-free assets. Extremely risk-averse individuals may prefer to invest their entire wealth in risk-free assets.

Step-by-step solution

Graph the outcomes of the two investments

To graph the outcomes of the two investments, plot the investment in risk-free assets on the x-axis and the investment in risky assets on the y-axis. The risk-free asset has a return of \(r\), which yields \(W^{*}(1+r)\) in both states. Therefore, its payoff is a straight line with a slope of \((1+r)\). On the other hand, the risky asset has differing returns (\(r_g\) in good times and \(r_b\) in bad times), yielding payoffs of \(W^{+}\left(1+r_{g}\right)\) and \(W^{*}\left(1+r_{b}\right)\). Consequently, the payoff will be two separate points on the graph corresponding to the good and bad states.

Illustrate a mixed portfolio containing both risk-free and risky assets

The mixed portfolio can be represented as a line segment connecting the risk-free asset and the risky asset on the graph. Each point on the line segment represents a different mix of the risk-free and risky assets. The fraction of wealth invested in the risky asset can be shown by the distance of a specific point on the line segment from the risk-free asset point, divided by the total length of the line segment.

Determine the mix of risk-free and risky assets based on individuals' attitudes towards risk

An individual's risk tolerance will influence how much they invest in risk-free and risky assets. The more risk-tolerant they are, the larger the fraction of wealth they will allocate towards risky assets. The less risk-tolerant they are, the larger the fraction of wealth they will allocate towards risk-free assets. In the case where a person holds no risky assets, it means they are extremely risk-averse and prefer to invest their entire wealth in risk-free assets.

Explain the constant relative risk aversion form of an individual's utility

According to the constant relative risk aversion (CRRA) form of utility - Equation 7.42, an individual's utility function takes the following form: \(u(W) = \frac{W^{1-\rho}}{1-\rho}\), where \(\rho\) is the CRRA coefficient (a measure of risk aversion). In this case, the elasticity of the utility function remains constant as wealth increases, meaning the proportion of wealth allocated to risky assets does not change. This is because the individual is assumed to maintain the same level of risk aversion, regardless of changes in wealth.

Key concepts

State-Preference Framework

In the context of investment, the state-preference framework is a tool that helps to analyze how different investment choices perform under varying economic states. Imagine the world can be in a "good" or "bad" state at any time. Each state represents a different scenario in which investments might produce different returns. In this framework:

• A risk-free asset provides a consistent return regardless of the state.
• A risky asset's return fluctuates depending on whether the world is in a good or bad state.

By considering these different states, investors can visualize potential outcomes and decide on their investment strategy based on their preferences and expectations.
Understanding this framework allows investors to simulate potential scenarios and make informed decisions about their asset allocations under uncertainty.

Risk-Free Asset

A risk-free asset is an investment that offers a guaranteed return with no risk of financial loss. In practice, "risk-free" is more of a theoretical concept, since almost all investments carry some risk. However, government bonds from stable governments are often treated as risk-free due to their low chance of default.
The return from a risk-free asset is stable and predictable. For example, if someone invests in a risk-free asset with an interest rate of 5%, they can expect their investment to grow by 5% over the specified period, regardless of economic conditions.
Risk-free assets are vital building blocks in investment portfolios, providing stability and a safe haven for portion of the capital that investors do not wish to expose to market volatility.

Risk Aversion

Risk aversion refers to the preference of investors to choose investments with more predictable, less risky outcomes. Different individuals have varying levels of risk aversion, influencing their investment choices.
For example:

• Risk-averse individuals prefer safer investments, like government bonds, often with lower returns.
• Risk-tolerant individuals are more inclined to invest in stocks or other high-risk assets for the potential of higher returns.
• Highly risk-averse individuals may invest solely in risk-free assets to avoid any possibility of financial loss.

Understanding risk aversion helps in creating investment strategies that align with an individual's comfort with risk, ensuring they are more likely to stick with their investments during turbulent market periods.

Mixed Portfolio

A mixed portfolio combines different types of assets, often blending risk-free and risky investments to balance potential returns with risk exposure. By diversifying, investors aim to improve their overall portfolio performance while managing risk levels.
Here’s how it works:

• A point on the line connecting the risk-free to the risky asset on a graph represents a certain mix of the two.
• The distance from the risk-free asset point indicates the proportion invested in risky assets.
• Investors can adjust their mix according to their risk tolerance and investment goals, moving along this line to achieve the desired balance.

By maintaining a mixed portfolio, investors can better weather financial fluctuations while still potentially benefiting from the higher returns that risky assets might offer.