Question

Investment in risky assets can be examined in the state-preference framework by assuming that \(W_{0}\) dollars invested in an asset with a certain return \(r\) will yield \(W_{0}(1+r)\) in both states of the world, whereas investment in a risky asset will yield \(W_{0}\left(l+r_{g}\right)\) in good times and \(W_{0}\left(l+r_{b}\right)\) in bad times (where \(\left.r_{g}>r>r_{b}\right)\). 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. \(^{26}\)

Short answer

A mixed portfolio consists of both risk-free and risky assets. Let the fraction of wealth invested in the risky asset be \(x\). Then, the fraction of wealth invested in the risk-free asset is \((1-x)\). In good times, the mixed portfolio yields \(W_{0}[x(1+r_{g})+(1-x)(1+r)]\) and in bad times, it yields \(W_{0}[x(1+r_{b})+(1-x)(1+r)]\). On the graph from part (a), plot these mixed portfolio outcomes as a convex combination of the risk-free and risky asset outcomes. For any value of \(x\) between 0 and 1, connect the good and bad state outcomes indicating the mixed portfolio outcomes in both states. #tag_title# 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? Individuals' attitudes toward risk affect their preferences for holding risk-free and risky assets in their mixed portfolio. Risk-loving individuals are willing to take on higher risks to obtain higher potential returns, causing them to have a higher fraction of risky assets in their portfolio. Conversely, risk-averse individuals prefer more stability and lower risks, resulting in a portfolio with a higher fraction of risk-free assets. In the case where a person is extremely risk-averse, they will hold no risky assets, meaning \(x=0\), and their entire portfolio will consist of risk-free assets. #tag_title# 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. If an individual's utility function exhibits constant relative risk aversion, it means that their risk aversion remains constant as their wealth (\(W_{0}\)) changes. This is represented by the utility function \(U(W_{0}) = \frac{W_{0}^{1-\gamma}}{1-\gamma}\), where \(\gamma\) represents the relative risk aversion parameter. Since the utility function remains constant, an individual with constant relative risk aversion will maintain the same risk preferences regardless of wealth level changes. This implies that the proportion of risky assets in their portfolio (denoted by \(x\)) will not change as their wealth (\(W_{0}\)) increases or decreases.

Step-by-step solution

a. Graph the outcomes from the two investments.

To graph the outcomes of the two investments, first consider the risk-free asset with return \(r\), which yields \(W_{0}(1+r)\) in both good and bad times. The risky asset yields \(W_{0}(1+r_{g})\) in good times and \(W_{0}(1+r_{b})\) in bad times. Since \(r_{g}>r>r_{b}\), the risky asset's outcome in good times will be higher than the risk-free asset, and its outcome in bad times will be lower than the risk-free asset. On the graph, plot wealth on the horizontal axis, and put the probabilities of both states on the vertical axis. Represent the risk-free asset with an upward-sloping line connecting the points \((W_{0}(1+r), 0)\) and \((W_{0}(1+r), 1)\). For the risky asset, represent good times with a steeper line segment connecting the points \((W_{0}(1+r_{g}), 0)\) and \((W_{0}(1+r_{g}), p)\), and bad times with a less steep line segment connecting \((W_{0}(1+r_{b}), 0)\) and \((W_{0}(1+r_{b}), 1-p)\).

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?

A mixed portfolio consists of both risk-free and risky assets. Let the fraction of wealth invested in the risky asset be \(x\). Then, the fraction of wealth invested in the risk-free asset is \((1-x)\). In good times, the mixed portfolio yields \(W_{0}[x(1+r_{g})+(1-x)(1+r)]\) and in bad times, it yields \(W_{0}[x(1+r_{b})+(1-x)(1+r)]\). On the graph from part (a), plot these mixed portfolio outcomes as a convex combination of the risk-free and risky asset outcomes. For any value of \(x\) between 0 and 1, connect the good and bad state outcomes indicating the mixed portfolio outcomes in both states.

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?

Individuals' attitudes toward risk affect their preferences for holding risk-free and risky assets in their mixed portfolio. Risk-loving individuals are willing to take on higher risks to obtain higher potential returns, causing them to have a higher fraction of risky assets in their portfolio. Conversely, risk-averse individuals prefer more stability and lower risks, resulting in a portfolio with a higher fraction of risk-free assets. In the case where a person is extremely risk-averse, they will hold no risky assets, meaning \(x=0\), and their entire portfolio will consist of risk-free 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.

If an individual's utility function exhibits constant relative risk aversion, it means that their risk aversion remains constant as their wealth (\(W_{0}\)) changes. This is represented by the utility function \(U(W_{0}) = \frac{W_{0}^{1-\gamma}}{1-\gamma}\), where \(\gamma\) represents the relative risk aversion parameter. Since the utility function remains constant, an individual with constant relative risk aversion will maintain the same risk preferences regardless of wealth level changes. This implies that the proportion of risky assets in their portfolio (denoted by \(x\)) will not change as their wealth (\(W_{0}\)) increases or decreases.