Chapter 4, Problem 4.9P

a)

To determine

To calculate: The certainty equivalent yield for stocks using the three utility functions and the conclusions about whether this person will invest $100,000 in stocks or bonds.

Explanation of Solution

The individual has $100,000 to invest in stocks. There is a 50 percent chance that the stocks will yield 16 percent over the next year, and a 50 percent chance that they will yield -2 percent. That is, the probabilities are,

  $p_{16\%}=p_{-2\%}=0.5$

If the stocks yield 16%, the returns will be as follows:

  $\begin{array}{c}{I}_{16\%}=\$100,000\left(1+0.16\right)\\ =\$100,000\left(1.16\right)\\ =\$116,000\end{array}$

And, if the stock yields -2%, the returns will be as follows:

  $\begin{array}{c}{I}_{-2\%}=\$100,000\left(1+0.02\right)\\ =\$100,000\left(0.98\right)\\ =\$98,000\end{array}$

Calculating the expected utility or certainty equivalent as follows:

  $CE=p_{16\%}.U\left(I_{16\%}\right)+p_{-2\%}.U\left(I_{-2\%}\right)$

For, $U\left(I\right)=\sqrt{I}$ $U\left(I\right)=\sqrt{I}$ the certainty equivalent (CE) is,

  $\begin{array}{c}CE=p_{16\%}.\sqrt{I_{16\%}}+p_{-2\%}.\sqrt{I_{-2\%}}\\ =0.5\sqrt{116,000}+0.5\sqrt{98,000}\\ =0.5\left(341\right)+0.5\left(313\right)\\ =327\end{array}$

Certainty equivalent income $\left(I_{CE}\right)$ would be the amount that yields 3227 utils.

  $\begin{array}{c}\sqrt{I_{CE}}=327\\ I_{CE}={\left(223.5\right)}^2\\ =\$106,929\end{array}$

For the certainty equivalent income of $106,929, the certainty equivalent $\left(i_{CE}\right)$ is as below:

  $\begin{array}{c}{i}_{CE}=\frac{\$106,929-\$100,000}{\$100,000}\\ =\frac{\$6,929}{\$100,000}\\ =0.069\%\end{array}$

Thus, the certainty equivalent yield for $U\left(I\right)=\sqrt{I}$ is approximately $\boxed{i_{CE}=7\%}$ .

  $U\left(I\right)=\ln I$ , calculating the certainty equivalent (CE) as below:

  $\begin{array}{c}CE=p_{16\%.}\ln \left(I_{16\%}\right)+p_{-2\%}.\ln \left(I_{-2\%}\right)\\ =\left(0.5\right)\ln \left(116,000\right)+\left(0.5\right)\ln \left(98,000\right)\\ =0.5\left(11.7\right)+0.5\left(11.5\right)\\ =11.6\end{array}$

Certainty equivalent income $\left(I_{CE}\right)$ would be the amount that yields 11.6 utils.

  $\begin{array}{c}\ln \left(I_{CE}\right)=11.6\\ I_{CE}=e^{11.6}\\ =\$109,098\end{array}$

For certainty equivalent income of $109,098, the certainty equivalent yield $\left(i_{CE}\right)$ is

  $\begin{array}{c}{i}_{CE}=\frac{\$109,098-\$100,000}{\$100,000}\\ =\frac{\$9,098}{\$100,000}\\ =0.091\end{array}$

Thus, the certainty equivalent yield for $U\left(I\right)=\ln I$ is approximately $\boxed{i_{CE}=9\%}$

For, $U\left(I\right)=\frac{-1}{1}$ , the certainty equivalent (CE) is as follows:

  $\begin{array}{c}CE=p_{16\%}.\left(\frac{-1}{I_{16\%}}\right)+p_{-2\%}.\left(\frac{-1}{I_{-2\%}}\right)\\ =\left(0.5\right)\left(\frac{-1}{116,000}\right)+\left(0.5\right)\left(\frac{-1}{98,000}\right)\\ =\left(\frac{-1}{232,000}\right)+\left(\frac{-1}{196,000}\right)\\ =\frac{-428}{45,472,000}\end{array}$

Certainty equivalent income $\left(I_{CE}\right)$ would be the amount that yields $\frac{-428}{45,472,000}$ utils.

  $\begin{array}{c}\left(\frac{-1}{I_{CE}}\right)=\frac{-428}{45,472,000}\\ I_{CE}=\frac{45,472,000}{428}\\ =\$106,242\end{array}$

For the certainty equivalent income of $106,242, the certainty equivalent yield $\left(i_{CE}\right)$ is:

  $\begin{array}{c}{i}_{CE}=\frac{\$106,242-\$100,000}{\$100,000}\\ =\frac{\$6,242}{\$100,000}\\ =0.062\end{array}$

Thus, the certainty equivalent yield for $U\left(I\right)=\frac{-1}{I}$ is approximately $\boxed{i_{CE}=6\%}$ .

Economics Concept Introduction

Introduction: Stocks and bonds are certificates that are traded to generate money to start a new business, or for expanding the improving an existing business. Bonds and stocks are also called to as securities, and those buying them are called investors.

b)

To determine

To calculate the certainty equivalent yield for stocks with $U\left(I\right)=-I^{-10}$ utility function.

Explanation of Solution

For, $U\left(I\right)=-I^{-10}$ , the certainty equivalent (CE) is as follows:

  $\begin{array}{c}CE=p_{16\%}.{\left(-I_{16\%}\right)}^{-10}+p_{-2\%}.{\left(-I_{-2\%}\right)}^{-10}\\ =\left(0.5\right){\left(-116,000\right)}^{-10}+\left(0.5\right){\left(-98,000\right)}^{-10}\\ =\left(0.5\right){\left(-116,000-98,000\right)}^{-10}\\ =\left(0.5\right){\left(-214,000\right)}^{-10}\end{array}$

Certainty equivalent income $\left(I_{CE}\right)$ would be the amount that yields $\left(0.5\right){\left(-214,000\right)}^{-10}$ utils.

  $\begin{array}{c}-I_{CE}{}^{-10}=\left(0.5\right){\left(-214,000\right)}^{-10}\\ I_{CE}=\left(0.5\right)\left(214,000\right)\\ =\$107,000\end{array}$

For the certainty equivalent income of $107,000, the certainty equivalent yield $\left(i_{CE}\right)$ is

  $\begin{array}{c}{i}_{CE}=\frac{\$107,000-\$100,000}{\$100,000}\\ =\frac{\$7,000}{\$100,000}\\ =0.07\end{array}$

Thus, the certainty equivalent yield for $U\left(I\right)=-I^{-10}$ is $\boxed{i_{CE}=7\%}$ .

This utility function yields exactly the same real return as believed by the individual.

Economics Concept Introduction

Introduction: Stocks and bonds are certificates that are traded to generate money to start a new business, or for expanding the improving an existing business. Bonds and stocks are also called to as securities, and those buying them are called investors.