91Ó°ÊÓ

The yields of two investments A and B (in thousands of dollars) are shown below, along with their probability.

Investment A

Investment B

Let Y be the yield of an investment.

The investment A's projected return is

$$\begin{gathered} {\mathrm{μ}}_A=\underset{}{∑}a·P(A=a) \\ {\mathrm{μ}}_A=0×0.1+1000×0.5+4000×0.4 \\ {\mathrm{μ}}_A=\$2100 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mathrm{μ}}_B=\underset{}{∑}b·P(B=b) \\ {\mathrm{μ}}_B=0×0.5+1000×0.3+16000×0.2 \\ {\mathrm{μ}}_B=\$3500 \end{gathered}$$

The predicted yield of the investment A is lower than the expected yield of the investment B, according to the aforementioned estimate.

The utility function is as follows:

$u\left(y\right)=\sqrt{Y}$

The investment A's projected return is

$$\begin{gathered} {\mathrm{μ}}_{\sqrt{A}}=\underset{}{∑}\sqrt{a}·P(A=a) \\ {\mathrm{μ}}_{\sqrt{A}}=\sqrt{0}×0.1+\sqrt{1000}×0.5+\sqrt{4000}×0.4 \\ {\mathrm{μ}}_{\sqrt{A}}=\$41.1096 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mathrm{μ}}_{\sqrt{B}}=\underset{}{∑}\sqrt{b}·P(B=b) \\ {\mathrm{μ}}_{\sqrt{B}}=0×0.5+\sqrt{1000}×0.3+\sqrt{16000}×0.2 \\ {\mathrm{μ}}_{\sqrt{B}}=34.7851 \end{gathered}$$

Investment A has an expected utility of 41.1096 while investment B has an expected utility of 34.7851. As a result, the predicted utility of investment A is greater than that of investment B.

The utility function is as follows:

$v\left(y\right)=y^{3/2}$

The investment A's projected return is

$$\begin{gathered} {\mathrm{μ}}_{A^{3/2}}=\underset{}{∑}{a}^{3/2}·P(A=a) \\ {\mathrm{μ}}_{A^{3/2}}=0^{3/2}×0.1+{1000}^{3/2}×0.5+{4000}^{3/2}×0.4 \\ {\mathrm{μ}}_{A^{3/2}}=117004.2734 \end{gathered}$$

The investment B's projected return is

$$\begin{gathered} {\mathrm{μ}}_{B^{3/2}}=\underset{}{∑}{b}^{3/2}·P(B=b) \\ {\mathrm{μ}}_{B^{3/2}}=0^{3/2}×0.5+{1000}^{3/2}×0.3+{16000}^{3/2}×0.2 \\ {\mathrm{μ}}_{B^{3/2}}=414258.3735 \end{gathered}$$

Investment A's expected utility is 117004.2734, while investment B's estimated utility is 414258.3735. As a result, investment A's predicted utility is lower than investment B's expected utility.