$\text { Investment }$ $ \quad $$ \quad $$ \quad $$ \quad $$\text { A }$ $ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$ \quad $$\text { B }$
$\begin{array}{|l|l|l|l|l|}
\hline\
\begin{array}{l}
\text { State } \
\text { description }
\end{array} & \text { Bad state } & \text { Good state } & \text { Bad state } & \text { Good state } \
\hline \text { Probability } & 0.5 & 0.5 & 0.5 & 0.5 \
\hline \text { Payoff } & 200 & 220 & 400 & 435 \
\hline
\end{array}$
-Assume an investor has log utility.The investor faces a choice between an asset with a utility of 6.250 and an investment that will pay $500 in a bad state and $525 in a good state (as there are only two possible future states).What does the probability of the good state need to be for the investor to be indifferent between the assets?

Question

$\text { Investment }$  $  \quad $$  \quad $$  \quad $$  \quad $$\text { A }$   $  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$  \quad $$\text { B }$
$\begin{array}{|l|l|l|l|l|}
\hline\
\begin{array}{l}
\text { State } \
\text { description }
\end{array} & \text { Bad state } & \text { Good state } & \text { Bad state } & \text { Good state } \
\hline \text { Probability } & 0.5 & 0.5 & 0.5 & 0.5 \
\hline \text { Payoff } & 200 & 220 & 400 & 435 \
\hline
\end{array}$
-Assume an investor has log utility.The investor faces a choice between an asset with a utility of 6.250 and an investment that will pay $500 in a bad state and $525 in a good state (as there are only two possible future states).What does the probability of the good state need to be for the investor to be indifferent between the assets?

Quizplus · Accepted Answer

Defining p as the unknown probability of state A.Hence, (1 - p)is the probability of state B occurring,given that there are only two possible states.Investment B has a utility in the bad state of ln(500)= 6.215 and in the good state of ln(525)= 6.263.For investment B to have the same utility as asset A,we know 6.250 = (p)(6.263)+ (1 - p)(6.215),which can be rewritten as 6.250 = 6.263p + 6.215 - 6.215p,which can be further simplified to 0.035 = 0.048p,and solving for p gives a probability of 0.729.

Investment \text { Investment } Investment \quad \quad

$\text { Investment }$ $\quad$ $\quad$