Question $3(25$ points$)$

A risk$-$averse investor is considering two options:

Investing in a High$-$Risk Stock $-$ The investment has a $50\%$ chance of yielding a $$100,000$ profit and a $50\%$ chance of resulting in a $$20,000$ loss.

Accepting a Guaranteed Payout $-$ A financial institution offers the investor a certain amount as an alternative to the risky investment.

The investor's utility function is given by:

$U(W)=W^{0\mathrm{.}5}$

where W represents wealth. Assume the investor currently has a wealth of $$50,000$ and is deciding whether to take the risky investment or accept a guaranteed payout.

Tasks:

Calculate the Expected Monetary Value $($EMV$)$ of the Risky Investment

Compute the Expected Utility $($EU$)$ of the Risky Investment:

Determine the Certainty Equivalent $($CE$)$:

Provide a Written Justification:

Explain the concept of Certainty Equivalent $($CE$)$ and how it reflects the decision$-$making of a risk$-$averse individual.

Compare the CE to the EMV and discuss why a risk$-$averse person might prefer a lower guaranteed payout over a risky alternative.