ECON 3330 Homework # 2 - The Economics of Risk

Homework # 2

The Economics of Risk (ECON 3330)

Fall 2020

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1. Suppose that Kayla has utility function u(w) = ln(w) and initial wealth w0 = 360. She has the opportunity to invest in two different assets, one risky and one safe.

ˆ The safe asset yields a return of 1 dollar for every dollar invested with probability 1 (so that r = 0).

ˆ The risky asset yields a return of 1 + x? for every dollar invested, where x? is the lottery

x? =

[ 1 2

1 2

0.45 −0.4

] .

(a) Solve Kayla’s portfolio problem to find the optimal amount of money invested in the risky asset, given by α, and the optimal amount of money invested in the safe asset, given by w0 − α.

(b) Kayla’s best friend, Layla, has utility function u(w) = √ w. She also has initial wealth w0 = 360.

Layla also has the opportunity to invest in the two different assets Kayla does. Solve Layla’s portfolio problem to find the optimal amount of money invested in the risky asset and in the safe asset.

(c) Now suppose that Kayla’s initial wealth increased and is now equal to 500. Find the optimal amount of money invested in the risky asset and in the safe asset in this case. What happened to α? Explain how you can use whether Kayla’s utility function exhibits DARA/IARA/CARA to make sense of what happens to α.

(d) Now suppose that Layla’s initial wealth increased and is now equal to 500. Find the optimal amount of money invested in risky and safe assets. What happened to α? Explain how you can use whether Layla’s utility function exhibits DARA/IARA/CARA to make sense of what happens to α.

(e) Mikayla, Kayla’s other best friend, has a utility function of u(w) = −e−.002w. She also has initial wealth of w0 = 360. Solve Mikayla’s portfolio problem to find the optimal amount of money invested in the risky asset and in the safe asset.

(f) Now Mikayla’s initial wealth increases to 500. Solve Mikayla’s portfolio problem to find the optimal amount of money invested in the risky asset and in the safe asset in this case. What happened to α? Explain how you can use whether Mikayla’s utility function exhibits DARA/IARA/CARA to make sense of what happens to α.

(g) Rank the friends according to their risk aversion for a given amount of wealth. Rank them as needed for their different wealth levels. Explain the rationale behind your choice.

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