Please explain qu1 and the concept behind the CARA function (a) What characterizes a CARA utility function? Give an example of such a utilityfunction and show why it is displays CARA.(b) It can be shown that an investor with a CARA utility function who faces normallydistributed risk has preferences represented by:aE [u(w)] = E [w] - Var [w]2where w is wealth and a is a risk-aversion parameter. Suppose the investorsoriginal wealth is given by wo and that there is a risky asset with rate of returndistribution r~ N(µ, σ²). There is also a risk-free asset with a rate of return rf.The investor seeks to allocate x to the risky asset and wo x to the risk-free one.What is the expected wealth and its variance resulting from this allocation?(c) Find the optimal allocation x* and discuss how it depends on the model parame-ters.

The utility function measures the consumer's satisfaction as a consumption function of real goods, such as food or clothing, etc. CARA is an abbreviation for Constant Absolute Risk Aversion. CARA utility function is characterized by utility function for consumers who prefer outcomes with low uncertainty. The utility function example = (1-e-ax)/a Here, the Utility function is differentiated: dU/dx = ae-ax / a d2 U/dx2 = -ae-ax Thus, Risk aversion = d2 U/dx2 / dU/dx = -a Absolute risk aversion = a Thus, when a<0 the utility function is risk aversion. Thus, this shows a CARA utility function.

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(a) What characterizes a CARA utility function? Give an example of such a utility function and show why it is displays CARA.

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ISBN:9780190931919

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Chapter1: Making Economics Decisions

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Question

Please explain qu1 and the concept behind the CARA function

(a) What characterizes a CARA utility function? Give an example of such a utility
function and show why it is displays CARA.
(b) It can be shown that an investor with a CARA utility function who faces normally
distributed risk has preferences represented by:
a
E [u(w)] = E [w] - Var [w]
2
where w is wealth and a is a risk-aversion parameter. Suppose the investors
original wealth is given by wo and that there is a risky asset with rate of return
distribution r~ N(µ, σ²). There is also a risk-free asset with a rate of return rf.
The investor seeks to allocate x to the risky asset and wo x to the risk-free one.
What is the expected wealth and its variance resulting from this allocation?
(c) Find the optimal allocation x* and discuss how it depends on the model parame-
ters.

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