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Q3 book 1

S

SABeauty

Member
On pg 77 the answer is given for part b - how are these numbers obtained?
 
Here we are maximising the expected utility by choosing the proportion of money 0<a<1 to investin the Asset A. So, the investor invests a*w0 in Asset A and the remaining (1-a)*w0 in Asset B.

Suppose Scenario A prevails.

If the investor invests aw0 in Asset A and it returns -100%, then the value of that investment will be zero. If she invests (1-a)w0 in Asset B and it returns +50%, then the value of that investment will be 1.5(1-a)w0.

So, if Scenario A arises she ends up with wealth of 0 + 1.5(1-a)w0, which gives her a utility of ln[0 + 1.5(1-a)w0].

Conversely if Scenario B occurs, then if she invests aw0 in Asset A and it returns +300%, then the value of that investment will 4aw0. If she invests (1-a)w0 in Asset B and it returns +50%, then the value of that investment will again be (1-a)w0.

So, if Scenario B arises she ends up with wealth of 4aw0 + 1.5(1-a)w0, which gives here a utility of ln[4aw0 + 1.5(1-a)w0].

Each scenario has a probability of 1/2 of arising and so her expected utility is:

EU = 1/2 * ln[0 + 1.5(1-a)w0] + 1/2 * ln[4aw0 + 1.5(1-a)w0]

We then find the value of "a" that maximises her expected utility by differentiating with repsect to a, setting the resulting expression to zero and solving for a.
 
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