3. Suppose that a risk averse individual has wealth I. She...

3. Suppose that a risk averse individual has wealth I. She has the opportunity to invest some of her $I in a risky asset. The rate of return on the asset is uncertain. With probability p, the return is $r_1$ and so $1 invested in the asset would be worth 1+$r_1$. With probability 1-p, the return is $r_2$ and so $1 invested in the asset would be worth 1+$r_2$. Suppose that the expected rate of return on the asset is strictly positive. Let a be the number of dollars invested in this asset. a) Set up the expected utility maximization problem for this investor where the choice variable is a. Write the first order conditions for this problem. b) Show that $a = 0$ is NOT the solution to the problem. In other words, every risk inverse investor will want to invest some money in this asset.

Transcript

00:01 Okay, so we first want to create a discrete probability distribution.

00:05 So for the first investment, we're going to have our x, which is an event, probability of that event, and then we're going to have the mean, which is going to be equivalent to this x times the probability of its event.

00:25 So for the first investment, we have a chance to make $70 million, but that's at a 20%.

00:34 Chance so to find the average there which is multiply those two so we should have 1 .4 million for the average or for the mean and then we also have the chance of making zero dollars and there's a 30 % chance of that so a mean be zero and then we have chance of losing a million dollars and there is a 50 % chance of that so therefore on average we're going to lose 500 so then for our second investment we have a chance of making three million dollars two million dollars or losing a million dollars the probability of those is 0 .1 0 .6 and 0 .3 so doing the multiplication our mean here would be 300 ,000 so 300k and then here would be 1 .2 million on average and then here we'll have negative 300k then our last investment which is a third investment right our events are neither we can make three million dollars we can not make any profit or we can lose a million probability of those is 0 .4 0 .5 and 0 .1.

02:41 So the mean of each situation, we can have an average of 1 .2 mil, be it to get 0 or negative 100 ,000.

02:55 So now we want to know the expected value for each one of the investments.

03:05 So the expected value of the first investment would just be the sum of our three investments.

03:13 So the sum of 1 .4 million of 0 and negative 500 ,000 well that's going to give us an expected value of 900 ,000 expected value of our second investment would give us 1 .2 million by just adding those up and then lastly the expected value of our third investment will just be adding those as well and it's to give us 1 .1 million so for c, we are asked which investment has the highest expected return...

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