An investor wants to invest a capital V = 1000 and chooses to make only one type of investment among the three: passbook (L), government bonds (O) or shares (A). It has to deal with different types of economic conjuncture (states of nature): unfavorable (d), average (m) and favorable (f). The following information matrix showing end-of-period capital summarizes this situation.

unfavorable average favorable

= 1/2 =3/10 =2/10

passbook 1022 1022 1022

g.bonds 1100 1060 1060

shares 800 1300 1600

a) Show that each decision can take the form of a lottery.

b) What are the mathematical expectations of these three lotteries?

c)What are the variances of these lotteries? What is the hierarchy of returns and risks?

d) We want to interpret these different actions in the context of a Markowitz function: (a) k = 2 (b) k = 0 (c) k = 1. Is there a degree of risk aversion that leads to prefer equity investment to savings account investment? Can we say that a risk-neutral person does not take risks? Same question for a slightly risk averse person (k > 0 very close to 0)?

We now consider a fourth lottery composed as follows: 1/2 placed on passbooks , and 1/4 on the other two placements. Write this lottery, denoted .

e) What is its expectation and variance?

f) Can the diversified investment be preferred to the others in the three previous cases: (a) k = 2 (b) k = 0 (c) k = 1? What is the degree of risk aversion for which we prefer the diversified investment to the booklet?

We now consider that our investor decides to place a part of his capital in shares and 1 in the passbooks.

g) Write the corresponding lottery. The same method is applied as for the diversified investment.

h) What equity share [0, 1] will an investor choose given that (a) k = 2 (b) k = 0 (c) k = 1?

i) How much risk aversion would it take for an investor to invest 1/4 of their capital in stocks?