An investor of a corporation has $$100$ million of excess funds to invest. The investor has been instructed to invest the entire amount for one year in either stocks or bonds $($but not both$)$ and then to reinvest the entire fund in either stocks or bonds $($but not both$)$ for one year more. The objective is to maximize the expected monetary value of the fund at the end of the second year.An investor of a corporation has $$100$ million of excess funds to invest. The investor has been

instructed to invest the entire amount for one year in either stocks or bonds $($but not both$)$ and then

to reinvest the entire fund in either stocks or bonds $($but not both$)$ for one year more. The objective

is to maximize the expected monetary value of the fund at the end of the second year.

The annual rates of return on these investments depend on the economic environment, as shown

in the table below.

The probabilities of growth, recession, and depression for the first year are $0\mathrm{.}7,0\mathrm{.}3,$ and $0,$

respectively. If growth occurs in the first year, these probabilities remain the same for the second

year. However, if a recession occurs in the first year, these probabilities change to $0\mathrm{.}2,0\mathrm{.}7,$ and

$0\mathrm{.}1,$ respectively, for the second year.

a$)$ Draw the corresponding decision tree for the above scenario. $(20$ points$)$

b$)$ Solve the decision tree. $(10$ points$)$

c$)$ Describe the optimal decision strategy verbally. $(3$ points$)$

The annual rates of return on these investments depend on the economic environment, as shown in the table below.

Investments

Economic Environment

Growth

Recession

Depression

Stocks

$20\%$

$-10\%$

$-50\%$

Bonds

$5\%$

$10\%$

$20\%$

The probabilities of growth, recession, and depression for the first year are $0\mathrm{.}7,0\mathrm{.}3,$ and $0,$ respectively. If growth occurs in the first year, these probabilities remain the same for the second year. However, if a recession occurs in the first year, these probabilities change to $0\mathrm{.}2,0\mathrm{.}7,$ and $0\mathrm{.}1,$ respectively, for the second year.

a$)$

Draw the corresponding decision tree for the above scenario. $(20$ points$)$

b$)$

Solve the decision tree. $(10$ points$)$

c$)$

Describe the optimal decision strategy verbally. $(3$ points$)$