Suppose there are only two periods, period $0$ and period $1,$ and three possible states

of the world in period $1$: a good weather state, a fair weather state, and a bad weather

state. Apples are the only product produced in this world, and they cannot be stored

from one period to the next.

The following abbreviations will be used: $PA=$ apple in the present period $($i$.$e$.,$

present apple$),GA=$ good weather apple in the next period, $FA=$ fair weather apple in

the next period, $BA=$ bad weather apple in the next period.

Suppose that an apple tree firm offers for sale a bond and stock:

The apple tree produces $160GA,100FA,$ and $50BA.$

The bond pays $40GA,40FA$ and $40BA.$

The stock pays $120GA,60FA$ and $10BA.$

In addition, securities $C,D,$ and E are available

Security C pays $140GA,80FA,$ and $30BA.$

Security D pays $60GA,30FA,$ and $5BA.$

Security E pays $80GA,20FA,$ and $0BA.$

The arbitrage$-$free price of the bond is $32PA,$ and the arbitrage$-$free price of the stock is

$44PA.$

Securities C$,$ D$,$ and E are also priced fairly at $60PA,22PA,$ and $20PA,$ respectively.

There are no arbitrage opportunities in this market.

Note: if you compute the determinant using computer, you may have not an exact result

due to numerical accuracy. For instance, if the true answer is $0,$ you may get a very small

number instead but not exactly $0.$

Round your answers to $4$ decimal digits.

a$)$ Are the stock, bond, and security C payoffs linearly independent?

b$)$ Find the price of the fair weather atomic security.

c$)$ Is the market complete?

d$)$ Calculate the arbitrage$-$free price of the apple tree.

e$)$ Verify that the principle of value additivity holds.

f$)$ Calculate the discount factor.

g$)$ Calculate the risk$-$free interest rate.

h$)$ Suppose a new security F is added to the market. Its pays $70GA,40FA$ and $15BA.$

Its price is $30PA.$ Are there any arbitrage opportunities?