$1.$ Explain in detail the following terms$/$concepts and provide intuitive examples.

How does prospect theory take them into account?

a$.$ Fourfold pattern of risk attitudes

b$.$ Risk aversion and loss aversion

c$.$ Willingness to pay and willingness to accept

d$.$ Framing as gains and framing as losses

$2.$ Three investors have bought ETFs at a value of $$40,000.$ The price went up to

$$70,000$ and then dropped to $$50,000,$ at which time they sold it$.$

All three are loss averse and have the following value function:

$v(w)=w$ when $w>0$

$=2w$ when $w<0$

The first investor uses the purchase price as her reference point. The second uses

the peak price as her reference point, and the third uses the sale price as reference

point. What is the change in value for each person? Who is happiest?

$3.$ Suppose that Amy has the following value function:

$v(w)=w^{\frac{3}{4}}$ when $w>0$

$=-2(-w{)}^{\frac{3}{4}}$ when $w<0$

$,$ where $x$ is the realized outcome relative to the reference point. Assume that the

Amy's reference point is the status quo and that she is offered the prospect P$1$:

$P1(0\mathrm{.}6,$$$100,-$$$100)$

a$)$ Will she participate in the prospect?

b$)$ Suppose that Amy participates in the prospect and loses $$100.$ She feels bad

about it and perceives it as a loss. Her reference point is unchanged at the

original status quo. She is offered the prospect again, does she accept this

second time?

c$)$ Amy is offered a new job for which she gets a $$10,000$ sign$-$on bonus. She

has in consequence a new wealth level and changes her reference point.

Would she accept now the prospect $1?$