Problem B:Alpha is deciding whether to invest $1,000 in a mutual fund. In one year,the mutual fund gives a return of 20% with probability 25%. It gives areturn of 10%with probability 50%. Finally, there is a 25% chance that the mutual fund will fall invalue and Alpha will lose 80% of his investment in that time period.

1. Suppose Alpha invests $1,000 in the fund. LetXbe the random variable denotingthe total value of Alphas investment after one year. (Note thatXis different fromthe random variable that denotes the return on Alphas investment.) What is theexpected value ofX?(0.5)2. Suppose Alpha evaluates his investment in accordance with prospect theory. Recallthat the value of any lottery (p1, x1;p2, x2;. . .;pn, xn) under prospect theory equals i=1wiv(xi) +ni=k+1wiv(xi), wherex1, . . . , xkare less than the reference pointrandxk+1, . . . , xnare greater thanor equal tor. In this problem, we assume that. v(x) ={(xr)0.8ifxr(rx)0.8ifr > x,where >1,r= $1,000 is the reference point, andxis the total value of Alphasinvestment after one year. We assume that the probability weighting functions aresuch that== 0.6. That is,g(z) =l(z) =z0.6(z0.6+ (1z)0.6)10.6,for allz[0,1].

(a) Show that Alpha is loss averse.(0.5)

(b) What is Alphas value of investing in the mutual fund if= 2?(2)

(c) Find the value ofsuch that Alpha is indifferent between investing and notinvesting in the mutual fund.(2)