Explain why you would use interviews vs. focus groups and when interviews are more appropriate than focus groups.
extension of the usual expected utility optimization frame- work that incorporates investors’ loss aversion and their behavioural tendency to distort the prob- abilities of gains and losses
This assignment is concerned with an extension of the usual expected utility optimization frame- work that incorporates investors’ loss aversion and their behavioural tendency to distort the prob- abilities of gains and losses.
Given the probabilitypof a gain and the probabilityqof a loss, so thatp+q= 1, it is assumed that the investor uses distorted ‘probabilities’,p+(p) andp-(q), in computing the expected utility where
p-(q) = and?, d >0. The utility function is assumed to take the form
wherea, ? >0. Suppose a representative agent has invested $1 in one unit of a project that will pay an amount? >1 if successful and 0 otherwise. At some timetprior to the completion of the project, the probability of the project succeeding is assessed to bep, and the investor is offered the amount
per $1 invested to sell part or all of their holding in the project, where 0
Throughout the assignment, denote byw+(f) andw-(f) the terminal wealth including the initial $1 investment in the event that the project is successful and unsuccessful respectively, assuming a fractionfhas been sold at timet, and assume that the risk free rater= 0.
Give a financial explanation of the amount?offered to the investor in (3), explaining the significance of?.[3 marks]
Assuming that the investor sells a fractionfof their ownership of the project, where 0=f=1, show that
U(w) =-?(-w)a, w
w+(f) =f?+ (1-f)?-1,
and conclude thatw-(f)=w+(f) for all 0=f=1.
3.Show that the only possible sign combinations for the pair (w-(f), w+(f)) are (-,-), (-,+),
and (+,+). Explain situations under which each sign combination may occur. 4.Letw-(f) andw+(f) be as defined in Question 2, and definef-andf+by
f-=1, f+=?-1. ? ?-?
[7 marks] [11 marks]
[2marks] [2marks] [3marks]
(e) Explain in words the significance off-andf+in relation to the terminal net profit for the investor.[3 marks]
5.In this question,assumef-=1 and 0
??p+(p)w+a(f)-?p-(1-p)(-w-(f))a, f=f-, v(f)=p+(p)w+a(f)+(1-p+(p))w-a(f), f >f-.
For future reference, letv1(f) andv2(f) be defined by
v2(f) =p+(p)w+a(f) + (1-p+(p))w-a(f),
so thatv(f) =v1(f) forf=f-, andv(f) =v2(f) forf > f-. Moreover, define
(?-1)?2+ 1 f2=(?-?)?2+?,
(a) Explain whyv(f) must be split into 2 cases.
[29 marks] [2 marks]
(b) Compute the derivativesw+'(f) andw-‘(f), and show thatw+'(f)0
for 0=f=1.[2 marks]
(c) Differentiatev(f) with respect tofto computev'(f), taking care to consider the two
regionsf=f-andf > f-separately, to show that[4 marks] ??a(?-?)p(p)wa-1(f) +a??p(1-p)(-w(f))a-1, f=f ,
v'(f)=++- – – a(?-?)p(p)wa-1(f)+a?(1-p(p))wa-1(f), f>f .
+++– (d) Using the results from part (c), show that iff=f-then
??f??f , ‘?1
e >0, e= 0, e
Note that you will need payclose attention to the signsof various quantities in deriving these inequalities.[12 marks]
(e) What is the significance of the pointsf1andf2?[2 marks]
attains its maximum value atf2.[4 marks]
(g) Ife >0,f-< f1, andf2>1, what is the optimal fraction at whichv(f) attains its
maximum value. You must provide a brief justification for your answer.[3 marks]
?= 0.95, and that the function to be maximized is
Create anExcelspreadsheet to plotv(f) as a function off. In creating the plot, construct, apart from other cells you may need to create, the following columns in the given order:
-amountf?received on selling the fractionf, -remaining fraction 1-f,
-net profit on successw+(f),
-net profit on project failurew-(f),
The plot should showv(f) fromf= 0 tof= 1 at steps of 0.025. Format theverticalaxis of your plot so that the minimum value is-0.32 and the maximum value is-0.2.[5 marks]
(a) Copy and paste the first five rows of the six columns specified above, with all values rounded to 5 decimal places.[2 marks]
(b) Copy and paste the plot, and make sure you have formatted the vertical axis as specified
(c) At what value offin your table doesv(f) attain its maximum value, and what is the corresponding value ofv(f)?[1 marks]