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? (p+q)
qd (p+q)
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
?=?p?(3)
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
?1/?and
??wa, w=0,
d1/d,(1)
p+(p) =?
U(w) =-?(-w)a, w
w+(f) =f?+ (1-f)?-1,
w-(f) =f?-1,
and conclude thatw-(f)=w+(f) for all 0=f=1.
(4)
(5)
[5 marks]
1
d
2
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. ? ?-?
(a) Showthatw-(f)>0ifandonlyiff>f-.
(b) Showthatw+(f)f+.
(c) Showthatf-1.
[7 marks] [11 marks]
(6)
[2marks] [2marks] [3marks]
(d) Showthatf+
(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
v1(f) =p+(p)w+a(f)-?p-(1-p)(-w-(f))a,
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=
ande= (?-?)?1-?.
??(?-?)p+(p)??1/(a-1) ??p-(1-p)
??(?-?)p+(p)??1/(a-1) ?(1-p+(p))
, ,
(?-1)?1-1 f1=(?-?)?1-?,
(?-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
v(f)??0?(?-1)?1-1??0, ??f??f1,
andiff>f-then
v'(f)??0?f??f2.
e >0, e= 0, e
3
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]
(f) Ife>0,f-
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]
6.Inthisquestion,assumethat?=2,p=1/2.4,a=0.88,?=1.2,?=0.69,d=0.61,
?= 0.95, and that the function to be maximized is
??p(p)wa(f)+(1-p(p))wa(f), w(f)=0andw(f)=0,
?+++–+
v(f)=p+(p)w+a(f)-?p-(1-p)(-w-(f))a, w-(f)
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:
-fractionf,
-amountf?received on selling the fractionf, -remaining fraction 1-f,
-net profit on successw+(f),
-net profit on project failurew-(f),
-function valuev(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
above.[2 marks]
(c) At what value offin your table doesv(f) attain its maximum value, and what is the corresponding value ofv(f)?[1 marks]
