ECOS3022 practice exam
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ECOS3022 practice exam
Q1 (25 marks)
Consider a pair of investors A and B: There is only one good - wealth in their economy. A or B only care only about date t = 1 wealth. (You can assume δ = 1). At date t = 1 the economy will be either in state 1, or in state 2. The investors agree that state 1 is twice as likely as state 2. In state 1 A will have wealth of 10 and B will have 0: In state 2 A will have 0 and B will have 5: At date 0 neither investor has any wealth but both have access to the Önancial markets with the following payo§ matrix
assets
states ┌ 3(1) 4(3) ┐
Investor B is risk-averse and values his wealth y according to v(y) = 2 ′y . Investor A is risk-neutral.
a: (1 mark) Are Önancial markets complete here? Explain what does this mean and verify
b: (1 mark) Denote with (q1 ;q2 ) the prices of the above assets. Let q1 = 1 and q2 = q: Use (z1(A);z2(A)); (z1(B);z2(B)) for the asset holdings of investors A an B correspondingly. In these notations write the t = 0 date budget constraints for investors A and B:
c: (2 marks) Denote with y 1(A) and y2(A) the state levels of wealth that investor A
can reach. Given her wealth endowment and the fact she trades at the markets above
what trade o§ between y 1(A) and y2(A) does A face? You should provide the equation that connects y2(A) to y 1(A) .
d: (2 marks) Denote with y1(B) and y2(B) the state levels of wealth available to investor B: Given B ís wealth endowment and the fact she trades at the markets above what trade o§ between y 1(B) and y2(B) does B face? You should provide the equation that connects y2(B) to y 1(B) .
e: (2 marks) Formulate the investorsíoptimization problems in this economy.
f: (3 marks) What price q will result in equilibrium? Your answer should, of course, respect the budget constraints and the preferences of the investors.
g: (2 marks) What equilibrium allocations of wealth over states would result for the risk-averse investor B? Provide the corresponding optimal ╱y1(B);y2(B)、:
h: (2 marks) What equilibrium allocations of wealth over states would result for the risk-neutral investor A? Provide the corresponding optimal ╱y1(A);y2(A)、:
i: (5 marks) How are these allocations achived through the assets holdings? Provide the corresponding optimal (z1(B);z2(B)) and (z1(A);z2(A)).
j: (2 marks) What are stochastic discount factors (SDF) in general and what role do they play?
k: (1 mark) Calculate the values of the stochastic discount factors for the two states. Hint: you need to recover the connection between the SDFs and Arrow security prices.
l: (2 marks) Who is the representative agent in this economy? Interpret the values for SDFs you have obtained in j: using the representative agent approach.
Q2 (20 marks)
Consider an economy with two assets and two states of the world such that the matrix of payo§s of the two assets is
assets
states ┌ 
3(2) ┐
Suppose that the prices of the assets are (q1 ;q2 ) = ( 
; 2) and the probabilities of the two states are (π; 1 _ π) = (1=2; 1=2). The initial wealth w〇 = 7. We further adopt the mean variance approach
a: (1 mark) Derive the expected return of the portfolio, μ(z1 ;z2 ). This should be a function of (z1 ;z2 ) :
b: (2 marks) Calculate the standard deviations σ 1 and σ 2 of assets 1 and 2 and the assetsícovariance σ 12 . Recall the notation σ 11 = σ 1(2) and σ 22 = σ 2(2) :
c: (2 marks) Derive the +tXndXod deniXtiОn of the portfolio σ (z1 ;z2 )? This should also be a function of (z1 ;z2 ) : Recall that the nXoiXnce of the portfolio is given by the expression σ2 (z1 ;z2 ) = σ 11 z 1(2) + 2σ12 z1 z2 + σ 22 z2(2) :
d: (1 mark) Write the investorís budget constraint?
e: (5 marks) Derive the e¢ ciency frontier of the portfolio (z1 ;z2 ). (The frontier should express μ(z1 ;z2 ) in terms of σ (z1 ;z2 )). Hint : you should use the expressions for μ and σ from a: and d: and the investorís budget constraint in e:
f: (5 marks) Suppose the investor has the utility function U (σ;μ) = μ _ 
μ2 _ 
σ 2 ; in the range where it is increasing in μ. What is this investorís optimal portfolio (σ;μ). Hint : you can formulate this optimization via the Lagrangean approach.
h: (4 marks) Plot the frontier you have derived in e on the graph in (σ;μ) space. Plot the optimal portfolio you have derived in f on this graph. Plot also the two portfolios where all the wealth is invested in one of the assets.
2022-05-17