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ECOS3022
S2 2022
Final exam
Time Limit: 120 minutes
This exam contains 3 pages and 3 questions. Total of points is 50.
1. Consider an asset economy with two agents A and B. They live for two periods t = 0 and 1. At t = 1, there are two possible states of the world: s1 and s2. They both agree that π 1 = 2/3 and π2 = 1/3. Agents’ endowed wealth are as follows: w0(A) = w0(B) = 0, w1(A) = 1, w2(A) = 4, w1(B) = 6 and w2(B) = 3.
Agents have access to a financial market that is complete and arbitrage-free.
The two agents seek to maximize their expected utilities at t = 1 with vA (y) = √y and vB (y) = lny.
(a) (1 point) Is there aggregate risk in this economy? Why or why not?
(b) (3 points) Suppose the Arrow security prices are (α1 ,α2 ). Let ys(i) denote investor i’s wealth level in state s. What is agent A’s budget constraint? What is agent B’s budget constraint?
(c) (3 points) Plot the Edgeworth box for this economy. Your graph should include the endowment, iso-expected value lines, and indifference curves for A and B.
(d) (3 points) Without solving agents’ maximization problems, what is the equilibrium allocation of wealth? Why?
(e) (2 points) If α1 = 1, what is the equilibrium α2 ?
(f) (3 points) Suppose the financial assets traded are as follows.
' r1 r2 '
r = 's1 0 1 '
's2 10 7 '
What are the asset prices?
2. Consider an asset economy with two agents A and B. They live for two periods t = 0 and 1. At t = 1, there are two possible states of the world: s1 and s2. They both agree that π1 = 1/2 and π2 = 1/2. Agents’ endowed wealth are as follows: w0(A) = 3, w0(B) = 3, w1(A) = 2, w2(A) = 6, w1(B) = 3 and w2(B) = 5.
Agent i = A,B seek to maximize
v(y0(i)) + δE[v(y1(i))],
where δ = 0.8 and v(y) = 2lny.
(a) (7 points) Compute the equilibrium prices of wealth and the equilibrium allocation of wealth.
If you cannot solve part (a), proceed with Arrow security prices α1 = 0.46,α2 = 0.29.
(b) (1 point) What is the equilibrium price of the risk-free bond?
(c) (1 point) Calculate the values of the risk-neutral probabilities (˜(α)1 , ˜(α)2 ).
(d) (3 points) Compare the values of (˜(α)1 , ˜(α)2 ) with those of (π1 ,π2 ). Explain the differ- ence.
(e) (2 points) What is the economic meaning of stochastic discount factors (SDF)?
(f) (1 point) Compute the stochastic discount factors for this economy.
(g) (2 points) State the relationship between discount factors and the mean endowment vectors.
(h) (2 points) Does the above relationship hold for either agent? Compare your answers in (a) and (f), and explain your observation.
3. Consider an asset economy with two states of the world and two financial assets. Assets’ payoffs are summarized in matrix r below. (Rows are states, and columns are assets.) The probability of the two states are (π1 ,π2 ) = (1/2, 1/2). The prices of the assets: q1 = 1.9 and q2 = 1.6. The investor’s initial wealth is w0 = 10.
' r1 r2 '
r = 's1 3 4 '
's2 5 2 '
Let (z1 , z2 ) be the portfolio held by the investor.
(a) (1 point) What is the mean of the portfolio µ(z1 , z2 )?
(b) (2 points) What is the standard deviation of the portfolio σ(z1 , z2 )? You need to show steps.
(c) (1 point) What is the investor’s budget constraint?
(d) (3 points) Derive the efficiency frontier.
(e) (2 points) Plot the efficiency frontier on the σ − µ plane.
(f) (5 points) Suppose the investor maximizes U(µ, σ) = µ−2/1σ2 . What is the optimal (µ, σ) pair?
(g) (2 points) What is the optimal portfolio (z1 , z2 )?