ECONOMICS OF FINANCIAL MARKETS

24-HOUR ONLINE EXAMINATION
ECON0001: ECONOMICS OF FINANCIAL MARKETS
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ECON0001 1 TURN OVER
PART A
Answer TWO questions from this section.
A.1 Consider an economy with three dates t = 0, 1, 2 and a single, all-purpose good at each date.
There is a continuum of ex-ante identical agents of measure 1. Each agent has an endowment of
one unit of the good at date t = 0 and nothing at dates t = 1, 2. At date 0 each agent is uncertain
about his preferences over the timing of consumption. With probability 1/2 he expects to be an
early consumer, who only values consumption at date 1. With the complementary probability
he expects to be a late consumer, who only values consumption at date 2. Note that he never
values consumption at date 0. Each agent has preferences represented by
u(c) = ln(c).
In order to provide for future consumption, each agent can invest in two assets, a short asset
and a long asset. The short asset produces r0 = 1.25 unit of the good at date t + 1 for every
unit invested at date t = 0, 1. The long asset produces R = 2 units of the good at date 2 for
every unit invested at date 0 and produces r = 0 at date 1. At date 0 each agent invests the
amount x in the long asset and the amount y in the short asset. The portfolio (x, y) must satisfy
the budget constraint x + y ≤ 1. Let c1 denote the amount consumed at date 1 by an early
consumer and c2 denote the amount consumed by a late consumer at date 2. Note that he will
consume either c1 or c2 but not both.
(a) Find the feasibility constraints in the case of a benevolent social planner who maximizes
the agents’ expected utility. What is the efficient (i.e., social planner) solution for the
investment portfolio (x, y) and the consumption plan (c1, c2)?
Suppose that a financial market opens at date 1, i.e., after they learn whether they are early or
late consumers.
(b) Find the equilibrium price P of the long asset at date 1. Write down the consumption plan
(c1, c2) and the investment portfolio (x, y) in equilibrium. Explain carefully why this is an
equilibrium.
(c) Illustrate the feasibility constraints of problems in (a) and (b) in a graph. In the same
graph, show the equilibrium market allocation and discuss why it is efficient or inefficient.
(d) Suppose, now, the agents have preference u(c) = c
1−σ
1−σ
. How would the consumption plans
in (a) and (b) change as the relative risk aversion σ changes? Use the graph in (c) to discuss
(in)efficiency of equilibrium in three cases: σ > 1, σ < 1, and σ = 1 (recall that u(c) = ln(c) when σ = 1). [You do not need to compute (c1, c2). Provide a clear explanation.] ECON0001 2 CONTINUED A.2 In our economy there are three dates t = 0, 1, 2 and a single, all-purpose good at each date. There is a continuum of ex-ante identical agents of measure 1. Each agent has an endowment of one unit of the good at date t = 0 and nothing at dates t = 1, 2. At date 0 each agent is uncertain about his preferences over the timing of consumption. With probability 1/2 he expects to be an early consumer, who only values consumption at date 1. With the complementary probability he expects to be a late consumer, who only values consumption at date 2. Note that he never values consumption at date 0. Each agent has preferences represented by u(c) = ln(c). There are two assets, a short asset and a long asset. The short asset produces one unit (r0 = 1) of the good at date t + 1 for every unit invested at date t = 0, 1. The long asset produces R = 2 units of the good at date 2 for every unit invested at date 0. If instead it is liquidated at date 1, it produces r = 0.75. Suppose there is a bank operating in a perfectly competitive sector (e.g., their payoffs are zero because of free entry). At date 0 the agents deposit their endowments in the bank. The bank allocates all agents’ endowments in a portfolio of x units of the long asset and y units of the short asset. The portfolio (x, y) must satisfy the budget constraint x + y ≤ 1. Let c1 denote the amount consumed at date 1 by an early consumer and c2 denote the amount consumed by a late consumer at date 2. Note that he will consume either c1 or c2 but not both. (a) What is the banking solution (where late consumers do not withdraw deposits in period 1)? That is, what portfolio and consumption levels will the bank choose? (b) Explain whether, given the portfolio and consumption levels obtained in (a), there exists an equilibrium with a bank run. (c) Suppose that the bank wants to avoid the possibility of a bank run. Write the feasibility constraints of the bank’s problem and decide whether the constraints are binding or not. Explain your intuitions. (d) Given your answer to (c), if the bank wants to avoid the possibility of a bank run, what level of consumption (c1, c2) will it offer to the consumers? (e) Would the optimal investment y in (d) be larger or smaller than its first best value in (a)? Explain your intuitions in details. A.3 Consider an economy with three dates t = 0, 1, 2 and a single, all-purpose good at each date. There are three regions in the economy. In each region there is a competitive banking sector. In each region there is a continuum of ex-ante identical “local” agents of measure 1 (agents cannot move to a different region). Each agent has an endowment of one unit of the good at date t = 0 and nothing at dates t = 1, 2. In order to provide for future consumption, each agent deposits his endowment in the representative bank of his region. The bank can invest the deposit in two ECON0001 3 TURN OVER assets, a short asset and a long asset. The short asset produces one unit (r0 = 1) of the good at date t + 1 for every unit invested at date t = 0, 1. The long asset produces R = 2 units of the good at date 2 for every unit invested at date 0. If instead it is liquidated at date 1, it produces r = 0.75. At date 0 each agent is uncertain about his preferences over the timing of consumption. The probability of being an early or late consumer depends on the state of nature that occurs. There are two, equally likely, states of nature, denoted by S1 and S2, one of which is realized at t = 1. The following table indicates the proportion of early consumers in each region depending on the state of nature (the letters A, B, C indicate the three regions). A B C S1 0.7 0.3 0.5 S2 0.3 0.7 0.5 Note that the average proportion of early (and late) consumers in the entire economy is 0.5 in either state of nature. The proportion of early consumers in region C is the same in both states. Each agent has preferences represented by u(c) = ln(c). (a) Determine the efficient solution (optimal allocation of risk) in the case of a benevolent social planner who maximizes the sum of the agents’ expected utility in all regions and can transfer the good across regions. (b) Suppose that the interbank network is complete and that all banks are allowed to exchange deposits at date 0. Write down the amounts of interbank deposits that each bank holds in other regions. (c) Suppose the state S1 is realized. In the complete network as in (b), describe the role of interbank deposits in achieving the optimal allocation at date 1 and 2. (d) If banks A and B are not allowed to exchange deposits, is the optimal allocation possible? Explain your intuitions in details. ECON0001 4 CONTINUED PART B Answer TWO questions from this section. B.1 Consider the Kyle (1985) model, but assume that instead of a single informed trader there are N > 1 informed traders, who all perfectly observe the final value of the security v but not the
equilibrium price at the time they determine their quantity demanded xi
.
(a) Suppose that market makers post the price schedule described by equation p(q) = µ + λq,
where q is the net order flow q =
PN
i=1 xi + u and µ = E[v]. Assuming that each informed
trader uses the following order submission strategy:
xi = Xi(v) = β(v − µ) for i ∈ {1, .., N},
find the value of β for which a Nash equilibrium exists, determine how β is affected by N,
and explain intuitively why.
(b) Suppose now that investors follow the order submission strategy derived in step (a). Show
that in this case the market makers’ pricing strategy is given by equation p(q) = µ + λq,
and find the value of λ that they optimally choose.
(c) What is the market depth in equilibrium, and how is it affected by an increase in the
number of informed traders, N? What is the economic intuition for this result?
B.2 Consider a modified version of the static model by Kyle (1985) where market makers conjecture
that the informed trader, in equilibrium, may have the incentive to include a random component
in his order, thus increasing the noise in the market. Hence, their conjecture about the trading
strategy of the insider is:
x = β(v − µ) + φ,
where φ is normally distributed with mean zero and variance σ
2
φ
, (that is φ ∼ N(0, σ2
φ
) and it is
such that Cov(φ, x) = Cov(φ, u) = Cov(φ, v) = 0).
(a) Assume that competitive market makers post the following price schedule:
p(q) = µ + λq,
where q = x + u is the net order flow. Find the competitive value of λ. Is market depth
higher or lower when compared to the baseline Kyle model? Explain intuitively why.
(b) Solve the maximization problem of the informed trader, that is the value of x that solves
max
x
E [x(v − p)|v] when the informed trader conjectures that the price schedule is p = µ + λq. What is the
optimal value of φ, in equilibrium? Explain your answer.
ECON0001 5 TURN OVER
B.3 Consider the static model by Kyle (1985), where (i) market makers are risk neutral and perfectly
competitive, (ii) the asset value is v ∼ N(µ, σ2
v
), (iii) the informed investor’s order is x = β(v−µ),
and the noise traders’ order is u ∼ N(0, σ2
u
), independent of v; and (iv) market makers only
observe the total net order q = x + u. Suppose that the insider is risk-averse (with constant
coefficient of absolute risk aversion b > 0) and that he liquidates any amount of the security
that he buys at a liquidation value v + , where  ∼ N(0, σ2

), independent of u and v. At the
time of trading the informed investor knows the realization of v but not that of . This noise in
his signal implies that in taking long or short positions based on his privileged information, he
bears some risk.
(a) Write down the mean-variance objective function of the insider, and obtain his demand
function from the solution of the first order condition associated to the insider’s expected
utility maximization problem. Identify the insider’s aggressiveness β
RA.
(b) Execution risk is the risk that a transaction will execute at a price that is very far away from
recent market prices. Denoting by β
K the insider’s trading aggressiveness in the baseline
Kyle (1985) case, show that β
RA < βK (you can assume that the equilibrium value of the market impact is positive: λ > 0). What is the intuition for this result? How does it relate
to execution risk? What happens to market liquidity?
B.4 In the lecture, we have investigated the situation in which order-processing costs are assumed
to be γ per share traded. Consider the following alternative assumptions:
(a) Assume that order-processing costs are k per transaction. Compute the bid-ask spread in
this case and show that it is decreasing with the size of the transaction. Which features of
the technology of trading would lead you to think that this is a realistic model of orderprocessing costs?
(b) Assume that order-processing costs are k per euro traded. Show that the absolute bid-ask
spread is increasing in the security’s underlying value and the relative bid-ask spread is
constant, in contrast with the expressions found in the lecture where order-processing costs
are a constant γ per euro traded, irrespective of the share value.
ECON0001 6 END OF PAPER