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Page Limit:  15 pages. Your answers, excluding the Cover Sheet, should not exceed this page limit. Please note that a page is one side of an A4 sheet with a minimum margin of 2 cm from the top, bottom, left and right borders of the page.  The submission can be handwritten or typed, but the font size should be no smaller than the equivalent to an 11pt font size. This page limit is generous to accommodate students with large handwriting. We expect most of the submissions to be significantly shorter than the set page limit. If you exceed the maximum number of pages, the mark will be reduced by 10 percentage points, but the penalised mark will not be reduced below the pass mark and marks already at or below the pass mark will not be reduced.

Number of Questions Answered Policy: In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers). All remaining answers will be ignored.

Answer one questions from Part A and one question from Part B.

The question in Part A will carry 50 per cent of the total mark, and the question in Part B will carry 50 per cent of the total mark.

PART A

Answer one question from this section.

A.1 Your fund is considering trading 10-year bonds issued by the Austrian government, and you see that at 9:30 a.m.  their lowest ask price is e102.31 and their highest bid price is e99.50.  Five seconds later a buy order for a block of e10 billion is executed at e102.76. At 10:30 a.m. you check the market again and see that the lowest ask price is e102.55 and their highest bid price is e100.02.

(a) Compute the (absolute and relative) quoted spread at 9:30 and at 10:30.  (b) Compute the (absolute and relative) effective ask-side half-spread at 9:30.

(c) Compare the quoted half-spread with the effective (relative) ask-side half-spread at 9:30. What may explain this difference between them?

(d) Compute the (absolute) realised spread in the 9:30-10:30 interval.

(e) Compare the realised spread computed under (d) with the (absolute) effective spread at

9:30 computed under (b). What may explain the difference between them?

A.2 Effect of fundamental volatility on liquidity and price discovery in the Glosten-Milgrom model.

Consider the multi-period Glosten-Milgrom model, where at time t the probability that market makers assign to the value of the security, v, being vH  is θt  and that of it being vL  is 1 − θt . Market makers are competitive and risk-neutral, and do not know v .  In each period, a single trader comes to the market: with probability 1 −π, he is a noise trader, who buys or sells 1 unit with probability 1/2 each; with probability π he is an informed trader, who knows security’s true value and so he buys when v = vH  and he sells when v = vL .  Notice that in this model we can measure the volatility of the fundamental value by vH − vL , i.e. by the range of the two possible values that the final value of the security can take.  Recall that, in this model, upon receiving a buy order at time t, the dealers’updated expectation of the security’s value is the weighted average of vH  and vL , where θ and 1 − θ are the updated probability weights:

µt(+) = θvH + (1 − θ)vL ,

while upon receiving a sell order at time t, the dealers’ updated expectation of the security’s value is the weighted average of vH  and vL , where θ and 1 − θ are the updated probability weights:

µt(−) = θvH + (1 − θ)vL ,

where

θ   =         (1 + π)       θt − 1 ,

θ   =                                      θt − 1 .

Moreover, recall that the transaction price at date t is

pt =    

Hence the transaction price can be written as

pt = µt = θtvH + (1 − θt)vL ,

where θt  = θ upon receiving a buy order at time t, and θt  = θ upon receiving a sell order at time t.

(a) Using only these expressions, compute an expression for the equilibrium bid-ask spread at any time t, and show how it varies as a function of fundamental volatility vH − vL . What is the intuitive explanation for your finding?

(b) Again using only these expressions, compute an expression for the squared pricing error at time t, first under the assumption that the true value of the security is high, i.e.‘pt− vH ) 2 , and then under the opposite assumption that the true value of the security is low, i.e. ‘pt− vL ) 2 .   In both cases, show how the squared pricing error varies as a function of fundamental volatility vH  − vL , for given θt .  What is the implication of your finding for the speed of price discovery? What is its intuitive explanation?

(c) Go back to the two expressions‘pt− vH ) 2 and‘pt− vL ) 2 that you have derived under point b, and recall that the prior probability of the security being high-valued or low valued is 1/2, i.e.  θ0  = 1/2.  If the true value of the security is vH , how will θt  and the squared pricing error‘pt− vH ) 2  behave over time? If instead the true value of the security is vL , how will θt  and the squared pricing error‘pt− vL ) 2  behave over time?

PART B

Answer one question from this section.

B.1  (Past Exam, 2018. Question B.1) 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 has an endowment of one unit of the good at time t = 0 and nothing at dates t = 1, 2. There are two assets, a short asset and a long asset.  The short asset produces one unit of the good at date t + 1 for every unit invested at date t = 0, 1.  The long asset produces R = 1.5 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 = 0.5.

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 a Von Neumann-Morgenstern utility function, with

u(c) = ln c.

Suppose there is a bank operating in a perfectly competitive sector (e.g., 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? That is, what portfolio and consumption levels will the bank

choose?

(b) Explain why with this solution a bank run is possible.

(c) Suppose the government wants to introduce a deposit insurance. What levels of consump- tion (c1 , c2 ) should it offer to the consumers to avoid a bank run?

(d) Finally, suppose agents’preferences are represented by u(c) = −  .  Is a bank run still possible?

B.2  (Past Exam, 2015. Question B.2) Consider the model of financial contagion of Allen and Gale.

Specifically, consider an economy with three dates t = 0, 1, 2 and a single, all-purpose good at each date. There are four regions in the economy. In each region there is a continuum of identical banks. In each region 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. 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 assets, a short asset and a long asset. The short asset produces one unit of the good at date t + 1 for every unit invested at date t = 0, 1. The long asset produces R = 1.5 units of the good at date 2 for every unit invested at date 0 (and produces the amount r < 1 at date 1). At date 0 each agent is uncertain about his preferences over the timing of consumption.  With some probability 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.

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 .

The following table indicates the proportion of early consumers in each region depending on the state of nature (the letters A, B , C , D indicate the four regions):

Note that the average proportion of early (and late) consumers in the entire economy is 0 .5 in either state of nature. Each agent has preferences represented by a Von Neumann-Morgenstern utility function, with

u(c) = ln c.

(a) Determine the efficient solution (optimal risk sharing), that is, the investment in the long

and in the short asset and the level of consumption for early and late consumers.  Is this solution incentive compatible?

(b) Now suppose that the representative bank in region A operates without any relation with

the banks in the other regions.   Suppose the bank decides to implement the first best, that is, to choose the same investments in the short and the long asset and the same levels of consumption as under part a.  Suppose that in this economy only essential crises (bankruns) occur (that is, whenever there are multiple equilibria, late consumers coordinate on the equilibrium in which they wait).  Suppose that state S1  occurs.  Find out whether there will be a bankrun for r = 0.4.

(c) Explain what financial contagion means in this economy.

(d) What is the role of interbank deposits in achieving an optimal allocation of risks?

(e) Do interbank deposits help provide liquidity when there is an aggregate excess demand for

liquidity in the system?