Exercises Flashcards
w1e1. An Economy is composed of 10 agents. Each agent is endowed with 1 thousand € today, and she
knows that she will need to consume either next year (t1) or the year after (t2). Agents find out whether
they need to consume in t1 or t2 only at t1. However, it is common knowledge that the share of people
who will need to consume at t1 is 50%. In the economy there is one investment opportunity today. The
investment has a 2-year expected rate of return of 60%. However, if liquidated in t1, only 80% of the
invested value can be recovered. It is possible to liquidate only part of the investment.
a) Describe what happens in case an efficient financial market but no financial intermediaries
exist. Compute the number of zero-coupon bonds with a face value of 100 € and their market
price in t1, and the level of consumption in t1 and t2 for each agent.
b) Knowing that agents consuming next year would like to spend 1.2 thousand € then, show how
a bank can allow for this, how much the bank invests today and how much will be consumed by
agents who wait t2 to consume.
Bond market. Every agent invest 50% of its wealth
The equilibrium price is 1/R = 1/1.6 = 0.625 → 0.625 x 100 € = 62.5 €
Agent 1 can return 0.51,0001.6=800 € in t=2 → she issues 8 bonds
She gets 62.5 € x 8 = 500 € → She consumes 500 +500 = 1,000 € in t1
Every Agent 2 buys 8 bonds, and consumes in t2: 5001.6 + 800 = 1,600 €
In total, there are 8 x 5 = 40 bonds in the market.
5,000 € are consumed in t1, and 8,000 € in t2
Banks
Each agent deposits 1,000 € with the bank.
The bank needs 1,200 € in t1 for 5 of them → it keeps 6,000 € liquid, it invest the rest (4,000 €)
In t1 there are 1,200 € for each Agent 1. In t2 there are 4,000 x 1.6 = 6,400 € /5 = 1,280 €*
w1e2. A risk-neutral firm is considering to make an investment, and needs to finance it via debt capital. There
are two possible projects; both need an initial investment of 1 million €. Project G will return 1.6
million € with a probability of 65% and 0 with a probability of 35%. Project B will return 2.4 million €
with a probability of 25% and 0 with a probability of 75%. The firm has two possibilities: it can issue
bonds, or it can ask a bank for a loan. The bank has monitoring costs equal to the 2% of the loan
amount; by monitoring the firm, it can make sure that the firm implement the project with the highest
expected value. Both banks and financial markets are risk-neutral and characterized by perfect
competition.
Compute:
a. The maximum interest rate on debt over which the firm will choose the inferior project
b. The coupon rate on bonds that financial markets would ask
c. The interest rate on loans that banks would ask
Does the company get financed? By whom? What is the expected return for the firm?
Expected return of G: 0.65 x 1.6 = 1.04 million €
Expected return of B: 0.25 x 2.4 = 0.6 million €
The firm implements G IFF 0.65 x (1.6 – R) > 0.25 x (2.4 – R) → Max R is 1.1 or 10% interest rate.
Financial markets want 0 expected return so: Rfm = 1/0.65 = 1.538, or 53.8% interest rate
With this level of interest rates, the firm would surely invest in the bad project → Financial markets
know it and thus would ask 1/0.25 = 4 or 300% interest rate! The project will never generate enough
returns to repay this amount → No financing possible via bonds issuance
If banks can force the firm to implement G at a cost of 0.02, they will ask a return of (1 + 0.02)/0.65 =
1.569, or 56.9% interest rate.
If the firm accepts, it will get €0 return with probability 35% and 1.6 – 1.569 = €0.031 million with
probability 65%. The firm’s expected return is 0.65 x 0.031 = € 0.02 mln
The company gets financed via bank lending with an interest rate of 56.9%
w1e3. A risk-neutral firm is considering to make an investment, and it needs to finance it via debt capital.
There are two possible projects, denominated X and Y; both need an initial investment of 10 million €.
Project X will return 40 million € with a probability of 20% and 0 otherwise. Project Y will return 25
million € with a probability of 50% and 0 otherwise. The firm has two possibilities to finance the
investment: it can issue bonds, or it can ask a bank for a loan. A bank always charge monitoring costs
equal to 2.6 million €; by monitoring the firm, the bank can make sure that the firm implements a
specific project (either X or Y). Both banks and financial markets are risk-neutral and characterized by
perfect competition. Under these assumptions:
a. Show whether there is the potential for asset substitution behaviour by the firm (i.e., whether
moral hazard is a concern)
b. Compute which interest rate financial markets would ask to finance this company
c. Compute which interest rate a bank would ask to finance this company
d. Does the company get financed? If yes, by whom?
e. Compute for which level of monitoring costs the firm would be indifferent as to undertake or
not an investment financed by the bank
a.
Project X → E[X] = 40 * 0.2 = € 8 mln < € 10 mln
Project Y → E[Y] = 25 * 0.5 = € 12.5 mln > € 10 mln
There is a potential for moral hazard problems because E[Y] > E[X] but max[X] > max[Y]
b.
Company undertakes Y IFF R < [12.5 – 8] / [0.5 – 0.2] = € 15 mln
Equilibrium R if Y undertaken → 10/0.5 = € 20 mln > 15 mln → Company would surely do X R with X would have to be 10/0.2 = € 50 mln, more than X can create. Company does not get financed
c.
R = (10 + 2.6) / 0.5 = € 25.2 mln. → interest rate = (25.2/10) – 100% = 152%
d.
Also the bank does not finance the company, since 25.2 > 25. No one finances the company
e.
Indifference when R = € 25 mln → (10 + C)/0.5 = 25 → C = € 2.5 mln
(Q10 of Mock Exam)
Consider the theoretical model for banking as a monitoring device. If both banks and the market are willing to
finance the company, then the company:
(A) Will ask for a bank loan because the bank knows the company better
(B) Will ask for a bank loan because the interest rate is lower
(C) Will issue bonds because it can then engage in asset substitution
(D) Will issue bonds because the interest rate is lower
Will issue bonds because the interest rate is lower
w2e1
An Economy is composed of 10 agents. Each agent is endowed with 1 thousand € today, and she
knows that she will need to consume either next year (t1) or the year after (t2). Agents find out whether
they need to consume in t1 or t2 only at t1. However, it is common knowledge that the share of people
who will need to consume at t1 is 50%. In the economy there is one investment opportunity today. The
investment has a 2-year expected rate of return of 60%. However, if liquidated in t1, only 80% of the
invested value can be recovered. It is possible to liquidate only part of the investment.
Show what happens in presence of a bank if the actual proportion of
agents who need to consume next year unexpectedly jumps to 60%. Compute the social loss.
Banks (recap)
C1 = 1,200 (5 agents → 6,000 € liquid)
Invest 4,000 € → x 1,6 = 6,400 € available in t2
C2 = 6,400 € /5 = 1,280 €
Bank run
Bank needs to pay 6 x 1,200 € = 7,200 €, it only has 6,000 € → needs 1,200 €
Needs to liquidate X% of the investment such that X 4,000 € 0.8 = 1,200 € →
X= 1,200/(4,000 x 0.8) = 37.5 % → Liquidate 4,000 x 0.375 = 1,500 €
Keep invested 4,000 € -1,500 € = 2,500 €
In t2 the bank gets 2,500 x 1.6 = 4,000 €
There are 4 impatient agents left → 4,000/4 = 1,000 € for every agent (< 1,200)
Since 1,000 is less than 1,200 (C1) → Bank run!
Bank liquidates everything, it has 6,000 € + 0.8 x 4,000 € = 6,000 + 3,200 = 9,200 €
Every agent gets 920 €, which does not cover the initial deposit → Bank defaults.
Social loss = (1.6 – 0.8) [1- (0.5 x 1.2)] = 0.32 for every € → 3,200 €
Equivalently, Social Loss = 6,000 + 6,400 – 9,200 = 3,200 €
w1e2
Consider an economy where 50% of agents will consume in t1 and the other 50% in t2. There are
several households in the economy, each with 1 thousand € in a bank account as of today. Each family
wants to consume 1.2 thousand € if it has to do it in t1. There are also several banks in the economy.
All banks are risk neutral and compete perfectly. Each bank has 10 clients, and expects to have the
same proportion of agents in need for consumption in each of the next two years as the whole
economy. Banks invest in 2-year illiquid projects with a rate of return of 60% and a liquidation value
equal to 80% of the amount invested. In t1, a bank called A-bank finds out that 8 of his clients want to
close their accounts. At the same time, another bank, called B-bank, finds out that only 3 of its clients
want to close their accounts. Finally a third bank, called C-bank, experiences 4 clients withdrawing all
their savings. All the other banks in the economy face the expected number of withdrawals. Show what
will happen in t1 and t2 if:
* Both B-bank and C-bank decide to make a loan to A-bank. What is the interest rate in the interbank
market?
* C-bank does not trust A-bank and refuses to lend to it.
Already computed in Ex1 of Exercises A (same data)
C1 = 1,200 → Bank keeps 6,000 liquid, invest 4,000
4,000 x 1.6 = 6,400 → C2 = 6,400/5 = 1,280
Case 1
* A-bank has invested 4 thousand €, and has liquidity for 6 thousand €.
* It needs 3 x 1,200€ = 3,600 € more, i.e. it needs to pay out 9.6 thousand € in t1
* B-bank and C-bank lend 3.6 thousand € to A-bank in t1.
* In t2, A-bank gets 4,000 x 1.6 = 6,400 € from the investment
* It pays 2 x 1,280 = 2,560 € to its depositors who waited
* It is thus left with 6.4 – 2.56 = 3.84 thousands €, which is what it can promise to pay to B-bank
and C-bank
* The rate of return on the interbank market is thus (3.84/3.6) – 1 = 6.67%
Case 2
* B-bank can still decide to lend to A-bank.
* But if it does, it can give no more than to 2.4 thousand €.
* A-bank still needs 1.2 thousand €, it has to liquidate part (37.5%) of the investment
* As in Ex1, the remaining investment is (1 – 0.375) x 4,000 = 2,500
* In t2 A-bank has only 4,000 € left (2,500 x 1.6). It has to give 2.56 thousand € to its depositors.
* Deposits have higher seniority than the debt contracted with B-bank
* A-bank pays depositors, only 4 – 2.56 = 1.44 thousand € left to repay the debt
* A-bank clearly cannot repay the debt → B-bank will not lend to it → A-bank would not be able
to repay its patient depositors → bank run
w2e3
A bank has lent out 100,000,000 € with a yearly interest rate of 10%. The loans have a 2.5%
probability of defaulting entirely. The bank as 80,000,000 € of deposits with no interest rate, and
Equity capital for 21,000,000 €.
Compute:
a. The Expected rate of return on Equity
b. The net subsidy from deposit insurance in €
c. The fair price for deposit insurance
Imagine that the price of insurance does not change next year. Do you think the bank will:
1) Issue more equity?
2) Try to attract more deposits?
3) Try to finance less risky projects?
Insurance premium: (80+21) – 100 = 1 mln €
Loans are expected to return 100,000,000 x (1+ 0.1) x (1 – 0.025) = 107.25 mln €
The Return (NPV) on loans is thus 107.25 – 100 = 7.25 mln €
The fair price of insurance would be 0.025 x 80 = 2 mln €
The expected return going to shareholders is thus 7.25 + (2 – 1) = 8.25 mln €
The expected ROE is = 8.25/21 = 39.3%
The bank receives an effective subsidy from the insurer. For a given level of assets, it has an incentive
to:
* increase deposits
* reduce equity
* increase the level of risk of its assets.
In a small country live 2 million people, each with a deposit of 10,000$ at a local bank. The bank has just made
a loan for a 10-year project which is expected to give a total return of 260% (i.e., a rate of return of 160%), all
paid at the end of the project period. The bank has the right to liquidate (part of) the project at any point in
time before the 10-year period expires; however, the recovery rate in that case is only 75% of the initial
investment, and no interests will be paid. 60% of the citizens are expected to withdraw their deposits in 5 years
from now; the bank pays a 5-year interest rate of 5% on deposits held with the bank for 5 years. The remaining
customers are expected to withdraw their deposits in 10 years from now. Assume that the hypotheses of the
Diamond-Dybvig model hold. Compute:
a) the amount promised to each depositor withdrawing in 5 years
b) the total amount invested by the bank and the amount kept as reserves
c) the amount promised to each depositor withdrawing in 10 years
Consider from now on that, after 5 years have passed, the bank finds out the actual proportion of its customers
willing to close their accounts already is not 60% but 80%.
d) Show what the bank would have to do
e) Determine whether a bank run occurs
If there is a Central Bank willing to act as a Lender of Last Resort (LLR) in case of liquidity shortages in the
banking system:
f) At what optimal 5-year interest rate should the LLR lend to this bank?
g) What would happen if the interest rate asked by the LLR is lower than the optimal one?
a) C1 = 10,000 € x 1.05 = 10,500 €
b)
Total deposits: €10,000 x 2 million = €20 billion
Impatient costumers = 60% x 2 million = 1.2 million people
Total kept as reserves = €10,500 x 1.2 million = €12.6 billion.
Total invested: 20 billion – 12.6 billion = € 7.4 billion
c)
Investment returns: 7.4 billion x 260% = €19.24 billion
Number of patient customer: 2 million – 1.2 million = 800,000 people
Each patient depositor gets €19.24 billion /800,000 = 24,050 €
→ interest rate of (24,050 – 10,000)/10,000 = 140.5%
d)
The bank needs €10,500 x 80% x 2 million = €16.8 billion
The capital shortage is thus = 16.8 – 12.6 = €4.2 billion
The bank needs to liquidate = 4.2/75% = €5.6 billion of the investment
The investment left is thus = 7.4 – 5.6 = €1.8 billion
e)
There are 20% x 2 million = 400,000 customers who have not withdrawn yet
The investment left returns = €1.8 billion x 260% = €4.68 billion
The bank can give to each customer left = €4.68 billion/400,000 = €11,700
Less than promised, but still more than with early withdrawal → No bank run!
f)
LLR: the bank doesn’t have to liquidate part of the investment. It can borrow the €4.2 billion it needs.
The investment will produce €19.24 billion
After 10 years, the bank has to pay 24,050€ to 400,000 customers = €9.62 billion
The remaining €19.24 – €9.62 = €9.62 billion should return to the central bank
The optimal interest rate is thus = (9.62 – 4.2)/4.2 = 129.05%
g)
If the CB asks for lower interest rate → extra profits for the bank. In the future banks may :
* reduce their reserves and lend more
* reduce the liquidity of their loans (lower L)
(Q2 of Mock exam)
Which of the following is more likely to be a rationale for introducing a deposit rate ceiling?
(A) It limits banks’ competition
(B) It favours alternative forms of liquid investments
(C) It favours banks’ investments in risk management
(D) It limits banks’ proprietary trading
It limits banks’ competition
(Q15 of Mock Exam)
Which of the following is not an element supporting the separation of commercial and investment banks?
(A) Banks can sell their own securities to their customers
(B) The price of traded securities is volatile
(C) Banks should have a well-diversified portfolio of assets
(D) Banks use their own capital for trading
Banks should have a well-diversified portfolio of assets
