Question

The government of Bolivia takes out a loan of fixed size (L) and agrees to pay r interest on it. The loan is extended by a group of large banks in New York. The loan is due in one year’s time. If Bolivia chooses to default, it will lose a fraction c of its income (Y ) due to trade sanctions and other financial distress. The rest will be consumed. Otherwise, the country can pay principal and interest on the loan and consume whatever is left. The level of income in one year’s time is unknown when the terms of the loan are decided. Both lenders and the borrower face the same uncertainty over its eventual realization.

(a) Write down expressions for the levels of consumption associated with defaulting and re- paying, respectively. Use a diagram to argue that default will happen if and only if income in one year’s time is below a certain threshold level. Solve for that threshold.

(b) Describe how the likelihood of default changes with the size of the loan, the share of income lost due to default, and the interest rate owed.

(c) The US banks extending this loan can borrow and lend freely at 3% per year, to each other and to other risk-free customers. Moreover, their analysts estimate Bolivia’s default probability, given the size of the loan, to be 2%. What is the lowest interest rate r that the banks can charge Bolivia and still break even in expectation?

(d) In no more than 1 paragraph, explain intuitively why this setup might have two equilibria, one in which the interest rate is high and another in which Bolivia borrows cheaply.

Answer 1

PART A

Use a diagram to argue that default will happen if and only if income in one year’s time is below a certain threshold level. Solve for that threshold.

expressions for the levels of consumption associated with defaulting and repaying

According to J. R. McCulloch, “Consumption … is, in fact, the object of industry” (Mc Culloch, 1824).

In the given case; In case of default

INCOME(Y) = LOSS DUE TO DEFAULT(c)+ CONSUMPTION (C) hence

level of CONSUMPTION (C) = INCOME(Y) – LOSS(c) [EQUATION A]

In case of repayment:

INCOME (Y)= PRINCIPAL PAID+ INTEREST PAID(R%) +CONSUMPTION(C)

CONSUMPTION (C)= INCOME(Y)- L(1 + R/100) [EQUATION B]

Since in case of default there is a fixed loss C as constant of the part of total income hence we have assumed it to be constant factor.

While in repayment the principal and interest are to be repayed out of the income which depend on the rate of interest.

Total amount to be repayed = L +LR/100 = L(1 + R/100). Which is time dependent hence variable

So by equation A and B IF INCOME (Y) and consumption are assumed to be equal the THRESHHOLD POINT will be:

Loss (c) = L(1+R100) [EQUATION 3]

At this point either default or repayment will have the same effect. Hence if the income levels are greater than this point the loan will never be Defaulted.

PART B

Description of how the likelihood of default changes with the size of the loan(L), the share of income lost due to default(c), and the interest rate owed(R).

Loss (c) = L(1+R100) [EQUATION 3]

As per equation 3, for constant income and consumption levels, the size of the loan is directly proportional to the fraction of income loss(c) . If c is kept constant in that case the likelihood of default will increase. Increasing the loan ticket size will also increase the interest paid (LR/100) and share of income lost will also increase(c).

PART C

1. The US banks extending this loan can borrow and lend freely at 3% per year, to each other and to other risk-free customers. Moreover, their analysts estimate Bolivia’s default probability, given the size of the loan, to be 2%. What is the lowest interest rate r that the banks can charge Bolivia and still break even in expectation?

BREAK EVEN POINT : assuming n=1

break even point (1+r)= (1+0.03) /(1+0.02)

1+r = 1.0098

r= 0.98 is the lowest interest rate r that the banks can charge Bolivia and still break even in expectation

PART D

This setup might have two equilibria,

EQUILIBIRIUM 1: the interest rate is high

When the ROI is high the share of income lost due to default is also high as these are directly proportional. So the repayment and default of the loan depend on the expected income

EQUILIBIRUM 2: Bolivia borrows cheaply.

When the ROI is low the share of income lost due to default is also low as these are directly proportional. So it is easy for Bolivia to repay the dues also.