Economics 2121 Pamela Labadie
“Risk, Return and Leverage.”
1. Securitization provides a way to diversify idiosyncratic risk. You are given an example in the notes
of a $ 100 loan with a default rate of 5 %. For 100 households, 95 pay $115 and 5 pay nothing. The
mortgage-backed security created by pooling the loans guaranteed a payment of $109.25 to each of the 100 lenders.
Now suppose that there is an unanticipated increase in the rate of default, from 5% to 10%. Determine the expected value of a loan and the standard deviation with the new default rate. The mortgagebacked security had guaranteed a payment of $109.25. Is that payment still possible with the higher default rate? Explain.
2. We will use the example above and interpret this as health insurance. Suppose there are 100 people: 95 people will be healthy and 5 will be sick, but at the time the insurance is purchased, no one knows who will be sick. Assume purchase of insurance is mandatory. If you are healthy, you incur no health costs. If you become sick, you incur costs of $115. The total health expenses for the 5 people will equal $575 (so 5 times 115). If all 100 people split the cost of the illness, what is the insurance premium paid by each person? Show that this payment is equal to the probability of illness times the cost per person, or the expected value of health costs for an individual.
An important part of the statement of the problem was the assumption insurance purchase is mandatory. Suppose that insurance purchase is not mandatory and there is a no pre-condition clause, meaning you cannot be denied insurance coverage because of a pre-condition. Explain how the absence of mandatory insurance plus the no pre-condition clause leads to an adverse selection problem.
3. The leverage ratio, which we have defined as
Assets
Capital
1
is sometimes defined as
Liabilities
Capital
Recall that
Assets = Liabilities + Capital
Divide both sides of this equation by capital. You are told
Liabilities
Capital = 33
Determine the leverage ratio when it is defined as
Assets
Capital
4. In a repurchase agreement an asset, often a Treasury bond, is used as collateral. Let P denote the
market price of the bond. The bond is often sold with a “haircut,” meaning the price of the bond
for the transaction is below the market price. Let p be the price of the bond in the repo agreement.
Then p < P. The haircut is the percentage based on p
P
. For example, if the market price of the bond
is $10,000, the borrower sells the bond for a discount, say $9,000 so the haircut is 10 percent (so the
ratio 9000
10000 = 0.9 so the value of the bond has been “shaved” by 10 percent). Recall in the repo, the
borrower sells a bond as collateral and then turns around an agrees to repurchase it at a higher price
at some point in the future. Provide some reasons why there might be a haircut in the transaction. In
way is the haircut like a down payment on a loan?
5. This is a question about amortization. Amortization is an accounting technique used to lower the book
value of a loan over the life of the loan. Suppose you take out a 3 year loan for $100,000 at interest
rate of 5 percent. This is a fixed rate loan.
(a) Determine the constant loan payment so the the loan is paid off in five years. Call this dollar
amount x.
(b) One time period goes by. The loan payment is made and there are two more payments of x dollars to be made. Determine the discounted present value of the remaining loan payments. This is the amount that will be recorded on the balance sheet after the first payment is received and this is
the process of amortizing a loan. 2
(c) At the start (time t) the interest rate is 5 percent. At time t + 1, the interest rate rises to 7
percent (assume this increase is unanticipated). Determine the discounted present value of the
two remaining payments of x. This would be the fair value or marked to market accounting value.
6. Use data from the Federal Reserve Bank of St. Louis to plot the 10 year minus the 2 year Treasury
rate. Describe what the current spread implies about the slope of the yield curve
