I’m working on a business exercise and need an explanation and answer to help me learn.
Chapter 10
1. Suppose the estimated linear probability model is PD = 0.3X1 + 0.2X2 – .05X3 + error, where X1 = 0.75 is the borrower’s debt/equity ratio; X2 = 0.25 is the volatility of borrower earnings; and X3 = 0.10 is the borrower’s profit ratio.
a. What is the projected probability of default for the borrower?
b. What is the projected probability of repayment if the debt/equity ratio is 2.5?
c. What is a major weakness of the linear probability model?
2. If the rate on one-year T-Bills currently is 6 percent, what is the repayment probability for each of the following two securities? Assume that if the loan is defaulted, no payments are expected. What is the market-determined risk premium for the corresponding probability of default for each security?
a. One-year AA rated bond yielding 9.5 percent?
b. One-year BB rated bond yielding 13.5 percent?
3. A bank has made a loan charging a base lending rate of 12 percent. It expects a probability of default of 5 percent. If the loan is defaulted, the bank expects to recover 40 percent of its principal and interest through the sale of its collateral. What is the expected return on this loan?
4. Assume a one-year T-Bill is currently yielding 6 percent, and an AAA-rated discount bond with similar maturity is yielding 9 percent. If the expected recovery from collateral in the event of default is 60 percent of principal and interest, what is the probability of repayment of the AAA-rated bond? What is the probability of default?
5. Calculate the term structure of default probabilities over three years using the following spot rates from the Treasury and corporate bond (pure discount) yield curves. Be sure to calculate both the annual marginal and the cumulative default probabilities.
Spot Spot Spot
1 Year 2 Year 3 Year
Treasury bonds 5.0% 6.2% 6.8%
BBB-rated bonds 7.0 8.0 9.0
6. A bank is planning to make a loan of $25,000,000 to a firm in the steel industry. It expects to charge a servicing fee of 50 basis points. The loan has a maturity of 8 years and a duration of 7.5 years. The cost of funds (the RAROC benchmark) for the bank is 10 percent. Assume the bank has estimated the maximum change in the risk premium on the steel manufacturing sector to be approximately 4.2 percent, based on two years of historical data. The current market interest rate for loans in this sector is 12 percent.
a. Using the RAROC model, determine whether the bank should make the loan?
b. What should be the duration in order for this loan to be approved?
c. Assuming that duration cannot be changed, how much additional interest and fee income would be necessary to make the loan acceptable?
d. Given the proposed income stream and the negotiated duration, what adjustment in the risk premium would be necessary to make the loan acceptable?
Chapter 11
1. The Bank of Tinytown has two $20,000 loans that have the following characteristics: Loan A has an expected return of 10 percent and a standard deviation of returns of 10 percent. The expected return and standard deviation of returns for loan B are 12 percent and 20 percent, respectively.
a. If the correlation coefficient between loans A and B is .15, what are the expected return and standard deviation of this portfolio?
b. What is the standard deviation of the portfolio if the correlation is -.15?
2. Suppose that an FI holds two loans with the following characteristics.
Annual
Spread Between Loss to FI Expected
Loan Rate and FI=s Annual Given Default
Loan Xi Cost of Funds Fees Default Frequency
1 .6 4.5% 2.5% 30% 3% ρ12 = -.4
2 .4 3.5% 2% 20% 5%
Calculate of the return and risk on the two-asset portfolio using KMV Portfolio Manager.
3. Information concerning the allocation of loan portfolios to different market sectors is given below:
Allocation of Loan Portfolios in Different Sectors (%)
Sectors National Bank A Bank B
Commercial 30% 50% 10%
Consumer 40% 30% 40%
Real Estate 30% 20% 50%
Bank A and Bank B would like to estimate how much their portfolios deviate from the national average. Which bank is further away from the national average?
4. Over the last ten years, a bank has experienced the following loan losses on its C&I loans, consumer loans, and total loan portfolio.
Year C&I Loans Consumer Loans Total Loans
2012 0.0080 0.0165 0.0075
2011 0.0088 0.0183 0.0085
2010 0.0100 0.0210 0.0100
2009 0.0120 0.0255 0.0125
2008 0.0104 0.0219 0.0105
2007 0.0084 0.0174 0.0080
2006 0.0072 0.0147 0.0065
2005 0.0080 0.0165 0.0075
2004 0.0096 0.0201 0.0095
2003 0.0144 0.0309 0.0155
Using regression analysis on these historical loan losses, the bank has estimated the following:
XC = 0.002 + 0.8XL and Xh = 0.003 + 1.8XL
where XC = loss rate in the commercial sector, Xh = loss rate in the consumer (household) sector, XL = loss rate for its total loan portfolio.
If the bank’s total loan loss rates increase by 15 percent, what are the expected loss rate increases in the commercial and consumer sectors?
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