Practice Exam 2 Bank Management-Solution

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Jan 9, 2024

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Solution of Practice Exam 2 of Bank Management 1. Suppose the estimated linear probability model is PD = 0.3X1 + 0.2X2 - .05X3 + error, where X1 = 0.8 is the borrower's debt/equity ratio; X2 = 0.3 is the volatility of borrower earnings; and X3 = 0.5 is the borrower’s profit ratio. What is the projected probability of default for the borrower? PD = 0.3 * 0.8 + 0.2 * 0.3 – 0.05 * 0.5 = 0.24 + 0.06 – 0.025 = 0.275 = 27.5% 2. If the rate of 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, 60% of 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 10 percent? P * 1.1 + (1 – P) * 0.6 * 1.1 = 1.06 => P= 90.91% b. One-year BB rated bond yielding 12 percent? P * 1.12 + (1 – P) * 0.6 * 1.12 = 1.06 => P = 86.61% 3. A bank has made a loan charging a base lending rate of 10 percent. It expects a probability of default of 5 percent. If the loan is defaulted, it expects to recover 40 percent of its money through the sale of its collateral. What is the expected return on this loan? 1+ E(r) = (0.95 * 1.1 + 0.05 * 0.4 * 1.1) E(r)= 1.045 + 0.022 – 1 = 0.067 = 6.7% 4. 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 6.0% 7.0% 8.0% BBB rated Bonds 7.5% 8.5% 9.5% P 1 (1 + 0.075) = 1 + 0.06 => P 1 = 0.986, MP 1 = 1 – P 1 = 0.014, CP 1 = 1 – P 1 = 0.014 E 2 r 1 = ( 1 + t R 2 ) 2 1 + t R 1 1 = ¿ E 2 r 1 T = 1.07 2 1.06 1 = 0.08, E 2 r 1 C = 1.085 2 1.075 1 = 0.0951
P 2 (1 + 0.0951) = 1 + 0.08 => P 2 = 0.9862, MP 2 = 1 – P 2 = 0.0138, CP 2 = 1 – P 1 P 2 = 0.0276 E 3 r 1 = ( 1 + t R 3 ) 3 ( 1 + t R 2 ) 2 1 = ¿ E 3 r 1 T = 1.08 3 1.07 2 1 = 0.1003, E 3 r 1 C = 1.095 3 1.085 2 1 = 0.1153 P 3 (1 + 0.1153) = 1 + 0.1003 => P 3 = 0.9866, MP 3 = 1 – P 3 = 0.0134, CP 3 = 1 – P 1 P 2 P 3 = 0.0406 5 . A bank is planning to make a loan of $10,000,000 to a firm in the steel industry. It expects to charge a servicing fee of 75 basis points. The loan has a maturity of 10 years and a duration of 9 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 3.5 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? 1-yr Net Income = (0.12 – 0.1 + 0.0075) * 10,000,000 = 275,000 ln = D ln ∆CR 1 + R = 9 10,000,000 0.035 1.12 =− 2,812,500 RAROC = 1 yr Net Income ln = 275,000 2,812,500 = 9.78% < 10% ,reject b. What should be the duration in order for this loan to be approved? ln = 1 yr Net Income RAROC = 275,000 0.1 = 2,750,000 ln = D 10,000,000 0.035 1.12 = 2,750,000 = ¿ D = 8.8 ( years ) c. Assuming that duration cannot be changed, how much additional interest and fee income would be necessary to make the loan acceptable? 1-yr Net Income = RAROC * ΔLN = 0.1 * 2,812,500 = 281,250 Additional interest and fee income = 281,250 – 275,000 = 6,250
d. Given the proposed income stream and the negotiated duration, what adjustment in the risk premium would be necessary to make the loan acceptable? 9 10,000,000 ∆CR 1.12 = 2,750,000 = ¿ ∆CR = 3.42% 6. The Bank of Tinytown has a $1 million loan portfolio, in which $400,000 goes to Loan A, and the other $600,000 goes to Loan B. Loan A has an expected return of 12 percent and a standard deviation of returns of 8 percent. The expected return and standard deviation of returns for loan B are 9 percent and 6 percent, respectively. If the covariance between A and B is 0.05, what are the expected return and standard deviation of this portfolio? What if the correlation between A and B is -0.18? E(R p ) = 0.4 * 0.12 + 0.6 * 0.09 = 0.048 + 0.054 = 0.102 = 10.2% 0.001024 + 0.001296 + 0.024 ¿ ¿ σ p = ( 0.4 2 0.08 2 + 0.6 2 0.06 2 + 2 0.4 0.6 0.05 )= ¿ 0.001024 + 0.001296 0.00041472 ¿ ¿ σ p = [ 0.4 2 0.08 2 + 0.6 2 0.06 2 + 2 0.4 0.6 ( 0.18 ) 0.08 0.06 ]= ¿ 7. A bank holds a loan portfolio with the following characteristics: Loan i Xi Annual spread between loan rate and bank’s cost of funds Annual fees Loss to bank given default Expected default frequency Correlatio n 1 .8 5% 2% 20% 3% -0.2 2 .2 4% 3% 15% 4% a. what is the return and risk on loan 1? R 1 = AIS – EDF * LGD = 0.05 + 0.02 – 0.03 * 0.2 = 6.4% σ 1 =[ EDF ( 1 EDF ) ] 1 2 LGD =( 0.03 0.97 ) 1 2 0.2 = 3.41% b. what is the return and risk on loan 2?
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R 2 = 0.04 + 0.03 – 0.04 * 0.15 = 6.4% σ 2 =( 0.04 0.96 ) 1 2 0.15 = 2.94 % c. what is the return and risk on loan portfolio? R p = 0.8 * 6.4% + 0.2 * 6.4% = 6.4% 0.000064162 ¿ 0.000744198 + 0.000034574 + ¿ ¿ σ p = [ 0.8 2 0.0341 2 + 0.2 2 0.0294 2 + 2 0.8 0.2 ( 0.2 ) 0.0341 0.0294 ]= ¿ 8. 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? σ A = ( 0.5 0.3 ) 2 +( 0.3 0.4 ) 2 +( 0.2 0.3 ) 2 3 = 14.14 % σ B = ( 0.1 0.3 ) 2 +( 0.4 0.4 ) 2 +( 0.5 0.3 ) 2 3 = 16.33% 9. Follow Bank has a five-year, zero-coupon bond with a face value of $2 million. The bond is trading at a yield to maturity of 7.00 percent. The historical mean change in daily yields is 15 basis points, and the standard deviation is 25 basis points. a. What is the modified duration of the bond? MD = D 1 + R = 5 1.07 = 4.6729 b. What is the maximum adverse daily yield move given that we desire no more than a 2.5 percent chance that yield changes will be greater than this maximum? Potential adverse move in yield at 2.5 percent =µ+1.96σ = .0015+ 1.96*.0025 =.0064
c. What is the price volatility of this bond? Price volatility = MD x potential adverse move in yield = 4.6729 x .0064 = 0.0299 or 2.99 percent d. What is the daily earnings at risk for this bond? P B = 2000000 1.07 5 = 1,425,972.36 DEAR = 1425972.36 * 0.0299 = 42645.9279 10. Bank of Southern Vermont has determined that its inventory of 20 million euros (€) and 25 million British pounds (£) is subject to market risk. The spot exchange rates are $1.40/€ and $1.60/£, respectively. The ’s of the spot exchange rates of the € and £, based on the daily changes of spot rates over the past six months, are 65 bp and 45 bp, respectively. Determine the bank’s DEAR for both currencies. Use adverse rate changes in the 95 th percentile. $ position of euro € = 20 * 1.4 = 28(mil) $ position of ₤ = 25 * 1.6 = 40 (mil) FX volatility =µ+1.96σ FX volatility € = 1.96 * 0.0065 = 0.0127 FX volatility ₤ = 1.96 * 0.0045 = 0.0088 DEAR = $ position * price volatility DEAR = 28000000 * 0.0127 = 355,600 DEAR = 40000000 * 0.0088 = 352,800 11. Bank of Alaska’s stock portfolio has a market value of $10 million. The standard deviation on the market portfolio ( m ) has been estimated at 2%. What is the DEAR of this portfolio, using adverse rate changes in the 99 th percentile? Price volatility =2.58σ =2.58 * 0.02 = 0.0516 DEAR = 10000000 * 0.0516 = 516,000 Use the following to answer questions 12-13: On December 31, 2001 Historic Bank had long positions of 1,800,000 Japanese Yen and 660,000 Swiss Francs. The closing exchange rates were ¥90/$ and Sf1.65/$. (keep 4 decimal places in your computation) 12. What is the value of delta for the respective positions of the two currencies in dollars? $ position of ¥ = 1800000 90 = 20000, $ position of sf = 660000 1.65 = 400000 With 1% increase in ER: ¥90.9/$, sf1.6665/$
new $ position of ¥ = 1800000 90.9 = 19801.9802, new $ positionof sf = 660000 1.6665 = 396039.604 Delta ¥ = 19801.98 – 20000 = -198.02, Delta sf = 396039.604 – 400000 = -3960.396 13. Over the past 500 days, the 25 th worst day for adverse exchange rate changes saw a change in the exchange rates of 0.58 percent for the Yen and 0.60 percent for the Swiss Franc. What is the expected VAR exposure on December 31? VAR = delta ¥ * ΔER ¥ + delta sf * ΔER sf = -198.02 * 0.58 + (-3960.396) * 0.60 = -114.8516 + 2376.2376 = -2491.0892 14. Sumitomo Bank's risk manager has estimated that the DEARs of two of its major assets in its trading portfolio, foreign exchange and bonds, are -$1000 and -$2000, respectively. What is the total DEAR of Sumitomo's trading portfolio if the correlation among assets is assumed to be -0.5 ? DEAR 2 = 1000 2 + 2000 2 + 2 * (-0.5) * (-1000)*(-2000) = 1000000 + 4000000 – 2000000 = 3000000 DEAR = 1732.05
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