Financial-Institutions Management. Solutions A financial institution has the following market value balance sheet structure:

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1 FIN 683 Professor Robert Hauswald Financial-Institutions Management Kogod School of Business, AU Solutions 1 Chapter 7: Bank Risks - Interest Rate Risks 6. A financial institution has the following market value balance sheet structure: Assets Liabilities and Equity Cash $1,000 Certificate of deposit $10,000 Bond $10,000 Equity $1,000 Total assets $11,000 Total liabilities and equity $11,000 a. The bond has a 10-year maturity, a fixed-rate coupon of 10 percent paid at the end of each year, and a par value of $10,000. The certificate of deposit has a 1-year maturity and a 6 percent fixed rate of interest. The FI expects no additional asset growth. What will be the net interest income (NII) at the end of the first year? Note: Net interest income equals interest income minus interest expense. Interest income $1,000 $10,000 x 0.10 Interest expense 600 $10,000 x 0.06 Net interest income (NII) $400 b. If at the end of year 1 market interest rates have increased 100 basis points (1 percent), what will be the net interest income for the second year? Is the change in NII caused by reinvestment risk or refinancing risk? Interest income $1,000 $10,000 x 0.10 Interest expense 700 $10,000 x 0.07 Net interest income (NII) $300 The decrease in net interest income is caused by the increase in financing cost without a corresponding increase in the earnings rate. Thus, the change in NII is caused by refinancing risk. The increase in market interest rates does not affect the interest income because the bond has a fixed-rate coupon for ten years. Note: this answer makes no assumption about reinvesting the first year s interest income at the new higher rate. c. Assuming that market interest rates increase 1 percent, the bond will have a value of $9,446 at the end of year 1. What will be the market value of the equity for the FI?

2 Assume that all of the NII in part (a) is used to cover operating expenses or is distributed as dividends. Cash $1,000 Certificate of deposit $10,000 Bond $9,446 Equity $ 446 Total assets $10,446 Total liabilities and equity $10,446 d. If market interest rates had decreased 100 basis points by the end of year 1, would the market value of equity be higher or lower than $1,000? Why? The market value of the equity would be higher ($1,600) because the value of the bond would be higher ($10,600) and the value of the CD would remain unchanged. e. What factors have caused the changes in operating performance and market value for this firm? The operating performance has been affected by the changes in the market interest rates that have caused the corresponding changes in interest income, interest expense, and net interest income. These specific changes have occurred because of the unique maturities of the fixed-rate assets and liabilities. Similarly, the economic or market value of the firm has changed because of the effect of the changing rates on the market value of the bond. 7. How does a policy of matching the maturities of assets and liabilities work (a) to minimize interest rate risk and (b) against the asset-transformation function for FIs? A policy of maturity matching will allow changes in market interest rates to have approximately the same effect on both interest income and interest expense. An increase in rates will tend to increase both income and expense, and a decrease in rates will tend to decrease both income and expense. The changes in income and expense may not be equal because of different cash flow characteristics of the assets and liabilities. The asset-transformation function of an FI involves investing short-term liabilities in long-term assets. Maturity matching clearly works against successful implementation of this process. 12. A bank invested $50 million in a two-year asset paying 10 percent interest per year and simultaneously issued a $50 million, one-year liability paying 8 percent interest per year. The liability will be rolled over after one year at the current market rate. What will be the bank s net interest income if at the end of the first year all interest rates have increased by 1 percent (100 basis points)? Net interest income is not affected in the first year, but NII will decrease in the second year. Year 1 Year 2 Interest income $5,000,000 $5,000,000 Interest expense $4,000,000 $4,500,000 Net interest income $1,000,000 $500,000

3 The bank s net interest income decreases in year 2 by $500,000 as the result of refinancing risk. Chapter 7: Bank Risks Credit Risks 15. What is the difference between firm-specific credit risk and systematic credit risk? How can an FI alleviate firm-specific credit risk? Firm-specific credit risk refers to the likelihood that a single asset may deteriorate in quality, while systematic credit risk involves macroeconomic factors that may increase the default risk of all firms in the economy. Thus, if S&P lowers its rating on IBM stock and if an investor is holding only this particular stock, he may face significant losses as a result of this downgrading. However, portfolio theory in finance has shown that firm-specific credit risk can be diversified away if a portfolio of well-diversified stocks is held. Similarly, if an FI holds a well-diversified portfolio of assets, the FI will face only systematic credit risk that will be affected by the general condition of the economy. The risks specific to any one customer will not be a significant portion of the FIs overall credit risk. Chapter 7: Foreign Exchange Risks 22. If an FI has the same amount of foreign assets and foreign liabilities in the same currency, has that FI necessarily reduced the risk involved in these international transactions to zero? Explain. Matching the size of the foreign currency book will not eliminate the risk of the international transactions if the maturities of the assets and liabilities are mismatched. To the extent that the asset and liabilities are mismatched in terms of maturities, or more importantly durations, the FI will be exposed to foreign interest rate risk. 24. Assume that a bank has assets located in London worth 150 million on which it earns an average of 8 percent per year. The bank has 100 million in liabilities on which it pays an average of 6 percent per year. The current spot exchange rate is 1.50/$. a. If the exchange rate at the end of the year is 2.00/$, will the dollar have appreciated or devalued against the mark? The dollar will have appreciated, or conversely, the will have depreciated. b. Given the change in the exchange rate, what is the effect in dollars on the net interest income from the foreign assets and liabilities? Note: The net interest income is interest income minus interest expense.

4 Measurement in Interest received = 12 million Interest paid = 6 million Net interest income = 6 million Measurement in $ before devaluation Interest received in dollars = $8 million Interest paid in dollars = $4 million Net interest income = $4 million Measurement in $ after devaluation Interest received in dollars = $6 million Interest paid in dollars = $3 million Net interest income = $3 million Thus, net interest income decreases by $1 million as a result of foreign exchange risk. c. What is the effect of the exchange rate change on the value of assets and liabilities in dollars? The assets were worth $100 million ( 150m/1.50) before depreciation, but after devaluation they are worth only $75 million. The liabilities were worth $66.67 million before depreciation, but they are worth only $50 million after devaluation. Since assets decline by $25 million and liabilities by $16.67 million, net worth decreases by $8.33 million using spot rates at the end of the year. 25. Six months ago, Qualitybank, issued a $100 million, one-year maturity CD denominated in euros. On the same date, $60 million was invested in a -denominated loan and $40 million was invested in a U.S. Treasury bill. The exchange rate on this date was /$. Assume no repayment of principal and an exchange rate today of /$. a. What is the current value of the CD principal (in euros and dollars)? The current principal value on the CD is m and $125m ( m/1.3905). b. What is the current value of the euro-denominated loan principal (in euros and dollars)? The current principal value on the loan is m and $75m ( m/1.3905). c. What is the current value of the U.S. Treasury bill (in euros and dollars)?

5 The current principal value on the U.S. Treasury bill is $40m and ($40m x ). For a U.S. bank this does not change in value. d. What is Qualitybank s profit/loss from this transaction (in euros and dollars)? Qualitybank's loss is m (($40m x ) m) or $10m (($125m - $100m) ($75m - $60m)). Solution matrix for problem 25: At Issue Date: Dollar Transaction Values (in millions) Euro Transaction Values (in millions) Euro Euro Euro Euro Loan $60 CD $100 Loan CD U.S T-bill $40 U.S. T-bill $100 $ Today: Dollar Transaction Values (in millions) Transaction Values (in millions) Euro Euro Euro Euro Loan $75 CD $125 Loan CD U.S. T-bill $40 U.S. T-bill $115 $ Loss -$ 10 Loss Suppose you purchase a 10-year, AAA-rated Swiss bond for par that is paying an annual coupon of 6 percent. The bond has a face value of 1,000 Swiss francs (SF). The spot rate at the time of purchase is SF1.50/$. At the end of the year, the bond is downgraded to AA and the yield increases to 8 percent. In addition, the SF appreciates to SF1.35/$. a. What is the loss or gain to a Swiss investor who holds this bond for a year? What portion of this loss or gain is due to foreign exchange risk? What portion is due to interest rate risk? Beginning of the Year Price of Bond SF * PVA End of the Year Price of Bond = 60 i = = 10 = 6, n= 10 + SF1,000 * PVi= 6, n SF1,000 = = SF 60 * PVAi = 8, n= 9 + SF1,000 * PVi= 8, n 9 = SF The loss to the Swiss investor (SF SF60 - SF1,000)/SF1,000 = percent. The entire amount of the loss is due to interest rate risk.

6 b. What is the loss or gain to a U.S. investor who holds this bond for a year? What portion of this loss or gain is due to foreign exchange risk? What portion is due to interest rate risk? Price at beginning of year = SF1,000/SF1.50 = $ Price at end of year = SF875.06/SF1.35 = $ Interest received at end of year = SF60/SF1.35 = $44.44 Gain to U.S. investor = ($ $ $666.67)/$ = +3.89%. The U.S. investor had an equivalent loss of 6.49 percent ([((SF SF60)/SF1.5) - $666.67]/$666.67) from interest rate risk, but had a gain of percent (3.89% - (-6.49%)) from foreign exchange risk.

7 Chapter 8 Repricing models 14. Consider the following balance sheet for WatchoverU Savings, Inc. (in millions): Assets Liabilities and Equity Floating-rate mortgages 1-year time deposits (currently 10% annually) $50 (currently 6% annually) $70 30-year fixed-rate loans 3-year time deposits (currently 7% annually) $50 (currently 7% annually) $20 Equity $10 Total assets $100 Total liabilities & equity $100 a. What is WatchoverU s expected net interest income at year-end? Current expected interest income: $50m(0.10) + $50m(0.07) = $8.5m. Expected interest expense: $70m(0.06) + $20m(0.07) = $5.6m. Expected net interest income: $8.5m - $5.6m = $2.9m. b. What will net interest income be at year-end if interest rates rise by 2 percent? After the 200 basis point interest rate increase, net interest income declines to: 50(0.12) + 50(0.07) - 70(0.08) - 20(.07) = $9.5m - $7.0m = $2.5m, a decline of $0.4m. c. Using the cumulative repricing gap model, what is the expected net interest income for a 2 percent increase in interest rates? Wachovia s' repricing or funding gap is $50m - $70m = -$20m. The change in net interest income using the funding gap model is (-$20m)(0.02) = -$.4m. d. What will net interest income be at year-end if interest rates on RSAs increase by 2 percent but interest rates on RSLs increase by 1 percent? Is it reasonable for changes in interest rates on RSAs and RSLs to differ? Why? After the unequal rate increases, net interest income will be 50(0.12) + 50(0.07) - 70(0.07) - 20(.07) = $9.5m - $6.3m = $3.2m, an increase of $0.3m. It is not uncommon for interest rates to adjust in an unequal manner on RSAs versus RSLs. Interest rates often do not adjust solely because of market pressures. In many cases the changes are affected by decisions of management. Thus, you can see the difference between this answer and the answer for part a.

8 15. Use the following information about a hypothetical government security dealer named M.P. Jorgan. Market yields are in parenthesis, and amounts are in millions. Assets Liabilities and Equity Cash $10 Overnight repos $170 1-month T-bills (7.05%) 75 Subordinated debt 3-month T-bills (7.25%) 75 7-year fixed rate (8.55%) year T-notes (7.50%) 50 8-year T-notes (8.96%) year munis (floating rate) (8.20% reset every 6 months) 25 Equity 15 Total assets $335 Total liabilities & equity $335 a. What is the repricing gap if the planning period is 30 days? 3 months? 2 years? Recall that cash is a non-interest-earning asset. Repricing gap using a 30-day planning period = $75 - $170 = -$95 million.repricing gap using a 3-month planning period = ($75 + $75) - $170 = -$20 million. Reprising gap using a 2-year planning period = ($75 + $75 + $50 + $25) - $170 = +$55 million. b. What is the impact over the next 30 days on net interest income if interest rates increase 50 basis points? Decrease 75 basis points? If interest rates increase 50 basis points, net interest income will decrease by $475,000. NII = CGAP( R) = -$95m.(.005) = -$0.475m. If interest rates decrease by 75 basis points, net interest income will increase by $712,500. NII = CGAP( R) = -$95m.(-.0075) = $0.7125m. c. The following one-year runoffs are expected: $10 million for two-year T-notes and $20 million for eight-year T-notes. What is the one-year repricing gap? The repricing gap over the 1-year planning period = ($75m. + $75m. + $10m. + $20m. + $25m.) - $170m. = +$35 million. d. If runoffs are considered, what is the effect on net interest income at year-end if interest rates increase 50 basis points? Decrease 75 basis points? If interest rates increase 50 basis points, net interest income will increase by $175,000. NII = CGAP( R) = $35m.(0.005) = $0.175m.

9 If interest rates decrease 75 basis points, net interest income will decrease by $262,500. NII = CGAP( R) = $35m.( ) = -$0.2625m. 16. A bank has the following balance sheet: Assets Avg. Rate Liabilities/Equity Avg. Rate Rate sensitive $550, % Rate sensitive $375, % Fixed rate 755, Fixed rate 805, Nonearning 265,000 Nonpaying 390,000 Total $1,570,000 Total $1,570,000 Suppose interest rates rise such that the average yield on rate-sensitive assets increases by 45 basis points and the average yield on rate-sensitive liabilities increases by 35 basis points. a. Calculate the bank s repricing GAP and gap ratio. Repricing GAP = $550,000 - $375,000 = $175,000 Gap ratio = $175,000/$1,570,000 = 11.15% b. Assuming the bank does not change the composition of its balance sheet, calculate the resulting change in the bank s interest income, interest expense, and net interest income. II = $550,000(.0045) = $2,475 IE = $375,000(.0035) = $1, NII = $2,475 - $1, = $1, c. Explain how the CGAP and spread effects influenced the change in net interest income. The CGAP affect worked to increase net interest income. That is, the CGAP was positive while interest rates increased. Thus, interest income increased by more than interest expense. The result is an increase in NII. The spread effect also worked to increase net interest income. The spread increased by 10 basis points. According to the spread affect, as spread increases, so does net interest income. 17. A bank has the following balance sheet: Assets Avg. Rate Liabilities/Equity Avg. Rate Rate sensitive $550, % Rate sensitive $575, %

10 Fixed rate 755, Fixed rate 605, Nonearning 265,000 Nonpaying 390,000 Total $1,570,000 Total $1,570,000 Suppose interest rates fall such that the average yield on rate-sensitive assets decreases by 15 basis points and the average yield on rate-sensitive liabilities decreases by 5 basis points. a. Calculate the bank s CGAP and gap ratio. CGAP = $550,000 - $575,000 = -$25,000 Gap ratio = -$25,000/$1,570,000 = -1.59% b. Assuming the bank does not change the composition of its balance sheet, calculate the resulting change in the bank s interest income, interest expense, and net interest income. II = $550,000(-.0015) = -$825 IE = $575,000(-.0005) = -$ NII = -$825 (-$287.50) = -$ c. The bank s CGAP is negative and interest rates decreased, yet net interest income decreased. Explain how the CGAP and spread effects influenced this decrease in net interest income. The CGAP affect worked to increase net interest income. That is, the CGAP was negative while interest rates decreased. Thus, interest income decreased by less than interest expense. The result is an increase in NII. The spread effect, on the other hand, worked to decrease net interest income. The spread decreased by 10 basis points. According to the spread affect, as spread decreases, so does net interest income. In this case, the increase in NII due to the CGAP effect was dominated by the decrease in NII due to the spread effect. 18. The balance sheet of A. G. Fredwards, a government security dealer, is listed below. Market yields are in parentheses, and amounts are in millions. Assets Liabilities and Equity Cash $20 Overnight repos $340 1-month T-bills (7.05%) 150 Subordinated debt 3-month T-bills (7.25%) year fixed rate (8.55%) year T-notes (7.50%) year T-notes (8.96%) 200

11 5-year munis (floating rate) (8.20% reset every 6 months) 50 Equity 30 Total assets $670 Total liabilities and equity $670 a. What is the repricing gap if the planning period is 30 days? 3 month days? 2 years? Repricing gap using a 30-day planning period = $150 - $340 = -$190 million.repricing gap using a 3-month planning period = ($150 + $150) - $340 = -$40 million. Repricing gap using a 2-year planning period = ($150 + $150 + $100 + $50) - $340 = $110 million. b. What is the impact over the next three months on net interest income if interest rates on RSAs increase 50 basis points and on RSLs increase 60 basis points? II = ($150m. + $150m.)(.005) = $1.5m. IE = $340m.(.006) = $2.04m. NII = $1.5m. ($2.04m.) = -$.54m. c. What is the impact over the next two years on net interest income if interest rates on RSAs increase 50 basis points and on RSLs increase 75 basis points? II = ($150m. + $150m. + $100 + $50)(.005) = $2.25m. IE = $340m.(.0075) = $2.04m. NII = $2.25m. ($2.04m.) = $.21m. d. Explain the difference in your answers to parts (b) and (c). Why is one answer a negative change in NII, while the other is positive? For the 3-month analysis, the CGAP affect worked to decrease net interest income. That is, the CGAP was negative while interest rates increased. Thus, interest income increased by less than interest expense. The result is a decrease in NII. For the 3-year analysis, the CGAP affect worked to increase net interest income. That is, the CGAP was positive while interest rates increased. Thus, interest income increased by more than interest expense. The result is an increase in NII. 19. A bank has the following balance sheet: Assets Avg. Rate Liabilities/Equity Avg. Rate Rate sensitive $225, % Rate sensitive $300, % Fixed rate 550, Fixed rate 505, Nonearning 120,000 Nonpaying 90,000 Total $895,000 Total $895,000

12 Suppose interest rates rise such that the average yield on rate-sensitive assets increases by 45 basis points and the average yield on rate-sensitive liabilities increases by 35 basis points. a. Calculate the bank s repricing GAP. Repricing GAP = $225,000 - $300,000 = -$75,000 b. Assuming the bank does not change the composition of its balance sheet, calculate the net interest income for the bank before and after the interest rate changes. What is the resulting change in net interest income? NII b = ($225,000(.0635) +$550,000(.0755)) ($300,000(.0425) + $505,000(.0615)) = $55, $43, = $12,005 NII a = ($225,000( ) +$550,000(.0755)) ($300,000( ) + $505,000(.0615)) = $56,825 - $44, = $11, NII = $11, $12,005 = -$37.5 c. Explain how the CGAP and spread effects influenced this increase in net interest income. The CGAP affect worked to decrease net interest income. That is, the CGAP was negative while interest rates increased. Thus, interest income increased by more than interest expense. The result is an decrease in NII. In contrast, the spread effect worked to increase net interest income. The spread increased by 10 basis points. According to the spread affect, as spread increases, so does net interest income. However, in this case, the increase in NII due to the spread effect was dominated by the decrease in NII due to the CGAP effect. 22. Nearby Bank has the following balance sheet (in millions): Assets Liabilities and Equity Cash $60 Demand deposits $140 5-year Treasury notes 60 1-year certificates of deposit year mortgages 200 Equity 20 Total assets $320 Total liabilities and equity $320

13 What is the maturity gap for Nearby Bank? Is Nearby Bank more exposed to an increase or decrease in interest rates? Explain why? M A = [0*60 + 5* *200]/320 = years, and M L = [0* *160]/300 = Therefore, the maturity gap = MGAP = = years. Nearby Bank is exposed to an increase in interest rates. If rates rise, the value of assets will decrease much more than the value of liabilities. 23. County Bank has the following market value balance sheet (in millions, all interest at annual rates). All securities are selling at par equal to book value. Assets Liabilities and Equity Cash $20 Demand deposits $ year commercial loan at 10% 5-year CDs at 6% interest, interest, balloon payment 160 balloon payment year mortgages at 8% interest, 20-year debentures at 7% interest, 120 balloon payment 300 balloon payment Equity 50 Total assets $480 Total liabilities & equity $480 a. What is the maturity gap for County Bank? M A = [0* * *300]/480 = years. M L = [0* * *120]/430 = 8.02 years. MGAP = = years. b. What will be the maturity gap if the interest rates on all assets and liabilities increase by 1 percent? If interest rates increase one percent, the value and average maturity of the assets will be: Cash = $20 Commercial loans = $16*PVIFA n=15, i=11% + $160*PVIF n=15,i=11% = $ Mortgages = $24*PVIFA n=30,i=9% + $300*PVIF n=30,i=9% = $ M A = [0* * *30]/( ) = years The value and average maturity of the liabilities will be: Demand deposits = $100 CDs = $12.60*PVIFA n=5,i=7% + $210*PVIF n=5,i=7% = $ Debentures = $8.4*PVIFA n=20,i=8% + $120*PVIF n=20,i=8% = $ M L = [0* * *108.22]/( ) = 7.74 years

14 The maturity gap = MGAP = = years. The maturity gap increased because the average maturity of the liabilities decreased more than the average maturity of the assets. This result occurred primarily because of the differences in the cash flow streams for the mortgages and the debentures. c. What will happen to the market value of the equity? The market value of the assets has decreased from $480 to $437.67, or $ The market value of the liabilities has decreased from $430 to $409.61, or $ Therefore, the market value of the equity will decrease by $ $20.39 = $21.94, or percent. 24. If a bank manager is certain that interest rates were going to increase within the next six months, how should the bank manager adjust the bank s maturity gap to take advantage of this anticipated increase? What if the manager believes rates will fall? Would your suggested adjustments be difficult or easy to achieve? When rates rise, the value of the longer-lived assets will fall by more the shorter-lived liabilities. If the maturity gap is positive, the bank manager will want to shorten the maturity gap. If the repricing gap is negative, the manager will want to move it towards zero or positive. If rates are expected to decrease, the manager should reverse these strategies. Changing the maturity or repricing gaps on the balance sheet often involves changing the mix of assets and liabilities. Attempts to make these changes may involve changes in financial strategy for the bank which may not be easy to accomplish. Later in the text, methods of achieving the same results using derivatives will be explored. 25. Consumer Bank has $20 million in cash and a $180 million loan portfolio. The assets are funded with demand deposits of $18 million, a $162 million CD, and $20 million in equity. The loan portfolio has a maturity of 2 years, earns interest at the annual rate of 7 percent, and is amortized monthly. The bank pays 7 percent annual interest on the CD, but the principal will not be paid until the CD matures at the end of 2 years. a. What is the maturity gap for Consumer Bank? M A = [0*$20 + 2*$180]/$200 = 1.80 years M L = [0*$18 + 2*$162]/$180 = 1.80 years MGAP = = 0 years. b. Is Consumer Bank immunized or protected against changes in interest rates? Why or why not? It is tempting to conclude that the bank is immunized because the maturity gap is zero. However, the cash flow stream for the loan and the cash flow stream for the CD are different

15 because the loan amortizes monthly and the CD pays annual interest. Thus, any change in interest rates will affect the earning power of the loan more than the interest cost of the CD. c. Does Consumer Bank face interest rate risk? That is, if market interest rates increase or decrease 1 percent, what happens to the value of the equity? The bank does face interest rate risk. If market rates increase 1 percent, the value of the cash and demand deposits does not change. However, the value of the loan will decrease to $178.19, and the value of the CD will fall to $ Thus, the value of the equity will be ($ $20 - $18 - $159.11) = $ In this case, the increase in interest rates causes the market value of equity to increase because of the reinvestment opportunities on the loan payments. If market rates decrease 1 percent, the value of the loan increases to $181.84, and the value of the CD increases to $ Thus the value of the equity decreases to $ d. How can a decrease in interest rates create interest rate risk? The amortized loan payments would be reinvested at lower rates. Thus, even though interest rates have decreased, the different cash flow patterns of the loan and the CD have caused interest rate risk. 28. The following is a simplified FI balance sheet: Assets Liabilities and Equity Loans $1,000 Deposits $850 0 Equity $150 Total assets $1,000 Total liabilities & equity $1,000 The average maturity of loans is four years and the average maturity of deposits is two years. Assume loan and deposit balances are reported as book value, zero-coupon items. a. Assume that interest rate on both loans and deposits is 9 percent. What is the market value of equity? The value of loans = $1,000/(1.09) 4 = $ , and the value of deposits = $850/(1.09) 2 = $ The net worth = $ $ = -$ (That is, net worth is negative.) b. What must be the interest rate on deposits to force the market value of equity to be zero? What economic market conditions must exist to make this situation possible?

16 In this case the deposit value should equal the loan value. Thus, $850/(1 + x) 2 = $ Solving for x, we get %. That is, deposit rates will have to increase more because they have a shorter maturity. Note: for those using calculators, you need to compute I/YEAR after entering 850 = FV, = PV, 0 = PMT, 2 = N. c. Assume that interest rate on both loans and deposits is 9 percent. What must be the average maturity of deposits for the market value of equity to be zero? In this case, we need to solve the equation in part (b) for N. The result is years. If interest rates remain at 9 percent, then the average maturity of deposits has to be higher in order to match the value of a 4-year loan. 30. Scandia Bank has issued a one-year, $1million CD paying 5.75 percent to fund a one-year loan paying an interest rate of 6 percent. The principal of the loan will be paid in two installments, $500,000 in six months and the balance at the end of the year. a. What is the maturity gap of Scandia Bank? According to the maturity model, what does this maturity gap imply about the interest rate risk exposure faced by Scandia Bank? The maturity gap is 1 year 1 year = 0. The maturity gap model would state that the bank is immunized against changes in interest rates because assets and liabilities are of equal maturity. b. What is the expected net interest income at the end of the year? Principal received in six months $500,000 Interest received in six months (.03 x $1,000,000) 30,000 Total $530,000 Principal received at the end of the year $500,000 Interest received at the end of the year (.03 x $500,000) 15,000 Future value of interest received in six months ($530,000 x 1.03*) 545,900 Total principal and interest received $1,060,900 Principal and interest paid on deposits ($1,000,000 x ) $1,057,500 Net interest income received $3,400 * It is assumed that the money will be reinvested at current loan rates. Note that the principal is also included in the analysis because interest expense is based on $1,000,000.

17 c. What would be the effect on annual net interest income of a 2 percent interest rate increase that occurred immediately after the loan was made? What would be the effect of a 2 percent decrease in rates? If interest rates increase 2 percent, then the reinvestment benefits of cash flows in six months will be higher: Principal received in six months $500,000 Interest received in six months (.03 x $1,000,000) 30,000 Total $530,000 Principal received at the end of the year $500,000 Interest received at the end of the year (.03 x $500,000) 15,000 Future value of interest received in six months ($530,000 x 1.04) 551,200 Total principal and interest received $1,066,200 Principal and interest paid on deposits ($1,000,000 x ) $1,057,500 Net interest income received $8,700 If interest rates decrease by 2 percent, then reinvestment income is reduced. Principal received in six months $500,000 Interest received in six months (.03 x $1,000,000) 30,000 Total $530,000 Principal received at the end of the year $500,000 Interest received at the end of the year (.03 x $500,000) 15,000 Future value of interest received in six months ($530,000 x 1.02) 540,600 Total principal and interest received $1,055,600 Principal and interest paid on deposits ($1,000,000 x ) $1,057,500 Net income received -$1,900 d. What do these results indicate about the ability of the maturity model to immunize portfolios against interest rate exposure? The results indicate that just matching assets and liabilities by maturity is not sufficient to immunize a portfolio against interest rate risk. If the timing of the cash flows within a period is different for assets and liabilities, the effects of interest rate changes are different. For a truly effective immunization strategy, one also needs to account for the timing of cash flows.

18 31. EDF Bank has a very simple balance sheet. Assets consist of a two-year, $1 million loan that pays an interest rate of LIBOR plus 4 percent annually. The loan is funded with a two-year deposit on which the bank pays LIBOR plus 3.5 percent interest annually. LIBOR currently is 4 percent, and both the loan and the deposit principal will be paid at maturity. a. What is the maturity gap of this balance sheet? Maturity gap = 2-2 = 0 years b. What is the expected net interest income in year 1 and year 2? Interest received in year 1 $80,000 Interest received in year 2 $80,000 Interest paid in year 1 75,000 Interest paid in year 2 75,000 Net interest income in year 1 $5,000 Net interest income in year 2 $5,000 c. Immediately prior to the beginning of year 2, LIBOR rates increased to 6 percent. What is the expected net interest income in year 2? What would be the effect on net interest income of a 2 percent decrease in LIBOR? Year 2: If interest rates increase 2 percent Year 2: If interest rates decrease 2 percent Interest received in year 2 $100,000 Interest received in year 2 $60,000 Interest paid in year 2 95,000 Interest paid in year 2 55,000 Net interest income in year 2 $5,000 Net interest income in year 2 $5, What are the weaknesses of the maturity model? First, the maturity model does not consider the degree of leverage on the balance sheet. For example, if assets are not financed entirely with deposits, a change in interest rates may cause the assets to change in value by a different amount than the liabilities. Second, the maturity model does not take into account the timing of the cash flows of either the assets or the liabilities, and thus reinvestment and/or refinancing risk may become important factors in profitability and valuation as interest rates change.

19 Chapter 9 Duration and Convexity 3. A one-year, $100,000 loan carries a coupon rate and a market interest rate of 12 percent. The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the year. a. What will be the cash flows at the end of six months and at the end of the year? CF 1/2 = ($100,000 x.12 x ½) + $50,000 = $56,000 interest and principal. CF 1 = ($50,000 x.12 x ½) + $50,000 = $53,000 interest and principal. b. What is the present value of each cash flow discounted at the market rate? What is the total present value? PV of CF 1/2 = $56, = $52, PV of CF 1 = $53,000 (1.06) 2 = 47, PV Total CF = $100, c. What proportion of the total present value of cash flows occurs at the end of 6 months? What proportion occurs at the end of the year? X 1/2 = $52, $100,000 =.5283 = 52.83% X 1 = $47, $100,000 =.4717 = 47.17% d. What is the duration of this loan? Duration =.5283(1/2) (1) =.7358 OR t CF PVof CF PV of CF x t ½ $56,000 $52, $26, ,000 47, , $100, $73, Duration = $73,584.91/$100, = years 4. Two bonds are available for purchase in the financial markets. The first bond is a two-year, $1,000 bond that pays an annual coupon of 10 percent. The second bond is a two-year, $1,000, zero-coupon bond. a. What is the duration of the coupon bond if the current yield-to-maturity (R) is 8 percent?10 percent? 12 percent? (Hint: You may wish to create a spreadsheet program to assist in the calculations.)

20 Coupon Bond: Par value = $1,000 Coupon rate = 10% Annual payments R = 8% Maturity = 2 years 1 $100 $92.59 $ , , $1, $1, Duration = $1,978.74/$1, = R = 10% Maturity = 2 years 1 $100 $90.91 $ , , $1, $1, Duration = $1,909.09/$1, = R = 12% Maturity = 2 years 1 $100 $89.29 $ , , $ $1, Duration = $1,843.11/$ = b. How does the change in the current yield to maturity affect the duration of this coupon bond? Increasing the yield to maturity decreases the duration of the bond. c. Calculate the duration of the zero-coupon bond with a yield to maturity of 8 percent, 10 percent, and 12 percent. Zero Coupon Bond: Par value = $1,000 Coupon rate = 0% R = 8% Maturity = 2 years 2 $1,000 $ $1, $ $1, Duration = $1,714.68/$ = R = 10% Maturity = 2 years 2 $1,000 $ $1, $ $1, Duration = $1,652.89/$ = R = 12% Maturity = 2 years 2 $1,000 $ $1, $ $1, Duration = $1,594.39/$ = d. How does the change in the yield to maturity affect the duration of the zero-coupon bond? Changing the yield to maturity does not affect the duration of the zero coupon bond.

21 e. Why does the change in the yield to maturity affect the coupon bond differently than it affects the zero-coupon bond? Increasing the yield to maturity on the coupon bond allows for a higher reinvestment income that more quickly recovers the initial investment. The zero-coupon bond has no cash flow until maturity. 5. What is the duration of a five-year, $1,000 Treasury bond with a 10 percent semiannual coupon selling at par? Selling with a yield to maturity of 12 percent? 14 percent? What can you conclude about the relationship between duration and yield to maturity? Plot the relationship. Why does this relationship exist? Five-year Treasury Bond: Par value = $1,000 Coupon rate = 10% Semiannual payments R = 10% Maturity = 5 years 0.5 $50 $47.62 $ $50 $45.35 $ $50 $43.19 $ $50 $41.14 $ $50 $39.18 $ $50 $37.31 $ $50 $35.53 $ $50 $33.84 $ $50 $32.23 $ $1,050 $ $3, $1, $4, Duration = $4,053.91/$1, = R = 12% Maturity = 5 years 0.5 $50 $47.17 $ $50 $44.50 $ $50 $41.98 $ $50 $39.60 $ $50 $37.36 $ $50 $35.25 $ $50 $33.25 $ $50 $31.37 $ $50 $29.59 $ $1,050 $ $2, $ $3, Duration = $3,716.03/$ = R = 14% Maturity = 5 years 0.5 $50 $46.73 $ $50 $43.67 $43.67

22 1.5 $50 $40.81 $ $50 $38.14 $ $50 $35.65 $ $50 $33.32 $ $50 $31.14 $ $50 $29.10 $ $50 $27.20 $ $1,050 $ $2, $ $3, Duration = $3, /$ = Duration and YTM Years Yield to Maturity 6. Consider three Treasury bonds each of which has a 10 percent semiannual coupon and trades at par. a. Calculate the duration for a bond that has a maturity of four years, three years, and two years? Four-year Treasury Bond: Par value = $1,000 Coupon rate = 10% Semiannual payments R = 10% Maturity = 4 years 0.5 $50 $47.62 $ $50 $45.35 $ $50 $43.19 $ $50 $41.14 $ $50 $39.18 $ $50 $37.31 $ $50 $35.53 $ $1,050 $ $2, $1, $3, Duration = $3,393.19/$1, = R = 10% Maturity = 3 years 0.5 $50 $47.62 $ $50 $45.35 $45.35

23 1.5 $50 $43.19 $ $50 $41.14 $ $50 $39.18 $ $1,050 $ $2, $1, $2, Duration = $2,664.74/$1, = R = 10% Maturity = 2 years 0.5 $50 $47.62 $ $50 $45.35 $ $50 $43.19 $ $1,050 $ $1, $1, $1, Duration = $1,861.62/$1, = b. What conclusions can you reach about the relationship of duration and the time to maturity? Plot the relationship. As maturity decreases, duration decreases at a decreasing rate. Although the graph below does not illustrate with great precision, the change in duration is less than the change in time to maturity. Change in Duration and Maturity Duration Maturity Duration Years Time to Maturity 7. A six-year, $10,000 CD pays 6 percent interest annually and has a 6 percent yield to maturity. What is the duration of the CD? What would be the duration if interest were paid semiannually? What is the relationship of duration to the relative frequency of interest payments? Six-year CD: Par value = $10,000 Coupon rate = 6% R = 6% Maturity = 6 years Annual payments 1 $600 $ $ $600 $ $1,068.00

24 3 $600 $ $1, $600 $ $1, $600 $ $2, $10,600 $ $44, $10, $52, Duration = $52,123.64/$1, = R = 6% Maturity = 6 years Semiannual payments 0.5 $300 $ $ $300 $ $ $300 $ $ $300 $ $ $300 $ $ $300 $ $ $300 $ $ $300 $ $ $300 $ $1, $300 $ $1, $300 $ $1, $10,300 $7, $43, $10, $51, Duration = $51,263.12/$10, = Duration decreases as the frequency of payments increases. This relationship occurs because (a) cash is being received more quickly, and (b) reinvestment income will occur more quickly from the earlier cash flows. 8. What is a consol bond? What is the duration of a consol bond that sells at a yield to maturity of 8 percent? 10 percent? 12 percent? Would a consol trading at a yield to maturity of 10 percent have a greater duration than a 20-year zero-coupon bond trading at the same yield to maturity? Why? A consol is a bond that pays a fixed coupon each year forever. A consol Consol Bond trading at a yield to maturity of 10 percent has a duration of 11 years, R D = 1 + 1/R while a 20-year zero-coupon bond trading at a YTM of 10 percent, or years any other YTM, has a duration of 20 years because no cash flows occur years before the twentieth year years 9. Maximum Pension Fund is attempting to balance one of the bond portfolios under its management. The fund has identified three bonds which have five-year maturities and which trade at a yield to maturity of 9 percent. The bonds differ only in that the coupons are 7 percent, 9 percent, and 11 percent. a. What is the duration for each bond? Five-year Bond: Par value = $1,000 Maturity = 5 years Annual payments

25 R = 9% Coupon rate = 7% 1 $70 $64.22 $ $70 $58.92 $ $70 $54.05 $ $70 $49.59 $ $1,070 $ $3, $ $4, Duration = $4,019.71/$ = R = 9% Coupon rate = 9% 1 $90 $82.57 $ $90 $75.75 $ $90 $69.50 $ $90 $63.76 $ $1,090 $ $3, $1, $4, Duration = $4,239.72/$1, = R = 9% Coupon rate = 11% 1 $110 $ $ $110 $92.58 $ $110 $84.94 $ $110 $77.93 $ $1,110 $ $3, $1, $4, Duration = $4,459.73/$1, = b. What is the relationship between duration and the amount of coupon interest that is paid? Plot the relationship. Years Duration and Coupon Rates % 9% 11% Coupon Rates Duration decreases as the amount of coupon interest increases. Duration Change in Coupon Duration % % %

26 10. An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a yield to maturity of 10 percent, have $10,000 par values, and have five years to maturity. The bonds differ only in the amount of annual coupon interest that they pay: 8, 10, and 12 percent. a. What is the duration for each five-year bond? Five-year Bond: Par value = $10,000 R = 10% Maturity = 5 years Annual payments Coupon rate = 8% 1 $800 $ $ $800 $ $1, $800 $ $1, $800 $ $2, $10,800 $6, $33, $9, $39, Duration = $39,568.14/9, = Coupon rate = 10% 1 $1,000 $ $ $1,000 $ $1, $1,000 $ $2, $1,000 $ $2, $11,000 $6, $34, $10, $41, Duration = $ /10, = Coupon rate = 12% 1 $1,200 $1, $1, $1,200 $ $1, $1,200 $ $2, $1,200 $ $3, $11,200 $6, $34, $10, $43, Duration = $43,829.17/10, = b. What is the relationship between duration and the amount of coupon interest that is paid? Duration and Coupon Rates Duration decreases as the amount of coupon interest increases. Years % 10% 12% Change in Duration Coupon Duration % % % Coupon Rates

27 11. You can obtain a loan of $100,000 at a rate of 10 percent for two years. You have a choice of i) paying the interest (10 percent) each year and the total principal at the end of the second year or ii) amortizing the loan, that is, paying interest (10 percent) and principal in equal payments each year. The loan is priced at par. a. What is the duration of the loan under both methods of payment? Two-year loan: Interest at end of year one; Principal and interest at end of year two Par value = $100,000 Coupon rate = 10% Annual payments R = 10% Maturity = 2 years 1 $10,000 $9, $9, $110,000 $90, $181, $100, $190, Duration = $190,909.09/$100,000 = Two-year loan: Amortized over two years Par value = $100,000 Coupon rate = 10% Annual amortized payments R = 10% Maturity = 2 years = $57, $57, $52, $52, $57, $47, $95, $100, $147, Duration = $147,619.05/$100,000 = b. Explain the difference in the two results? Duration decreases dramatically when a portion of the principal is repaid at the end of year one. Duration often is described as the weighted-average maturity of an asset. If more weight is given to early payments, the effective maturity of the asset is reduced. 15. Calculate the duration of a two-year, $1,000 bond that pays an annual coupon of 10 percent and trades at a yield of 14 percent. What is the expected change in the price of the bond if interest rates decline by 0.50 percent (50 basis points)? Two-year Bond: Par value = $1,000 Coupon rate = 10% Annual payments R = 14% Maturity = 2 years 1 $100 $87.72 $ $1,100 $ $1, $ $1, Duration = $1,780.55/$ = R The expected change in price = - dollar duration x R = D P = - MD x R = 1+ R - (1.9061/1.14) x (-.005) x $ = $7.81. This implies a new price of $ ($ $7.81). The actual price using conventional bond price discounting would be $ The difference of $0.05 is due to convexity, which is not considered in the duration elasticity measure.

28 16. The duration of an 11-year, $1,000 Treasury bond paying a 10 percent semiannual coupon and selling at par has been estimated at years. a. What is the modified duration of the bond? What is the dollar duration of the bond? Modified duration = D/(1 + R/2) = 6.763/(1 +.10/2) = years Dollar duration = MD x P = x $1,000 = 6441 b. What will be the estimated price change on the bond if interest rates increase 0.10 percent (10 basis points)? If rates decrease 0.20 percent (20 basis points)? For interest rates increase of 0.10 percent: Estimated change in price = - dollar duration x R = x = -$6.441 => new price = $1,000 - $6.441 = $ For interest rates decrease of 0.20 percent: Estimated change in price = x = $ => new price = $1,000 + $ = $1, c. What would the actual price of the bond be under each rate change situation in part (b) using the traditional present value bond pricing techniques? What is the amount of error in each case? Rate Price Actual Change Estimated Price Error $ $ $ $1, $1, $ Suppose you purchase a six-year, 8 percent coupon bond (paid annually) that is priced to yield 9 percent. The face value of the bond is $1,000. a. Show that the duration of this bond is equal to five years. Six-year Bond: Par value = $1,000 Coupon rate = 8% Annual payments R = 9% Maturity = 6 years 1 $80 $73.39 $ $80 $67.33 $ $80 $61.77 $ $80 $56.67 $ $80 $51.99 $ $1,080 $ $3, $ $4, Duration = $4,743.87/ = years

29 b. Show that if interest rates rise to 10 percent within the next year and your investment horizon is five years from today, you will still earn a 9 percent yield on your investment. Value of bond at end of year five: PV = ($80 + $1,000) 1.10 = $ Future value of interest payments at end of year five: $80*FVIF n=4, i=10% = $ Future value of all cash flows at n = 5: Coupon interest payments over five years $ Interest on interest at 10 percent Value of bond at end of year five $ Total future value of investment $1, Yield on purchase of asset at $ = $1,470.23*PVIV n=5, i=?% i = %. c. Show that a 9 percent yield also will be earned if interest rates fall next year to 8 percent. Value of bond at end of year five: PV = ($80 + $1,000) 1.08 = $1,000. Future value of interest payments at end of year five: $80*FVIF n=5, i=8% = $ Future value of all cash flows at n = 5: Coupon interest payments over five years $ Interest on interest at 8 percent Value of bond at end of year five $1, Total future value of investment $1, Yield on purchase of asset at $ = $1,469.33*PVIV n=5, i=?% i = percent. 18. Suppose you purchase a five-year, 15 percent coupon bond (paid annually) that is priced to yield 9 percent. The face value of the bond is $1,000. a. Show that the duration of this bond is equal to four years. Five-year Bond: Par value = $1,000 Coupon rate = 15% Annual payments R = 9% Maturity = 5 years 1 $150 $ $ $150 $ $ $150 $ $ $150 $ $ $1,150 $ $3, $1, $4, Duration = $ /1, = years b. Show that if interest rates rise to 10 percent within the next year and your investment horizon is four years from today, you will still earn a 9 percent yield on your investment.

30 Value of bond at end of year four: PV = ($150 + $1,000) 1.10 = $1, Future value of interest payments at end of year four: $150*FVIF n=4, i=10% = $ Future value of all cash flows at n = 4: Coupon interest payments over four years $ Interest on interest at 10 percent Value of bond at end of year four $1, Total future value of investment $1, Yield on purchase of asset at $1, = $1,741.60*PVIV n=4, i=?% i = 9.00%. c. Show that a 9 percent yield also will be earned if interest rates fall next year to 8 percent. Value of bond at end of year four: PV = ($150 + $1,000) 1.08 = $1, Future value of interest payments at end of year four: $150*FVIF n=4, i=8% = $ Future value of all cash flows at n = 4: Coupon interest payments over four years $ Interest on interest at 8 percent Value of bond at end of year four $1, Total future value of investment $1, Yield on purchase of asset at $1, = $1,740.73*PVIV n=4, i=?% i = 9.00 percent. 19. Consider the case in which an investor holds a bond for a period of time longer than the duration of the bond, that is, longer than the original investment horizon. a. If interest rates rise, will the return that is earned exceed or fall short of the original required rate of return? Explain. In this case the actual return earned would exceed the yield expected at the time of purchase. The benefits from a higher reinvestment rate would exceed the price reduction effect if the investor holds the bond for a sufficient length of time. b. What will happen to the realized return if interest rates decrease? Explain. If interest rates decrease, the realized yield on the bond will be less than the expected yield because the decrease in reinvestment earnings will be greater than the gain in bond value. c. Recalculate parts (b) and (c) of problem 18 above, assuming that the bond is held for all five years, to verify your answers to parts (a) and (b) of this problem. The case where interest rates rise to 10 percent, n = five years: Future value of interest payments at end of year five: $150*FVIF n=5, i=10% = $ Future value of all cash flows at n = 5: Coupon interest payments over five years $ Interest on interest at 10 percent

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