Solutions Chapter 10: Market Risk Fixed Income Instruments and DEAR 4. Follow Bank has a $1 million position in a five-year, zero-coupon bond with a face value of $1,40,55. The bond is trading at a yield to maturity of 7.00 percent. The historical mean change in daily yields is 0.0 percent, and the standard deviation is 1 basis points. a. What is the modified duration of the bond? MD = D/(1 + R) = 5/(1.07) = 4.679 years b. What is the maximum adverse daily yield move given that we desire no more than a 5 percent chance that yield changes will be greater than this maximum? Potential adverse move in yield at 5 percent = 1.65σ = 1.65 x 0.001 =.001980 c. What is the price volatility of this bond? Price volatility = MD x potential adverse move in yield = 4.679 x.00198 = 0.0095 or 0.95 percent d. What is the daily earnings at risk for this bond? DEAR = ($ value of position) x (price volatility) = $1,000,000 x 0.0095 = $9,5 1
5. What is meant by value at risk (VAR)? How is VAR related to DEAR in J.P. Morgan s RiskMetrics model? What would be the VAR for the bond in problem (4) for a 10-day period? What statistical assumption is needed for this calculation? Could this treatment be critical? Value at risk or VAR is the cumulative DEAR over a specified period of time and is given by the formula VAR = DEAR x [N] ½. VAR is a more realistic measure when it requires a longer period for an FI to unwind a position, that is, if markets are less liquid. The value for VAR in problem 4 above is $9,5 x [10] ½ = $9,58. According to the above formula, the relationship assumes that yield changes are independent. This means that losses incurred one day are not related to losses incurred the next day. Recent studies have indicated that this is not the case, but that shocks are autocorrelated in many markets over long periods of time. 6. The DEAR for a bank is $8,500. What is the VAR for a 10-day period? A 0-day period? Why is the VAR for a 0-day period not twice as much as that for a 10-day period? For the 10-day period: VAR = 8,500 x [10] ½ = 8,500 x 3.163 = $6,879 For the 0-day period: VAR = 8,500 x [0] ½ = 8,500 x 4.471 = $38,013 The reason that VAR 0 ( x VAR 10 ) is because [0] ½ ( x [10] ½ ). The interpretation is that the daily effects of an adverse event become less as time moves farther away from the event. 7. The mean change in the daily yields of a 15-year, zero-coupon bond has been five basis points (bp) over the past year with a standard deviation of 15 bp. Use these data and assume that the yield changes are normally distributed. a. What is the highest yield change expected if a 90 percent confidence limit is required; that is, adverse moves will not occur more than 1 day in 0? If yield changes are normally distributed, 90 percent of the area of a normal distribution will be 1.65 standard deviations (1.65σ) from the mean for a one-tailed distribution. In this example, it means 1.65 x 15 = 4.75 bp. Thus, the maximum adverse yield change expected for this zero-coupon bond is an increase of 4.75 basis points, or 0.475 percent, in interest rates.
b. What is the highest yield change expected if a 95 percent confidence limit is required? If a 95 percent confidence limit is required, then 95 percent of the area will be 1.96 standard deviations (1.96σ) from the mean. Thus, the maximum adverse yield change expected for this zero-coupon bond is an increase of (1.96 x 15 =) 9.40 basis points, or 0.94 percent, in interest rates. 8. In what sense is duration a measure of market risk? Market risk calculations are typically based on the trading portion of an FIs fixed-rate asset portfolio because these assets must reflect changes in value as market interest rates change. As such, duration or modified duration provides an easily measured and usable link between changes in the market interest rates and changes in the market value of fixed-income assets. 9. Bank Alpha has an inventory of AAA-rated, 15-year zero-coupon bonds with a face value of $400 million. The bonds currently are yielding 9.5 percent in the over-the-counter market. a. What is the modified duration of these bonds? MD = D/(1 + R) = 15/(1.095) = 13.6986. b. What is the price volatility if the potential adverse move in yields is 5 basis points? Price volatility = (MD) x (potential adverse move in yield) = (13.6986) x (.005) = 0.0345 or 3.45 percent. c. What is the DEAR? Daily earnings at risk (DEAR) = ($ value of position) x (Price volatility). Dollar value of position = $400m./(1 + 0.095) 15 = $10,59,350. Therefore, DEAR = $10,59,350 x 0.0345 = $3,511,79. 3
d. If the price volatility is based on a 90 percent confidence limit and a mean historical change in daily yields of 0.0 percent, what is the implied standard deviation of daily yield changes? The potential adverse move in yields = confidence limit value x standard deviation value. Therefore, 5 basis points = 1.65 x σ, and σ =.005/1.65 =.001515 or 15.15 basis points. 10. Bank Beta has an inventory of AAA-rated, 10-year zero-coupon bonds with a face value of $100 million. The modified duration of these bonds is 1.5 years, the DEAR is $,150,000, and the potential adverse move in yields is 35 basis points. What is the market value of the bonds, the yield on the bond, and the duration of the bond? Price volatility = (MD) x (potential adverse move in yield) = (1.5) x (.0035) = 0.04375 or 4.375 percent. Daily earnings at risk (DEAR) = ($ value of position) x (Price volatility) DEAR = $,150,000 = ($ value of position) x 0.04375 = > ($ value of position) = $,150,000/0.04375 = $49,14,857 = market value Dollar value of position = $00m./(1 + yield) 10 = $49,14,857. = > yield = ($100m/$49,14,857) 1/10 1 = 7.36% Therefore, the bonds currently are yielding 7.36 percent in the over-the-counter market. MD = D/(1 + R) = 1.5 = D/(1.0736) = > D = 1.5 x 1.0736 = 13.4 4
Chapter 10: Market Risk VaR 11. Bank Two has a portfolio of bonds with a market value of $00 million. The bonds have an estimated price volatility of 0.95 percent. What are the DEAR and the 10-day VAR for these bonds? Daily earnings at risk (DEAR) = ($ value of position) x (Price volatility) = $00 million x.0095 = $1,900,000 Value at risk (VAR) = DEAR x N = $1,900,000 x 10 = $1,900,000 x 3.163 = $6,008,38 1. Bank of Southern Vermont has determined that its inventory of 0 million euros ( ) and 5 million British pounds ( ) is subject to market risk. The spot exchange rates are $0.40/ and $1.8/, respectively. The σ 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 10-day VAR for both currencies. Use adverse rate changes in the 90 th percentile. FX position of = 0m x 0.40 = $8 million FX position of = 5m x 1.8 = $3 million FX volatility = 1.65 x 65bp = 107.5bp, or 1.075% FX volatility = 1.65 x 45bp = 74.5bp, or 0.745% DEAR = ($ value of position) x (Price volatility) DEAR of = $8m x.01075 = $85,800 DEAR of = $3m x.00745 = $37,600 VAR of = $85,800 x 10 = $85,800 x 3.163 = $71,33 5
VAR of = $37,600 x 10 = $37,600 x 3.163 = $751,357 13. Bank of Bentley has determined that its inventory of yen ( ) and Swiss franc (SF) denominated securities is subject to market risk. The spot exchange rates are 95.50/$ and SF1.075/$, respectively. The σ s of the spot exchange rates of the and SF, based on the daily changes of spot rates over the past six months, are 75 bp and 55 bp, respectively. Using adverse rate changes in the 90 th percentile, the 10-day VARs for the two currencies, and SF, are $350,000 and $500,000, respectively. Calculate the yen and Swiss franc-denominated value positions for Bank of Bentley. Value at risk (VAR) = DEAR x N => VAR of = $350,000 = DEAR x 10 = > DEAR = $350,000/ 10 = $110,680 VAR of SF = $500,000 = DEAR x 10 = > DEAR = $500,000/ 10 = $158,114 FX volatility = 1.65 x daily changes of spot rates over the past six months => FX volatility = 1.65 x 75bp =.01375, or 1.375% FX volatility SF = 1.65 x 55bp =.009075, or 0.9075% DEAR = ($ value of position) x (Price volatility) DEAR of = $110,680 = ($ value of position) x.01375 => ($ value of position) = $110,680/.01375 = $8,943,816 DEAR of SF = $158,114 = ($ value of position) x.009075 => ($ value of position) = $158,114/.009075 = $17,43,017 FX position in = Yen position/95.50 = $8,943,816 = > Yen position = 95.50 x $8,943,816 = 854,134,390 FX position in SF = SF position/1.075 = $17,43,017 = > SF position = 1.075 x $17,43,017 = SF18,79,744 14. Bank of Alaska s stock portfolio has a market value of $10 million. The beta of the 6
portfolio approximates the market portfolio, whose standard deviation (σ m ) has been estimated at 1.5 percent. What is the five-day VAR of this portfolio using adverse rate in the 99 th percentile? changes DEAR = ($ value of portfolio) x (.33 x σ m ) = $10m x (.33 x.015) = $10m x.03495 = $349,500 VAR = $349,500 x 5 = $349,500 x.361 = $781,506 15. Jeff Resnick, vice president of operations at Choice Bank, is estimating the aggregate DEAR of the bank s portfolio of assets consisting of loans (L), foreign currencies (FX), and common stock (EQ). The individual DEARs are $300,700, $74,000, and $16,700 respectively. If the correlation coefficients (ρ ij ) between L and FX, L and EQ, and FX and EQ are 0.3, 0.7, and 0.0, respectively, what is the DEAR of the aggregate portfolio? ( DEARL ) + (ρ L, DEAR portfolio = + (ρ L, + (ρ FX + ( DEAR FX EQ, EQ x DEAR x DEAR FX L L x DEAR ) FX + ( DEAR x DEAR x DEAR FX EQ x DEAR ) ) EQ EQ ) ) 0.5 $300,700 + $74,000 + $16,700 + (0.3)($300,700)($74,000) = + (0.7)($300,700)($16,700) + (0.0)($74,000)($16,700) 0.5 = 0.5 [ $84,3,66,000 ] = $533, 19 16. Calculate the DEAR for the following portfolio with the correlation coefficients and then with perfect positive correlation between various asset groups. What is the amount of risk reduction resulting from the lack of perfect positive correlation between the various assets groups? Estimated Assets DEAR (ρ S,FX ) (ρ S,B ) (ρ FX,B ) 7
Stocks (S) $300,000-0.10 0.75 0.0 Foreign Exchange (FX) 00,000 Bonds (B) 50,000 ( DEARS ) + (ρ S, DEAR portfolio= + (ρ S, + (ρ B FX, B + ( DEAR FX x DEAR x DEAR S x DEAR FX S FX ) + ( DEAR x DEAR x DEAR B FX ) x DEAR B ) ) B ) 0.5 $300,000 + $00,000 + $50,000 + ( 0.1)($300,000)($00,000) = + (0.75)($300,000)($50,000) + (0.0)($00,000)($50,000) 0.5 = 0.5 [ $313,000,000,000 ] = $559, 464 DEAR portfolio ( correlationcoefficients = 1) = $300,000 + $00,000 + $50,000 + (1.0)($300,000)($00,000) = + (1.0)($300,000)($50,000) + (1.0)($00,000)($50,000) 0.5 = 0.5 [ $56,500,000,000] = $750, 000 The DEAR for a portfolio with perfect correlation would be $750,000. Therefore, the risk reduction is $750,000 - $559,464 = $190,536. 8
Chapter 10: Market Risk Foreign Exchange Risk 18. Export Bank has a trading position in Japanese yen and Swiss francs. At the close of business on February 4, the bank had 300 million and SF10 million. The exchange rates for the most recent six days are given below: Exchange Rates per U.S. Dollar at the Close of Business /4 /3 / /1 1/9 1/8 Japanese yen 11.13 11.84 11.14 115.05 116.35 116.3 Swiss francs 1.4140 1.4175 1.4133 1.417 1.4157 1.413 a. What is the foreign exchange (FX) position in dollar equivalents using the FX rates on February 4? Japanese yen: 300,000,000/ 11.13 = $,675,466 Swiss francs: SF10,000,000/SF1.414 = $7,07,136 b. What is the definition of delta as it relates to the FX position? Delta measures the change in the dollar value of each FX position if the foreign currency depreciates by 1 percent against the dollar. c. What is the sensitivity of each FX position; that is, what is the value of delta for each currency on February 4? Japanese yen: 1.01 x current exchange rate = 1.01 x 11.13 = 113.513/$ Revalued position in $s = 300,000,000/113.513 = $,648,976 Delta of $ position to Yen = $,648,976 - $,675,466 = -$6,490 Swiss francs: 1.01 x current exchange rate = 1.01 x SF1.414 = SF1.4814 9
Revalued position in $s = SF10,000,000/1.4814 = $7,00,115 Delta of $ position to SF = $7,00,115 - $7,07,136 = -$70,01 d. What is the daily percentage change in exchange rates for each currency over the five-day period? Day Japanese yen: Swiss franc /4-0.691% -0.4691% % Change = (Rate t /Rate t-1 ) - 1 * 100 /3 0.64% 0.9718% / -.5934% -0.59084% /1-1.1173% 0.438% 1/9 0.0579% 0.4074% e. What is the total risk faced by the bank on each day? What is the worst-case day? What is the best-case day? Japanese yen Swiss francs Total Day Delta % Rate Risk Delta % Rate Risk Risk /4 -$6,490-0.691% $16,668 -$70,01-0.4691% $17,89 $33,957 /3 -$6,490 0.64% -$16,536 -$70,01 0.9718% -$0,809 -$37,344 / -$6,490 -.5934% $67,00 -$70,01-0.59084% $41,371 $108,373 /1 -$6,490-1.1173% $9,598 -$70,01 0.438% -$9,676 -$78 1/9 -$6,490 0.0579% -$683 -$70,01 0.4074% -$16,857 -$17,540 The worst-case day is February 3, and the best-case day is February. 10
f. Assume that you have data for the 500 trading days preceding February 4. Explain how you would identify the worst-case scenario with a 95 percent degree of confidence? The appropriate procedure would be to repeat the process illustrated in part (e) above for all 500 days. The 500 days would be ranked on the basis of total risk from the worst-case to the best-case. The fifth percentile from the absolute worst-case situation would be day 5 in the ranking. g. Explain how the 5 percent value at risk (VAR) position would be interpreted for business on February 5. Management would expect with a confidence level of 95 percent that the total risk on February 5 would be no worse than the total risk value for the 5 th worst day in the previous 500 days. This value represents the VAR for the portfolio. h. How would the simulation change at the end of the day on February 5? What variables and/or processes in the analysis may change? What variables and/or processes will not change? The analysis can be upgraded at the end of the each day. The values for delta may change for each of the assets in the analysis. As such, the value for VAR may also change. 19. Export Bank has a trading position in euros yen and Australian dollars. At the close of business on October 0, the bank had 0 million and A$30 million. The exchange rates for the most recent six days are given below: Exchange Rates per U.S. Dollar at the Close of Business 10/0 10/19 10/18 10/17 10/16 10/15 Euros 1.3900 1.3870 1.3675 1.3775 1.3850 1.4015 Australian $s 0.7800 0.7650 0.7900 0.7755 0.7605 0.7560 a. What is the foreign exchange (FX) position in dollar equivalents using the FX rates on October 0? Euros: 0 million/ 1.3900 = $14,388,489 Australian $s: A$30 million/a$0.7800 = $38,461,538 11
b. What is the sensitivity of each FX position; that is, what is the value of delta for each currency on October 0? Euros: 1.01 x current exchange rate = 1.01 x 1.3900 = 1.4039/$ Revalued position in $s = 0 million/1.4039 = $14,46,09 Delta of $ position to Yen = $14,46,09- $14,388,489 = -$14,460 Australian $s: 1.01 x current exchange rate = 1.01 x SF0.7800 = SF0.7878 Revalued position in $s = SF30 million/0.7878 = $38,080,731 Delta of $ position to SF = $38,080,731- $38,461,538 = -$380,807 c. What is the daily percentage change in exchange rates for each currency over the five-day period? Day Euro: Australian $s 10/0 0.169% 1.96078% % Change = (Rate t /Rate t-1 ) - 1 * 100 10/19 1.4596% -3.16456% 10/18-0.7595% 1.86976% 10/17-0.5415% 1.9739% 10/16-1.17731% 0.5954% d. What is the total risk faced by the bank on each day? What is the worst-case day? What is the best-case day? Euro Australian $s Total Day Delta % Rate Risk Delta % Rate Risk Risk 1
10/0 -$14,460 0.169% -$308.13 -$380,807 1.96078% -$7,466.79 -$7,774.9 10/19 -$14,460 1.4596% -$,031.43 -$380,807-3.16456% $1,050.88 $10,019.45 10/18 -$14,460-0.7595% $1,034.19 -$380,807 1.86976% -$7,10.18 -$6,085.99 10/17 -$14,460-0.5415% $771.45 -$380,807 1.9739% -$7,511.01 -$6,739.56 10/16 -$14,460-1.17731% $1,677.0 -$380,807 0.5954% -$,66.7 -$589.5 The worst-case day is October 0, and the best-case day is October 19. 13
Chapter 10: Market Risk Regulatory Standards 3. An FI has the following bonds in its portfolio: long 1-year U.S. Treasury bills, short 3½year Treasury bonds, long 3-year AAA-rated corporate bonds, and long 1-year B-rated (nonqualifying) bonds worth $40, $10, $5, and $10 million, respectively (market values). Using Table 10-8, determine the following: a. Charges for specific risk. Specific risk charges = $1.0 million (See below.) AAA = Qualifying bonds; B = Nonqualifying bonds Time Specific Risk General Market Risk band Issuer Position Weight% Charge Weight% Charge 1 year Treasury bill $40m 0.00 0.00 1.5 0.5000 3½-year Treasury bond ($10m) 0.00 0.00.5 (0.50) 3-year AAA-rated $5m 1.60 0.40.5 0.565 1-year B-rated $10m 8.0 0.80 4.50 0.4500 1.0 1.875 b. Charges for general market risk. General market risk charges = $1.875 million (From table above.) c. Charges for basis risk: vertical offsets within same time bands only (i.e., ignoring horizon effects). Time-band Longs Shorts Residuals Offset Disallowance Charge 3-year $0.565m ($0.5m) $0.3375m $0.50m 5% $0.0115m 14
d. The total capital charge, using the information from parts (a) through (c)? Total capital charges = $1.0m + $1.875 + $0.0115m = $,498,750 4. An FI has the following bonds in its portfolio. Bank Holdings (in millions) (1) () (3) (4) (5) (6) (7) Market Risk. Specific Risk General Time Band Issuer Position ($) Weight (%) Charge ($) Weight(%) Charge($) 1B3 months Treasury,000 0.00 0.00 0.0 4.00 3B6 months Qual Corp (5,000) 0.5 1.50 0.40 (0.00) 6B1 months Qual Corp 6,000 1.00 60.00 0.70 4.00 1B years Treasury,500 0.00 0.00 1.5 31.5 B3 years Treasury 1,500 0.00 0.00 1.75 6.5 3B4 years Treasury (4,000) 0.00 0.00.5 (90.00) 4B5 years Treasury 6,500 0.00 0.00.75 178.75 4B5 years Qual Corp (5,500) 1.60 88.00.5 (13.75) 5B7 years Qual Corp (5,000) 1.60 80.00 3.5 (16.50) 7B10 years Treasury 6,000 0.00 0.00 3.75 5.00 10B15 years Treasury (4,500) 0.00 0.00 4.50 (0.50) 15B0 years Treasury 4,000 0.00 0.00 5.5 10.00 15B0 years Non Qual (,000) 8.00 160.00 4.50 (90.00) >0 years Qual Corp 1,000 1.60 16.00 6.00 60.00 Specific risk 416.50 15
Residual general market risk 88.50 Using Table 10-8, determine the following: a. Charges for specific risk. From the table above (in italics), the specific risk charge is $416.50m. b. Charges for general market risk. From the table above (in italics), the general market risk charge is $88.50m. c. Charges for basis risk: vertical and horizontal offsets within and between time bands. Calculation of vertical and horizontal offsets (7) (1) () (3) (4) (5) (6) Charge($) 1. Specific risk 416.50 General Market Risk 88.50. Vertical offsets within same time bands Time band Longs Shorts Residual Offset Disallowance Charge 4B5 years 178.75 (13.75) 55.00 13.75 5.00% 6.1875 15B0 years 10.00 (90.00) 10.00 90.00 5.00 4.50 $10.6875 16
3. Horizontal offsets within same time zones Zone 1 1B3 months 4.00 3B6 months (0.00) 6B1 months 4.00. 8.00 Total zone 1 46.00 (0.00) 6.00 0.00 40.00% Zone 1B years 31.5 B3 years 6.5 3B4 years (90.00) 17.5 Total zone 57.50 (90.00) (3.50) 57.50 30.00% Zone 3 4B5 years 55.00 5B7 years (16.50) 7B10 years 5.00 10B15 years (0.50) 15B0 years 10.00 >0 years 60.00 Total zone 3 460.00 (365.00) 95.00 365.00 30.00% 109.50 134.75 17
4. Horizontal offsets between time zones Zones 1 and 6.00 (3.50) (6.50) 6.00 40.00% 10.40 Zones and 3 95.00 (6.50) 88.50 6.50 40.00%.60 13.00 d. The total capital charge, using the information from parts (a) through (c)? Total capital charge Specific risk 416.50 Vertical disallowances 10.6875 Horizontal disallowances 134.75 13.00 88.50 Offsets within same time zones Offsets between time zones Residual general market risk after all offsets 663.4375 Total Total capital charges = $663,437,500 7. An FI has an $160 million long position in yen, a $180 million short position in British pounds, a $80 million long position in Canadian dollars, and a $15 million short position in Swiss francs. The FI also holds various amounts of equities in its portfolio, as listed below. What would be the total capital charge required for the FI to cushion against FX and stock market risk? 18
Company Long Short IBM $15 million $75 million Xerox $65 million $10 million ExxonMobil $90 million KeyCorp $35 million FX risk: Total long position = $160m in yen + $80m in Canadian dollars = $40 million Total short position = $180m in British pounds + $15m in Canadian dollars = $305 million Higher of long or short positions = $305 million Capital charge = 0.08 x $305 = $4.4 million Common stock: Charges against unsystematic risk or firm-specific risk: Gross position in all stocks = $15m + $65m + $35m + $75m + $10m + $90m = $400m Capital charges = 4 percent x $400m = $16.0m Charges against systematic risk or market risk: Net Positions IBM $50m Xerox ExxonMobil KeyCorp 55m 90m 35m Total $30m 19
Capital charges = 8 percent x $30m = $18.4m Total capital charges = $16.0m + $18.4m = $34.4m Total capital charge required for the FI to cushion against FX and stock market risk = $4.4 million + $34.4m = $58.8 m 9. Dark Star Bank has estimated its average VAR for the previous 60 days to be $35.5 million. DEAR for the previous day was $30. million. a. Under the latest BIS standards, what is the amount of capital required to be held for market risk? Under the latest BIS standards, the proposed capital charge is the higher of: Previous day s VAR = DEAR x 10 = $30.m x 10 = $95,500,785 Average VAR x 3 = $35.5m x 3 = $106,500,000 => Capital charge = $106,500,000 b. Dark Star has $15 million of Tier 1 capital, $37.5 million of Tier capital, and $55 million of Tier 3 capital. Is this amount of capital sufficient? If not, what minimum amount of new capital should be raised? Of what type? Total capital needed = $106,500,000 Tier 1 + Tier + Tier 3 = $15m + $37.5 + $54m = $106.5m However, the capital is not sufficient because Tier 3 capital cannot exceed 50% of Tier 1 capital. Thus, Tier 1 capital (X) needs to be: X +.5X = $106.5m - $37.5 = $69m X = 69/3.5 = $19.7143m 0
If Tier 1 capital is increased by $19.7143m - $15m = $4.7143m and Tier 3 capital is decreased by $54m. - $4.7143m = $49.857m, then the capital charge will be met. That is, at this point, $19.7143m + $37.5m + $49.857m = $106.5m. 30. Bright Bank has estimated its average VAR for the previous 60 days to be $48.7 million. DEAR for the previous day was $50.3 million. a. Under the latest BIS standards, what is the amount of capital required to be held for market risk? Under the latest BIS standards, the proposed capital charge is the higher of: Previous day s VAR = DEAR x 10 = $48.7m x 10 = $154,00,9 Average VAR x 3 = $50.3m x 3 = $150,900,000 => Capital charge = $154,00,9 b. Bright Bank has $30 million of Tier 1 capital, $40,00,9 of Tier capital, and $84 million of Tier 3 capital. Is this amount of capital sufficient? If not, what minimum amount of new capital should be raised? Of what type? Total capital needed = $154,00,9 Tier 1 + Tier + Tier 3 = $30m + $40,00,9 + $84m = $154,00,9 However, the capital is not sufficient because Tier 3 capital cannot exceed 50% of Tier 1 capital. Thus, Tier 1 capital (X) needs to be: X +.5X = $154,00,9 - $40,00,9 = $114m X = 114/3.5 = $3.57143m 1
If Tier 1 capital is increased by $3.57143m - $30m = $.57143m and Tier 3 capital is decreased by $84m. - $.57143m = $81.4857m, then the capital charge will be met. That is, at this point, $3,571,430 + $40,00,9 + $81,48,570 = $154,00,9
Chapter 17: Liquidity Risk Depository Institutions 8. A DI with the following balance sheet (in millions) expects a net deposit drain of $15 million. Assets Liabilities and Equity Cash $10 Deposits $68 Loans 50 Equity 7 Securities 15 Total assets $75 Total liabilities and equity $75 Show the DI's balance sheet if the following conditions occur: a. The DI purchases liabilities to offset this expected drain. If the DI purchases liabilities, then the new balance sheet is: Cash $10 Deposits $53 Loans 50 Purchased liabilities 15 Securities 15 Equity 7 Total assets $75 Total liabilities and equity $75 b. The stored liquidity management method is used to meet the expected drain. If the DI uses reserve asset adjustment, a possible balance sheet may be: Loans $50 Deposits $53 Securities 10 Equity 7 Total assets $60 Total liabilities and equity $60 DIs will most likely use some combination of these two methods. 3
9. AllStarBank has the following balance sheet (in millions): Assets Liabilities and Equity Cash $30 Deposits $110 Loans 90 Borrowed funds 40 Securities 50 Equity 0 Total assets $170 Total liabilities and equity$170allstarbank s largest customer decides to exercise a $15 million loan commitment. How will the new balance sheet appear if AllStar uses the following liquidity risk strategies? a. Stored liquidity management. Assets Liabilities and Equity Cash $15 Deposits $110 Loans 105 Borrowed funds 40 Securities 50 Equity 0 Total assets $170 Total liabilities and equity $170 b. Purchased liquidity management. Assets Liabilities and Equity Cash $30 Deposits $110 Loans 105 Borrowed funds 55 Securities 50 Equity 0 Total assets $185 Total liabilities and equity $185 10. A DI has assets of $10 million consisting of $1 million in cash and $9 million in loans. The DI has core deposits of $6 million, subordinated debt of $ million, and equity of $ million. Increases in interest rates are expected to cause a net drain of $ million in core deposits over the year? a. The average cost of deposits is 6 percent and the average yield on loans is 8 percent. The DI decides to reduce its loan portfolio to offset this expected decline in deposits. What will be the effect on net interest income and the size of the DI after the implementation of this strategy? Assuming that the decrease in loans is offset by an equal decrease in deposits, the change in net interest income = (0.06 0.08) x $ million = -$40,000. The average size of the firm will be $8 million after the drain. 4
b. If the interest cost of issuing new short-term debt is expected to be 7.5 percent, what would be the effect on net interest income of offsetting the expected deposit drain with an increase in interest-bearing liabilities? Change in net interest income = (0.06 0.075) x $ million = -$30,000. c. What will be the size of the DI after the drain if the DI uses this strategy? The average size of the firm will be $10 million after the drain. d. What dynamic aspects of DI management would further support a strategy of replacing the deposit drain with interest-bearing liabilities? Purchasing interest-bearing liabilities may cost significantly more than the cost of replacing the deposits that are leaving the DI. However, using interest-bearing deposits protects the DI from decreasing asset size or changing the composition of the asset side of the balance sheet. 1. A DI has $10 million in T-bills, a $5 million line of credit to borrow in the repo market, and $5 million in excess cash reserves (above reserve requirements) with the Fed. The DI currently has borrowed $6 million in fed funds and $ million from the Fed discount window to meet seasonal demands. a. What is the DI s total available (sources of) liquidity? The DI s available resources for liquidity purposes are $10m + $5m + $5m = $0 million. b. What is the DI s current total uses of liquidity? The DI s current uses of liquidity are $6m + $m = $8 million. c. What is the net liquidity of the DI? The DI s net liquidity is $0m - $8m = $1 million. 5
d. What conclusions can you derive from the result? The net liquidity of $1 million suggests that the DI can withstand unexpected withdrawals of $1 million without having to reduce its less liquid assets at potential fire-sale prices. 13. A DI has the following assets in its portfolio: $0 million in cash reserves with the Fed, $0 million in T-bills, and $50 million in mortgage loans. If the assets need to be liquidated at short notice, the DI will receive only 99 percent of the fair market value of the T-bills and 90 percent of the fair market value of the mortgage loans. Estimate the liquidity index using the above information. Thus: I = ($0m/$90m)(1.00/1.00) + ($0m/$90m)(0.99/1.00) + ($50m/$90m)(0.90/1.00) = 0.94 14. Conglomerate Corporation has acquired Acme Corporation. To help finance the takeover, Conglomerate will liquidate the overfunded portion of Acme s pension fund. The face values and current and one-year future liquidation values of the assets that will be liquidated are given below: Liquidation Values Asset Face Value t = 0t = 1 yearibm stock $10,000 $9,900 $10,500 GE bonds 5,000 4,000 4,500 Treasury securities 15,000 13,000 14,000 Calculate the 1-year liquidity index for these securities. Thus, I = ($10,000/$30,000)($9,900/$10,500) + ($5,000/$30,000)($4,000/$4,500) + ($15,000/$30,000)($13,000/$14,000) = 0.97 15. Plainbank has $10 million in cash and equivalents, $30 million in loans, and $15 in core deposits. 6
a. Calculate the financing gap. Financing gap = average loans average deposits = $30 million - $15 million = $15 million b. What is the financing requirement? Financing requirement = financing gap + liquid assets = $15 million + $10 million = $5 m c. How can the financing gap be used in the day-to-day liquidity management of the bank? A rising financing gap on a daily basis over a period of time may indicate future liquidity problems due to increased deposit withdrawals and/or increased exercise of loan commitments. Sophisticated lenders in the money markets may be concerned about these trends and they may react by imposing higher risk premiums for borrowed funds or stricter credit limits on the amount of funds lent. 18. The following is the balance sheet of a DI (in millions): Assets Liabilities and Equity Cash $ Demand deposits $50 Loans 50 Premises and equipment 3 Equity 5 Total $55 Total $55The asset-liability management committee has estimated that the loans, whose average interest rate is 6 percent and whose average life is three years, will have to be discounted at 10 percent if they are to be sold in less than two days. If they can be sold in 4 days, they will have to be discounted at 8 percent. If they can be sold later than a week, the DI will receive the full market value. Loans are not amortized; that is, principal is paid at maturity. a. What will be the price received by the DI for the loans if they have to be sold in two days. In four days? 7
Price of loan = PVA n=3,k=10 ($3m) + PV n=3, k=10 ($50m) = $45.03m if sold in two days. Price of loan = PVA n=3,k=8 ($3m) + PV n=3, k=8 ($50m) = $47.4m if sold in four days. b. In a crisis, if depositors all demand payment on the first day, what amount will they receive? What will they receive if they demand to be paid within the week? Assume no deposit insurance. If depositors demand to withdraw all their money on the first day, the DI will have to dispose of its loans at fire-sale prices of $45.03 million. With its $ million in cash, it will be able to pay depositors on a first-come basis until $47.03 million has been withdrawn. The rest will have to wait until liquidation to share the remaining proceeds. Similarly, if the run takes place over a four-day period, the DI may have more time to dispose of its assets. This could generate $47.4 million. With its $ million in cash it would be able to satisfy on a first-come basis withdrawals up to $49.4 million. 8
Chapter 17: Liquidity Risk Other Financial Institutions. A mutual fund has the following assets in its portfolio: $40 million in fixed-income securities and $40 million in stocks at current market values. In the event of a liquidity crisis, the fund can sell the assets at a 96 percent of market value if they are disposed of in two days. The fund will receive 98 percent if the assets are disposed of in four days. Two shareholders, A and B, own 5 percent and 7 percent of equity (shares), respectively. a. Market uncertainty has caused shareholders to sell the shares back to the fund. What will the two shareholders receive if the mutual fund must sell all of the assets in two days? In four days? Value of fixed-income securities if sold in two days $40m x 0.96 = $38.4m Value of stocks if sold in two days $40m x 0.96 = $38.4m Total $76.8m Shareholder A will receive $76.8m x 0.05 = $3.84m down from the current value of $4.00m. Shareholder B will receive $76.8m x 0.07 = $5.376m down from the current value of $5.60m. Value of fixed-income securities if sold in four days $40m x 0.98 = $39.m Value of stocks if sold in two days $40m x 0.98 = $39.m Total $78.4m Shareholder A will receive $78.4m x 0.05 = $3.9m down from the current value of $4.00m. Shareholder B will receive $78.4m x 0.07 = $5.488m down from the current value of $5.60m. b. How does this situation differ from a bank run? How have bank regulators mitigated the problem of bank runs? This differs from a run on a bank in that in the mutual fund the claimants of the assets all receive the same amount, as a percentage of their investments. In the case of bank runs, the first to withdraw receives the full amount, leaving the likelihood that some depositors may not receive any money at all. One way of mitigating this problem is for regulators to offer deposit insurance such as that provided by the FDIC. This reduces the incentive to engage in runs. 9
3. A mutual fund has $1 million in cash and $9 million invested in securities. It currently has 1 million shares outstanding. a. What is the net asset value (NAV) of this fund? NAV = Market value of shares/number of shares = $10m/1m = $10 per share b. Assume that some of the shareholders decide to cash in their shares of the fund. How many shares at its current NAV can the fund take back without resorting to a sale of assets? At the current NAV, it can absorb up to $1 million, or 100,000 shares. c. As a result of anticipated heavy withdrawals, the fund sells 10,000 shares of IBM stock currently valued at $40. Unfortunately, it receives only $35 per share. What is the net asset value after the sale? What are the cash assets of the fund after the sale? The loss by selling 10,000 shares of IBM at $35 instead of $40 = -$5 x 10,000 = -$50,000. New NAV = $9,950,000 /1m = $9.95 Cash = $1 million + $350,000 = $1.35 million and $8.60 million in securities. d. Assume that after the sale of IBM shares, 100,000 shares are sold back to the fund. What is the current NAV? Is there a need to sell more securities to meet this redemption? If 100,000 shares are redeemed, it needs to pay $9.95 x 100,000 = $995,000. Its NAV will remain the same, i.e., ($9,950,000 - $995,000)/900,000 = $9.95. The mutual fund does not need to sell any extra shares since it has $1.35 million in cash to pay the $995,000. 30