商业银行风险管理英.1课堂练习

1、Key to exercises陈晓静 2022/7/20陈晓静 2Chapter 11.17 Credit risk primarily affects loan losses. Non-interest e includes trading gains and losses. Market risk therefore affects non-interest e. It also affects net interest e if assets and liabilities are not matched. Operations risk primarily affects non-i

2、nterest expense.2022/7/20陈晓静 3Chapter 11.19 The bank has an asset-liability mismatch of $25billion. The profit after tax is currently 12% of $ 2 billion or $0.24 billion. If interest rates rise by X% the banks before-tax loss (in billions of dollars) is 250.01X=0.25X. After taxes this loss es 0.70.2

3、5X=0.175X. The banks return on equity would be reduced to zero when 0.175X=0.24 or X=1.37.A 1.37% rise in rates would therefore reduce the return. 2022/7/20陈晓静 4Chapter 11.20 If long-term rates were simply a reflection of expected future short-term rates, we would expect the long rates to be less th

4、an short rates as often as they are greater than short rates. (This is based on the assumption that half of the time investors expect rates to increase and half of the time investors expect rates to decrease). Liquidity preference theory argues that long term rates are high 2022/7/20陈晓静 5Chapter 1Re

5、lative to expected future short-term rates. This means that long rates are greater than short rates most of the time. When long rates are less than short rates, the market is expecting a relatively steep decline in rates.Chapter 3 Insurance Companies and Pension Plans3.16 on Page 59(1) 0.011858*5,00

6、0,000=5929059290/1.03=57,563(2)(1-0.011858)*0.012966=0.012812250.01281225*5,000,000=64,06164,061/1.033=58,625(3) (1-0.011858)*(1-0.012966)*0.014123=0.013774580.01377458*5,000,000=68872.968872.9/1.035=59410.3686Chapter 3 Insurance Companies and Pension Plans(1)+(2)+(3)=57563 =175,598The minimum premi

7、um is XX+(1-0.011858)*X/1.032+(1-0.011858)(1-0.012966)*X/1.034=175,5982.7979865X=175,598 X=62,758.7017So the minimum premium is $62,759.2022/7/20陈晓静 8Chapter 5 : 5.31 on Page 1085.31 The investment in call options entails higher risks but can lead to higher returns. If the stock price stays at $94,

8、an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who buys call options gains 2000(120-95)-9400=$40,6002022/7/20陈晓静 9Chapter 5An investor who buys shares gains 100(120-94)=$2,600The stra

9、tegies are equally profitable if the stock price rises to a level, S, where100(S-94)=2000(S-95)-9400Or S=100The option strategy is therefore more profitable if the stock price rises above $100.2022/7/20陈晓静 10Chapter 5 5.33 on Page 1085.33 The arbitrage could borrow money to buy 100 ounces of gold to

10、day and short futures contracts on 100 ounces of gold for delivery in one year. This means that gold is purchased for $500 per ounce and sold for $700 per ounce. The return (40% per annum) is far greater than the 10% cost of the borrowed funds. This is such a profitable opportunity that the arbitrag

11、eur should buy as many ounces of gold as possible and short futures contracts on the same number of ounces. Unfortunately arbitrage opportunities as profitable as this rarely arise in practice. 2022/7/20陈晓静 11Chapter 5: 5.34 on Page 1085.34 (a) By entering into a three-year swap where it receives 6.

12、21% and pays LIBOR the company earns 5.71% for three years.(b) By entering into a five-year swap where it receives 6.47% and pays LIBOR the company earns 5.97% for five years.(c) By entering into a swap where it receives 6.83% and pays LIBOR for ten years the company ears 6.33% for ten years.2022/7/

13、20陈晓静 12Chapter 22.31 (a) The company should short (1.2-0.5)100,000,0001000250Or 280 contracts.(b) The company should take a long position in (1.5-1.2)100,000,0001000250Or 120 contracts.2022/7/20陈晓静 13Chapter 5:5.2 on Page 1065.2 Explain the difference between hedging, speculation and arbitrage.2022

14、/7/20陈晓静 14Key to 5.2A trader is hedging when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure. In a speculation the trader has no exposure to offset. She is betting on the future movements in the price of the asset. Arbitrage involves taking a

15、 position in two or more different markets to lock in a profit.Chapter 5: Exercise 5.10 on Page 1062022/7/20陈晓静 16Chapter 5:5.10You should buy 5,000 put options (or 50 contracts) with a strike price of $25 and an expiration date in 4 months. This provides a type of insurance. If at the end of 4 mont

16、hs the stock price proves to be less than $25, you can exercise the options and sell the shares for $25 each. The cost of this strategy is the price you pay for the put option. Chapter 5 5.14 on Page 107A long position in a four-month put option can provide insurance against the exchange rate fallin

17、g below the strike price. It ensures that the foreign currency can be sold for at least the strike price.2022/7/20陈晓静 18Chapter 5: 5.21 on Page 107 A corn farmer argues: “I do not use futures contracts for hedging. My real risk is not the price of corn. It is that my whole crop gets wiped out by the

18、 weather.” discuss this viewpoint. Should the farmer estimates his or her expected production of corn and hedge to try to lock in a price for expected production?2022/7/20陈晓静 19Key to 5.21Suppose that the weather is bad and the farmers production is lower than expected. Other farmers are likely to h

19、ave been affected similarly. Corn production overall will be low and as a consequence the price of corn will be relatively high. The farmer is likely to be over-hedged relative to actual production. The farmers problems arising from the bad harvest will be made worse by losses on the short futures p

20、osition. This problem emphasizes the 2022/7/20陈晓静 20Chapter 5:5.21Importance of looking at the big picture when hedging. The farmer is correct to question whether hedging price risk while ignoring other risks is a good strategy.2022/7/20陈晓静 21Chapter 5:5.22 on Page 107An airline executive has argued

21、: “ There is no point in our hedging the price of jet fuel. There is just as much chance that we will lose from doing this as that we will gain.” Discuss the executives viewpoint.2022/7/20陈晓静 22Key to 5.22It may well be true that there is just as much chance that the company will lose as that it wil

22、l gain. This means that the use of a futures contract for speculation would be like betting on whether a coin comes up heads or tails. But it might make sense for the airline to use futures for hedging rather than speculation. The futures contract then has the effect of reducing risks. It can be arg

23、ued that an airline should not expose its shareholders to risks associated with the future price of oil when there are contracts available to hedge the risks.2022/7/20陈晓静 23Chapter 5: 5.23 on Page 107Why is the cost of an Asian basket put option to Microsoft considerably less than the cost of a port

24、folio of put options, one for each currency and each maturity( see Business Snapshot 5.3)?2022/7/20陈晓静 24Key to 5.23 Microsoft is choosing an option on a portfolio of assets instead of the corresponding portfolio of options. The former is always less expensive because there is the potential for an i

25、ncrease in the price of one asset to be netted off against a decrease in the price of another. Compare (a) an option with a strike price of $20 on a portfolio of two assets each worth $10 and (b) a portfolio of two options2022/7/20陈晓静 255.23With a strike price of $10, one on each of assets. If both

26、assets increase in price or both assets decrease in price, the payoff from (a) is less than that from (b). Both the Asian feature and the basket feature in Microsofts options help to reduce the cost of the options because of the possibility of gains and loss being netted.Chapter 5: 5.25 on Page 107I

27、t means that the price of the energy source can go up or down but will tend over time to get pulled back to its long-run average level. Electricity has the highest rate of mean reversion; oil has the lowest.Chapter 5: 5.29 on Page 108The cost of shares is 500*50, or $25,000. When shares are shorted,

28、 the proceeds of the sale form part of the margin. In this case the total margin required is 1.6*25,000, or $40,000. The extra margin required is therefore $15,000. This can be in the form of cash or marginable securities. When the share price rises to S, the value of the underlying stock is 500S. T

29、here is a margin call when 40,00061.54.2022/7/20陈晓静 28Quiz (chapter 1-chapter 3)1.What does it mean to assert that the theta of an option position is -100 per day? If a trader feels that neither a stock price nor its implied volatility will change, what type of option position is approciate?(10%)202

30、2/7/20陈晓静 29Quiz (chapter 1-chapter 3)2.The expected return on the market portfolio is 12% and the risk-free rate is 6%. What is the expected return on an investment with a beta of (a) 0.2, (b) 0.5, and (c) 1.4? (10%)2022/7/20陈晓静 30Quiz (chapter 1-chapter 3)3. Explain why there are economies of scal

31、e in hedging options.(10%)2022/7/20陈晓静 31Quiz (chapter 1-chapter 3)4. Suppose that you own 5,000 shares worth $25 each. How can put options be used to provide you with insurance against a decline in the value of your holding over the next four months?(20%)2022/7/20陈晓静 32Quiz (chapter 1-chapter 3)5.

32、The price of gold is currently $500 per ounce. The forward price for delivery in one year is &700. An arbitrager can borrow money at 10% per annum. What should the arbitrager do? Assume that the cost of storing gold is zero and that gold provides no e.(20%)2022/7/20陈晓静 33Quiz (chapter 1-chapter 3)6.

33、 3.17 on p.78 （30）Change Delta 0f 0.6 to 0.7Key to Chapter 44.15He should pay tax for four parts:(1) dividends $2*100=$200 of 2009; $3*100=$300 of 2010(2) capital gains $5*100=$500 of 2009; $3*100=$300 0f 2010(3)(59-50)*100=$900 of 2011(4)reinvestment gain 4.17 of Chapter 4(1) overall return=(-5%+1%

34、+10%+15%+20%)/5=8.2%(2)return for hedge funds:-5%: 2%/5=0.4%1%: 2%+1%*20%=0.44%10%: 2%+10%*20%=0.8%15%: 2%+15%*20%=1%20%: 2%+20%*20%=1.2%4.17 of Chapter 4(3) return for fund of funds=1%+10%*(8.2%-0.4%-0.44%-0.8%-1%-1.2%)=1.436%(4) return for investors=8.2%-1.436%-(0.4%+0.44%+0.8%+1%+1.2%) =2.924%202

35、2/7/20陈晓静 37Key to Q6The delta of the portfolio is -1,000X0.50+-500X0.80-2,000X(-0.40)-500X0.70=-450The gamma of the portfolio is -1,000X2.2-500X0.6-2,000X1.3-500X1.8=-6,000The vega of the portfolio is -1,000X1.8-500X0.2-2,000X0.7-500X1.4=-4,0002022/7/20陈晓静 38Key to Q6(a) A long position in 4,000 tr

36、aded options will give a gamma-neutral portfolio since the long position has a gamma of 4,000X1.5=+6,000. The delta of the whole portfolio (including traded options) is then: 4,000X0.7-450=2,350Hence, in addition to the 4,000 traded options, a short position in 2,350 is necessary so that the portfol

37、io is both gamma and delta neutral.2022/7/20陈晓静 39Key to Q6(b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a vega of 5,000X0.8=+4,000. The delta of the whole portfolio (including traded options) is then 5,000X0.7-450=3,050Hence, in addition

38、to the 5,000 traded options, a short position in 3,050 is necessary so that the portfolio is both vega and delta neutral.2022/7/20陈晓静 40Quiz (chapter 1-chapter 3)The arbitrageur could borrow money to buy 100 ounces of gold today and short futures contracts on 100 ounces of gold for delivery in one y

39、ear. This means that gold is purchased for $500 per ounce and sold for $700 per ounce. The return (40% per annum) is far greater than the 10% cost of the borrowed funds. This is such a profitable opportunity that the arbitrageur 2022/7/20陈晓静 41Quiz (chapter 1-chapter 3)Should buy as many ounces of g

40、old as possible and short futures contracts on the same number of ounces. Unfortunately arbitrage opportunities as profitable as this rarely arise in practice.2022/7/20陈晓静 42Chapter 77.13 A five-year bond with a yield of 11% (continuously compounded) pays an 8% coupon at the end of each year. (a) wh

41、at is the bonds price? (b) what is the bonds duration? (c) Use the duration to calculate the effect on the bonds price of 0.2% decrease in its yield. (d) Recalculate the bonds price on the basis of a 10.8% per2022/7/20陈晓静 43Chapter 7Annum yield and verify that the result in agreement with your answe

42、r to (c).Chapter 7 7.13(a) B=8e-11%*1+8e-11%*2+8e-11%*3+8e-11%*4+108e-11%*5=86.80(b) D=2022/7/20陈晓静 44Chapter 7=1*8e-11%*1/86.80+2*8e-11%*2/86.80+3*8e-11%*3/86.80+4*8e-11%*4/86.80+5*108e-11%*5/86.80=4.256(c) B/B=-Dy B=86.80*(-4.256)*(-0.2%)=0.74(d) B=8e-10.8%*1+8e-10.8%*2+8e-10.8%*3+8e-10.8%*4+108e-

43、10.8%*5=87.54=86.80+0.742022/7/20陈晓静 452022/7/20陈晓静 46Chapter 77.19 (a) The duration of Portfolio A is 12000e-0.11+106000e-0.110 2000e-0.11 e-0.110 =5.95Since this is also the duration of Portfolio B, the two portfolios do have the same duration.(b) The value of Portfolio A is 2000e-0.1 e-0.110=4016

44、.952022/7/20陈晓静 47Chapter 7When yields increase by 10 basis points its value es2000e-0.101 e-0.10110=3993.18The percentage decrease in value is23.77/4016.95=0.59%The value of Portfolio B is 5000e-0.15.95=2757.81When yields increase by 10 basis points its value es2022/7/20陈晓静 48Chapter 75000e-0.1015.

45、95=2741.45The percentage decrease in value is16.36/2757.81=0.59%The percentage changes in the values of the two portfolios for a 10 basis point increase in yields are therefore the same.(c) When yields increase by 5% the value of Portfolio A es 2022/7/20陈晓静 49Chapter 72000e-0.15 e-0.1510=3060.20And

46、the value of portfolio B es 5000e-0.155.95=2048.15 The percentage reduction in the values of the two portfolios are: Portfolio A: 956.75/4016.95=23.82% Portfolio B: 709.66/2757.81=25.73%2022/7/20陈晓静 50Key to 7.22The delta with respect to the first factor is 0.215+0.26 (-3)+0.32 (-1)+0.35 2+0.36 5+0.

47、36 7+0.36 8=7.85Similarly the delta with respect to the second and third factors are 1.18 and -1.24.2022/7/20陈晓静 51Chapter 1111.21 ( page 243)Using table 11.2 the credit equivalent amount under Basel I ( in millions of dollars) for the three transactions are(a) 3+0.005100=3.5(b) 0.01150=1.5(c) 7+0.0

48、150=7.5The total credit equivalent amount is 3.5+1.5+7.5=12.5. Because the corporation has a risk weight of 50% the risk weighted amount2022/7/20陈晓静 52Chapter 11Is also 6.25. The capital required is 0.086.25 or $0.5 million.If netting applies, the current exposure after netting is in millions of dol

49、lars 3-5+7=5. The NRR is therefore 5/10=0.5. The credit equivalent amount is in millions of dollars5+(0.4+0.60.5)(0.005100+0.01150+0.0150)=6.752022/7/20陈晓静 53Chapter 11The risk weighted amount is 6.75*50%=3.375 and the capital required is 0.083.375=0.27. In this case the netting amendment reduces th

50、e capital by 46%.Under Basel II when the standardized approach is used the corporation has a risk weight of 20% and the capital required is therefore one fifth of that required under Basel I or 0.20.27 or $0.054million.2022/7/20陈晓静 54Chapter 1111.22 (page 243)Under the Basel II advanced IRB approach

51、 P=0.121+e-500.003=0.2233b=0.11852-0.05478ln(0.003)2=0.1907MA=1+(3.0-2.5) 0.1907/1-1.50.1907=1.53and2022/7/20陈晓静 55Chapter 11WCDR=NN-1(0.003)+ (0.999)/ =0.0720The RWA is RWA=12.5EADLGD(WCDR-PD) MA5000.6(0.0720-0.003) 1.5312.5=397.13The total capital is 8% of this or $31.77 million. Half of this must

52、 be Tire I. Under both the Basel II standardized approach and under Basel I the risk weight is 100% and the total capital required is 8% of $500 or $40 million.2022/7/20陈晓静 56Chapter 88.12 (page 170)(a) A loss of $1 million extends from the 94 percentile point of the loss distribution to the 96 perc

53、entile point. The 95% VaR is therefore $1 million.(b) The expected shortfall for one of the investments is the expected loss conditional that the loss is in the 5 percent tail. Given that we are in the tail there is a 20% chance than the loss is $1 million and an 80% chance that the loss is $10 mill

54、ion. The expected loss is therefore $8.2 2022/7/20陈晓静 57Chapter 8Million. This is the expected shortfall.(c) For a portfolio consisting of the two investments there is a 0.040.04=0.0016 chance that the loss is $20 million; there is a 20.040.02=0.0016 chance that the loss is $11 million; there is a 2

55、0.040.94=0.0752 chance that the loss is $9 million; there is a 0.020.02=0.0004 chance that the loss is $ 2 million; there is a 20.020.94=0.0376 chance that the loss is zero; there is a 0.940.94=0.8836 chance that the profit is $2million. It follows that the 95% VaR is $9 million.2022/7/20陈晓静 58Chapt

56、er 8(d) The expected shortfall for the portfolio consisting of the two investment is the expected loss conditional that the loss is in the 5% tail. Given that we are in the tail, there is a 0.0016/0.05=0.032 chance of a loss of $20 million, a 0.0016/0.05=0.032 chance of a loss of $11million; and a 0

57、.936 chance of a loss of $9 million. The expected loss is therefore $9.416.2022/7/20陈晓静 59Chapter 8(e) VaR does not satisfy the subadditivity condition because 91+1. However, expected shortfall does because 9.4168.2+8.2.2022/7/20陈晓静 60Exercise for Chapter 11 11.21A bank has the following transaction

58、 with an AA-rated corporation: (a) a 3-year interest rate swap with a principal of $150 million worth $3million; (b) a 5-year foreign exchange forward contract with a principal of $120 million worth -$5 million; and (c) a 6-month long option on gold with a principal of $60 worth $7 million. What is

59、the capital requirement under Basel I if there is no netting? What difference does it make if the netting amendment applies? What is the capital required under Basel II when the standardized approach is used?2022/7/20陈晓静 61Chapter 11 11.21 on Page 243Using table 11.2 the credit equivalent amount und

60、er Basel I ( in millions of dollars) for the three transactions are(a) 3+0.005150=3.75(b) 0+0.05120=6(c) 7+0.0160=7.6The total credit equivalent amount is 3.75+6+7.6=17.35. Because the corporation has a risk weight of 100% the risk weighted amount2022/7/20陈晓静 62 Chapter 11 11.21Is also 17.35. The ca