江财金融机构管理的作业及答案

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1. 再定价缺口 期限缺口 有效期限

假设一位名为M.P.Jorgan的政府证券交易商的有关资料如下，阅读后回答问题。括号内的数字是市场收益率，金额的单位是百万美元。

资产 现金 1个月的国库券(7.05%) 3个月的国库券(7.25%) 2年期中期国库券(7.50%) 8年期中期国库券(8.96%) 75 75 50 100 负债与股东权益 $ 170 15 $ 335 $ 10 隔夜再回购协议 股权资本

7年期固定利率次级债务(8.55%) 150 5年期市政债券(浮动利率8.20%， 25 每六个月重定一次) 总资产 $ 335 总负债加股东权益 (a) 如果计划期限是30天，那么再定价缺口是多少？91天呢？(提示：现金是一项没有利息收入的资产)

(b) 如果所有利率都上升50个基本点，将对未来30内的净利息收入有何影响？如果都下降75个基本点呢？ 参考答案： (a)30天计划期限的利率敏感性资产有：1个月的国库券； 30天计划期限的利率敏感性负债有：隔夜再回购协议；

所以，计划期限为30的再定价缺口：GAP=资产-负债=75-175=-95.

91天计划期限的利率敏感性资产有：1个月的国库券和3个月的国库券； 91天计划期限的利率敏感性负债有：隔夜再回购协议；

所以，计划期限为91的再定价缺口：GAP=资产-负债=75+75-175=-20.

(b)如果所有利率都上升50个基本点，那么未来30内的净利息收入变化为：

?NII?GAP*?R??95*0.0050??0.475，净利息收入减少0.475

如果所有利率都下降75个基本点，那么未来30内的净利息收入变化为：

?NII?GAP*?R??95*(?0.0075)?0.7125，净利息收入增加0.7125

以下是County银行按市场价值记账的资产负债表(单位：百万美元，所有利率都是年利率) 资产 现金 负债与股东权益 $ 100 $ 20 活期存款 15年期商业贷款(利率为10%，期末160 还清贷款) 30年期抵押贷款（利率为8%，每月300 分期付款） 总资产 5年期大额可转让存单（利率为6%，期210 末还清贷款） 20年期无抵押债券（利率为7%） 股权资本 120 50 $ 480 $ 480 总负债加股东权益 (a)County银行的期限缺口是多少？

(b)如果所有资产和负债的利率都上升1%，那么期限缺口又是多少？ 参考答案：

Mi??j?1WijMij，Wij?nPij?nj?1Pij，i?资产组合、负债组合

(a）资产组合的平均期限MA?WA1MA1?WA2MA2?WA3MA3， 其中MA1?0,MA2?15,MA3?30，WA1?所以MA?23.75。

负债组合的平均期限ML?WL1ML1?WL2ML2?WL3ML3， 其中ML1?0,ML2?5,ML3?20，WL1?所以ML?8.02。

期限缺口=MA?ML?23.75?8.02?15.73

20160300。 ,WA2?,WA3?480480480100210120。 ,WL2?,WL3?430430430(b)如果所有资产和负债的利率都上升1%，那么首先计算各资产、负债的市场价值。

PA1?20，PA2??t?115cashflowt141616?160?()??148.49，?tt15t?1(1?10%?1%)(1?11%)(1?11%)PA3??t?1所以

360cashflowt3602.201294??t?1?273.581，

8%?1%t9%t(1?)(1?)1212 MA?23.60

Demand deposits = $100 CDs = $12.60*PVIFAn=5,i=7% + $210*PVIFn=5,i=7% = $201.39 Debentures = $8.4*PVIFAn=20,i=8% + $120*PVIFn=20,i=8% = $108.22 ML = [0*100 + 5*201.39 + 20*108.22]/(100 + 201.39 + 108.22) = 7.74 years

计算一张年息票率为10%、到期收益率为14%、价值$1000的两年期债券的有效期限。如果利率下降0.5%，债券价格将如何变化？ 参考答案：

Two-year Bond Par value = $1,000 Coupon = 0.10 Annual payments YTM = 0.14 Maturity = 2 Time Cash Flow PVIF PV of CF PV*CF*T 1 $100.00 0.87719 $87.72 $87.72 PVIF = 1/(1+YTM)^(Time) 2 $1,100.00 0.76947 $846.41 $1,692.83 Price = $934.13 Numerator = $1,780.55 Duration = 1.9061 = Numerator/Price

Expected change in price = ?D

?R?.005P??1.9061$934.13?$7.81. This implies a new price of 1?R1.14$941.94. The actual price using conventional bond price discounting would be $941.99. The difference of

$0.05 is due to convexity, which was not considered in this solution.

2. 日风险收益的计算

Alpha 银行持有AAA级的15年期的零息债券，面值为4亿美元。债券目前在场外市场的收益率为9.5%，请问：如果潜在不利的收益率的变动为25个基点，那么日风险收益是多少？ 参考答案：

a. What is the modified duration of these bonds?

b. What is the price volatility if the potential adverse move in yields is 25 basis points?

Price volatility = (-MD) x (potential adverse move in yield)

= (-13.6986) x (.0025) = -0.03425 or -3.425 percent. Modified duration = (MD) = D/(1 + r) = 15/(1.095) = -13.6986.

c. What is the DEAR?

Daily earnings at risk (DEAR) = ($ Value of position) x (Price volatility) Dollar

value of position = 400/(1 + 0.095)15 = $102.5293million. Therefore,

d. If the price volatility is based on a 90 percent confidence limit and a mean DEAR = $102.5293499 million x -0.03425 = -$3.5116 million, or -$3,511,630.

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 (PAMY) = confidence limit value x standard deviation value. Therefore, 25 basis points = 1.65 x ?, and ? = .0025/1.65 = .001515 or 15.15 basis points.

3. 违约概率期限结构的计算

利用下面的国债和公司债券(纯折价债券)收益曲线中的即期利率，来计算三年的违约概率期限结构。一定要计算出边际违约概率和累积违约概率。

长期国债 参考答案：

1年即期 2年即期 3年即期 5.0% 6.1% 8.2% 7.2% 9.3% BBB级债券 7.0% The notation used for implied forward rates is f12 = forward rate from period 1 to

period 2.

Treasury Securities

BBB Graded Debt

(1.082)2 = (1.07)(1 + f12 )

(1.061)2 = (1.05)(1 + f12 )

f12 = 7.21%

f12 = 9.41%

(1.07)3 = (1.061)2(1 + f23 ) f23 = 8.82%

(1.093)3 = (1.082)2(1 + f23 )

f23 = 11.53%

Using the implied forward rates, estimate the annual marginal probability of

repayment:

p01(1.07) = 1.05 => p1 = 98.13 percent p12(1.0941) = 1.0721 => p2 = 97.99 percent p23 (1.1153) = 1.0882

=> p3 = 97.57 percent

Using marginal probabilities, estimate the cumulative probability of default:

cp02 = 1 - (p1 )(p2 )

= 1 - (.9813)(.9799) = 3.84 percent

cp03 = 1 - (p1 )(p2 )(p3 )

= 1 - (.9813)(.9799)(.9757) = 6.18 percent

16. Using regression analysis on historical loan losses, a bank has estimated the

following: XC = 0.002 + 0.8XL, and Xh = 0.003 + 1.8XL where XC = loss rate in the commercial sector, Xh = loss rate in the consumer

(household) sector, XL = loss rate for its total loan portfolio.

a. If the bank’s total loan loss rates increase by 10 percent, what are the increases in the expected loss rates in the commercial and consumer sectors?

Commercial loan loss rates will increase by 0.002 + 0.8(0.10) = 8.20 percent.

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