CH 9 練習題
24.
The balance sheet for Gotbucks Bank, Inc. (GBI), is presented below ($ millions):
(duration 請計算至小數點以下第四位；金額請計算至元，請勿回答以百萬元為單
位)
Assets
Cash
Federal funds
Loans (floating)
Loans (fixed)
Total assets
Liabilities and Equity
Core deposits
$20
Federal funds
50
Euro CDs
130
Equity
20
Total liabilities and equity $220
$30
20
105
65
$220
Notes to the balance sheet: The fed funds rate is 8.5 percent, the floating loan rate is
LIBOR (London Interbank Offered Rate) + 4 percent, and currently LIBOR is 11 percent.
Fixed-rate loans have four-year maturities, are priced at par, and pay 12 percent semiannual interest. The principal is repaid at maturity. Core deposits are fixed rate for two
years at 8 percent paid annually. The principal is repaid at maturity. Euro CDs currently
yield 9 percent.
a. What is the duration of the fixed-rate loan portfolio of Gotbucks Bank?
Five-year Loan (values in millions of $s)
Par value = $65
Coupon rate = 12%
R = 12%
Maturity = 4 years
t
CFt
DFt
CFt x DFt
Annual payments
1
3.9
1.06
CFt x DFt x t
3.679245 1.839622642
2
3.9
1.1236
3.470986 3.470986116
3
3.9
1.191016
3.274515 4.911772806
4
3.9
1.262477
3.089165 6.178330573
5
3.9
1.338226
2.914307 7.285767185
6
3.9
1.418519
2.749346 8.248038323
7
3.9
1.50363
2.593723 9.078029601
8
68.9
1.593848
43.22871 172.9148495
65 213.9273968
Duration = $213.9273968/$65.000 = 3.2912
1
3.29119072
b. If the duration of the floating-rate loans and fed funds is 0.36 year, what is the
duration of GBI’s assets?
DA = [$30(0) + $20(0.36) + $105(0.36) + $65(3.2912)]/$220 = 258.928/220 = 1.1769 years
c. What is the duration of the core deposits if they are priced at par?
Two-year Core Deposits (values in millions of $s)
Par value = $20
Coupon rate = 8%
Annual payments
R = 8%
Maturity = 2 years
t
CFt
DFt
CFt x DFt
CFt x DFt x t
1
1.6
0.9259
1.481
1.481
2
21.6
0.8573
18.519
37.037
20.000
38.519
Duration = $38.519/$20.000 = 1.9259
d. If the duration of the Euro CDs and fed funds liabilities is 0.401 year, what is the
duration of GBI’s liabilities?
DL = [$20(1.9259) + $50(0.401) + $130(0.401)]/$200 = 0.5535 years
e. What is GBI’s duration gap? What is its interest rate risk exposure?
GBI’s leveraged adjusted duration gap is: 1.1769 - 200/220 x (0.5535) = 0.6737 years
Since GBI’s duration gap is positive, an increase in interest rates will lead to a decrease in the
market value of equity.
f. What is the impact on the market value of equity if the relative change in all interest
rates is an increase of 1 percent (100 basis points)? Note that the relative change in
interest rates is ∆R/(1+R) = 0.01.
For a 1 percent increase, the change in equity value is:
ΔE = -0.6737 x $220,000,000 x (0.01) = -$1,482,140
g. What is the impact on the market value of equity if the relative change in all interest
rates is a decrease of 0.5 percent (-50 basis points)?
2
For a 0.5 percent decrease, the change in equity value is:
ΔE = -0.6737 x $220,000,000 x (-0.005) = $741,070
h. What would the duration of its assets need to change to get DGAP equal to zero?
What would the duration of its liabilities need to change to get DGAP equal to zero?
Immunization requires the bank to have a leverage adjusted duration gap of 0. Therefore, GBI
could reduce the duration of its assets to 0.5032 (0.5535 x 200/220) years by using more fed
funds and floating rate loans. Or GBI could increase the duration of its liabilities to 1.2946
(1.1769 x 220/200) years.
26.
The following balance sheet information is available (amounts in thousands of dollars
and duration in years) for a financial institution: (duration 請計算至小數點以下第四
位；金額請計算至千元)
Amount
$90
55
180
2,724
2,092
242
715
T-bills
T-notes
T-bonds
Loans
Deposits
Federal funds
Equity
Duration
0.50
0.90
x
7.00
1.00
0.01
Treasury bonds are five-year maturities paying 6 percent annually and selling at par.
a. What is the duration of the T-bond portfolio? 4.4651
Five-year Treasury Bond
Par value = $176 Coupon rate = 6%
Semiannual payments
R = 6%
Maturity = 5 years
t
CFt
DFt
CFt x DFt
CFt x DFt x t
1
10.8
1.06
10.18868
10.18867925
2
10.8
1.1236
9.611962
19.2239231
3
10.8
1.191016
9.067888
27.20366477
4
10.8
1.262477
8.554612
34.21844625
5
190.8
1.338226
142.5769
712.8842969
3
180
803.7190103
4.46510561
Duration = $803.7190103/$180.00 = 4.4651
b. What is the average duration of all the assets?
[(0.5)($90) + (0.9)($55) + (4.4651)($180) + (7)($2,724)]/$3,049 = 6.5484years
c. What is the average duration of all the liabilities?
[(1)($2,092) + (0.01)($242)]/$2,334 = 0.8974 years
d. What is the leverage adjusted duration gap? What is the interest rate risk exposure?
DGAP = DA - kDL = 6.5484 - ($2,334/$3,049)(0.8974) = 5.8614 years
The duration gap is positive, indicating that an increase in interest rates will lead to a
decrease in the market value of equity.
e. What is the forecasted impact on the market value of equity caused by a relative
upward shift in the entire yield curve of 0.5 percent [i.e., ∆R/(1+R) = 0.0050]?
The market value of the equity will change by:
ΔE = -DGAP x A x ΔR/(1 + R) = -5.8614 x $3,049 x (0.0050) = -$89.3570
f. If the yield curve shifts downward by 0.25 percent [i.e., ∆R/(1+R) = -0.0025], what
is the forecasted impact on the market value of equity?
The change in the value of equity is ΔMVE = -5.8614 x $3,049 x (-0.0025) = $44.6785
g. What variables are available to the financial institution to immunize the balance
sheet? How much would each variable need to change to get DGAP equal to 0?
Immunization requires the bank to have a leverage adjusted duration gap of 0. Therefore, the
FI could reduce the duration of its assets to 0.6870 years by using more T-bills and floating
rate loans. Or, the FI could try to increase the duration of its deposits possibly by using fixedrate CDs with a maturity of 3 or 4 years. Finally, the FI could use a combination of reducing
asset duration and increasing liability duration in such a manner that DGAP is 0. This
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duration gap of 5.8614 years is quite large and it is not likely that the FI will be able to reduce
it to zero by using only balance sheet adjustments. For example, even if the FI moved all of
its loans into T-bills, the duration of the assets still would exceed the duration of the liabilities
after adjusting for leverage. This adjustment in asset mix would imply foregoing a large yield
advantage from the loan portfolio relative to the T-bill yields in most economic
environments.
CH 10 練習題
23.
MNO Inc., a publicly traded manufacturing firm in the United States, has provided the
following financial information in its application for a loan. All numbers are in
thousands of dollars. (回答以千元為單位)解答有錯
Assets
Cash
Accounts receivables
Inventory
Plant and equipment
Total assets
Liabilities and Equity
Accounts payable
$ 30
Notes payable
40
Accruals
30
Long-term debt
150
Equity (ret. earnings = $300)
450
Total liabilities and equity
$700
$ 20
90
90
500
$700
Also assume sales = $500,000; cost of goods sold = $360,000; and the market value of
equity is equal to the book value.
a. What is the Altman discriminant function value for MNO Inc.? Recall that:
Net working capital = Current assets - Current liabilities.
Current assets = Cash + Accounts receivable + Inventories.
Current liabilities = Accounts payable + Accruals + Notes payable.
EBIT = Revenues - Cost of goods sold.
Altman’s discriminant function is given by: Z = 1.2X1 + 1.4X2 + 3.3X3 + 0.6X4 +
1.0X5
All numbers are in $000s.
X1 = (20 + 90 + 90 – 30 – 40 – 30) / 700 = 0.1429
(TA)
X2 = 300 / 700 = 0.4286
X3 = (500 – 360) / 700 = 0.20
5
X1 = Working capital/total assets
X2 = Retained earnings/TA
X3 = EBIT/TA
X4 = 450 / (30+40+30+150) = 1.80
X5 = 500 / 700 = 0.7143
Z
X4 = Market value of equity/Book value of liab
X5 = Sales/TA
= 1.2(0.1429) + 1.4(0.4286) + 3.3(0.20) + 0.6(1.80) + 1.0(0.7143) = 3.2258
= 0.17148 + 0.60004 + 0.6600 + 1.0800 + 0.7143 = 3.2258
b. Based on the Altman’s Z score only, should you approve MNO Inc.'s application to
your bank for a $500,000 capital expansion loan?
Since the Z score of 3.2258 is greater than 2.99, ABC Inc.’s application for a capital
expansion loan should be approved.
c. If sales for MNO were $250,000, the market value of equity was only half of book
value, and all other values are unchanged, would your credit decision change?
ABC’s EBIT would be $300,000 - $360,000 = -$60,000.
X1 = (20 + 90 + 90 - 30 - 40 - 30) / 700 = 0.1429
X2 = 300 / 700 = 0.4286
X3 = (250-360) / 700 = -0.1571
X4 = 225 / (30+40+30+150) = 0.9
X5 = 250 / 700 = 0.3571
Z
= 1.2(0.1429) + 1.4(0.4286) + 3.3(-0.1571) + 0.6(0.9000) + 1.0(0.3571) = 1.1502
Since ABC's Z-score falls to 1.1502 < 1.81, credit should be denied.
d. Would the discriminant function change for firms in different industries? Would the
function be different for manufacturing firms in different geographic sections of the
country? What are the implications for the use of these types of models by FIs?
Discriminant function models are very sensitive to the weights for the different variables.
Since different industries have different operating characteristics, a reasonable answer would
be yes with the condition that there is no reason that the functions could not be similar for
different industries. In the retail market, the demographics of the market play a big role in the
value of the weights. For example, credit card companies often evaluate different models for
different areas of the country. Because of the sensitivity of the models, extreme care should
be taken in the process of selecting the correct sample to validate the model for use.
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31.
Calculate the term structure of default probabilities over three years using the following
spot rates from the Treasury strip and corporate bond (pure discount) yield curves. Be
sure to calculate both the annual marginal and the cumulative default probabilities. (計
算至 % 小數點以下第二位)
Treasury strips
BBB-rated bonds
Spot 1 Year
5.0%
8.0
Spot 2 Year
6.1%
9.2
Spot 3 Year
7.0%
10.3
The notation used for implied forward rates on Treasuries is f1 = forward rate from period 1
to period 2 and on corporate bonds is c1 = forward rate from period 1 to period 2.
Treasury strips
BBB-rated debt
(1.061)2 = (1.05)(1 + f1)
(1.092)2 = (1.08)(1 + c1)
f1 = 7.21%
c1 = 10.41%
(1.07)3 = (1.061)2(1 + f2)
(1.103)3 = (1.092)2(1 + c2)
f2 = 8.82%
c2 = 12.53%
Using the implied forward rates, estimate the annual marginal probability of repayment:
p1(1.08) = 1.05
p2(1.1041) = 1.0721
p3 (1.1253) = 1.0882
=> p1 = 97.22%
=> p2 = 97.10%
=> p3 = 96.70 %
=> 1-p1 = 2.78%
=> 1-p2 = 2.90%
=> 1-p3 = 3.30%
Using marginal probabilities, estimate the cumulative probability of default:
Cp2
= 1 - (p1)(p2)
Cp3
= 1 - (0.9722)(0.9710) = 5.60 percent
= 1 - (p1)(p2)(p3)
= 1 - (0.9722)(0.9710)(0.9670) = 8.71 percent
34. The following is a schedule of historical defaults (yearly and cumulative) experienced
by an FI manager on a portfolio of commercial and mortgage loans. (計算至 % 小數
點以下第二位)
7
Loan Type
Commercial:
Annual default
Cumulative default
Mortgage:
Annual default
Cumulative default
1 Year
Years after Issuance
2 Years
3 Years
4 Years
5 Years
0.02%
______
______
0.03%
______
0.09%
______
0.23%
0.15%
______
0.30%
______
0.61%
______
0.72%
______
______
2.47%
0.82%
______
a. Complete the blank spaces in the table.
Commercial: Annual default
Cumulative default:
Mortgage: Annual default
Cumulative default
0.02%, 0.01%, 0.06%, 0.14%, and 0.15%
0.02%, 0.03%, 0.09%, 0.23%, and 0.38%
0.30%, 0.61%, 0.72%, 0.86%, and 0.82%
0.30%, 0.91%, 1.62%, 2.47%, and 3.27%
The annual survival rate is pt = 1 – annual default rate, and the cumulative default rate for:
CP2 = 1 – (p1 x p2) = 1 - (0.9998 x P2) = 0.03% , P2= 0.9999, 1-P2 = 0.01%
CP3 = 1 – (p1 x p2 x p3) = 1 - (0.9998 x 0.9999 x P3) = 0.09%, P3= 0.9994, 1-P3 = 0.06%
CP4 = 1 – (p1 x p2 x p3 x p4) = 1 - (0.998 x 0.9999 x 0.9994 x P4) = 0.0023, P4= 0.9986, 1P4 = 0.14%
CP5 =1 – (p1 x p2 x p3 x p4 x p5) = 1 - (0.9998 x 0.9999 x 0.9994 x 0.9986 x 0.9985) =
0.003795 = 0.38%
CP2 = 1 – (p1 x p2) = 1 - (0.997 x 0.9939) = 0.00908 = 0.91%
CP3 = 1 – (p1 x p2 x p3) = 1 - (0.997 x 0.9939 x 0.9928) = 0.01622 = 1.62%
CP4 = 1 – (p1 x p2 x p3 x p4) = 1 - (0.997 x 0.9939 x 0.9928 x P4) = 0.0247, P4= 0.9914, 1P4 = 0.86%
CP5 =1 – (p1 x p2 x p3 x p4 x p5) = 1 - (0.997 x 0.9939 x 0.9928 x 0.9914 x 0.9918) = 0.03267
= 3.27%
b. What are the probabilities that each type of loan will not be in default after five
years?
The cumulative survival rate is = (1 - MMR1) x (1 - MMR2) x (1 - MMR3) x (1 - MMR4) x (1 MMR5) where MMR = marginal mortality rate
8
Commercial loan = (1 - 0.0002) x (1 - 0.0001) x (1 - 0.0006) x (1 - 0.0014) x (1 - 0.0015) =
0.9962 or 99.62%.
Mortgage loan = (1 - 0.003) x (1 - 0.0061) x (1 - 0.0072) x (1 - 0.0086) x (1 - 0.0082) = 0.9673 or
96.73%.
c. What is the measured difference between the cumulative default (mortality) rates for
commercial and mortgage loans after four years?
Looking at the table, the cumulative rates of default in year 4 are 0.23% and 2.47%,
respectively, for the commercial and mortgage loans. Another way of estimation is:
Cumulative mortality rate (CMR)
For commercial loan
= 1- (1 - MMR1)(1 - MMR2)(1 - MMR3)(1 - MMR4)
= 1- (1 - 0.0002)(1 - 0.0001)(1 - 0.0006)(1 - 0.0014)
= 0.002299 or 0.23%
For mortgage loan
= 1- (1 - 0.0003)(1 - 0.0061)(1 - 0.0072)(1 - 0.0086)
= 0.02468 or 2.47%
The difference in cumulative default rates is 2.47% - 0.23% = 2.24%.
CH 11 練習題
7. Suppose that an FI holds two loans with the following characteristics: (請計算至%小數點
以下第三位)
Loan i
1
2
Xi
0.55
0.45
Ri
8%
10
σi
8.55%
9.15
σi2
0.731025%
0.837225%
a. If the correlation coefficient between the returns on loans A and B is 0.15, what are
the expected return and standard deviation of this portfolio?
b. What is the standard deviation of the portfolio if the correlation is -0.15?
c. What role does the covariance, or correlation, play in the risk reduction attributes of
modern portfolio theory?
解答：
a.The return on the loan portfolio is:
Rp = 0.55 (8%) + 0.45 (10%) = 8.90%
The risk of the portfolio is:
9
σp2 = (0.55)2 (0.00731025) + (0.45)2 (0.00837225) + 2 (0.55) (0.45) (0.15) (8.55%)
(9.15%) = 0.004487607
and
σp = 6.699%
b. Rp = 0.55 (8%) + 0.45 (10%) = 8.90%
σp2 = (0.55)2 (0.00731025) + (0.45)2 (0.00837225) + 2 (0.55) (0.45) (-0.15) (8.55%)
(9.15%) = 0.003325855
and
σp = 5.767%
c. What role does the covariance, or correlation, play in the risk reduction attributes of
modern portfolio theory?
The risk of the portfolio as measured by the standard deviation is reduced when the
covariance is reduced. If the correlation is less than +1.0, the standard deviation of the
portfolio will always be less than the weighted average of the standard deviations of the
individual assets.
13. Suppose that an FI holds two loans with the following characteristics.
Annual Spread
between Loan
Loan
Xi
Loss to FI
Expected
Rate and FI’s
Annual
Given
Default
Cost of Funds
Fees
Default
Frequency
1
0.40
4.0%
1.5%
18.75%
4%
2
0.60
2.5%
1.15%
15%
1.5%
ρ12 = -0.10
Calculate of the return and risk on the two-asset portfolio using Moody’s Analytics
Portfolio Manager. (請計算至%小數點以下第二位)
解答：
The return and risk on loan 1 are:
R1 = (0.04 + 0.015) - [0.04 x 0.1875] = 4.75%
σ1 = [0.04 x (1 - 0.04)]1/2 x 0.1875 = 3.674%
The return and risk on loan 2 are:
R2 = (0.025 + 0.0115) - [0.015 x 0.15] = 3.425%
σ2 = [0.015 x (1 - 0.015)]1/2 x 0.15 = 1.823%
The return and risk of the portfolio is then:
Rp = 0.4 (4.75%) + 0.6 (3.425%) = 3.955%
10
σp2 = (0.4)2 (3.674%)2 + (0.6)2 (1.823%)2 + 2 (0.4) (0.6)(-0.1)(3.674%)(1.823%) = 0.000302688
and, σp = (0.000302688)1/2 = 1.742%
17.
WallsFarther Bank has the following balance sheet (in millions of dollars). (金額與
LCR%計算至小數點以下第二位)
Cash inflows over the next 30 days from the bank’s performing assets are $5.5 million.
Calculate the LCR for WallsFarther Bank.
The liquidity coverage ratio for WallsFarther Bank is calculated as follows:
Level 1 assets = $12 + $19 + $125 =
Level 2A assets = ($94 + $138) x 0.85 = $197.20
Capped at 40% of high-quality liquid assets = $156 x 0.40 =
Stock of high-quality liquid assets
Level 2B assets = $106 x 0.50 = $53.00
40% cap on Level 2 assets already met
Stock of high-quality liquid assets
Cash outflows:
Stable retail deposits
$55 x 0.03 = $ 1.65
Less stable retail deposits
$20 x 0.10 = 2.00
Stable small business deposits
$80 x 0.05 = 4.00
Less stable small business deposits
$49 x 0.10 = 4.90
Non-financial corporates
$250 x 0.75 = 187.50
Total cash outflows over next 30 days
$200.05
Total cash inflows over next 30 days
Total net cash outflows over next 30 days
156
62.4
$218.4
0.0
$218.4
5.50
$194.55
Liquidity coverage ratio = $218.4m/$194.55m = 112.26%. The bank is in compliance with
liquidity requirements based on the LCR.
11