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FINM7407 –Financial Institutions and Markets
Semester 2 2023 Sample Final Examination Solution
This exam has 55 marks in total and carries 55% of the total mark of this course. There are five
(5) questions. Answer all questions.
Question 1 [11 marks]
A bank is planning to make a loan of $5,000,000 to a firm in the steel industry. It expects to charge
a servicing fee of 50 basis points. The loan has a maturity of 8 years with a duration of 7.5 years.
The cost of funds (the RAROC benchmark) for the bank is 10 percent. The bank has estimated the
maximum change in the risk premium on the steel manufacturing sector to be approximately 4.2
percent, based on two years of historical data. The current market interest rate for loans in this sector
is 12 percent.
a. Using the RAROC model, determine whether the bank should make the loan?
RAROC = Fees and interest earned on loan/Loan or capital risk
Loan risk, or ∆LN = -DLN x LN x (∆R/(1 + R)) = -7.5 x $5m x (0.042/1.12) = -$1,406,250
Expected interest = 0.12 x $5,000,000 = $600,000
Servicing fees = 0.0050 x $5,000,000 = $25,000
Less cost of funds = 0.10 x $5,000,000 = -$500,000
Net interest and fee income = $125,000
RAROC = $125,000/1,406,250 = 8.89 percent. Since RAROC is lower than the cost of funds to
the bank, the bank should not make the loan.
b. What should be the duration in order for this loan to be approved?
For RAROC to be 10 percent, loan risk should be:
$125,000/∆LN = 0.10 ⇒ ∆LN = 125,000 / 0.10 = $1,250,000
⇒ -DLN x LN x (∆R/(1 + R)) = 1,250,000
DLN = 1,250,000/(5,000,000 x (0.042/1.12)) = 6.67 years.
Thus, this loan can be made if the duration is reduced to 6.67 years from 7.5 years.
c. Assuming that duration cannot be changed, how much additional interest and fee income
will be necessary to make the loan acceptable?
Necessary RAROC = Income/Risk ⇒ Income = RAROC x Risk
= $1,406,250 x 0.10 = $140,625
Therefore, additional income = $140,625 - $125,000 = $15,625, or
$15,625/$5,000,000 = 0.003125 = 0.3125%.
Thus, this loan can be made if fees are increased from 50 basis points to 81.25 basis points.
d. Given the proposed income stream and the negotiated duration, what adjustment in the
loan rate would be necessary to make the loan acceptable?
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Need an additional $15,625 => $15,625/$5,000,000 = 0.003125 or 0.3125%
Expected interest = 0.123125 x $5,000,000 = $615,625
Servicing fees = 0.0050 x $5,000,000 = $25,000
Less cost of funds = 0.10 x $5,000,000 = -$500,000
Net interest and fee income = $140,625
RAROC = $140,625/1,406,250 = 10.00 percent = cost of funds to the bank. Thus, increasing the
loan rate from 12% to 12.3125% will make the loan acceptable
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Question 2 [14 marks]
Consider $100 million of 30-year mortgages with a coupon of 5 percent per year paid quarterly.
a. What is the quarterly mortgage payment?
There are 120 quarterly payments over 30 years. The quarterly mortgage payments are
$100m = PVAn=120, k=1.25% x PMT => PMT = $1,613,350.
b. What are the interest and principal repayments over the first year of life of the
mortgages?
(Fixed) Interest Principal Remaining
Quarter Balance Payment Payment Payment Principal
1 $100,000,000 $1,613,350 $1,250,000 $363,350 $99,636,650
2 99,636,650 1,613,350 1,245,458 367,891 99,268,759
3 99,268,759 1,613,350 1,240,859 372,490 98,896,269
4 98,896,269 1,613,350 1,236,203 377,146 98,519,123
c. Construct a 30-year CMO using this mortgage pool as collateral. The pool has three
tranches, where tranche A offers the least protection against prepayment and tranche C
offers the most protection against prepayment. Tranche A of $25 million receives
quarterly payments at 4 percent per year, tranche B of $50 million receives quarterly
payments at 5 percent per year, and tranche C of $25 million receives quarterly
payments at 6 percent per year.
Tranche A Tranche B Tranche C Total Issue
Principal amount $25 million $50 million $25 million $100 million
Interest rate 4 percent 5 percent 6 percent 5 percent
Quarterly interest on
initial balance $250,000 $625,000 $375,000 $1,250,000
Quarterly amortization $1,613,350
d. Assume nonamortization of principal and no prepayments. What are the total promised
coupon payments to the three classes? What are the principal payments to each of the
three classes for the first year?
Regular tranche A interest payments are $250,000 quarterly. If there are no
prepayments, then the regular GNMA quarterly payment of $1,613,350 is distributed
among the three tranches. Five million is the total coupon interest payment for all three
tranches. Therefore, $363,350 of principal is repaid each quarter. Tranche A receives all
principal payments. Tranche A cash flows are $250,000 + $363,350 = $613,350
quarterly. The cash flows to tranches B and C are the scheduled interest payments.
Tranche A amortization schedule
Fixed Interest Principal Remaining
Quarter Balance Payment Payment Payment Principal
1 25,000,000 613,350 250,000 363,350 24,636,650
2 24,636,650 613,350 246,367 366,983 24,269,667
3 24,269,667 613,350 242,697 370,653 23,899,014
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4 23,899,014 613,350 238,990 374,359 23,524,655
Tranche B amortization schedule
Fixed Interest Principal Remaining
Quarter Balance Payment Payment Payment Principal
1 50,000,000 625,000 625,000 0 50,000,000
2 50,000,000 625,000 625,000 0 50,000,000
3 50,000,000 625,000 625,000 0 50,000,000
4 50,000,000 625,000 625,000 0 50,000,000
Tranche C amortization schedule
Fixed Interest Principal Remaining
Quarter Balance Payment Payment Payment Principal
1 25,000,000 375,000 375,000 0 25,000,000
2 25,000,000 375,000 375,000 0 25,000,000
3 25,000,000 375,000 375,000 0 25,000,000
4 25,000,000 375,000 375,000 0 25,000,000
e. If, over the first year, the trustee receives quarterly prepayments of $5 million on the
mortgage pool, how are these funds distributed?
The quarterly prepayments of $5 million will be credited entirely to tranche A until
tranche A is completely retired. Then prepayments will be paid entirely to tranche B.
The amortization schedule for tranche A for the first year is shown below. This
amortization schedule assumes that the trustee has a quarterly payment amount from the
mortgage pool of $1,613,350.
Tranche A amortization schedule
Fixed Interest Principal Remaining Mortgage
Quarter Balance Payment Payment Payment Principal Payment
1 25,000,000 5,613,350 250,000 5,363,350 19,636,650 1,613,350
2 19,636,650 5,613,350 196,367 5,416,983 14,219,667 1,613,350
3 14,219,667 5,613,350 142,197 5,471,153 8,748,514 1,613,350
4 8,748,514 5,613,350 87,485 5,525,864 3,222,650 1,613,350
However, since some of the mortgages will be paid off early, the actual payment
received by the trustee from the mortgage pool will decrease each quarter. Thus, the
payment for the second quarter will decrease from $1,613,350 to $1,532,388 (n = 119
quarters, i = 5 percent, mortgage principal = $94,636,650). The CMO amortization
schedule for tranche A given that the mortgage payments decrease with the prepayments
is given below. The revised mortgage payment for each quarter is shown in the last
column.
Fixed Interest Principal Remaining Mortgage
Quarter Balance Payment Payment Payment Principal Payment PVA n r
1 25,000,000 5,613,350 250,000 5,363,350 19,636,650 1,613,350 100,000,000 120 0.0125
2 19,636,650 5,532,388 196,367 5,336,021 14,300,629 1,532,388 94,636,650 119 0.0125
3 14,300,629 5,451,344 143,006 5,308,338 8,992,291 1,451,344 89,300,629 118 0.0125
4 8,992,291 5,370,213 89,923 5,280,290 3,712,001 1,370,213 83,992,291 117 0.0125
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Question 3 [10 marks]
A. A manager decides not to lend to any firm in sectors that generate losses in excess of 5 percent
of capital.
a. If the average historical losses in the automobile sector total 8 percent, what is the
maximum loan a manager can lend to firms in this sector as a percentage of total capital?
Concentration limit = (Maximum loss as a percent of capital) x (1/Loss rate) = 0.05 x
1/0.08 = 62.5 percent of capital is the maximum amount that can be lent to firms in the
automobile sector.
b. If the average historical losses in the mining sector total 15 percent, what is the
maximum loan a manager can lend to firms in this sector as a percentage of total capital?
Concentration limit = (Maximum loss as a percent of capital) x (1/Loss rate) = 0.05 x
1/0.15 = 33.3 percent of capital is the maximum amount that can be lent to firms in the
mining sector.
B. Suppose that an FI holds two loans with the following characteristics.
Annual
Spread between Loss to FI Expected
Loan Rate and FI’s Annual Given Default
Loan Xi Cost of Funds Fees Default Frequency
1 0.45 5.5% 2.25% 30% 3.5% ρ12 = -0.15
2 0.55 3.5 1.75 20 1.0
Calculate of the return and risk on the two-asset portfolio using Moody’s Analytics
Portfolio Manager.
The return and risk on loan 1 are:
R1 = (0.055 + 0.0225) - [0.035 x 0.30] = 0.0670 or 6.70%
σ1 = [0.035 x (1 - 0.035)]1/2 x 0.30 = 0.05513 or 5.513%
The return and risk on loan 2 are:
R2 = (0.035 + 0.0175) - [0.01 x 0.20] = 0.0505 or 5.05%
σ2 = [0.01 x (1 - 0.01)]1/2 x 0.20 = 0.01990 or 1.990%
The return and risk of the portfolio is then:
Rp = 0.45 (6.70%) + 0.55 (5.05%) = 5.7925%
σp2 = (0.45)2 (5.513%)2 + (0.55)2 (1.990%)2 + 2 (0.45) (0.55)(-0.15)(5.513%)(1.990%) = 6.53876%
and, σp = (6.53876%)1/2 = 2.56%
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Question 4 [10 marks]
Hedge Row Bank has the following balance sheet (in millions):
Assets $150 Liabilities $135
Equity 15
Total $150 Total $150
The duration of the assets is six years and the duration of the liabilities is four years.
The interest rate on both the assets and the liabilities is currently at 10 percent.
The bank is expecting that the interest rates will increase in the near future.
a. What is the duration gap for Hedge Row Bank?
DGAP = DA – kDL = 6 – (135/150)*(4) = 6 – 3.6 = 2.4 years
(k=135/150=0.9)
b. What will be the effect on net worth if next year interest rates increase by 110
basis points?
∆E = -DGAP[∆R/(1 + R)]A = -2.4(0.011/1.10)$150 = -$3.6 million.
c. Suppose that the FI plans to use futures contracts with the following
characteristics to hedge its position: a one-year futures is currently price at $97
per $100 of face value for a 20-year, 8 percent coupon bond underlying the
futures contract, the minimum contract size is $100,000, and the duration of the
bond underlying the futures contract is 9.5 years. That is: PF = $97,000 and DF =
9.5 years. How many futures contracts that the FI should use and whether the FI
should buy or sell these future contracts?
Number of futures contracts for hedging this position is:
= −( − ) × ×
=-(6-0.9*4)*150/(9.5*0.097) =-390.67 contracts
The FI should sell ~391 futures contracts.
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Question 5 [10 marks]
National Bank has the following balance sheet (in millions) and has no off-balance-sheet
activities.
Assets Liabilities and Equity
Cash $20 Deposits $960
Treasury bills 40 Subordinated debentures 25
Residential mortgages Common stock 45
(category 1; loan-to-value Retained earnings 40
ratio = 70%) 600 Total liabilities and equity $1,090
Business loans 430
Total assets $1,090
a. What is the CET1 risk-based ratio?
Risk-weighted assets = $20x0.0 + $40x0.0 + $600x0.5 + $430x1.0 = $730
The CET1 risk-based ratio is ($45 + $40)/$730 = 0.11644 or 11.644 percent.
b. What is the Tier I risk-based capital ratio?
Risk-weighted assets = $20x0.0 + $40x0.0 + $600x0.5 + $430x1.0 = $730
Tier I capital ratio = ($45 + $40)/$730 = 0.11644 or 11.644 percent.
c. What is the total risk-based capital ratio?
The total risk-based capital ratio = ($45 + $40 + $25)/$730 = 0.15068 or 15.068 percent.
d. What is the leverage ratio?
The leverage ratio is ($45 + $40)/$1,090 = 0.07798 or 7.798 percent.
e. In what capital risk category would the bank be placed?
The bank would be place in the well-capitalized category.