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無代寫-CHAPTER 16:
時間:2021-07-12
PRACTICE PROBLEMS
CHAPTER 16: SWAPS

Exercise 16.1: Companies A and B have been offered the following rates per annum
on a $20 million five-year loan:

Fixed Rate Floating Rate
Company A 12.0% LIBOR + 0.1%
Company B 13.4% LIBOR + 0.6%

Company A requires a floating-rate loan; company B requires a fixed-rate loan.
Design a swap that will net a bank acting as intermediary 0.1 percent per annum
and be equally attractive to both companies. Write the swap in such a way that
the floating rate is LIBOR; e.g. for each firm you’ll write Firm X pays (or
receives) LIBOR to (from) the financial intermediary in exchange for __%.

Exercise 16.2: Companies X and Y have been offered the following rates per annum
on a $5 million ten-year loan:

Fixed Rate Floating Rate
Company X 7.0% LIBOR + 0.5%
Company Y 8.8% LIBOR + 1.5%

Company X requires a floating-rate loan; company Y requires a fixed-rate loan.
Design a swap that will net a bank acting as intermediary 0.2 percent per annum
and be equally attractive to both companies. Write the swap in such a way that
the floating rate is LIBOR.

Exercise 16.3: Explain the difference between the credit risk and the market risk in a
financial contract. Can these risks be hedged?

Exercise 16.4: Explain why a bank is subject to credit risk when it enters into two
offsetting swap contracts. (You can think of the bank as being the financial
intermediary here.)

Exercise 16.5: Under the terms of an interest-rate swap, a financial institution has
agreed to pay 10 percent per annum compounded quarterly and to receive three-
month LIBOR in return on a notional principal of $100 million with payments
being exchanged every three months. The swap has a remaining life of 14
months. The current LIBOR rate is 12 percent per annum (compounded
quarterly) for all maturities. The three-month LIBOR rate one month ago was
11.8 percent per annum (compounded quarterly). What is the value of the swap?

Exercise 16.6: A financial institution has entered into an interest-rate swap with
company X. Under the terms of the swap, it receives 10 percent per annum
(compounded semiannually) and pays six-month LIBOR on a principal of $10
million for five years. Payments are made every six months. Suppose that
company X defaults on the sixth payment date (end of year 3) when the interest
rate (with semiannual compounding) is 8 percent per annum for all maturities.
2
What was the value of the swap to the financial institution at the time of default?
Assume that six-month LIBOR was 9 percent per annum halfway through year 3.

Exercise 16.7: A bank finds that its assets are not matched with its liabilities. It is
taking floating-rate deposits and making fixed-rate loans. How can swaps be used
to offset the risk?

Exercise 16.8: A firm has a portfolio of two bonds: a 20-year 5% coupon bond with
total face value $500,000, price 95.62 (in decimal) and duration 12.7 years, and a
12-year 10% coupon bond with total face value $200,000, price 140.68 and
duration 8.0 years. What is the duration of the portfolio? If the firm swaps the
5% bond for LIBOR, what is the new duration of the portfolio? Assume that all
payments are made semiannually.

SOLUTIONS

Exercise 16.1:
Fixed Rate Floating Rate
Company A 12.0% LIBOR + 0.1%
Company B 13.4% LIBOR + 0.6%
Spread 1.4% 0.5%
The total gain is 1.4 ? 0.5 = 0.9%. The Financial Intermediary gets 0.1%, and the
firms share the remaining 0.8% evenly, so each firm should be 0.4% better off
than they would be without the swap.
Company A requires a floating-rate loan; company B requires a fixed-rate loan.
Company A has a comparative advantage in fixed; B has a comparative advantage in
floating. So, A borrows fixed, and swaps for floating
A ends up paying LIBOR + 0.1 ? 0.4 = LIBOR ? 0.3%;
B ends up paying 13.4 ? 0.4 = 13.0%.






Or, writing the floating-rates as LIBOR, (as we did in class) we have






So, in terms of the swap, A pays the F.I. LIBOR and receives 12.3%, while B pays the
F.I. 12.4% and receives LIBOR.

A
A
F.I.
F.I.
B
B
12%
12%
LIBOR ? 0.3%
LIBOR
12%
12.3%
LIBOR ? 0.3%
LIBOR
12.1%
12.4%
LIBOR + 0.6%
LIBOR + 0.6%
3
Exercise 16.2:
Fixed Rate Floating Rate
Company X 7.0% LIBOR + 0.5%
Company Y 8.8% LIBOR + 1.5%
Spread 1.8% 1.0%
The total gain is 1.8 ? 1.0 = 0.8% The F.I. gets 0.2%, and the firms share the
remaining 0.6% evenly, so each firm should be 0.3% better off than they would be
without the swap.
Company X requires a floating-rate loan; company Y requires a fixed-rate loan.
Company X has a comparative advantage in fixed; Y has a comparative advantage in
floating. So, X borrows fixed and swaps for floating.
X ends up paying LIBOR + 0.5 ? 0.3% = LIBOR + 0.2%.
Y ends up paying 8.8 ? 0.3 = 8.5%.






Or, writing the floating-rates as LIBOR, (as we did in class) we have






So, in terms of the swap, A pays the F.I. LIBOR and receives 6.8%, while B pays the
F.I. 7.0% and receives LIBOR.

Exercise 16.3: Credit risk arises from the possibility of a default by the counterparty.
Market risk arises from movements in market variables such as interest rates.
Market risks can be hedged; and, with credit derivatives, it is now possible to
hedge credit risk, too.

Exercise 16.4: At the start of the swap, both contracts have a value of approximately
zero. As time passes it is likely that this will change so that one swap has a
positive value to the bank and the other has a negative value to the bank. If the
counterparty on the other side of the positive-value swap defaults, the bank still
has to honour its contract with the other counterparty. It loses an amount equal to
the positive value of the swap.

Exercise 16.5: The swap can be regarded as a long position in a floating-rate bond
(i.e., we’ve bought a floating rate bond, so that we receive floating rate payments)
combined with a short position in a fixed-rate bond (i.e., we’ve sold a fixed-rate
bond, so we pay fixed rate payments). The correct discount rate is 12% per
annum with quarterly compounding, i.e., 3% every 3 months. The next payment
is in 2 months.
Immediately after the next payment, the floating-rate bond will be worth $100
million. The next floating payment is (in millions)
0.118?100?0.25 = 2.95.
A
A
F.I.
F.I.
B
B
7%
7%
LIBOR ? 0.2%
LIBOR
7%
6.8%
LIBOR ? 0.2%
LIBOR
7.2%
7.0%
LIBOR + 1.5%
LIBOR + 1.5%
4
The value of the floating-rate bond is therefore
PV(100 + 2.95) = (102.95)
24
120.121
4
? ?? ??? ?? ?
= (102.95)(1.03)?2/3 = 100.94.
The value of the fixed-rate bond is (the payments are 100?0.1?0.25 = 2.5 million)
(1.03)1/3[2.5(1.03)?1 + 2.5(1.03)?2 + 2.5(1.03)?3 + 2.5(1.03)?4 + 102.5(1.03)?5]
= (1.03)1/3{(2.5)[1 ? (1.03)?5]/0.03 + 100(1.03)?5}
= 98.68 million.
Here are the detailed calculations, if this helps:
This calculation is similar to the calculation on slide 16-41.
In this case, the interest rate is 12% per year, compounded quarterly, or 3%
compounded every three months. So, it’s 3% per period, where each period is of
length three months.
So, the direct calculation would look like this:
2.5(1.03)?4*2/12 + 2.5(1.03)??4*5/12 + 2.5(1.03)??4*8/12 + 2.5(1.03)??4*11/12 +
102.5(1.03)??4*14/12
= 2.5(1.03)?8/12 + 2.5(1.03)?20/12 + 2.5(1.03)?32/12 + 2.5(1.03)?44/12 + 102.5(1.03)??56/12
= 2.5(1.03)?2/3 + 2.5(1.03)?5/3 + 2.5(1.03)?8/3 + 2.5(1.03)?11/3 + 102.5(1.03)?14/3
= (1.03)1/3[2.5(1.03)?1 + 2.5(1.03)?2 + 2.5(1.03)?3 + 2.5(1.03)?4 + 102.5(1.03)?5]
= (1.03)1/3PLCD.

The value of the swap to the financial institution is therefore
100.94 ? 98.68 = $2.26 million.

Exercise 16.6: Here, we have effectively bought a fixed-rate bond and sold a
floating-rate bond. The correct discount rate is 8 percent per annum compounded
semiannually (or 4% every six months). The next payment would be in six
months.
At time of default, the financial institution was due to receive $500,000 (= 0.5?10%
of 10 million) and pay $450,000 (= 0.5?9% of 10 million). So, the immediate
loss to the financial institution was $50,000.
Immediately after the current payment, the floating-rate bond would be worth the face
value, $10 million. Since there were 4 payments remaining (after the current
payment) the fixed-rate bond would be worth (in millions)
0.5(1.04)?1 + 0.5(1.04)?2 + 0.5(1.04)?3 + 10.5(1.04)?4
= 0.5[1 ? (1.04)?4]/0.04 + 10(1.04)?4 = 10.363 million.
The value of the swap to the F.I. immediately after the current payment was
10.363 ? 10 = 0.363 million = 363,000.
This positive value is what they fail to receive, so it is a loss to the F.I.
The total value of the swap including the current payment was a loss to the F.I. of
363,000 + 50,000 = 413,000.

Exercise 16.7: The bank is paying floating on the deposits and receiving fixed on the
loans. It can offset its risk by entering into interest rate swaps (with other
financial institutions or corporations) in which it contracts to pay fixed and
receive floating.

Exercise 16.8:
The total value of the 5% bond is 0.9562?500,000 = 478,100;
5
the value of the 10% bond is 1.4068?200,000 = 281,360;
and the total value of the portfolio is 478,100 + 281,360 = 75,9460.
The proportion invested in the 5% bond is x1 = 478,100/759,460 = 0.63;
the proportion invested in the 10% bond is x2 = 281,360/759,460 = 0.37.
The duration of the portfolio is
x1D1 + x2D2 = 0.63?12.7 + 0.37?8 = 10.96.
If the 5% bond is swapped for a floating-rate bond with its first payment in 6 months,
then its duration is now D1* = 0.5 years.
Since swaps have an initial value of 0, the floating-rate bond must have the same
price as the 5% bond, so x1 and x2 are unchanged.
The new duration of the portfolio is
x1D1* + x2D2 = 0.63?0.5 + 0.37?8 = 3.28.

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