Download Final Exam Preparation

Final Exam Preparation (modified on 0812 2008)
Final Exam Preparation (In addition to Solnik Chapters 7, 9, 10 and 11 Questions and
Solutions)
Final Exam covers chapters 11 through 17. There will be 10 questions on the
exam. You may pick 8 questions to answer. Each question counts 25 points.
Again, you only need to fully understand Levich for an excellent grade.
text first! Have enough rest before the exam being essential.
Read your
Credit Structures: CDO, a CDO with independent default, a CDO with
correlated default, CDS, CDS Indexes, CDX, Tranched CDX, CDOs and the
Subprime Crisis [Find answers yourself, from e.g., Fabozzi Chapter 29: Credit
Derivatives.]
What is a Barbell strategy in bond portfolio?
What is a Bullet strategy in bond
portfolio? What is a ladder strategy in bond portfolio?
What is a dual currency bond?
What’s its advantage over regular bond?
SPREAD RISK IN THE EUROCURRENCY MARKET
4.
The Portfolio Manager of the WXYZ pension fund wants to protect herself
against a decline in future interest rates. The fund’s planned short-term
investments are placed in 3-month Eurodollar deposits at the LIBID rate. The
current LIBID-LIBOR spread in the interbank market is 7.375-7.500%, and
the current price of a CME futures contract (which settles on the basis of
three-month Eurodollar LIBOR) is 92.50 reflecting a 7.500% interest rate.
a.
How could the WXYZ fund use the futures market to hedge itself?
What is the minimum interest that the firm locks in?
b.
Suppose that at maturity, Eurodollar rates have fallen to 6.375-6.500%
in the interbank market. Evaluate the hedge. What deposit rate has the
fund secured?
c.
Suppose that at maturity, Eurodollar rates have increased to
8.375-8.625% in the interbank market. Assume that the LIBID-LIBOR
spread has widened because of greater interest rate and
macroeconomic uncertainty. Now, evaluate the hedge. What deposit
rate has the fund secured?
SOLUTIONS (from Guillaume Helie and Octavian Afilipoai):
a.
The fund manager should use the money to buy the CME futures
contract at 92.50 to lock in the 7.50% interest rate.
The usual hedging strategy consists in taking a long interest rate
futures position. The profit would be:
N x LIBID t+3m x (1/4) + N x [(1 – LIBOR t+3m) – (1 – LIBOR t, 3m)] x
(1/4)
Note that (1 – LIBOR t+3m) is (1/100) x price of future at
maturity and (1 – LIBOR t, 3m) is (1/100) x price of future today
= N/4 x (LIBOR t, 3m – spread t+3m) where spread t+3m = LIBOR t+3m –
LIBID t+3m
= N/4 x (0.075 – spread t+3m)
We are not exposed to the risk of the interest rate moving up or moving
down, but we are exposed to the rise of the LIBID – LIBOR spread
widening.
The firm has not secured a minimum interest rate. However, if the
spread doesn’t widen, the interest rate will be at least LIBOR t,3m –
spread t = 7.375%
Earnings = interest earnings + gain/loss on long futures
Earnings = (100 – S i, t+n – spread) + (S i, t+n – F i, t, n)
Earnings = 100 – F i, t, n – spread = 7.5% - spread
So the minimum interest rate locked in is 7.5% minus the (LIBID –
LIBOR) spread on the maturity date.
b.
In this case, the hedge caused a net gain and the locked-in deposit rate
of 7.5% is higher than the Eurodollar deposit rate of 6.375% at
maturity.
The spread is of 0.125%, so the firm has secured an interest rate of
7.375%, which is a gain compared to 6.375% (actual LIBID rate).
c.
In this case, the hedge caused a net loss and the locked-in deposit rate
of 7.5% is lower than the Eurodollar deposit rate of 8.375% at
maturity.
The spread has widened to 0.25%, so the firm has secured an interest
rate of 7.25%, which is a loss compared to 8.375% (actual LIBID rate).
Appendix 12.3 Introduction to Exotic Options in Levich Chapter 12, Pages 469-70
Average Rate Options, Barrier Options, Basket Options
Asian Options with Arithmetic Mean has a higher price than one based on the
Geometric Mean. Why is that? Explain.
What is a “down-and-out” call? What is a “up-and-in” put?
What is a “basket” option?
Read solved examples in Eun: Futures and Options on Foreign Exchange
http://highered.mcgraw-hill.com/sites/dl/free/0072521279/91312/eun21279_ch09_dr.pdf
Read the boxed article titled Fearless Dealers in Eun: The Market for Foreign
Exchange
http://highered.mcgraw-hill.com/sites/dl/free/0072521279/91312/eun21279_ch04_dr.pdf
ARBITRAGE IN THE INTEREST RATE FUTURES MARKET
3.
Suppose the interest rate futures contract for delivery in three months is currently
selling at 110. The deliverable bond for that particular contract is a 25-year bond,
currently traded at 100 with a coupon rate of 10%. The current 3-month rate is
7%.
a.
Is there any arbitrage opportunity? If yes, what would you do and what
would be your potential gain from an arbitrage transaction?
b.
What is the theoretical price of the futures contract?
c.
Suppose the price was 95 instead of 110. What would you do to take
advantage of arbitrage opportunities?
SOLUTIONS:
a.
Yes, there is an arbitrage opportunity. Here is how:
Sell Futures contracts at 110; Purchase the bond at 100 Borrow 100 at
7%.
Profit = Proceeds - Outlays
Profit = (Price of Bond + Accrued Interest) - (Principal Repayment +
Interest Payment);
Profit = 110 + (100 * 10% / 4) - (100 - 100 * 7% / 4)
= 110 +2.5 - 100 - 1.75;
Profit = 10.75
b.
The correct price is determined so that there are no arbitrage
opportunities.
0 = (F + 2.5) - (100 + 1.75); F = 101.75 - 2.5 = 99.25
c.
Buy the futures at 95; Sell Bond at 100; Lend at 7% for 3 months.
Profit = (Principal + Interest Payment) - (Price of Bond + Accrued
Interest); Profit = 100 + 1.75 - 95 - 2.5; Profit = 4.25
4. If the Eurodollar CD futures contract is quoted at 91.75, what is the
annualized futures three-month LIBOR?
The three-month Eurodollar CD is the underlying instrument for the Eurodollar CD
futures contract. As with the Treasury bill futures contract, this contract is for $1
million of face value and is traded on an index price basis. The index price basis in
which the contract is quoted is equal to 100 minus the annualized futures LIBOR. In
our problem, a Eurodollar CD futures price of 91.75 means a futures three-month
LIBOR of 100 – 91.75 = 8.25. This translates into a rate of return of 8.25%. Thus, the
annualized futures three-month LIBOR is 8.25%.
Calculation on Page 529 Bond Portfolio Return Calculations
Suppose coupon is paid at the end of each year
Bond Price
1009128
1000000 Cash Flow
40000
Swiss Franc Return
0.049128
1009128=40000/1.0375+40000/(1.0375)^2+40000/(1.0375)^3+1040000/(1.0375)^4
1000000=40000/1.04+40000/(1.04)^2+40000/(1.04)^3+1040000/(1.04)^4
0.049128=((C3-D3)+G3)/D3
Bond Price
630705.3
US Dollar Return
630705.3=C3*0.625
650000 Cash Flow
0.008777
25000
650000=D3*0.650
0.008777=((C6-D6)+G6)/D6
Computing the Swap Rate
Also See Page 511, Appendix 13.1 of Levich
And Exercise #11 on Pages 514-5 of Levich
Table 8.4 Three-month LIBOR forward rates and swap rates implied by Eurodollar
futures prices with maturity dates given in the first column. Prices are from
November 8, 2007. Source: Wall Street Journal online.
The Swap rate in June 2008 row is the fixed quarterly interest rate for a loan initiated
in December that matures in September, with swap payments made in March, June,
and September. Calculate with equation (8.2), we have
0.01201 x 0.98814 + 0.01082 x 0.97756 + 0.01020 x 0.96769
0.98814 + 0.97756 + 0.96769
x 4 = 0.044059 = 4.4059%  Make sure you know how to calculate the Swap Rates.
1. Consider an interest-rate swap with these features: maturity is five years,
notional principal is $100 million, payments occur every six months, the
fixed-rate payer pays a rate of 9.05% and receives LIBOR, while the
floating-rate payer pays LIBOR and receives 9%. Now suppose that at a
payment date, LIBOR is at 6.5%. What is each party’s payment and receipt at
that date?
[NOTE. The below answers assume the time period of six months is from January 1st
to June 30th which is a period of 181 days. The answer will vary if the number of days
per six-month period changes based upon if the first month is a month other than
January.]
Fixed-rate payer pays: (notional amount)(fixed-rate)(number of days in period / 360)
= ($100,000,000)(0.0905)(181 / 360) = $4,550,138.89.
Fixed-party receives: (notional amount)(three-month LIBOR)(days in period / 360)
= ($100,000,000)(0.065)(181 / 360) = $3,268,055.56.
Floating-rate payer pays: (notional amount)(three-month LIBOR)(days in period /
360) = ($100,000,000)(0.065)(181 / 360) = $3,268,055.56.
Floating-rate payer receives: (notional amount)(fixed-rate)(number of days in period
/ 360) = ($100,000,000)(0.09)(181 / 360) = $4,525,000.00.
2. Suppose that a dealer quotes these terms on a five-year swap: fixed-rate payer
to pay 9.5% for LIBOR and floating-rate payer to pay LIBOR for 9.2%.
Answer the following questions.
(a) What is the dealer’s bid-asked spread?
Dealer’s bid-asked spread =
(offer price dealer quotes fixed-rate payer) – (bid price dealer quotes floating-rate
payer)
 Dealer’s bid-offer spread = 9.50% – 9.20% = 0.3% or 0.003 or 30 basis points.
(b) How would the dealer quote the terms by reference to the yield on five-year
Treasury notes?
The fixed rate is some spread above the Treasury yield curve with the same term to
maturity as the swap. Suppose the five-year Treasury yield is 9.0%. Then the offer
price that the dealer would quote to the fixed-rate payer is the five-year Treasury rate
plus 50 basis points versus receiving LIBOR flat. For the floating-rate payer, the bid
price quoted would be LIBOR flat versus the five-year Treasury rate plus 20 basis
points. The dealer would quote such a swap as 20-50, meaning that the dealer is
willing to enter into a swap to receive LIBOR and pay a fixed rate equal to the
five-year Treasury rate plus 20 basis points; it would also be willing to enter into a
swap to pay LIBOR and receive a fixed rate equal to the five-year Treasury rate plus
50 basis points. The difference between the Treasury rate paid and received is the
bid-offer spread.
3. Give two interpretations of an interest-rate swap.
There are two ways that a swap position can be interpreted: (i) as a package of
forward/ futures contracts, and (ii) as a package of cash flows from buying and selling
cash market instruments.
4. In determining the cash flow for the floating-rate side of a LIBOR swap,
explain how the cash flow is determined.
Assume a swap of 12 quarterly floating-rate payments for three years with the first
quarter consisting of 90 days from January 1st of year 1 to March 31st of year 1
assuming a non-leap year. The cash flow for this period is:
floating-rate payment = notional amount × three-month LIBOR ×
90
.
360
Note that each futures contract is for $1 million and hence 100 contracts have a
notional amount of $100 million. Let’s assume $100 million notional amount and a
LIBOR of 5%. The cash flow for period 1 is:
payment = $100,000,000 × 0.05 × 0.25 = $1,250,000.
While this first quarterly payment is known, the next 11 are not. The second
quarterly payment, from April 1 of year 1 to June 30 of year 1, has 91 days. The
floating-rate payment is determined by three-month LIBOR on April 1 of year 1 and
paid on June 30 of year 1. This is achieved by looking at the three-month Eurodollar
CD futures contract for settlement on June 30 of year 1. That futures contract provides
the rate that can be locked in for three-month LIBOR on April 1 of year 1. We refer to
that rate for three-month LIBOR as the forward rate. Therefore, if the fixed-rate payer
bought 100 three-month Eurodollar CD futures contracts on January 1 of year 1 (the
inception of the swap) that settle on June 30 of year 1, then the payment that will be
locked in for the second quarter (April 1 to June 30 of year 1) is
payment = notional amount × annual forward rate ×
number of days in period
.
360
Given that the notional amount is $100 million and the number of days is 91, let us
assume the annual forward rate is 5.2%. Using these numbers, the payment is:
fixed-rate payment = $100,000,000 × 0.052 ×
91
= $1,314,444.44.
360
Similarly, the Eurodollar CD futures contract can be used to lock in a
floating-rate payment for each of the next 10 quarters. It is important to emphasize
that the reference rate at the beginning of period t determines the floating rate that will
be paid for the period. However, the floating-rate payment is not made until the end of
period t.
5. How is the swap rate calculated?
To compute the swap rate, we begin with the basic relationship for no arbitrage to
exist:
present value of fixed-rate payments = present value of floating-rate payments.
For the fixed-rate payment for period t, we have:
fixed-rate payment = notional amount × swap rate ×
number of days in period t
.
360
The present value of the fixed-rate payment for period t is found by multiplying the
fixed-rate payment expression by the forward discount factor for period t. That is, we
have:
present value of the fixed-rate payment for period t =
notional amount × swap rate ×
days in period t
× forward discount factor for period
360
t.
Summing up the present value of the fixed-rate payment for each period gives the
present value of the fixed-rate payments. Letting N be the number of periods in the
swap, we have:
present value of the fixed-rate payments =
N
swap rate ×
 notional amount
×
t =1
days in period t
× forward discount factor for
360
period t.
The condition for no arbitrage is that the present value of the fixed-rate payments as
given by the expression above is equal to the present value of the floating-rate
payments. That is, we have:
present value of floating-rate payments =
N
swap rate ×
 notional amount
t =1
×
days in period t
× forward discount factor for
360
period t.
Solving for the swap rate, we have:
swap rate =
present value of floating-rate payments
.
days in period t
notional amount 
 forward discount factor for period t

360
t 1
N
See Page 511, Appendix 13.1 of Levich.
example?
What is the Swap Rate in that
6. How is the value of a swap determined?
Once the swap transaction is completed, changes in market interest rates will change
the payments of the floating-rate side of the swap. The value of an interest-rate
swap is the difference between the present value of the payments of the two sides of
the swap. The three-month LIBOR forward rates from the current Eurodollar CD
futures contracts are used to (i) calculate the floating-rate payments and (ii) determine
the discount factors at which to calculate the present value of the payments.
7. What factors affect the swap rate?
For the swap rate, we have:
swap rate =
present value of floating-rate payments
.
days in period t
notional amount 
 forward discount factor for period t

360
t 1
N
From this equation, we see that the swap rate is determined by the present value of
floating-rate payments, the notional amount, the number of periods in a year, and the
forward discount factor.
The present value of floating-rate payments is also determined by the notional amount,
the number of periods in a year, and the forward discount factor. It is also determined
by the reference rate (such as LIBOR) and forward rates based on this benchmark. It
is important to emphasize that the reference rate at the beginning of period t
determines the floating rate that will be paid for the period. However, the floating-rate
payment is not made until the end of period t. We should also point out that the same
forward rates that are used to compute the floating-rate payments—those obtained
from the Eurodollar CD futures contract—are used in computing the forward discount
factors for each period t.
8. Describe the role of an intermediary in a swap.
The role of an intermediary in a swap is that of a broker. More details are supplied
below.
The role of the intermediary in an interest-rate swap sheds some light on the evolution
of the market. Intermediaries in these transactions have been commercial banks and
investment banks who in the early stages of the market sought out end users of swaps.
That is, they found in their client bases those entities that needed the swap to
accomplish a funding or investing objective, and they matched the two entities. In
essence, the intermediary in this type of transaction performed the function of a
broker.
The only time that the intermediary would take the opposite side of a swap (i.e.,
would act as a principal) was to balance out the transaction. For example, if an
intermediary had two clients that were willing to do a swap but one wanted the
notional principal amount to be $100 million and the other wanted it to be $85 million,
the intermediary might become the counterparty to the extent of $15 million. That is,
the intermediary would warehouse or take a position as a principal to the transaction
to make up the $15 million difference between client objectives. To protect itself
against an adverse interest-rate movement, the intermediary would hedge its position.
The parties to swaps have to be concerned that the other party might default on its
obligation. Although a default would not mean any principal was lost because the
notional principal amount had not been exchanged, it would mean that the objective
for which the swap was entered into would be impaired. As the early transactions
involved a higher- and a lower-credit-rated entity, the former would be concerned
with the potential for default of the latter. To reduce the risk of default, many early
swap transactions required that the lower-credit-rated entity obtain a guarantee from a
highly rated commercial bank (intermediary).
As the frequency and the size of the transactions increased, many intermediaries
became comfortable with the transactions and became principals instead of acting as
brokers. As long as an intermediary had one entity willing to do a swap, the
intermediary was willing to be the counterparty. Consequently, interest-rate swaps
became part of an intermediary’s inventory of product positions. Advances in
quantitative techniques and futures products for hedging complex positions such as
swaps made the protection of large inventory positions feasible.
Binomial Option Pricing
See Page 466 of Levich: Appendix 12.1 Determination of the Replicating Portfolio in the
Two-Period Binomial Model
Figure 10.11 Construction of a binomial tree depicting
stock-price paths, along with risk-neutral probabilities of reaching
the various terminal prices.
Currency Linked Bonds
A Real World Way to Manage Real Options—The
Copano Chemical Plant Example in a High
Inflation Era, r = 13%
Forward Rate Agreement
Interest Rate Strips and Stacks
Bond Basics
Duration