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. Read your text first! Have enough rest before the exam being essential.
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 yearBond 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 / 650000 / Cash Flow / 25000
US Dollar Return / 0.008777
630705.3=C3*0.625
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
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% ç 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)
è 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 × .
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 × .
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 × = $1,314,444.44.
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 × .
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 × × forward discount factor for period 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 =
swap rate × × × forward discount factor for 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: