Advanced Problems in Arbitrage

Fina 5210 Problem Set 5 Fall 2014

Advanced Problems in Arbitrage

For preparation prior to problems 1, 2 & 3, see Problem Set 4, Problem 8. Problems 1, 2 & 3 are “warm-up” for the subsequent problems that illustrate how the yield curve is shaped. Problems 1, 2 & 3 illustrate that an upwardsloping yield curve is associated with expectations that short-term interest rates will be rising in the future (under the rational expectations theory). It is also the case that a downward-sloping yield curve is associated with expectations that short-term interest rates will decline in the future.

1. The following prices are observed. Formulate an arbitrage strategy to profit from the situation.

• Interest rate is 7.25% compounded daily, for 270-day T-bills in the spot market.


• The futures rate is 7.50% for T-bills with 180 days to maturity, to be delivered 90 days from now.









For preparation prior to problem 4, see lecture notes from Topic 10, Slides 2, 3 & 4. This problem begins the series of problems that illustrate how the yield curve is shaped. In this example, there is a “kink” in the yield curve that is the source of the arbitrage opportunity (in this example, the yield curve slopes down, then turns up).

4. The following prices are observed. Calculate the yield to maturity for each bond, and formulate a trading strategy to profit from the situation.

• Treasury bonds with 6% coupon and 10 years remaining to maturity are selling at $75.08 per $100 of face value (net of accrued interest).


• Equivalent risk bonds with 0% coupon and 10 years remaining to maturity are selling at $36.80 per $100 of face value.

For preparation prior to problem 5, see lecture notes from Topic 10, Slide 9. This problem continues the series of problems that illustrate how the yield curve is shaped. In this example, there is another variety of “kink” in the yield curve that is the source of the arbitrage opportunity (in this example, the yield curve slopes up, then turns down).





Prof. Kensinger page 1


For preparation prior to problems 6 through 10, see lecture notes from Topic 10, Slides 10 through 13. This problem continues the series of problems that illustrate how the yield curve is shaped. In this example, there is a

“kink” in the yield curve that is the source of the arbitrage opportunity (in problem 6, the yield curve slopes down, then turns up).

6. The following prices are observed. Formulate a trading strategy to profit from the situation.




In problem 7, the yield curve slopes up, then turns down. It can’t change direction. If it is upward sloping anywhere, it must be upward sloping over the entire range (until it becomes flat).





In problem 8, the yield curve slopes down, then turns up. It can’t change direction. If it is downward sloping anywhere, it must be downward- sloping over the entire range (until it becomes flat).





In problem 9, the yield curve is flat, then turns up. It can’t change direction. Once it becomes flat anywhere, it must remain flat over the entire remaining range of increasing convexity.







In problem 10, the yield curve is downward sloping, but the slope becomes steeper. The yield curve can’t do this. If it is downward sloping, it must continue downward at a decreasing rate until it becomes flat. (It is also true that if the yield curve is upward sloping, it must continue upward at a decreasing rate until it becomes flat.)





For preparation prior to problem 11, see Problem Set 4, Problem 5. Problem 11 provides further practice, and illustrates that the arbitrage can originate from anywhere in the world.

11. You are an expatriate working for Bank America in Hong Kong, and observe the following prices. Formulate an arbitrage strategy to profit from the situation.

• Swiss Franc per Dollar exchange rate is 1.15 spot and 1.14 for 180-day forward.

• Swiss interest rate is 6.00% compounded daily.

• U.S. stock market index is 1500 today.

• At today's level of the index, the average annual dividend yield on the stocks in the index is 2% (for simplicity, assume the dividends for your six-month holding period will all be paid at the end of 180 days).

• The U.S. stock market index 180-day futures price is 1575

For preparation prior to problem 12, see lecture notes from Topic 11, Slides 5 through 10. This problem illustrates how to create a hedge in order to profit from advance information.

12. Suppose you work for Goldman Sachs. Through diligent research you have discovered information that you believe is not widely available, indicating that there will soon be bad news about Exxon earnings. You want to sell Exxon short but also want to hedge against the possibility that a general market upswing might pull Exxon up along with the rest of the market and cause you to lose money. Assuming that the CAPM adequately captures the real world and that Exxon has a beta of 0.95, describe how you would construct an arbitrage portfolio to offset your short position in Exxon.

13. Suppose you work for Goldman Sachs. Through diligent research you have discovered information that you believe is not widely available, indicating that there will soon be bad news about General Motors earnings.

You want to sell GM short but also want to hedge against the possibility that a general market upswing might pull GM up along with the rest of the market and cause you to lose money. Assuming that the CAPM adequately captures the real world and that GM has a beta of 1.05, describe how you would construct an arbitrage portfolio to offset your short position in GM.

This is a true case, and is covered in the text. It illustrates a violation of the value additivity principle. The best explanation for this violation is that the creation of Primes and Scores made the market more complete.

14. The Americus Trust primes and scores are an interesting example of arbitrage in the equities markets. The trust would receive shares of a stock (the first was Exxon) and in return issue one prime and one score. The primes and scores were then separately traded on the New York Stock Exchange. The primes received all the dividends paid on the stock held in the trust. The trust was scheduled to dissolve after two years, at which time the primes would recieve capital appreciation up to a defined limit (20% in several cases) and the scores would receive any additional capital appreciation. Thus the scores were like call options, while the primes were like stock with a short call. (The primes and scores are described in more detail in the text—look in the index under Americus Trust securities.)

As it turned out, the market value of a prime plus a score was consistently higher than the value of a share of stock. Discuss the implications concerning value additivity and capital market efficiency.



For preparation prior to problems 15 & 16, see lecture notes in Topic 8, slide 42. This problem is an application of a standard fixed to floating rate swap. Using the swap, an underwriter can arrange an arbitrage that benefits all parties. The source of the arbitrage is a violation of market equilibrium in which the “quality gap” increases for longer maturities (the quality gap is the space between the yield curve for a high quality credit risk, compared with the yield curve for a lower quality credit risk). The quality gap must be the same for all maturities.

15. JunkCo has a low bond rating because it is small and new. JunkCo needs to finance some new expansion and would like to borrow at a fixed rate for five years, but the lowest rate available is 15%—which management considers too high. So, JunkCo borrows for five years at a variable rate pegged at 3% over the rate for 1-year

Treasury Notes (which is now 8%). Meanwhile AAA Corp needs money for only a year, and because of its high rating can borrow for that maturity at 9%. If it wanted to, AAA Corp could borrow for five years at a fixed rate of 11%. Suppose you work for Bankers Trust. Can you figure out an alternative borrowing and swap arrangement that would make both JunkCo and AAA Corp better off?

Now, let’s discuss what this means about the so-called “quality gap”—which refers to the difference between the yield curve for “quality” bonds and the yield curve for “junk” bonds. Knowledge of this could be important for developing investment strategies.

16. Myron Labs is a British company producing pharmaceuticals and doing research into new medicines. Myron

Labs needs to finance some new expansion and would like to borrow at a fixed rate for five years, but the lowest rate available to them in England is 11%—which management considers too high. So, Myron Labs decides to borrow for five years at a variable rate 2% over the rate for British Treasury Bills (which is now

5%). Meanwhile Advanced Devices, an American company, needs money for only one year; and can borrow in the U.S. for that maturity at 6%. If it wanted, Advanced Devices could borrow for five years at a fixed rate of 8% in the U.S. market. Currency exchange rate is £1= $2.00 spot, and also £1= $2.00 in the 1-year forward market. Suppose you work for CitiCorp. Can you figure out an alternative borrowing and swap arrangement that would make both Myron Labs and Advanced Devices Corp better off?

For preparation prior to problem 17, see lecture notes concerning equity call swaps covered in Topic 8, slide 52

(also see Topic 8, slide 51). Using an equity call swap, an underwriter can arrange an arbitrage that benefits all parties. The source of the arbitrage is a restriction that prevents the counterparty from selling stock that the underwriter fears will decline.

17. Suppose you are an analyst working for the Sacramento County Public Employees’ Pension Fund (SCPEP).

The fund has $10 million invested in a Japanese stock market index mutual fund. Currently the dividend yield on the stocks in the Japanese index is 0.5% annually. SCPEP has been approached by Bankers Trust with an offer to arrange a bond issue with principal of $5 million, a 1% annual coupon, and a maturity payment at the end of five years that will be determined by the appreciation in the Nikkei Dow Index of 225 Japanese stocks.

The formula for calculating the maturity payment is as follows:

Maturity payment = Principal * (I

T declines.

/I

0

) if the index increases, with guaranteed return of principal if the index

• I

T

• I

0

= the Nikkei index at maturity and

= the Nikkei index at origination

The bond will be issued by Private Export Funding Corporation, whose debt is guaranteed by the

Import/Export Bank of the United States. PEFCO has a AAA bond rating, and would arrange a swap through

Bankers Trust that would allow it to pay a fixed annual interest rate over the five-year period and receive a variable amount equal to the notional principal times (I

T

/I

0

– 1). Thus there is virtually no risk of default. So, it appears that this so-called “Protected Equity Note” (PEN for short) is a better investment than the Japanese index mutual fund. Your boss wants you to analyze this PEN and report back with a recommendation and an explanation of what is going on.


Fina 5210 Solutions: Problem Set 5 Fall 2014

Advanced Problems in Arbitrage

1. In this problem, there are two ways to invest for a 270-day period. One way is to buy 270-day bills, which yields 7.25%.

The other is to buy 180-day bills and then contract to roll them over, which yields

7.33%. Clearly, the roll-over strategy is more attractive. An arbitrage to take advantage of this involves the following steps: a. Borrow $1,000,000 for 270 days at

7.25% compounded daily b. Invest it in 90-day bills. c. Contract to buy 180-day bills in 90 days to yield 7.50%.

The risk-free profit from this arbitrage will be $1,055,739.13 – $1,055,088.67=

$650.46. Repeat this 1,000 times and then retire.


The other is to buy 180-day bills and then contract to roll them over, which yields only 7.23%. Clearly, the 270-day bills are more attractive. An arbitrage to take advantage of this involves the following steps: a. Borrow $1,000,000 for 180 days at

7.10% compounded daily b. Invest it in 270-day bills. c. Contract to sell 90-day bills in 180 days (by which time the bills you bought in the previous step will have

90 days left to maturity).

The risk-free profit from this arbitrage will be $1,017,634.17 – $1,017,408.41 =


4. YTM does not follow the normal pattern.

It is 11.60% for the 8% coupon, 10.00% for the 6% coupon, and 10.24% for the 0% coupon. Therefore sell four of the 6%, while buying three of the 8% coupon and one of the 0% coupon bonds. Net cash flow is +$26.52 in the present, and zero in all future periods. Arbitrage examples like this prove that the yield curve cannot change direction.


It is 12.10% for the 10% coupon, 12.22% for the 8% coupon, and 11.68% for the 6% coupon. Therefore sell one each of the 6% and the 10% coupon, while buying two of the 8% coupon bonds. Net cash flow is

+$3 in the present, and zero in all future periods. Arbitrage examples like this prove that the yield curve cannot change direction.


It is 8% for both the 6 and 8 year maturities, but 7.75% for the 7 year. If you sell two 7-year bonds and buy one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any discount rate. Cash flows are +2.40 in period 0, 0 in periods 1-11, 100 in period

12, –3 in period 13, –203 in period 14, +3 in period 15, and +103 in period 16.

Arbitrage examples like this prove that the yield curve cannot be kinked.

The risk-free profit from this arbitrage will be $1,035,758.04 – $1,035,630.37 =



The other is to buy 90-day bills and then contract to roll them over in 90 days, which yields only 7.075%. Clearly, the 180-day bills are more attractive. An arbitrage to take advantage of this involves the following steps: a. Borrow $1,000,000 for 90 days at 7% compounded daily b. Invest it in 180-day bills. c. Contract to sell 90-day bills in 90 days (by which time the bills you bought in the previous step will have

90 days left to maturity).


It is 8% for both the 6 and 8 year maturities, but 8.50% for the 7 year. If you buy two 7-year bonds and sell one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any discount rate. Cash flows are +4.96 in period 0, 0 in periods 1-11, –100 in period

12, +3 in period 13, +203 in period 14, –3 in period 15, and –103 in period 16.


Prof. Kensinger

Fina 5210 Solutions: Problem Set 5


It is 8.50% for the 6-year maturity, 8.00% for the 7-year, but 8.38% for the 8-year. If you sell two 7-year bonds and buy one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any discount rate. Cash flows are +4.09 in period 0, 0 in periods 1-11, +100 in period

12, -3 in period 13, -203 in period 14, +3 in period 15, and +103 in period 16.



It is 8.50% for the 8-year maturity, but

8.00% for both the 7 year and 6-year. If you sell two 7-year bonds and buy one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any discount rate. Cash flows are +2.57 in period 0, 0 in periods 1-11, +100 in period

12, -3 in period 13, -203 in period 14, +3 in period 15, and +103 in period 16.

Arbitrage examples like this prove that the yield curve cannot change direction.


It is 8% for the 6-year maturity, 7.75% for the 7-year but 7.00% for the 8-year. If you buy two 7-year bonds and sell one each of the 6 and 8 year bond, you will obtain a cash flow stream that has positive NPV at any discount rate. Cash flows are +3.20 in period 0, 0 in periods 1-11, –100 in period

12, +3 in period 13, +203 in period 14, –3 in period 15, and –103 in period 16.


11. An arbitrage to take advantage of this involves the following steps: a. Borrow SF 1,725,000 at 6% compounded daily. b. Convert it to $1,500,000 and invest in the stocks in the U.S. stock market index (a million dollars worth of stock is 1,000 times the index). c. Sell stock index futures to cover the position. d. Contract to exchange Dollars back to

SF at the rate of 1.14 SF per dollar.

At the termination of the hedge, collect dividends of $15,000 and sell the stock to net $1,575,000 (1,000 times the index futures price). Convert your $1,590,000 into SF at 1.14, to yield SF 1,812,600. You will have to pay only SF 1,776,799.41 to settle your debt, leaving a profit of

SF35,800.59. The only uncovered source

Fall 2013 of risk arises from the dividends. Although individual company dividends are somewhat unpredictable, however, the average dividend yield for five hundred companies is reasonably stable, so the risk is minimal.

12. Randomly select 15 stocks and create an equally-weighted portfolio. Then use lending to adjust the portfolio beta to 0.95.

Sell Exxon and buy equal value in this hedge portfolio. For example, if you sell

$1000,000 of Exxon stock, you would need to buy $950,000 of the stock portfolio and put the remaining $50,000 into Treasury

Bills.

13. Randomly select 15 stocks and create an equally-weighted portfolio. Then use leverage to adjust the portfolio beta to 1.05.

Sell GM and buy equal value in this hedge portfolio. For example, if you sell

$1000,000 of Exxon stock, you would need to buy $1,050,000 of the stock portfolio and so would need to borrow $50,000 of

Treasury Bills.

14. Value additivity is violated, but not because of market inefficiency. The reason is that the creation of primes and scores makes the market more nearly complete, thereby enhancing value.


Fina 5210 Solutions: Problem Set 5

15. AAA Corp borrows fixed for 5 years at

11% and JunkCo borrows floating. Then

AAA Corp swaps to receive 11% fixed and pay T-bill, while Junkco swaps to receive

T-bill and pay 11% fixed. Thus during the first year AAA Corp gets to borrow at the

T-bill rate. Then instead of paying off its debt (as originally planned) it invests in T-

Bills and continues to roll over for the remaining 4 years, while paying interest on the fixed-rate loan. AAA Corp’s annual cash flow stream for the remaining life of the arrangement is as follows:

Principal ×

#

%

%

%

$

+

−

11%

TBill

+ TBill

− 11%

&

(

(

(

'

= 0

Meanwhile, Junkco in effect borrows at

14% fixed, a full percentage point lower than it could do on its own. Junkco’s annual cash flow steam is as follows:

Principal ×

# + TBill

%

%

$

− 11%

− ( TBill + 3% )

&

(

(

'

= − Principal × 14%

Now, let’s discuss what this means about the “quality gap.” This, along with 16&17, will be done in class discussion.

Fall 2013


Advanced Problems in Arbitrage

Advanced Problems in Arbitrage

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