Portfolio Management【 Reading 58】Sample equivalent, 2012

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A cap on a floating rate note, from the bondholder’s perspective, is equivalent to:

A) owning a series of calls on fixed income securities.

B) writing a series of interest rate puts.

C) writing a series of puts on fixed income securities.

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For a bondholder, a cap, which puts a maximum on floating rate interest payments, is equivalent to writing a series of puts on fixed income securities. These would require the buyer to pay when rates rise and bond prices fall, negating interest rate increases above the cap rate. Writing a series of interest rate calls, not puts, would be an equivalent strategy. Calls on fixed income securities would pay when rates decrease, not when they increase.

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A floor on a floating rate note, from the bondholder’s perspective, is equivalent to:

A) owning a series of calls on fixed income securities.

B) owning a series of puts on fixed income securities.

C) writing a series of interest rate puts.

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A floor, which puts a minimum on floating rate interest payments is equivalent to owning calls on fixed income securities which will pay when interest rates fall. Owning interest rate puts, rather than writing them, would be equivalent to the floor. Puts on fixed income securities pay when interest rates increase.

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An issuer who wishes to issue a floating rate note with a collar would be equivalently issuing the note and:

A) buying a cap and selling a floor.

B) selling a cap and buying a floor.

C) buying a cap and a floor.

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Issuing a floating rate note with a collar (a cap and a floor) is equivalent to issuing the note, buying a cap to put an upper limit on the interest cost, and selling a floor which would put a minimum on interest expense and offset the cost of the cap to some extent.

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Steve Miller is a senior fixed income trader for a large hedge fund based in New York. Miller has recently hired C.D. Johnson to assist Miller in implementing some derivative-based trades. Miller would like to ensure that Johnson understands the basics of interest rate derivatives before allowing him to be involved into some more complicated trading strategies. Miller creates a hypothetical bond scenario for Johnson to analyze in order for him to evaluate Johnson’s expertise in the area. Miller instructs Johnson to consider the London Interbank Offered Rate (LIBOR) interest rate environment in Table 1.

Table 1 90-Day LIBOR Forward Rates and Implied Spot Rates
Period (in months)	LIBOR Forward Rates	Implied Spot Rates
0 × 3	5.500%	5.5000%
3 × 6	5.750%	5.6250%
6 × 9	6.000%	5.7499%
9 × 12	6.250%	5.8749%
12 × 15	7.000%	6.0997%
15 × 18	7.000%	6.2496%
48 × 51	8.100%	7.1228%
51 × 54	8.200%	7.1826%
54 × 57	8.300%	7.2413%
57 × 60	8.400%	7.2992%
60 × 63	8.500%	7.3563%
63 × 66	8.600%	7.4127%
66 × 69	8.700%	7.4686%
69 × 72	8.800%	7.5240%
72 × 75	8.900%	7.5789%
75 × 78	9.000%	7.6335%
78 × 81	9.100%	7.6877%
81 × 84	9.200%	7.7416%
84 × 87	9.300%	7.7953%
87 × 90	9.400%	7.8487%

Miller suggests to Johnson to examine the instruments shown in Table 2 using the information in Table 1. Miller instructs Johnson to use 0.25 years for each quarter and to not concern himself with actual day counts.

Table 2 Interest Rate Instruments
Dollar Amount of Floating Rate Bond	$30,000,000
Floating Rate Bond Spread over LIBOR	0.50%
Time to Maturity (years)	1
Cap Strike Rate	6.00%
Floor Strike Rate	5.00%
Interest Payments	quarterly

Johnson wants to evaluate the effect of an increase in rates on the inception value of a plain vanilla pay, fixed interest rate swap. Specifically, if interest rates increase across all maturities in Table 1, how would the inception value of the swap be affected? The inception value of the swap would:

A) decrease.

B) increase.

C) stay the same.

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The value stays the same because the inception value of all plain vanilla interest rates swaps is zero by design.
An increase would, however, be correct for an existing pay fixed swap. The counterparty receives the floating rate while paying the fixed rate. Therefore, it would receive a higher interest rate but would still have to pay the same fixed interest rate. Therefore, the value of the swap would increase. (Study Session 17, LOS 57.a, c)

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Miller asks Johnson to hedge a hypothetical short position in the floating rate bond in Table 2. Which of the following is the best hedge for this position?

A) Buy an interest rate cap.

B) Sell an interest rate cap.

C) Buy an interest rate floor.

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An interest rate cap provides a positive payoff when interest rates are above the cap strike rate. Therefore, the buyer of this instrument is able to hedge himself against rising interest rates. Incorrect answer explanations:

• Selling an interest rate cap is not a hedge against rising interest rates.
• Buying an interest rate floor hedges the risk of decreasing interest rates.

(Study Session 17, LOS 58.a)

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Miller now asks Johnson to compute the payoff of the cap and floor in Table 2 assuming that LIBOR has risen to 7% at expiration. Specifically, Miller wants Johnson to determine the net payoff of the corresponding short collar (buying the floor and selling the cap) for the total outstanding amount of the floating rate bond. Which of the following is the closest to Johnson's answer?

A) $300,000.

B) −$300,000.

C) −$450,000.

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The floor expires worthless while the cap is exercised and the seller has to pay the difference between the cap strike rate and LIBOR which is 1% in this case. Hence the calculation is as follows:

Net Payoff = (6.00% − (7.00%)) × $30,000,000 = −$300,000

The answer −$450,000 is incorrect because the payoff is determined by the LIBOR rate, not by the spread over LIBOR for the floating rate bond. (Study Session 17, LOS 58.b)

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Next, Miller asks Johnson to determine the net payoff of the corresponding long collar (buying the cap and selling the floor) for the total outstanding amount of the floating rate bond. Assume that LIBOR has risen to 8% at expiration. Which of the following is the closest to Johnson's answer?

A) −$600,000.

B) $600,000.

C) $900,000.

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The floor expires worthless while the cap is exercised and the seller has to pay the difference between the cap strike rate and LIBOR which is 2% in this case. Hence the calculation is as follows:

Net Payoff = (8.00% − (6.00%)) × $30,000,000 = $600,000

(Study Session 17, LOS 58.b)

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Miller asks Johnson which of the following strategies allows an investor to benefit from both increasing and decreasing interest rates?

A) Buy an at the money cap and sell an at the money floor.

B) Buy an at the money cap and an at the money floor.

C) Sell an at the money cap and an at the money floor.

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This is a straddle on interest rates. The cap provides a positive payoff when interest rates rise and the floor provides a positive payoff when interest rates fall. Incorrect answer explanations:

• Sell an at the money cap and an at the money floor. In this case the investor would suffer from increasing and decreasing interest rates since the caplets and floorlets would be exercised against him.
• Buy an at the money cap and sell an at the money floor. In this case the investor would suffer from decreasing interest rates since the floorlets would be exercised against him.

(Study Session 17, LOS 58.a)

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Johnson now considers the floating rate bond shown in Table 2. Specifically, Johnson considers this note from the perspective of the issuer. If the issuer decided to hedge the interest rate risk associated with this liability which of the following is the most appropriate hedge?

A) Selling an interest rate floor.

B) Buying an interest rate floor.

C) Selling Eurodollar futures.

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If a short position in Eurodollar futures is added to the existing liability in the correct amount, the interest risk is hedged.
Incorrect answer explanations:

• Buying an interest rate floor is a hedge against declining interest rates if one has a long position in a floating rate bond.
• Selling an interest rate floor is not a hedge against changing interest rates.

(Study Session 17, LOS 58.a)

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Which of the following best describes an interest rate cap? An interest rate cap is a package or portfolio of interest rate options that provide a positive payoff to the buyer if the:

A) reference rate exceeds the strike rate.

B) T-Bond futures exceeds the strike price.

C) reference rate is below the strike rate.

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An interest rate cap is a package of European-type call options (called caplets) on a reference interest rate.

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Jacob Bower is a bond strategist who would like to begin using fixed-income derivatives in his strategies. Bower has a firm understanding of the properties fixed-income securities. However, his understanding of interest rate derivatives is not nearly as strong. He decides to train himself on the valuation and sensitivity of interest rate derivatives using various interest rate scenarios. He considers the forward London Interbank Offered Rate (LIBOR) interest rate environment shown in Table 1. Using a rounded daycount (i.e., 0.25 years for each quarter) he has also computed the corresponding implied spot rates resulting from these LIBOR forward rates. These are included in Table 1.

Table 1 90-Day LIBOR Forward Rates and Implied Spot Rates
Period (in months)	LIBOR Forward Rates	Implied Spot Rates
0 × 3	5.500%	5.5000%
3 × 6	5.750%	5.6250%
6 × 9	6.000%	5.7499%
9 × 12	6.250%	5.8749%
12 × 15	7.000%	6.0997%
15 × 18	7.000%	6.2496%

Bower has also estimated the LIBOR forward rate volatilities to be 20%. The particular fixed instruments that Bower would like to examine are shown in Table 2. He also wants to analyze the strategy shown in Table 3.

Table 2 Interest Rate Instruments
Dollar Amount of Floating Rate Bond			$42,000,000
Floating Rate Bond paying LIBOR +			0.25%
Time to Maturity (years)			8
Cap Strike Rate		7.00%
Floor Strike Rate		6.00%
Interest Payments		quarterly

Table 3 Initial Position in 90-day LIBOR Eurodollar Contracts
Contract Month (from now)	Strategy A (contracts)	Strategy B (contracts)
3 months	300	100
6 months	0	100
9 months	0	100

Bower is a bit puzzled about how to use caps and floors. He wonders how he could benefit both from increasing and decreasing interest rates. Which of the following trades would most likely profit from this interest rate scenario?

A) Buy at the money cap and at the money floor.

B) Sell at the money cap and at the money floor.

C) Buy at the money cap and sell at the money floor.

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This is a straddle on interest rates. The cap provides a positive payoff when interest rates rise and the floor provides a positive payoff when interest rates fall. (Study Session 17, LOS 58.a)

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Bower shorts the floating rate bond given in Table 2. Which of the following will best reduce Bower's interest rate risk?

A) Shorting Eurodollar futures.

B) Buying an interest rate floor.

C) Shorting an interest rate floor.

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If he adds a short position in Eurodollar futures to the existing liability in the correct amount, he is able to lock in a specific interest rate. A short Eurodollar position will increase in value if interest rates rise because the contract is quoted as a discount instrument so increases in rates reduce the futures price. (Study Session 17, LOS 58.a)

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Bower has studied swaps extensively. However, he is not sure which of the following is the swap fixed rate for a one-year interest rate swap based on 90-day LIBOR with quarterly payments. Using the information in Table 1 and the formula below, what is the most appropriate swap fixed rate for this swap?

A) 5.65%.

B) 6.01%.

C) 5.75%.

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The swap fixed rate is computed as follows:

Z90-day =	1
	1 + (0.055 × 90 / 360)
=	0.98644

Z180-day =	1
	1 + (0.05625 × 180 / 360)
=	0.97264

Z270-day =	1
	1 + (0.057499 × 270 / 360)
=	0.95866

Z360-day =	1
	1 + (0.058749 × 360 / 360)
=	0.94451

The quarterly fixed rate on the swap =	1 − 0.94451
	0.98644 + 0.97264 + 0.95866 + 0.94451

= 0.05549 / 3.86225 = 0.01437 = 1.437%

The fixed rate on the swap in annual terms is:

1.437% × 360 / 90 = 5.75%

(Study Session 17, LOS 57.c)

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Bower would like to perform some sensitivity analysis on a one year collar to changes in LIBOR. Specifically, he wonders how the price of a collar (buying a cap and selling a floor) is affected by an increase in the LIBOR forward rate volatility. Using the information in Tables 1 and 2 which of the following is most accurate? The price of the collar will:

A) increase.

B) stay the same.

C) decrease.

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The price of the floor will increase more than the price of the cap since the floor is closer to being at the money than the cap. Therefore, the floor price is more sensitive to volatility changes in the LIBOR forward rate. Since the price of the collar is equal to the price of the cap minus the price of the floor, the net effect is a price decrease for the collar. (Study Session 17, LOS 58.a)

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Bower computes the implied volatility of a one year caplet on the 90-day LIBOR forward rates to be 18.5%. Using the given information what does this mean for the caplet's market price relative to its theoretical price? The caplet's market price is:

A) undervalued or overvalued.

B) undervalued.

C) overvalued.

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Volatility and option prices are always positively related. Therefore, since the option implied volatility is lower than the estimated volatility, this implies that the caplet is undervalued relative to its theoretical value. (Study Session 17, LOS 58.a)

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For this question only, assume Bower expects the currently positively sloped LIBOR curve to shift upward in a parallel manner. Using a plain vanilla interest rate swap, which of the following will allow Bower to best take advantage of his expectations? Purchase a:

A) receive fixed interest rate swap.

B) floating rate bond and enter into a receive fixed swap.

C) pay fixed interest rate swap.

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Since the interest rates are expected to rise for all maturities, one can benefit from this rise by receiving a floating rate (LIBOR) and borrowing at a fixed rate (i.e. a pay fixed swap). (Study Session 17, LOS 57.c)

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To the issuer of a floating rate note, a cap is equivalent to:

A) writing a series of interest rate calls.

B) owning a series of calls on a fixed income security.

C) owning a series of interest rate calls.

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The issuer of the note is borrowing at a floating rate, and will have higher interest expenses if rates increase. A cap is equivalent to owning a series of interest rate calls at the cap rate that will pay the difference between the market rate and the cap rate. If interest rates increase, the payoff from the calls will compensate the borrower for the higher interest expenses.

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Assume that a three-year semi-annually settled cap with a strike rate of 8% and a notional amount of $100 million is being analyzed. The reference rate is six-month LIBOR. LIBOR for the next four semi-annual periods is as follows:

Period	LIBOR
1	7.5%
2	8.2%
3	8.1%
4	8.7%

What is the payoff for the cap for period 4?

A) $700,000.

B) $350,000.

C) $0.

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The payoff for each semi-annual period is computed as follows:
Payoff = notional amount × (six-month LIBOR – cap rate)/2    so for period 4:
         = $100 million × (8.7% - 8.0%)/2 = $350,000.

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Mark Austin is a risk management consultant specializing in fixed income securities. He has been asked to evaluate the interest rate risk situation of an intermediate size bank. The bank has a large floating rate liability of $100,000,000 on which it pays the London Inter Bank Offered Rate (LIBOR) on a quarterly basis. Austin is concerned about the significant interest rate risk the bank incurs because of this liability. Since most of the bank's assets are invested in fixed rate instruments there is a considerable duration mismatch. Some of the bank's assets are floating rate notes tied to LIBOR. However, the total par value of these securities is significantly less than the liability position.
Austin considers both swaps and interest rate options. The interest rate options are 2-year caps and floors with quarterly exercise dates. He wishes to hedge the entire liability.
Austin has the technology to be able to generate an interest rate tree resulting in the quarterly incremented interest rate tree and the corresponding pricing tree for a two-year caplet shown in Table 1.
He has also obtained the prices for an at-the-money 6 month cap and floor with quarterly exercise. These are shown in Table 2.

Table 1: Interest Rate Tree and Price Tree for Two-Year Caplet ($100M notional)
							12.23%
						10.66%
					9.68%		10.01%
				8.79%		8.72%
			7.32%		7.92%		8.19%
		6.65%		7.19%		7.14%
	6.05%		5.99%		6.48%		6.70%
5.00%		5.45%		5.89%		5.84%
	4.95%		4.91%		5.31%		5.48%
		4.46%		4.82%		4.78%
			4.02%		4.34%		4.48%
				3.95%		3.91%
					3.55%		3.67%
						3.20%
							3.00%

							$1,557,206
						$1,214,622
					$933,050		$1,001,459
				$697,982		$741,758
			$507,549		$524,260		$546,738
		$357,476		$354,271		$348,281
	$244,490		$231,189		$209,772		$174,677
$163,310		$146,294		$121,961		$84,861
	$90,294		$69,369		$41,334		$0
		$38,766		$20,180		$0
			$9,892		$0		$0
				$0		$0
					$0		$0
						$0
							$0

Table 2: At-the-Money 0.5 year Cap and Floor Values
Price of at-the-money Cap	$133,377
Price of at-the-money Floor	$258,510

Austin analyzes alternative hedging strategies. Which of the following is the most appropriate transaction to most efficiently hedge the interest rate risk for a floating rate liability without sacrificing potential gains from interest rate decreases?

A) Selling a cap.

B) Buying a cap.

C) Buying a collar.

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Buying a cap combined with a floating rate liability limits the exposure to interest rate increases (no exposure to interest rate increases above strike rate) while the floating rate borrower is still able to benefit from interest rate decreases.

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Austin now wants to compute the cost to convert the bank's floating rate liability to a fixed rate liability for 6 months. Using the information in Tables 1 and 2, what would be the cash flow to Austin to implement this hedge using at the money interest rate options?

A) $125,133.

B) −$246,894.

C) −$125,133.

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This is the difference between the 0.5 year cap and the 0.5 year floor both with a strike rate of 5.000% with the values shown in Table 2.
So we have cash flow to convert floating to fixed = −$133,377 + $258,510 = $125,133

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Austin is now considering some of the bank's floating rate assets. Which of the following transactions is the most appropriate to minimize the interest rate risk of these assets without sacrificing upside gains?

A) Buy a collar.

B) Buy a floor.

C) Buy a cap.

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Buying a floor combined with a floating rate holding limits the exposure to interest rate decreases (no exposure to interest rate decreases below strike rate) while the floating rate holder is still able to benefit from interest rate increases. Ideally, Patrick should consider matching this asset position against the liability position.

invic

Which of the following best represents an interest floor?

A) A portfolio of put options on an interest rate.

B) A portfolio of call options on an interest rate.

C) A put option on an interest rate.

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A long floor (floor buyer) has the same general expiration-date payoff diagram as that for long interest rate put position.