SWAP CONTRACTS, CONVERTIBLE SECURITIES, AND OTHER EMBEDDED DERIVATIVES
1. CFA Examination III (1994)
l(a). An interest rate swap is a customized risk management vehicle. In a pension portfolio (i.e., investment) context, an interest rate swap would be represented by an agreement between two parties to exchange a series of interest money cash flows for a certain period of time (term) based on a stated (notional) amount of principal. For example, one party will agree to make a series of floating rate coupon payments to another party in exchange for receipt of a series of fixed rate coupon payments (or vice versa, in which case the swap would work in reverse). No exchange of principal payments is made.
l(b). Strategies using interest rate swaps to affect duration or improve return in a domestic fixed income portfolio can be divided into two categories:
Duration modification. Swapping floating for fixed rate interest payments increases portfolio duration (and vice versa, decreases duration when the portfolio is the floating rate recipient). This method of modifying duration can be used either to control risk (e.g., keep it within policy guidelines/ranges) or to enhance return (e.g., to profit from a rate anticipation bet while remaining within an allowed range).
Seeking profit opportunities in the swap market. Opportunities occur in the swap market, as in the cash markets, to profit from temporary disequilibrium between demand and supply. If, in the process of exploiting such opportunities, portfolio duration would be moved beyond a policy guideline/range, it can be controlled by using bond futures contracts or by making appropriate cash market transactions.
If a strategy calls for a large-scale reorientation of the portfolio’s characteristics in a manner that swaps could achieve, their use for implementation of the strategy would act to reduce transaction costs (thus improving portfolio return) and might also permit transactions to be effected more quickly or completely than through conventional trading mechanisms.
2. An interest rate swap is an agreement to exchange a series of cash flows based on the difference between a fixed interest rate and a floating interest rate on some notional amount. A fixed rate receiver would get the difference between a fixed rate and a floating rate if the fixed rate was above the floating rate, and pay the difference if floating was above fixed. The fixed rate is set so that no cash changes hands upon initiation of the deal. This can be thought of as:
i. A series of forward contracts on the floating rate because forward contracts also have no initial cash flow and will net the difference between the floating rate and the forward rate (which acts like a fixed rate). To make the analogy precise, only one forward rate is chosen but it is chosen such that the sum of the values of all the contracts are zero. The fixed rate receiver is like the short position in the interest rate forwards, because if interest rates go up, she loses.
ii. A pair of bonds, one with a floating rate coupon the other with a fixed-rate coupon (both selling at par with the same face value and maturity). Being the fixed-rate receiver in the swap is the same as being long the fixed-rate bond and short the floating-rate bond. As long as the floating rate is less than the fixed rate, the coupon payment from the fixed-rate bond will cover the interest due on the floating rate bond and the difference is profit. If the floating rate is above the fixed rate then the fixed-rate receiver must make up the difference. Since the bonds are of the same face amount, there is no net cash flow at the beginning or end of the agreement.
3. To have zero value at origination, the present vale of the expected cash flows from the swap must be zero. This implies that if there is an upward sloping yield curve, the expected cash flows to the floating payer will come at the beginning of the swap and be offset by expected payments by the floating payer at the end of the swap. The only way this is possible is if the fixed rate is somewhere between the current floating rate (low) and the implied floating rate later in the contract (high).
4. You are essentially holding a two-year swap agreement that requires you to pay 7% in exchange for floating. Since the market rate for the same swap is 6.5%, if your counterparty defaulted you would be able to replace the swap at a lower rate. Thus, it would be to your benefit for your counterparty to default, and you would realize an economic gain.
5. You have created an off-market swap where you are paying a fixed rate of 7%. This is seen by analyzing the portfolio of long a 7% cap and short a 7% floor. If interest rates are above 7%, then the cap pays the difference between 7% and the floating rate, and the floor is out-of-the money. If interest rates are below 7%, then you must pay the difference between floating and 7%, and the cap is out-of-the-money. Since the market rate is a fixed rate of 8% for the same maturity and you only have to pay 7%, you will have to pay money up-front to get the cap and floor. In other words, the cap costs more than you will get from selling the floor.
6. CFA Examination III (1995)
6(a). The problem here is to sell equities and reinvest the proceeds with the skilled fixed-income manager, without changing the split between the existing allocations to the two asset classes. The solution is to turn to derivative financial instruments as the means to the end: selling enough of the existing fixed-income exposure and bringing in enough of the equity exposure to get the desired mix result.
The following are distinct derivatives strategies that the board could use to increase the Fund’s allocation to the fixed-income manager without changing the present fixed-income/equity proportions:
Strategy 1 – Use futures: One strategy would be to sell futures on a fixed-income index and buy futures on an equity index.
Selling the futures eliminates the fixed income index return and risk, while keeping the skilled fixed income manager’s extra return. By being long the equity index, the portfolio obtains the index return and risk, keeping its exposure to the equity market.
Strategy 2 – Use swaps: A second strategy would be to use over-the-counter (OTC) swaps.
BI would swap a fixed-income index return for an equity index return in a notional amount large enough to keep the skilled manager’s extra return while eliminating the fixed income market return and replacing it with the equity market return.
Strategy 3 – Use option combinations: A third strategy would use put and call options to create futures-like securities.
Buying put options and selling call options on a fixed-income index, while selling put options and buying call options on a stock index, would achieve the same result as the appropriate futures position.
6(b). The following are advantages and disadvantages of each strategy identified and explained in Part A:
Strategy 1 – Use futures
1. Futures contracts are liquid instruments.
2. Transaction costs are low.
3. Credit risk is negligible because the securities are marked to market daily.
1. If the holding period is long, rollover (transaction) costs are incurred.
2. Standard contract forms are limited, so contracts may not exist on the index or instrument needed. Tracking errors may create basis risk between the index and the performance benchmark.
Strategic 2 – Use swaps
1. Swaps can be tailored to fit the desired investment horizon, eliminating (or reducing) rollover costs.
2. Swaps can be contracted for a specific index (like the performance benchmark) even if there is no futures contract on it.
3. The desired adjustment goal can be accomplished through a single transaction.
1. A counterparty credit risk is created that can be much larger than with other types of instruments.
2. Swap agreements are illiquid instruments, and disposals can be both difficult and expensive.
3. Transaction costs are large because of typical “tailoring” of a given swap.
Strategy 3 – Use option combinations
Buying put and call options on fixed income index (synthetic short position) and buying call and selling puts on stock index (synthetic long position)
1. Transaction costs are low.
2. Credit risks are small.
1. Rollover(s) may be necessary.
2. The “right” option may not be available when needed or at all.
3. Holders may exercise the put option and end the hedge.
A generic disadvantage of any strategy is that returns are automatically eroded by the costs of establishing and maintaining the strategies, of meeting margin requirements, if any, and of unwinding them, if necessary.
7. CFA Examination III (1995)
7(i). From BI’s perspective, the major risk it would eliminate under this transaction is represented by participation in the EAFE Index (to the extent that this participation has been reduced), the return on which is now being paid to the counterparty bank.
BI has reduced its international equity exposure. The market risk, formerly present in that part of the total participation that has now been swapped out in exchange for S&P Index exposure, has been eliminated.
7(ii). From BI’s perspective, the major risk that is retained after this transaction (i.e., not eliminated) is the risk of tracking error between BI’s international equity portfolio and the EAFE Index. This potential tracking error arises from differences between the portfolio and the index in terms of country weights and security selection. Thus, as the correlation coefficient between BI’s portfolio and the index is probably less than 1.0, a basis risk is retained.
To the extent that significant differences in composition now exist (either in terms of country weights or security selection), BI’s return experience could be quite different from that expected when the decision to change the exposure was made. In effect, the transaction is a bet that U.S. stocks will outperform EAFE stocks in U.S. return terms. The market risks inherent in all such exposures still apply, including exposure to global equity markets. Because the EAFE return is dollar denominated, the BI international equity portfolio is still exposed to currency risk either hedged or unhedged.
7(iii). From BI’s perspective, risks created under this transaction include:
1. Counterparty credit risks. BI now has contractual relationships with two banks that it did not have before, creating a new risk dimension that will require monitoring and ongoing evaluation. This credit risk is, in effect, an added element of cost and uncertainty that must be considered in assessing the cost/benefit outcomes expected to result from the transaction.
2. Return risks arising from spread changes between LIBOR and T-bill rates. If the initial or expected spread between these two markets moves against BI, it will find its realized return to be different from its expected return. This will, in effect, increase the cost of the move and reduce its efficacy.
3. Return risks arising from differential S&P 500 Index and EAFE Index performance. Future relative performance levels are uncertain. Thus, BI’s new risks include not only a spread risk in terms of the S&P Index (added) versus the EAFE Index (reduced) but also spread risk between the actual portfolio and the index portfolios.
4. Risk related to the one year term of the transaction. If BI wants to reverse or modify the original move away from international to U.S. equity exposure or to change the LIBOR/T bill markers, new complications arise. Liquidity risk is now present. Changing or undoing the contracts with the counterparty banks may be impossible or unduly costly. A risk of being “frozen in place” and being unable to take advantage of new circumstances or avoid new dangers has been introduced. A corollary risk is that the one year term does not equate with the “temporary” reduction intent of the bond’s proviso. A risk of overstaying a temporary reduction has been introduced and may manifest itself in liquidity or additional expense terms.
5. Rollover risk. BI may be unable to renew the contracts on satisfactory terms if the board wants to do so at the expiration of the original one-year term.
6. Lower diversification. BI has temporarily reduced its diversification because it has decreased its international equity exposure and increased its domestic equity exposure.
7. Risk of mismatched notional amounts. If the swaps have variable notional amounts, there is risk that the notional amounts upon which the swap payments are based may become mismatched as the return experience of the EAFE and S&P 500 indexes diverge.
8. Regulatory, legal, or accounting risk. BI has added the risk that the regulatory, legal, or accounting treatment of these swaps may be changed unfavorably during the term of the contracts.
9. Benchmark measurement. The benchmark should measure the portfolio manager’s execution of the strategies for which he or she is responsible. If the board changes the strategy, it may need to change the benchmark.
8. CFA Examination III (1997)
8(a). Transactions Needed to Restore Balance
i. Futures. The most straightforward way to restore the portfolio’s 50 percent bonds/50 percent equity allocation and restore (lengthen) the duration is for the plan to purchase a $200 million equity index (i.e., S&P 500 Index) futures position and a $200 million fixed (i.e., long-duration Treasury) bond futures position.
ii. Swaps. Entering into two swaps will restore the 50 percent bond/50 percent equity allocation. On the fixed-income side, the floater’s impact on portfolio duration (shortening) will be offset if the plan enters into a $200 million receiver interest rate swap in which it pays T-bill (plus 50 basis points [bps]) and receives fixed (long-bond) exposure. On the equity side, rebalancing also requires a $200 million swap in which the plan pays T-bill (plus 50 bps) returns and receives equity index return. This transaction could be accomplished with one swap – that is, swapping $400 million of floating-rate returns for a 50/50 blend of fixed-income and equity returns.
8(b). Advantages and Disadvantages of Using Futures
• Futures contracts do not expose the plan to credit (counterparty) risk, as swaps do.
• Although volume and liquidity in futures contract trading vary among markets and indexes, futures pricing for many major market indexes is generally tight enough that exchange-traded futures offer flexibility, observable prices, and ease of entry and exit. Buying or selling the contract is simpler and more certain in the open market than is identifying a willing counterparty to negotiate a reverse swap.
• Implementing a futures position is easier than implementing a swap because of the smaller denominations possible for futures.
• Because of regulations, swaps cannot be used for certain cases for which futures may be used.
• Futures may be mispriced (below fair value).
• A futures contract involves the buyer in rollover risk during the two-year period. The swap approach transfers the rolling risk to the counterparty.
• Futures must be marked to market and involve maintenance margins.
• The buyer incurs the risk of tracking error – related to the futures contract vs. the underlying security (basis risk) or the futures contract vs. the hedged asset. Standard futures contracts do not lend themselves to customization in terms of underlying assets or maturities.
• Futures may be mispriced (above fair value).
• Futures require two transactions to achieve the same result as one swap.
9. The difference between warrants and regular options comes from the difference in issuer. Unlike a regular call option, when a warrant is exercised the shares purchased are new shares created by the company. Since the shares have identical rights as existing shares and have been purchased at a discount to existing shares (otherwise the warrants wouldn’t be exercised) the value of existing shares is watered down. This depresses the value of the stock. Consequently warrants are less valuable than regular options since regular options have no affect on the capital structure of the firm.
Firms may wish to issue warrants if the floatation costs are lower or if they are worried about not being able to sell enough new stock. Warrants can be used to “force” the issuance of new shares.
10. Convertible bonds and preferred stock are both very similar to an ordinary bond (or perpetuity) and a call option on the firm’s common stock. This is because these instruments give the holder the option but not the obligation to trade in the existing asset for common stock, much the way a call option gives its holder the right but not the obligation to purchase shares at a prospected price. Since call options have upside potential and a limited downside, these traits are passed on to the convertible bonds and preferred stock. The pricing of these securities must take into account the optionality embedded in the issue.
11. CFA Examination II (2000)
11(a).i. Equity index-linked note
Unlike traditional debt securities that pay a scheduled rate of coupon interest on a periodic basis and the par amount of principal at maturity, the equity uindex-linked note typically pays no or little coupon interest; at maturity, however, a unit holder receives the orginal issue price plus a supplemental redemption amount, the value of which depends on where the equity index settled relative to a predetermined initial level.
11(a).ii. Commodity-linked bear bond
Unlike traditional debt securities that pay a scheduled rate of coupon interest on a periodic basis and thee pr amount of principal at maturity, the commodity-linked bear bond allows for an investor to participate in a delcine in a commodity’s price. In exchange for a lower than market coupon, buyers of a bear tranche will receive a redemption value that exceeds the puchase price if the commodity price has declined by the maturity date.
11(b).i. A dual currency bond is a debt instrument that has coupons denominated in a different currency than its principal amount. These bonds can be viewed as a combination of two simpler financial instruments: (1) a single-currency fixed-coupon bond in the home currency, and (2) a forward contracct to exchange the bond’s principal into a predetermined amount of a foreign currency. Dual currency bonds are generally sold to investors who are willing to “take a view” over the longer term in the foreign exchange market.
11(b).ii. First the effective forward exchange rate could be valued appropriately through the price premium, so the bond is properly priced. For a dual currency bond to be properly priced, the nominal forward exchange rate implied by the promised exchange of principalat maturity is actually favorable, and the price premium brings the effective forward exchange rate to a fairly priced level.
Alternatively, the investor might be “paying up” for the convenience and the desirable foreign currency exposure that cannot be acquired in any other way. That is, the USD fixed income investor is willing to pay a premium to acquire a forward exposure in CHF, taking a longer term view that the CHF will appreciate against the USD.
11(b).iii. If the bond’s coupon pays in USD and the principal pays in CHF. Then the annual coupon payment will remain unchanged over the life of the bond. The principal payment due at maturity (in CHF) would have been fixed by the forward contract rate at the time the dual currency bond was purchased, but the principal payment revalued in USD terms will be higher to the USD investor because of the strengthening of the CHF against the USD. In other words, the CHF principal payment due at maturity, expressed in USD, will be higher because of the CHF appreciation.
Answers to Problems
1. From Exhibit 24.4, we find that the fixed rate receiver will get the bid rate for the two year swap, viz., 3.34%.
Date Fixed- LIBOR Fixed-rate Floating-rate
Rate Receipt Payment
2/19/02 3.34% 3.60% – –
8/19/02 3.34% 3.90% $375,750 $407,250
2/19/03 3.34% 4.45% $375,750 $448,500
8/19/03 3.34% 4.60% $375,750 $503,406
2/19/04 3.34% 4.20% $375,750 $529,000
8/19/04 3.34% 3.95% $375,750 $477,750
2/19/05 3.34% 4.05% $375,750 $454,250
The values for the receipts and payments are found using:
Fixed-rate receipt = Swap fixed rate x (180/360) x Notional Principal
Floating-rate payment = reference ratet-1 x (#days/360) x Notional Principal
The number of days is given in Exhibit 24.6.
2(a). Company W will want to enter into a receive-fixed pay-floating swap with semiannual payments for 5 years with a notional principal of USD 35M. The current quote on this swap is 4.82%. Company X will want to enter into a pay-fixed receive- floating swap with semi-annual payments for 4 years with a notional principal of USD 50M. The current quote on this swap is 4.51%.
2(b). The two swaps are not matched perfectly. The first mismatch is the size of the notional principal. Since Company W’s swap is for USD 15M less than Company X’s the dealer is exposed to a USD 15M receive-fixed swap for the first four years. The second mismatch is maturity. Company W’s swap is one year longer, so after four years, the dealer will be exposed to a USD 35M pay-fixed swap.
3. The key is to recognize that the combination of buying the cap and writing a floor at the same strike rate generates the same settlement cash flows as a pay fixed swap. The fixed rate on the swap would equal the strike rate on the cap and floor.
Consider first the 8 percent cap-floor combination. The treasurer could buy the cap for 413 basis points (the market maker’s offer) and sell the floor for 401 basis points (the market maker’s bid). The net is an up-front outflow of 12 basis points (times the notional principal). Because the 8 percent pay-fixed swap would not entail an initial payment, the 8 percent cap floor combination can be rejected.
Consider next the 7 percent cap floor combination. The treasurer could buy the cap for 597 basis points and sell the floor for 320 basis points, resulting in a net up front outflow of 277 basis points. The fixed rate on the synthetic swap would be 7 percent however, not 8 percent. The attraction of the cap-floor alternative turns on the trade-off of a present value of 277 basis points versus a three-year annuity of 100 basis points (actually a 12-period annuity of 25 basis points per quarterly period). Using the three year fixed rate of .20 percent, the present value of the savings is 263.57 basis points; that is,
t=1 (1+ .00820/4)t
Because the 263.57 basis points are less than the up-front cost of 277 basis points, the 7 percent cap floor combination can be rejected as well.
Consider finally the 9 percent cap-floor combination. The treasurer could buy the cap for 220 basis points and sell the floor for 502 basis points, resulting in a net up-front inflow of 282 basis points. The fixed rate on the synthetic swap would be 9 percent. Because the initial receipt exceeds the present value of the higher swap coupon (i.e. 282 > 263.57), this combination should be considered. Is it definitely better? Perhaps so in terms of cash flow and the time value of money, but the treasurer would also have to consider the tax and accounting treatment of the difference in the options premiums to confirm the benefit.
4(a). Issue the Traditional FRN and enter one pay fixed swap:
[(LIBOR + 0.10%) x (365/360)] + [6.38% LIBOR x 365/360)]
= 6.38% + (0.10% x 365/360) = 6.3814%
Issue the Bull Floater and enter one receive-fixed swap:
[(12.75% – LIBOR) x (365/360)] + [(LIBOR x 365/360) – 6.34%)]
= (12.75% x 365/360) – 6.34% = 12.9271% – 6.34% = 6.5871%
Issue the Bear Floater and enter two pay fixed swaps (or one with twice the notional principal):
[(2 x LIBOR 6.40%) x (365/360)] + [2 x 6.38% (LIBOR x 365/360)}]
= -6.4889% + 12.74% = 6.2511%
4(b). Note that the Bull/swap combination is fixed only for LIBOR < 12.75%, since the coupon on the Bull Floater would be zero if LIBOR >12.75% while the net settlement payout on the swap increases. The Bear/swap combination is fixed only for LIBOR > 3.20%. If LIBOR < 3.20%, the coupon of the Bear Floater remains zero while the net settlement payout on the swap increases. 4(c). One would need caps and floors to offset the risk if LIBOR rises above 12.75% or falls below 3.20%. 4(d). The bear floater has the lowest total cost. 4(e). Investors in a Bear Floater are “bearish” on bonds, expecting bond prices to be falling. So, as market rates (and LIBOR) rise, the coupon rate on the Bear Floater rises even faster, in fact, at twice the rate. An investor who buys the Bear Floater as a hedge must be exposed to higher levels for LIBOR. Event Exposure Hedge (Buy Bear floater) LIBOR rises Lose Gain LIBOR falls Gain Lose This could occur, for instance, if the investor had long-term, fixed-rate assets funded by short-term or variable-rate liabilities. The duration gap would be positive. 5. First of all, recognize that at the current exchange rates, USD 25 million translates in JPY 3,186,750,000 (the product of USD 25,000,000 and JPY 127.47/USD) and GBP 12,736,264 (USD 25,000 divided by USD 1.82/GBP). These become the principal amounts governing the transaction. Then: 5(a). The desired swap could be accomplished by combining the following transactions: • “pay 4.92% Japanese yen, receive US dollar LIBOR” currency swap • “receive 9.83% British sterling, pay US dollar LIBOR” currency swap, resulting in a “pay 4.92% yen, receive 9.83% sterling,” which represents your offer swap rate in yen and your bid rate in sterling. 5(b). After swapping principal amounts at the origination date, the cash flow exchanges on first settlement date from the counterparty’s standpoint are calculated as follows: Yen payment = (0.0492) x (183/365) x (JPY 3,186,750,000) = JPY 78,608,828 and Pound receipt = (0.0983) x (183/365) x (GBP 13,736,264) = GBP 686,390 The full set of cash exchanges, which will be known at the inception of the deal, is: Settlement Date Days in Payments Receipts Settlement (Millions) Millions) Initial GBP 13.736 JPY3,186.75 1 183 JPY 78.61 GBP 0.69 2 182 JPY 78.18 GBP 0.68 3 183 JPY 78.61 GBP 0.69 4 182 JPY 78.18 GBP 0.68 5 183 JPY 78.61 GBP 0.69 Final 182 JPY 3,186.75 GBP 13.736 and JPY 78.18 and GBP 0.68 6. CFA Examination (2001) 6(a). World would assume both counterparty risk and currency risk. Counterparty risk is the risk that Bishop’s counterparty will default on payment of principal or interest cash flows in the swap. Currency risk is the currency exposure risk associated with all cash flows. If the US$ appreciates (Euro depreciates), there would be a loss on funding of the coupon payments; however, if the US$ depreciates, then the dollars will be worth less at the swap’s maturity. 6(b). YEAR 0 YEAR 1 YEAR 2 YEAR 3 World Pays Notional Principal $3 billion € 3.33 billion Interest Payment € 193.14 million1 € 193.14 million € 193.14 million World Receives Notional Principal € 3.33 billion $3 billion Interest Payment $219 million2 $219 million $219 million 1 € 193.14 million = € 3.33 billion x 5.8% 2 $219 million = $3 billion x 7.3% 6(c). World would not reduce its borrowing cost, because what Bishop saves in the Euro market, she loses in the dollar market. The interest rate on the Euro pay side of her swap is 5.80 percent, lower than the 6.25 percent she would pay on her Euro debt issue, an interest savings of 45 bps. But Bishop is only receiving 7.30 percent in U.S. dollars to pay on her 7.75 percent U.S. debt interest payent, and interest shortfall of 45 bps. Given a constant currency exchange rate, the 45 bps shortfall exactly offsets the savings from paying 5.80 percent versus the 6.25 percent. Thus there is no interest cost savings by selling the U.S. dollar debt issue and entering into the swap arrangement. 7(a). With these estimates, the settlement payments can be calculated as follows: March 2: Floating rate payment = (0.0350 0.0010) x ($50,000,000) x (90/360) = $425.000 Equity index receipt =[(477.51 463.11)/463.111 x ($50,000,000) = $1,554,706 So the net receipt the fund expects would be ($1,554,706 $425,000) = $1,129,706 June 2: Floating rate payment = (0.0325 0.0010) x ($50,000,000) x (92/360) = $402,500 Equity index receipt = [(464.74 477.51)/477.51] x ($50,000,000) = $1,337,145 So the net payment the fund owes would be ($1,337,145 + $402,500) = $1,739,645 September 2: Floating rate payment = (0.0375 0.0010) x ($50,000,000) x (92/360) = $466.389 Equity index receipt =[(480.86 464.74)/464.74] x ($50,000,000) = $1,734,303 So the net receipt the fund expects would be ($1,734,303 $466,389) = $1,267,914 December 2: Floating rate payment = (0.0400 0.0010) x ($50,000,000) x (91/360) = $492,917 Equity index receipt = [(482.59 480.86)/480.86] x ($50,000,000) = $179,886 So the net payment the fund owes would be ($492,917 $179,886) = $313,031 7(b). It is also quite common for equity swaps to be based on a notional principal amount that varies directly with the level of the underlying index. If, for instance, the swap participants had agreed to let the initial notional principal of $50 million vary over time, it would have been adjusted up on March 2 to $51.555 million. This adjustment is calculated as [$50 million (1 + [(477.51 463.11)/463.11]). That is, each settlement date, the notional principal is adjusted up (down) by the percentage of capital appreciation (depreciation) in the starting level of the index. This adjustment process, which is equivalent to adding the gross equity settlement payment to the initial notional principal, simulates the return that investors with direct stock positions would obtain inasmuch as their actual equity exposure would rise or fall with market conditions. In contrast, a fixed notional principal in an equity swap is equivalent to an asset allocation strategy by which the equity exposure is kept constant. 8. CFA Examination III (1996) 8(a). To achieve Ames’ goal, HSF would enter into a two year swap agreement as the fixed rate payer. In return for agreeing to make quarterly cash flow payments based on the two year swap fixed rate of 5.5%, HSF would receive a cash flow based on the three month LIBOR that prevailed at the beginning of the settlement period. Each quarterly swap settlement payment would be made on a ret basis, the direction of which would depend on the difference between that period’s LIBOR and 5.5%. There would be no exchange of (notional) principal. 8(b). If the candidate uses the interest rate information provided in the question, the amount of the cash flow would be the difference in the fixed rate and the Eurodollar rate multiplied times the notional amount times the fraction of annual time period. The calculation would be (.055 – .051) x $ 10,000,000 x 1/4 = $10,000. The direction: the endowment pays this cash flow at the end of the first period. [NOTE: Candidates may use different practices in calculating the holding period of the swap (e.g., 92 days/360 or 365 days) which may produce slightly different solutions.] [IMPORTANT NOTE: Technically, insufficient information was provided in market convention of determining LIBOR at the start of the settlement period but paying the cash flow at the end. Thus, the September 30, 1996 settlement payment will be based on the three month spot LIBOR that prevailed on June 30, 1996. The September 1996 Eurodollar futures rate of 5.1% (as established on June 30, 1996) will be floating rate for the second swap settlement payment, which will be made on December 30, 1996. Since the problem does not provide the current (i.e., June 30, 1996) three month LIBOR, there is no way to answer the question correctly as it is written.] (c). A pay fixed swap can be viewed as either: (1) lengthening the duration of a floating rate liability, or (2) shortening the duration of a fixed rate asset. In HFS’s case, the second interpretation is appropriate. By reducing asset duration, HFS has moved to a more defensive posture against the anticipated increase in short term rates. Said differently, by converting its fixed rate asset into a synthetic floating rate note, HFS is now in a position to receive larger coupon cash flows if Ames’ interest rate forecast is correct. 8(d). A plain vanilla interest rate swap is an agreement that requires two counterparties to exchange cash flows on a periodic basis, with one of these cash flows based on a fixed interest rate and the other tied to a variable (i.e., floating) reference rate index. In general, the cash flow for either side of the swap on a given settlement date is calculated as [(Rate) x (Notional Principal) x (% of Year covered by Settlement Period)]. For two risk neutral counterparties to agree to such an exchange without requiring (or being willing to make) a front end premium payment, the sum of the discounted expected values of these cash flow streams must be equal. Although all but the first floating rate cash flow settlement payment on the swap will not be known at the time the agreement is originated, they can be locked in using a strip of Eurodollar futures contracts. For a two year, quarterly settlement swap negotiated in June 1996, which has a total of eight settlement dates, the following sequence of LIBOR can be established: Payment #1: three month spot LBOR in June 1996; Payment #2: September 1996 Eurodollar futures rate, as established in June 1996; Payment #3: December 1996 Eurodollar futures rate, as established in June 1996; etc. Thus, the floating rate side of the swap contract can be fully hedged in June 1996 with Eurodollar futures positions extending out to the March 1998 contract. With these future LIBOR levels established, the swap fixed rate can then be calculated as the time weighted average of this sequence. That is, the fixed rate can be thought of as the yield that generates an annuity equal (in present value terms) to the variable cash flows associated with the future LIBOR payments tied to the strip of Eurodollar futures contracts. (By way of an analogy, the swap fixed rate is to the strip of forward LIBOR what a bond’s yield to maturity is to the underlying term structure of zero coupon discount rates.) 8(e). HFS’s pay fixed swap position (with a fixed rate of 5.5%) could be replicated with an interest rate collar by: (1) purchasing an interest rate cap, which can be viewed as a portfolio of European style call options on LIBOR, at a strike rate of 5.5%; and (2) selling an interest rate floor, which is a portfolio of European-style put options on LIBOR, at the same strike rate. If, on a given settlement date, LIBOR is above 5.5%, HFS will receive a settlement check from the cap seller. Conversely, if LIBOR is less than 5%, HFS will have to make a settlement payment to the floor buyer. If the cap sell/floor buyer is the same company, these arc the identical settlement exchanges that would occur on a swap with a fixed rate of 5.5%. In general any “long cap, short floor” combination (i.e., collar) done with the same strike rate, maturity date, and settlement terms will recreate a pay-fixed swap under those same conditions. However, only when that common strike rate is equal to the prevailing swap fixed rate will the collar have no net front-end cost to either counterparty. This result is the swap equivalent of the put-call parity condition. 9. The Spanish pension fund would buy El Oso Grande, enter three swaps (each for a notional principal equal to the par value of the FRN or, equivalently, one swap with three times the notional value) to receive the fixed rate of 13.35 percent and write three floors at a strike rate of 8 percent (or, again, one with three times the notional amount). The role of the floors is to create a fixed rate asset even when LIBOR is less than 8 percent. Neglecting the floors, if LIBOR > 8 percent, the return will be (3 x LIBOR 24 percent) + (3 x 13.35 percent 3 x LIBOR). The LIBOR flows net out by design, and the return is 16.05 percent. If LIBOR < 8 percent, the return will be (3 x 13.35 percent 3 x LIBOR) = 16.05 percent. Writing the floors in effect sells off the potential for a higher return if LIBOR is low. The receipt of 420 basis points (3 x 140 basis points for each floor) raises the fixed rate of return above 16.05 percent.
The net cost of the position is only 95.8 percent of par, the cost of the FRN less the premium received on writing the floors. So the all in, semiannual fixed rate of return is 17.34 percent, calculated as
10 8.025 100
95.80 = +
t=1 (1 + y/2)t (1 + y/2)10
Payoff to EOG is,
Coupon = max(0, 3*LIBOR – 24%)
So the bondholder’s exposure is the equivalent of long 3 floaters and long a floor. We can rewrite the payoff as
Coupon = max[0, 3*(LIBOR – 8%)]
Coupon = 3*max[0, LIBOR – 8%]
So to convert this to straight fixed, the fund would need to undertake 3 receive fixed swaps and sell 3 floors at 8%.
This gives a new coupon of (where AFP = amortized floor premium)
New Coupon= 3*max[0, LIBOR-8%] + 3*13.35 – 3*LIBOR + 3* min[0,LIBOR-8%]+AFP
New Coupon= 3*[LIBOR 8%] + 3*13.35 3*LIBOR + AFP
New Coupon= 3*LIBOR – 24% + 3*13.35 3*LIBOR + AFP
New Coupon= 16.05% + AFP
PVIFA(AFP,13.5%) = 140
AFP = 40bp
New Coupon = 16.05% + .40% = 16.45%
10(a). Conversion value = 48.852 shares x 12.125 = $592.33
The conversion option embedded is this bond is currently out of the money since the conversion value is below the current market price of the bond.
10(b). Conversion parity price = Bond price/conversion ratio = 965/48.852 = $19.75
Bond Price – Conversion Value
10(c). Payback =
Bond Income – Income from equal investment in common stock
= (965 – 592.33)/(76.25 – 0) = 372.67/76.25 = 4.88 years
Sold the bond for $965.00 and used the proceeds to purchase 79.588 shares
(=$965.00/$12.125) of Bildon Enterprise stocks, the payback period would be 4.88 years.
14 38.125 1000
10(d). $965.00 = +
t=1 (1 + y/2)t (1 + y/2)14
Solving for y = 4.147% (semiannual) or 8.29% (annual)
14 38.125 1000
. $917.61 = +
t=1 (1 + .04625)t (1 + .04625)14
This means that the net value of the combined option is $47.39, $965.00 minus $917.61.
11. CFA Examination II (2001)
11(a)i. Conversion value of a convertible bond is the value of the security if it is converted immediately. That is:
= market price of the common stock x conversion ratio
= $40 x 22
11(a)ii. Market conversion price is the price that an investor effectively pays for the common stock if the convertible bond is purchased.
= market price of the convertible bond/conversion ratio
= $47.7273 ≈ $47.73
11(a)iii. Premium payback is the period of time that it takes the investor to recover the premium paid for the convertible bond Because the investor generally receives higher coupon interest from the convertible bond than would be received from the common stock dividends (based on the number of shares equal to the conversion ratio, the period of time to recover the premium can be determined.
Premium payback period = conversion premium per share/income differential per share
Conversion premium/share = $47.7273 – $40 = $7.7273
Income differential/share = ($1,000 x 6.5%)/22 – $1.20 = $2.9545 – $1.20 = $1.7545
Premium payback period = $7.7273/$1.7545 = 4.4043 ≈ 4.40 years
11(b). The value of the convertible bond = value of the straight bond
+ value of conversion option
– value of call option on the bond
Change Determine whether the value will Increase, Decrease or remain Unchanged
Justify your response with one reason
An increase in stock price volatility
Increase The conversion option on the stock becomes more valuable
An increase in interest rate volatility
Decrease The chance of the bond being called increases, causing the value of the call option on the bond to become more valuable.
12(a). SEK issued a two-year silver-linked note. The bull tranche has its principal redemption amount increase directly with silver price movement. The investor is able to purchase a fixed-income security that also allows for participation in silver price movements. In exchange for accepting a lower-than-market coupon, buyers of the bull tranche will receive a redemption value that exceeds their purchase price if the gold index increases.
12(b). Bull Redemption:
USD 1,000 + [(spot silver price per ounce – USD 4.46) x USD 224.21525)]
12(b)(i). If spot silver = USD 4.96 per ounce
= 1000 + (4.96 – 4.56)(224.21525) = 1000 + 89.69 = $1,089.69
12(b)(ii). If spot silver = USD 3.96 per ounce
= 1000 + (3.96 – 4.56)(224.21525) = 1000 + (-134.53) = $865.47
12(c). Since SEK is effectively short gold across the bull segment, to hedge the position SEK would long gold in the futures market.
SEK If silver = USD 4.35 per ounce
= 1000 + (4.35 – 4.56)(224.21525) = 1000 + (-47.09) = $953.91
1001.25 = 65/(1 + y)t + 952.91/(1 + y)2
t = 1
y = 4.42%
13(a). The bond’s cash flows can be broken down depending on two mutually exclusive possibilities; either the NYSE index exceeds 166 after three years or it does not:
NYSE Index CF1 CF2 CF3
< or = 166 3 3 103+0 > 166 3 3 103 + [100 x (NY3 – 166)/166]
Thus, you receive the same cash flows as a straight bond which pays a 3% coupon and is redeemable at par plus the additional cash flows associated with 0.60241 (=100/166) units of a call option on the NYSE index which pays a dollar for each point the index exceeds 166 at maturity.
13(b). A regular bond without the embedded index option would have to sell for 71.65%, or:
P = [3/(1.0765)]+ [3/(1.0765)2 ] + [103/(1.0765)3] = $87.94
Guinness was able to sell the SPEL for 100.625, meaning that the value of the option feature must satisfy:
100.625 = 87.94 + (Bond’s Option Value)
so that the bond’s option feature is worth $12.685. (Recall that this represents only about 60% of a regular NYSE index call option struck at 166.) The bond’s option value is purely a time premium since the option is currently out of the money (i.e., the index value of 134 is less than the exercise price of 166).
d1 = ln(35/50) + [.052 + (.342/2)] (5) = -0.3567 + [.052 +.0578](5) = 0.1923/.7603
.34 (5).5 .7603
d2 =.25 .34 (5).5 = -.51
c(t) = 35(.5987)- 50[e-.052(5)(.3050)] = 20.9545- 50[.7711)(.3050)]
= 20.9545 – 11.7593 = 9.195
q = 10,000/100,000= 0.1
w(t) = 9195/1.1 = 8.36
14(b). Since the stock price before exercise must be $5,200,000/100,000 = $52, then all warrants will be exercised. Then the warrant holder will purchase 10,000 shares at $50 per share, thus injecting $500,000 of new capital into the firm. Then the firm will be worth $5,700,000/10,000 = $51.82.
14(c). The warrant was originally worth $8.36. At expiration, the warrant was only 2 points in-the-money and with the dilution effect, it was worth only 2/1.1 = 1.82. This is a percentage decrease in value of (1.82/8.36) – 1 = -0.7823. Meanwhile the stock price increased from $35 to $52, an increase of $8.57. Obviously, the warrant lost considerable value even though the stock price gained significantly in value. The reason is that when the warrant was purchased all of its price was its time value (it was not in-the-money). During the life of any warrant or option, it will lose all of its time value. The large time value reflects the volatility and long time to expiration. Ultimately at expiration, the warrant finished barely in-the-money. The warrant holder gained a little in intrinsic value but lost so much time value because the stock price, in spite of its large percentage move, still did not change by enough over the warrant’s life.
15(a). The payoff to the mine is shown as:
Price Cash Flow
$300 $4,000,000 (= (300 – 260)x100,000)
Spend $1 million
$230 $0 – No production
The profit is either $300 – $268.40 = $32.60 or $240 – $268.40 = -$38.40. The difference between the payoffs is $70 per ounce, and the difference in cash flows is $4,000,000. Therefore the number of 1-ounce forward contracts needed is:
$4,000,000/$70 = 57,143 contracts (rounded up).
If the price of gold falls to $230 next year, the loss on the forwards would be
-$38.40×57,143 = -$2,285,720.
To offset this loss, one would need to invest $2,285,720/1.05 = $2,176,876 in T-bills. This is the value of the replicating portfolio (and the lease) and is in excess of the cost of opening the mine.
15(b). The situation is similar to the one in part (a), except the cash flow from production at a price of $280 would be $2,000,000, and the profit or loss from the gold forward contracts would be $12.60 and -$18.40, respectively.
The number of forward contracts would therefore be
$2,000,000/$30 = 66,667 contracts (rounded up).
If the gold price were to fall to $250 next year, the loss on the forwards would be
-$18.40×66,667 = -$1,226,672.
To offset this loss, one would need to invest $1,226,672/1.05 = $1,168,260 in T-bills. Again, this would be the value of the replicating portfolio and the value of the lease, and is again in excess of the cost of reopening the mine.
15(c). Increased volatility in the, price of gold increases the value of the lease, exactly as increased volatility would increase the value of an option.