Reading 26 Risk Management Applications of Forward and Futures Strategies

Question 1

General formula for FRA payoff

Answer 1

Question 2

How can a borrower lock in the rate that will be set at a future date on a single-payment loan?

Answer 2

A borrower can lock in the rate that will be set at a future date on a single-payment loan by entering into a long position in an FRA.

The FRA obligates the borrower to make a fixed interest payment and receive a floating interest payment, thereby protecting the borrower if the loan rate is higher than the fixed rate in the FRA but also eliminating gains if the loan rate is lower than the fixed rate in the FRA.

Question 3

Duration of a bond futures contract is?

Answer 3

The duration of a bond futures contract is determined as the duration of the bond underlying the futures contract as of the futures expiration, based on the yield of the bond underlying the futures contract.

The modified duration is obtained by dividing the duration by 1 plus the yield. The duration of a futures contract is implied by these factors and is called the implied (modified) duration.

Question 4

Implied yield of a futures contract is?

Answer 4

The implied yield of a futures contract is the yield implied by the futures price on the bond underlying the futures contract as of the futures expiration.

Question 5

Yield beta?

Answer 5

The yield beta is the sensitivity of the yield on a bond portfolio relative to the implied yield on the futures contract.

Question 6

The number of bond futures contracts required to change the duration of a bond portfolio?

Answer 6

The number of bond futures contracts required to change the duration of a bond portfolio is based on the ratio of the market value of the bonds to the futures price multiplied by the difference between the target or desired modified duration and the actual modified duration, divided by the implied modified duration of the futures.

The number of bond futures, denoted as N_bf, will be

N_bf = [(MDUR_T−MDUR_B)_/MDUR_f] * B/f_b

where MDUR_T is the target modified duration, MDUR_B is the modified duration of the existing bonds, MDUR_f is the implied modified duration of the futures

Question 7

The actual adjusted duration of a bond portfolio vs. the desired duration?

Answer 7

The actual adjusted duration of a bond portfolio may not equal the desired duration for a number of reasons, including that the yield beta may be inaccurate or unstable or the bonds could contain call features or default risk.

In addition, duration is a measure of instantaneous risk and may not accurately capture the risk over a long horizon without frequent portfolio adjustments.

Question 8

Beta, formula

Answer 8

β=cov_SI/σ²_I

where cov_SI is the covariance between the stock portfolio and the index and σ²_I is the variance of the index.

Question 9

If we wish to change the beta, we specify the desired beta as a target beta of β_T using futures?

Answer 9

N_f = [(β_T−β_S)/β_f]*(S/f)

• if we want to increase the beta, β_T will exceed β_S and the sign of N_f will be positive, which means that we must buy futures. If we want to decrease the beta, β_T will be less than β_S, the sign of N_f will be negative, and we must sell futures.
• need to remember that the futures contract will hedge only the risk associated with the relationship between the portfolio and the index on which the futures contract is based.
• recall also that dividends can interfere with how this transaction performs. Index futures typically are based only on price indices; they do not reflect the payment and reinvestment of dividends. Therefore, dividends will accrue on the stocks but are not reflected in the index. This is not a major problem, however, because dividends in the short-term period covered by most contracts are not particularly risky.

Question 10

Equity Analysts Inc. (EQA) is an equity portfolio management firm. One of its clients has decided to be more aggressive for a short period of time. It would like EQA to move the beta on its $65 million portfolio from 0.85 to 1.05. EQA can use a futures contract priced at $188,500, which has a beta of 0.92, to implement this change in risk.

• A. Determine the number of futures contracts EQA should use and whether it should buy or sell futures.
• B. At the horizon date, the equity market is down 2 percent. The stock portfolio falls 1.65 percent, and the futures price falls to $185,000. Determine the overall value of the position and the effective beta.

Answer 10

• Solution to A:

The number of futures contracts EQA should use is

N_f=((1.05−0.85)/0.92)($65,000,000/$188,500)=74.96

So EQA should buy 75 contracts.

• Solution to B:

The value of the stock portfolio will be $65,000,000(1 – 0.0165) = $63,927,500. The profit on the futures transaction is 75($185,000 – $188,500) = –$262,500. The overall value of the position is $63,927,500 – $262,500 = $63,665,000.

Thus, the overall return is $63,665,000/$65,000,000−1=−0.0205

Because the market went down by 2 percent, the effective beta is 0.0205/0.02 = 1.025.

Question 11

Creating Equity out of Cash, general approach

Answer 11

In simple terms, we say that:

Long stock + Short futures = Long risk-free bond

We can turn this equation around to obtain:

Long stock = Long risk-free bond + Long futures

This synthetic replication of the underlying asset can be a very useful transaction when we wish to construct a synthetic stock index fund, or when we wish to convert into equity a cash position that we are required to maintain for liquidity purposes

Question 12

Creating a Synthetic Index Fund, illustrate

Answer 12

To create this synthetic index fund, we must buy a certain number of futures. Let the following be the appropriate values of the inputs:

V = amount of money to be invested, £100 million

f = futures price, £4,000

T = time to expiration of futures, 0.25

δ = dividend yield on the index, 0.025

r = risk-free rate, 0.05

q = multiplier, £10

We would like to replicate owning the stock and reinvesting the dividends. How many futures contracts would we need to buy and add to a long bond position? We designate N_f as the required number of futures contracts and N_f* as its rounded-off value.

Now observe that the payoff of N_f* futures contracts will be N_f*q(S_T – f). This equation is based on the fact that we have N_f* futures contracts, each of which has a multiplier of q. The futures contracts are established at a price of f. When it expires, the futures price will be the spot price, S_T, reflecting the convergence of the futures price at expiration to the spot price.

The futures payoff can be rewritten as N_f*qS_T – N_f*qf. The minus sign on the second term means that we shall have to pay N_f*qf. The (implied) plus sign on the first term means that we shall receive N_f*qS_T. Knowing that we buy N_f* futures contracts, we also want to know how much to invest in bonds. We shall call this V* and calculate it based on N_f*. Below we shall show how to calculate N_f* and V*. If we invest enough money in bonds to accumulate a value of N_f*qf, this investment will cover the amount we agree to pay for the FTSE: N_f* × q × f. The present value of this amount is N_f*qf/(1 + r)^T.

Because the amount of money we start with is V, we should have V equal to N_f*qf/(1 + r)^T. From here we can solve for N_f* to obtain

N_f*=V(1+r)^T/(qf )    (rounded to an integer)

But once we round off the number of futures, we do not truly have V dollars invested. The amount we actually have invested is

V*=N_f*qf/(1+r)^T

We can show that investing V* in bonds and buying N_f* futures contracts at a price of f is equivalent to buying N_f*q/(1 + δ)^T units of stock.

• All this transaction does is capture the performance of the index. Because the index is a price index only and does not include dividends, this synthetic replication strategy can capture only the index performance without the dividends.
• Another concern that could be encountered in practice is that the futures contract could expire later than the desired date. If so, the strategy will still be successful if the futures contract is correctly priced when the strategy is completed. Consistent with that point, we should note that any strategy using futures will be effective only to the extent that the futures contract is correctly priced when the position is opened and also when it is closed. This point underscores the importance of understanding the pricing of futures contracts.

Question 13

Equitizing Cash

Answer 13

• take a given amount of cash and turn it into an equity position while maintaining the liquidity provided by the cash. This type of transaction is sometimes called equitizing cash
• There is one important aspect of this problem, however, over which the fund has no control: the pricing of the futures. Because the fund will take a long position in futures, the futures contract must be correctly priced. If the futures contract is overpriced, the fund will pay too much for the futures. In that case, the risk-free bonds will not be enough to offset the excessively high price effectively paid for the stock. If, however, the futures contract is underpriced, the fund will get a bargain and will come out much better.

Question 14

Economic exposure (risk)

Answer 14

Economic exposure is the loss of sales that a domestic exporter might experience if the domestic currency appreciates relative to a foreign currency.

Question 15

Credit risk VAR

Answer 15

Credit risk increases as the value of the position increases.

Since credit risk increases when the value of the position held increases, we should focus on the upper not lower tail of the distribution of gains on positions held when using VAR to evaluate credit risk.

Question 16

Index Advantage (INDEXA) is a money management firm that specializes in turning the idle cash of clients into equity index positions at very low cost. INDEXA has a new client with about $500 million of cash that it would like to invest in the small-cap equity sector. INDEXA will construct the position using a futures contract on a small-cap index. The futures price is 1,500, the multiplier is $100, and the contract expires in six months. The underlying small-cap index has a dividend yield of 1 percent. The risk-free rate is 3 percent per year.

• A. Determine exactly how the cash can be equitized using futures contracts.
• B. When the futures contract expires, the index is at ST. Demonstrate how the position produces the same outcome as an actual investment in the index.

Answer 16

• Solution to A:

INDEXA should purchase

N_f=$500,000,000(1.03)^0.5/($100*1,500)=3,382.96

futures contracts. Round this amount to N_f* = 3,383. Then invest

(3,383*$100*1,500)/(1.03)^0.5=$500,005,342

in risk-free bonds paying 3 percent interest. Note that this is not exactly an initial investment of $500 million, because one cannot purchase fractions of futures contracts. The bonds will grow to a value of $500,005,342(1.03)^0.5 = $507,450,000. The number of units of stock effectively purchased through the use of futures is

N_f*q/(1+δ)^T=(3,383*100)(/1.01)^0.5=336,621.08

If 336,621.08 shares were actually purchased, the accumulation and reinvestment of dividends would result in there being 336,621.08 (1.01)^0.5 = 338,300 shares at the futures expiration.

• Solution to B:

At expiration, the payoff on the futures is

3,383(100)(S_T – 1500) = 338,300S_T – $507,450,000

In other words, to settle the futures, INDEXA will owe $507,450,000 and receive the equivalent of 338,300 units of stock worth S_T.

Question 17

Synthetics Inc. (SYNINC) executes a variety of synthetic strategies for pension funds. One such strategy is to enable the client to maintain a liquid balance in cash while retaining exposure to equity market movements. A similar strategy is to enable the client to maintain its position in the market but temporarily convert it to cash. A client with a $100 million equity position wants to convert it to cash for three months. An equity market futures contract is priced at $325,000, expires in three months, and is based on an underlying index with a dividend yield of 2 percent. The risk-free rate is 3.5 percent.

• A. Determine the number of futures contracts SYNINC should trade and the effective amount of money it has invested in risk-free bonds to achieve this objective.
• B. When the futures contracts expire, the equity index is at S_T. Show how this transaction results in the appropriate outcome.

Answer 17

• Solution to A:

First note that no multiplier is quoted here. The futures price of $325,000 is equivalent to a quoted price of $325,000 and a multiplier of 1.0. The number of futures contracts is

N_f=−$100,000,000(1.035)^0.25/$325,000=−310.35

Rounding off, SYNINC should sell 310 contracts. This is equivalent to selling futures contracts on stock worth

(310*$325,000)/(1.035)^0.25=$99,887,229

and is the equivalent of investing $99,887,229 in risk-free bonds, which will grow to a value of $99,887,229(1.035)^0.25 = $100,750,000. The number of units of stock being effectively converted to cash is (ignoring the minus sign)

N_f*q/(1+δ)^T=310(1)/(1.02)^0.25=308.47

The accumulation and reinvestment of dividends would make this figure grow to 308.47(1.02)^0.25 = 310 units when the futures expires.

• Solution to B:

At expiration, the profit on the futures is –310(S_T – $325,000) = –310S_T + $100,750,000. That means SYNINC will have to pay 310S_T and will receive $100,750,000 to settle the futures contract. Due to reinvestment of dividends, it will end up with the equivalent of 310 units of stock, which can be sold to cover the amount –310S_T. This will leave $100,750,000, the equivalent of having invested in risk-free bonds.

Question 18

Q-Tech Advisors manages a portfolio consisting of $100 million, allocated 70 percent to stock at a beta of 1.05 and 30 percent to bonds at a modified duration of 5.5. As a tactical strategy, it would like to temporarily adjust the allocation to 60 percent stock and 40 percent bonds. Also, it would like to change the beta on the stock position from 1.05 to 1.00 and the modified duration from 5.5 to 5.0. It will use a stock index futures contract, which is priced at $280,000 and has a beta of 0.98, and a bond futures contract, which is priced at $125,000 and has an implied modified duration of 6.50.

• A. Determine how many stock index and bond futures contracts it should use and whether to go long or short.
• B. At the horizon date, the stock portfolio has fallen by 3 percent and the bonds have risen by 1 percent. The stock index futures price is $272,160, and the bond futures price is $126,500. Determine the market value of the portfolio assuming the transactions specified in Part A are done, and compare it to the market value of the portfolio had the transactions been done in the securities themselves.

Answer 18

• Solution to A:

To reduce the allocation from 70 percent stock ($70 million) and 30 percent bonds ($30 million) to 60 percent stock ($60 million) and 40 percent bonds ($40 million), Q-Tech must synthetically sell $10 million of stock and buy $10 million of bonds. First, assume that Q-Tech will sell $10 million of stock and leave the proceeds in cash. Doing so will require

N_sf=((0−1.05)/0.98)($10,000,000/$280,000)=−38.27

futures contracts. It should sell 38 contracts, which creates synthetic cash of $10 million. To buy $10 million of bonds, Q-Tech should buy

N_bf=((5.50−0.0)/6.50)($10,000,000/$125,000)=67.69

futures contracts, which rounds to 68. This transaction allows Q-Tech to synthetically borrow $10 million (selling a stock futures contract is equivalent to borrowing cash) and buy $10 million of bonds. Because we have created synthetic cash and a synthetic loan, these amounts offset. Thus, at this point, having sold 38 stock index futures and bought 68 bond futures, Q-Tech has effectively sold $10 million of stock and bought $10 million of bonds. It has produced a synthetically re-allocated portfolio of $60 million of stock and $40 million of bonds.

Now it needs to adjust the beta on the $60 million of stock to its target of 1.00. The number of futures contracts would, therefore, be

N_sf=((1.00−1.05)/0.98)($60,000,000/$280,000)=−10.93

So it should sell an additional 11 contracts. In total, it should sell 38 + 11 = 49 contracts.

To adjust the modified duration from 5.50 to its target of 5.00 on the $40 million of bonds, the number of futures contracts is

N_bf=((5−5.50)/6.50)($40,000,000/$125,000)=−24.62

So it should sell 25 contracts. In total, therefore, it should buy 68 – 25 = 43 contracts.

• Solution to B:

The value of the stock will be $70,000,000(1 – 0.03) = $67,900,000.

The profit on the stock index futures will be –49($272,160 – $280,000) = $384,160.

The total value of the stock position is therefore $67,900,000 + $384,160 = $68,284,160.

The value of the bonds will be $30,000,000(1.01) = $30,300,000.

The profit on the bond futures will be 43($126,500 – $125,000) = $64,500.

The total value of the bond position is, therefore, $30,300,000 + $64,500 = $30,364,500.

Therefore, the overall position is worth $68,284,160 + $30,364,500 = $98,648,660.

Had the transactions been done in the securities themselves, the stock would be worth $60,000,000(1 – 0.03) = $58,200,000. The bonds would be worth $40,000,000(1.01) = $40,400,000. The overall value of the portfolio would be $58,200,000 + $40,400,000 = $98,600,000, which is a difference of only $48,660 or 0.05 percent of the original value of the portfolio.

Question 19

Pre-Investing in an Asset Class

Answer 19

Futures contracts do not require a cash outlay but can be used to add exposure. We call this approach pre-investing.

An advisor to a mutual fund would like to pre-invest $10 million in cash that it will receive in three months. It would like to allocate this money to a position of 60 percent stock and 40 percent bonds. It can do this by taking long positions in stock index futures and bond futures. The trick is to establish the position at the appropriate beta and duration.

Long underlying + Loan = Long futures

Question 20

Total Asset Strategies (TAST) specializes in a variety of risk management strategies, one of which is to enable investors to take positions in markets in anticipation of future transactions in securities. One of its popular strategies is to have the client invest when it does not have the money but will be receiving it later. One client interested in this strategy will receive $6 million at a later date but wants to proceed and take a position of $3 million in stock and $3 million in bonds. The desired stock beta is 1.0, and the desired bond duration is 6.2. A stock index futures contract is priced at $195,000 and has a beta of 0.97. A bond futures contract is priced at $110,000 and has an implied modified duration of 6.0.

• A. Find the number of stock and bond futures contracts TAST should trade and whether it should go long or short.
• B. At expiration, the stock has gone down by 5 percent, and the stock index futures price is down to $185,737.50. The bonds are up 2 percent, and the bond futures price is up to $112,090. Determine the value of the portfolio and compare it with what it would have been had the transactions been made in the actual securities

Answer 20

• Solution to A:

The approximate number of stock index futures is

((1.00−0.0)/0.97)($3,000,000/$195,000)=15.86

So TAST should buy 16 contracts. The number of bond futures is

((6.2−0.0)/6.0)($3,000,000/$110,000)=28.18

So it should buy 28 contracts.

• Solution to B:

The profit on the stock index futures is 16($185,737.50 – $195,000) = –$148,200.

The profit on the bond futures is 28($112,090 – $110,000) = $58,520. The total profit is –$148,200 + $58,520 = –$89,680, a loss of $89,680. Suppose TAST had invested directly. The stock would have been worth $3,000,000(1 – 0.05) = $2,850,000, and the bonds would have been worth $3,000,000(1.02) = $3,060,000, for a total value of $2,850,000 + $3,060,000 = $5,910,000, or a loss of $90,000, which is about the same as the loss using only the futures.

Question 21

Foreign Currency Risk

Answer 21

• Transaction exposure - the risk associated with a foreign exchange rate on a specific business transaction such as a purchase or sale.
• Translation exposure - the risk associated with the conversion of foreign financial statements into domestic currency.
• Economic exposure - the risk associated with changes in the relative attractiveness of products and services offered for sale, arising out of the competitive effects of changes in exchange rates

The management of a single cash flow that will have to be converted from one currency to another is generally done using forward contracts. Futures contracts tend to be too standardized to meet the needs of most companies.

Question 22

Royal Tech Ltd. is a UK technology company that has recently acquired a US subsidiary. The subsidiary has an underfunded pension fund, and Royal Tech has absorbed the subsidiary’s employees into its own pension fund, bringing the US subsidiary’s defined-benefit plan up to an adequate level of funding. Soon Royal Tech will be making its first payments to retired employees in the United States. Royal Tech is obligated to pay about $1.5 million to these retirees. It can easily set aside in risk-free bonds the amount of pounds it will need to make the payment, but it is concerned about the foreign currency risk in making the US dollar payment. To manage this risk, Royal Tech is considering using a forward contract that has a contract rate of £0.60 per dollar.

• A. Determine how Royal Tech would eliminate this risk by identifying an appropriate forward transaction. Be sure to specify the notional principal and state whether to go long or short. What domestic transaction should it undertake?
• B. At expiration of the forward contract, the spot exchange rate is ST. Explain what happens.

Answer 22

• Solution to A:

Royal Tech will need to come up with $1,500,000 and is obligated to buy dollars at a later date. It is thus short dollars. To have $1,500,000 secured at the forward contract expiration, Royal Tech would need to go long a forward contract on the dollar. With the forward rate equal to £0.60, the contract will need a notional principal of £900,000. So Royal Tech must set aside funds so that it will have £900,000 available when the forward contract expires. When it delivers the £900,000, it will receive £900,000(1/£0.60) = $1,500,000, where 1/£0.60 ≈ $1.67 is the dollar-per-pound forward rate.

• Solution to B:

At expiration, it will not matter what the spot exchange rate is. Royal Tech will deliver £900,000 and receive $1,500,000.

Question 23

Hedging foreign stock market return and foreign currency risks

Answer 23

• It is not possible to invest in a foreign equity market and precisely hedge the currency risk only. To hedge the currency risk, one must know the exact amount of foreign currency that will be available at a future date. Without locking in the equity return, it is not possible to know how much foreign currency will be available.
• It is possible to hedge the foreign equity market return and accept the exchange rate risk or hedge the foreign equity market return and hedge the exchange rate risk. By hedging the equity market return, one would know the proper amount of currency that would be available at a later date and could use a futures or forward contract to hedge the currency risk. The equity return, however, would equal the risk-free rate.
• If only the foreign stock market return is hedged, the portfolio return is the foreign risk-free rate before converting to the domestic currency. If both the foreign stock market and the exchange rate risk are hedged, the return equals the domestic risk-free rate.

Question 24

FCA Managers (FCAM) is a US asset management firm. Among its asset classes is a portfolio of Swiss stocks worth SF10 million, which has a beta of 1.00. The spot exchange rate is $0.75, the Swiss interest rate is 5 percent, and the US interest rate is 6 percent. Both of these interest rates are compounded in the Libor manner: Rate × (Days/360). These rates are consistent with a six-month forward rate of $0.7537. FCAM is considering hedging the local market return on the portfolio and possibly hedging the exchange rate risk for a six-month period. A futures contract on the Swiss market is priced at SF300,000 and has a beta of 0.90.

• A. What futures position should FCAM take to hedge the Swiss market return? What return could it expect?
• B. Assuming that it hedges the Swiss market return, how could it hedge the exchange rate risk as well, and what return could it expect?

Answer 24

• Solution to A:

To hedge the Swiss local market return, the number of futures contracts is

Nf=((0−1.00)/0.90)(SF10,000,000/SF300,000)=−37.04

So FCAM should sell 37 contracts. Because the portfolio is perfectly hedged, its return should be the Swiss risk-free rate of 5 percent.

• Solution to B:

If hedged, the Swiss portfolio should grow to a value of SF10,000,000[1 + 0.05(180/360)] = SF10,250,000.

FCAM could hedge this amount with a forward contract with this much notional principal. If the portfolio is hedged, it will convert to a value of SF10,250,000($0.7537) = $7,725,425.

In dollars, the portfolio was originally worth SF10,000,000($0.75) = 7,500,000. Thus, the return is $7,725,425/$7,500,000−1≈0.03 , which is the US risk-free rate for six months.

Question 25

the primary differences between futures and forwards?

Answer 25

The primary differences between the two:

1. Futures contracts are standardized, with all terms except for the price set by the futures exchange. Forward contracts are customized. The two parties set the terms according to their needs.
2. Futures contracts are guaranteed by the clearinghouse against default. Forward contracts subject each party to the possibility of default by the other party.
3. Futures contracts require margin deposits and the daily settlement of gains and losses. Forward contracts pay off the full value of the contract at expiration. Some participants in forward contracts agree prior to expiration to use margin deposits and occasional settlements to reduce the default risk.
4. Futures contracts are regulated by federal authorities. Forward contracts are essentially unregulated.
5. Futures contracts are conducted in a public arena, the futures exchange, and are reported to the exchanges and the regulatory authority. Forward contracts are conducted privately, and individual transactions are not generally reported to the public or regulators.

Question 26

Futures vs. forwards when hedging

Answer 26

• Forward contracts are usually preferred over futures contracts when the risk is related to an event on a specific date, such as an interest rate reset.
• Forward contracts on foreign currency are usually preferred over futures contracts, primarily because of the liquidity of the market.
• Futures contracts require margins and daily settlements but are guaranteed against credit losses and may be preferred when credit concerns are an issue.
• Either contract may be preferred or required if there are restrictions on the use of the other. Dealers use both instruments in managing their risk, occasionally preferring one instrument and sometimes preferring the other.
• Forward contracts are preferred if privacy is important.

Question 27

Measuring betas and durations as an obstacle for hedging

Answer 27

Because they can be somewhat unstable, betas and durations are difficult to measure, even under the best of circumstances. Even when no derivatives transactions are undertaken, the values believed to be the betas and durations may not truly turn out to reflect the sensitivities of stocks and bonds to the underlying sources of risk. Therefore, if derivatives transactions do not work out to provide the exact hedging results expected, users should not necessarily blame derivatives.

Question 28

Main issues of using dervatives for hedging

Answer 28

• transaction costs are a major consideration in the use of derivatives
• Transacting in futures and forwards also has a major advantage of being less disruptive to the portfolio and its managers. For example, the asset classes of many portfolios are managed by different persons or firms. If the manager of the overall portfolio wants to change the risk of certain asset classes or alter the allocations between asset classes, he can do so using derivatives.
• In the matter of liquidity, however, futures and forwards do not always offer the advantages often attributed to them. They require less capital to trade than the underlying securities, but they are not immune to liquidity problems.
• Many organizations are not permitted to use futures or forwards. Futures and forwards are fully leveraged positions, because they essentially require no equity.