MANAGING RISK



MANAGING RISK

This handout explores ways in which management can reduce company risk and thereby raise firm value and share value. We begin by explaining how reducing risk can enhance value. This is followed by a discussion of the multitude of risk-management techniques in common practice by mid-sized and large companies in the United States and abroad.

HOW RISK MANAGEMENT CREATES VALUE

The typical large company spends heavily on insurance, forward contracts, swaps, and other risk management techniques in order to reduce (hedge against) uncertainty (risk) associated with firm asset values, cash flows, and taxable income, that is, to reduce total risk. [Note that we are not talking about beta risk. We are referring total risk, or variance (more specifically, coefficient of variation, which is variance divided by mean). The reduction in risk can produce savings that exceed the cost of implementing the risk management process. This can happen because risk management techniques can:

1. Reduce financing costs by lessening reliance on relatively expensive external capital.

2. Reduce the likelihood, and expected cost, of financial distress.

3. Reduce management costs and improve management decisions.

4. Reduce taxes.

We now consider each of the above benefits of hedging.

Hedging to Reduce Financing Costs

Internal financing is the use of cash generated by operations to finance expenses and capital outlays. External financing is the use of funds from borrowing or from the sale of new equity (issuing new shares). Internal financing is cheaper in two important ways. First, there are no fees to investment bankers, accountants, lawyers and others who are involved in accessing the capital markets. Second, unlike external equity finance, internal financing is not interpreted by the markets as a sign that management believes that the firm’s shares are overpriced. This effect explains the share price decline that often accompanies the sale of new shares.

Reducing the uncertainty associated with cash flow from operations (internally generated funds) lessens the probability that the firm will have an unexpected insufficiency of internal funds to finance investment, an insufficiency that would have to be remedied by resorting to expensive external financing. To illustrate, assume that, over the next 5 years, annual net cash flow from operations is expected to be anywhere from $450 million to $650 million and that capital outlays are forecasted to be a stable $500 million annually. If the company were to have several successive years in which cash flows were below $500 million, the firm would have to turn to the external capital markets to finance part of the $500 million capital budget. On the other hand, there would be no need for external financing if the annual cash flow from operations could, through hedging, be constrained to a range having a minimum of $500 million.

Hedging to Avoid Financial Distress Costs

Financial distress is the inability, or imminent inability, to service debts as scheduled. In extreme cases, this results in bankruptcy and takeover of the firm by creditors. Hedging stabilizes asset value and can reduce the probability that asset value will sink so far as to produce financial distress. These costs can take various forms.

Direct financial distress costs - direct expenditures to deal with the financial distress:

• Legal fees

• Accounting fees

• Investment bank fees

• Lenders’ time in seeking payment from borrower

• Management time used in bargaining with lenders

• Disruption of the business as lenders take control (if a change of control occurs)

Indirect financial distress costs:

• Under-management of the firm because management is distracted by the financial distress, and because shareholders’ have less incentive to maintain effective firm management because the company might have to be surrendered to creditors

• Reduced access to external capital

Problems made worse by the presence of financial distress:

• Loss of customers because they believe that the firm will be less likely to survive and therefore less likely to honor warranties, provide services, provide parts, etc.

• Departure of key employees because the firm is more likely to fail

• Worsening of terms offered by suppliers because the firm is more likely to fail (failing business are last in line for goods and services)

Direct and indirect costs of financial distress are in addition to loss due to the poor performance of the business. Furthermore, the financial distress (which arises because of the use of debt) can also intensify the loss of customers, employees and suppliers. To illustrate, suppose that United Bakers has suffered a decline in its business and a series of losses. If United were all equity, it would avoid the added costs of financial distress. But if United has debt, some or all of the above costs may be present. Debt makes a bad situation worse, sometimes much worse.

Hedging to Improve Management Decisions

Greater cash flow predictability facilitates analysis, planning and decision-making. For example, reducing (through hedging) the risk of a proposed project’s future cash flow will usually enable a more accurate estimate of the discount rate to be used in computing the present value of that cash flow. The more accurate discount rate estimate implies greater confidence in the present value figure and the resulting decision as to whether to adopt the project. The outcome is likely to be fewer mistakes in accepting and rejecting projects.

The forward or futures markets can be used to hedge and to improve decisions. A forward or futures contract is an agreement now to buy, or sell, a particular good or service at some date in future. For example, using futures contracts, farmers agree now to deliver wheat in the future for a price that is set now (the “futures price”). Since the currently prevailing wheat futures price reflects the market’s current expectation about future demand for wheat, the use of futures prices can help a farmer make better-informed production decisions (whether or not the farmer actually sells any wheat in the futures markets).

Insurance companies provide a hedge (insurance) and in the process gain specialized expertise in the costs of, and methods of preventing, the various categories of loss-creating events. There are economies of scale in gaining this specialized knowledge. To exploit this comparative advantage, British Petroleum insures against relatively frequent and common events such as on-the-job injuries, fire damage, and natural disasters. BP management believes that insurance companies have a better understanding of the probabilities and magnitudes of these potential losses (including knowledge of the cost of settling lawsuits). On the other hand, BP self-insures against very unusual losses about which it has the better understanding.

Hedging is also used to improve performance evaluation so that effective managers can be properly rewarded. For example, Disney Corporation hedges exchange rate risk and other sources of hedgeable risk in order to isolate those aspects of performance that are produced by management decisions, rather than by forces outside the control of the manager.

Hedging to Reduce Taxes

Gains and losses are usually treated asymmetrically under U.S. tax law. The tax on $100 million of taxable income might be $35 million, whereas the present value of the tax rebate on a $100 million tax-deductible loss might be only $25 million. This asymmetry produces the possibility of an advantage to reducing the uncertainty of taxable income.

Assume Exhibit 1a for Roy, Inc. There is a 50% probability of strong performance and 50% probability of weak performance over the coming year. If Roy does not hedge, its pretax income will be $100 million (strong) or ( $20 million (weak). If Roy hedges, pretax income will be $35 million for certain. We assume here that transaction costs of hedging are $5 million, so expected (average) pretax income is $5 million less with hedging than without hedging ($35 million versus $40 million). Exhibit 1b shows after-tax income. For simplicity, assume a 40 percent tax rate on positive taxable income, but no tax rebate if the firm incurs a loss. Expected (average) after-tax income is $20 million with no hedging and $21 million with hedging. This is so even taking into account the lower expected pretax income with hedging.

Exhibit 1a. Pretax Income for Two Equally Likely Outcomes (in $millions)

| |Strong Performance |Weak Performance |Average |

|Unhedged |$100 | ( $20 |$40 |

|Hedged |$35 |$35 |$35 |

Exhibit 1b. After-tax Income for Two Equally Likely Outcomes (in $millions)a

| |Strong Performance |Weak Performance |Average |

|Unhedged |$60 | ( $20 |$20 |

|Hedged |$21 |$21 |$21 |

a Assume that pre-tax income is taxed at a 40 percent tax rate (Federal, state and local tax combined), and that a loss produces no tax benefit.

WHICH FIRMS ENGAGE IN HEDGING?

Virtually every company, large and small, uses various kinds of insurance to “hedge” against insurable risks. Property and casualty insurance, and life insurance (e.g., to cover key employees), credit insurance, and health insurance for employees are all available at a price.

Non-insurance hedging includes hedging against input and output price fluctuations, financial hedging to reduce uncertainty about the rates on borrowing and lending, and currency hedging to protect against exchange rate changes. The use of hedging through means other than insurance depends on the type of firm. Large firms are more likely than are smaller firms to hedge, both because establishing the risk management function involves large fixed costs, and because sophistication of management practices and firm size are positively correlated. However, of those firms that do hedge, small firms tend to hedge more completely; this is likely because the small firms that hedge tend to be riskier than the large firms that hedge (and so have a greater need to hedge in order to prevent financial distress).

Hedging is also more common among companies with exceptional growth opportunities, particularly those with large R&D budgets. It appears that these companies hedge in order to ensure that they have sufficient internal funds to finance their capital outlays. Furthermore, high growth firms often have high business risk and have assets that are very susceptible to value shrinkage in the event of financial distress. Such companies have high potential financial distress costs and therefore benefit disproportionately from hedging activities that reduce the likelihood of financial distress.

The extent of hedging, and the type of hedging, depends on industry. Companies that produce or purchase basic commodities (oil, metals, and agricultural products) commonly hedge the price risk of those commodities through commodity futures contracts. Issuers and holders of debt instruments hedge interest rate risk through interest rate futures. For example, banks and other lenders that lend long term and borrow short-term hedge interest rate risk. Firms that have international operations (virtually every large company) usually hedge in the foreign exchange markets.

TECHNIQUES FOR RISK-MANAGEMENT

INSURANCE

Insurance is a contractual arrangement with an outside party, almost always an insurance company, who agrees to cover a particular kind of loss in exchange for a fixed payment, or premium. Insurance companies offer two kinds of coverage, life and health insurance and property and liability insurance. Health insurance covers some or all of the medical, prescription drugs, disability, etc.. Property and liability insurance protects against a multitude of risks, including loss due to natural disaster, fire, accidents, or theft, and legal liability arising from property damage, bodily injury or financial loss to others. Business firms purchase both, and the range of offerings within these two categories has proliferated over time. If a company does not purchase insurance to protect against a risk, it is “self insuring” against that risk.

Benefits of Firm-Provided Life and Health Insurance: Businesses provide or subsidize employee life and health insurance plans primarily for three reasons.

• Lower After-Tax Cost: Group plans are cheaper than individual policies to administer, and a tax benefit may result if the company purchases insurance for the employee (rather than the employee buying the insurance individually).

• Reduction of Absenteeism: The risk of employee health-related absenteeism is reduced if employees are diligent in obtaining needed medical examinations and treatment. Insurance makes this behavior more likely; in fact, some employees might not purchase adequate insurance coverage unless it is provided by the company.

The Decision to Insure or Self-Insure: In evaluating a particular risk, a good start is to ascertain whether to insure through an insurance company or to self-insure. Here are some reasons to purchase the insurance from an insurance company.

• For many risks, insurance companies have the greatest knowledge and expertise. They may have a comparative advantage in estimating the likelihood and potential level of loss, and they may be helpful in providing advice on how to control and limit risk.

• Purchasing insurance reduces cash flow risk and the likelihood of financial distress costs for the insured. If a company self-insures a large risk, it may face significant probability of financial distress and financial distress costs. If it purchases insurance from an insurance company, it pays a premium but knows that it will be compensated if the loss occurs. Furthermore, an insurance company has the benefit of diversification; the risk of a large theft may be high for an individual, but the average theft loss for 10,000 individuals may be highly predictable. This means that the risk does not expose the insurance company to financial distress, and can charge a relatively low premium. Outside insurance is a way of reducing the overall bankruptcy costs in the economy. Large risks that jeopardize the survival of the insured are good candidates for insurance.

• Insurance companies have lower in processing claims. Insurance companies process a far larger volume of claims than would most insureds. They have the benefits of economies of scale. Moreover, they better understand insurance management, and consequently are likely to have a lower cost for any given volume of claims.

Insurance also has several drawbacks relative to self-insurance.

• Insurance companies have additional costs. Insurance companies incur a multitude of costs (marketing, claims management, etc.) that must be covered by the premiums paid by the insured. Self-insurance avoids some of those costs (e.g., marketing and advertising).

• There is the “adverse selection” problem. In cases in which the insurance company cannot accurately discern the actual risk profile of those applying for the insurance, the applicants will tend to be the biggest risks. Insurance companies know this and so charge a premium that reflects the average risk of those being covered. Those not in the high-risk category end up paying a premium that covers those who are. Some low risk individual will opt not to buy insurance, which makes insurance for those who do even more costly.

• There is the “moral hazard” problem. This refers to the perverse incentives that insurance creates for the insured. One a reduced incentive for the insured to prevent loss because the insurance company is there to reimburse. In its worst manifestation, we have the perpetrator of insurance fraud, the insured who creates a loss in order to collect on the insurance. Moral hazard produces costs for insurance companies that are covered by premiums. The responsible and honest pay the toll for those less virtuous. Some of these may choose not to buy, or to buy less, insurance, which in turn further raises the rates.

The positives and negatives of insurance must be weighed in light of the particular circumstances. Almost invariably, companies decide to insure against many risks and self-insure against others. An alternative is to remove or mitigate the risk by changing the company’s way of doing business.

FORWARD CONTRACTS

A forward contract executed at time 0 is an agreement under which two parties agree to a future exchange under terms specified in the contract. Forward contracts can cover commodities, manufactured goods, services, financial assets, and just about anything that people buy and sell. For example, on April 20, 2004, a refrigerator manufacturer could strike an agreement with a department store chain to provide 500 units for $300 per unit on November 15, 2004. The contract requires that, on November 15, 2004, the manufacturer will deliver the 500 refrigerators to the department store, and the department store will pay $300 per refrigerator (the forward price per unit) to the manufacturer. This is a forward contract. It is a direct transaction between a particular buyer and a particular seller, both of which are obliged to perform on the contract. Performance occurs at the expiration date of the forward contract, in this case, November 15, 2004. Forward contracts have existed for thousands of years.

Financial Forward Contracts

A financial forward contract is an agreement now to borrow, or lend, funds at a future date under terms that are set now. For example, Corpus Corporation might be laying out its capital expansion plans for the next five years. Management has decided that in one year it will want to borrow $1,000,000 to fund the purchase of new equipment. It wants to repay the entire loan (interest and principal) in three years. Management is worried that interest rates will rise over the coming year and so wants to lock in the interest rate now. To accomplish this, the company has a variety of strategies that it might employ. Below we describe four alternative strategies that Corpus could employ to lock in the future borrowing rate.

In this handout, now will be referred to as time 0, one year from now as time 1, etc. In explaining the four strategies we will use the following symbols:

[pic] = current (time 0) nominal spot rate on a loan that is to be repaid at time t

[pic] = current forward rate on a loan that involves lending at time t with repayment at time t+j

(where t > 0 and j > 1)

Strategy 1: At time 0, obtain a $1 million forward loan from a bank (or from a dealer that is an intermediary between a bank and the borrower). The forward loan specifies that in one year (at time 1) the bank will lend $1 million to Corpus at a forward interest rate, [pic] (where [pic] is set now, at time 0). The loan will be repayable in three years, at time 3. Corpus may have to pay an up front fee to obtain this loan. The fee will be larger the lower is the borrowing interest rate provided for in the agreement. Exhibit 2a shows the cash flows from the forward loan under the assumption that the interest rate is that which would be charged if there were no extra up-front fee.

Exhibit 2a. Cash Flows to Corpus from Forward Loan Negotiated at Time 0

| |From Lending |From Borrowing |Net Cash Flow |

|Date |(1-year loan) |(2-year forward loan) |to Corpus [= (2) + (3)] |

|(1) |(2) |(3) |(4) |

|Time 0 (now) |$0 |$0 |$0 |

|Time 1 |$0. |$1 mil. |$1 mil. |

|Time 2 |$0 |$0 |$0 |

|Time 3 |$0 |( [pic] $1 mil. |( [pic] $1 mil. |

Keep in mind that rate [pic] is determined and negotiated at time 0, when the forward loan is agreed upon.

At this point it will be helpful to introduce the term “notional principal.” Notional principal refers to the basis for computing interest payments, not an amount owed or lent. For example, Mechanic National Bank might negotiate the following arrangement with Lang Corporation. Each year for the next three years (payments one year from now, two years from now, and three years from now), Lang will pay 7 percent per annum interest ($70,000) on a notional principal of $1 million, and Mechanic Bank will in return pay Lang Corporate [prevailing prime rate + 3%] on a notional principle of $1 million. Under this agreement, there is no actual principle owed or paid; the $1 million is simply a way of computing the interest owed. The cash flow is only interest payments.

Strategy 2: At time 0, establish a forward rate agreement with a bank; at time 1 take out a two-year spot loan. Suppose that Corpus can generally borrow spot at [LIBOR + 2.5%]. A forward rate agreement negotiated at time 0 might stipulate the following for the period from time 1 to time 3 using a $1 million notional principal to compute interest payments.

If LIBOR > 8%, the bank will pay Corpus [2-year LIBOR ( 8%] (1a)

If LIBOR < 8%, Corpus will pay the bank [8% ( 2-year LIBOR] (1b)

Using (1a) and (1b), the net cash flow to Corpus from the forward rate agreement will be:

Rate earned (paid if negative) by Corpus on forward rate contract = 2-year LIBOR ( 8% (2)

The right-hand side of (2) is positive for Corpus (inflow) if [2-year LIBOR ( 8%] > 0, and is negative Corpus (expense) if [2-year LIBOR ( 8%] < 0. In addition to this forward rate agreement negotiated at time 0, Corpus takes out at time 1 a two-year $1 million spot loan at an interest rate of [LIBOR + 2.5%]. The net interest expense on the combination of the two-year spot loan (negotiated at time 1) and the forward rate agreement (negotiated at time 0) will be the [LIBOR + 2.5%] rate on the 2-year spot loan minus the [2-year LIBOR ( 8%] interest earned to Corpus from the forward rate agreement (the right-hand side of (2)). That amount is:

Corpus’s net borrowing rate = [2-year LIBOR + 2.5%] ( [2-year LIBOR ( 8%] = 10.5% (3)

The forward rate agreement combined with the time 1 2-year spot loan enables Corpus to lock in a 2-year (time 1 to time 3) borrowing rate of 10.5% [This assumes that market conditions and Corpus’s credit rating continue to allow Corpus to borrow spot at rate [LIBOR + 2.5%.]. Exhibit 2b shows Corpus’s resulting cash flows. Exhibit 2b assumes that the 2-year LIBOR rate prevailing at time 1 is the basis for computing both the 2-year rate to be charged on the time 1 2-year spot loan, and the payment in (1a) or (1b) under the forward rate agreement. For simplicity, let the loan payments under the 2-year spot loan be interest only at time 2, and interest plus principal at time 3. The forward rate agreement involves payments at times 2 and 3.

Exhibit 2b. Cash Flows to Corpus from Combination of Forward Rate Agreement (Negotiated at time 0) and Two-Year Spot Loan (Negotiated at Time 1)

| | | |Net Cash Flow |

| |From Forward | |to Corpus |

|Date |Rate Agreement |From Time 1 Two-Year Spot Borrowing |[= (2) + (3)] |

|(1) |(2) |(3) |(4) |

|Time 0 |$0 |$0 |$0 |

|Time 1 |$0 |$1 mil. |$1 mil. |

|Time 2 |[2-year LIBOR ( 8%] $1 mil. |( [2-year LIBOR + 2.5%] $1 mil. |( (.105) $1 mil. |

|Time 3 |[2-year LIBOR ( 8%] $1 mil. |( [2-year LIBOR + 2.5%] $1 mil.( $1 mil. |( (1.105) $1 mil. |

For example, if the 2-year LIBOR spot rate at time 1 turns out to be 7%, column (2) of Exhibit 2b would be [2-year LIBOR ( 8%] $1 mil. = [( (.01) $1 million] at times 2 and 3, and column (3) would be [( (.01) $1 million] at time 2, and [( (1.095) $1 million] at time 3.

Strategy 3: At time 0, borrow spot (for 3 years) and lend spot (for 1 year). This strategy could be referred to as a “homemade forward contract.” Corpus now (at time 0) borrows amount [$1 mil./(1+[pic])] for three years at annual rate [pic], and then lends the [$1 mil./(1+[pic])] for one year at rate [pic]. Exhibit 2c shows the resulting cash flows for Corpus.

Exhibit 2c. Cash Flows to Corpus from Lending and Borrowing Transactions

| |From Borrowing |From Lending |Net Cash Flow |

|Date |(3-year loan) |(1-year loan) |to Corpus [= (2) + (3)] |

| |(2) |(3) |(4) |

|Time 0 |$1 mil./(1+[pic]) |( $1 mil./(1+[pic]) |$0 |

|Time 1 |$0 |$1 mil. |$1 mil. |

|Time 2 |$0 |$0 |$0 |

|Time 3 |( [pic][$1 mil./(1+[pic])] |$0 |( [pic][$1 mil./(1+[pic])] |

Column (4) of Exhibit 2c shows the net cash flows from the combination of the 1-year lending and 3-year borrowing transactions. The firm receives $1 million at time 1 (one year from now) and repays the debt at time 3 (three years from now). Compare column (4) of Exhibit 2c with column (4) of Exhibit 2a. The exhibits are the same in two important respects. First, both involve receiving $1 million at time 1 and repaying the $1 million at time 3. Second, both involve interest rates that are set (by contract) at time 0. We pointed out in our earlier discussion of interest rates and bond valuation that two ways of achieving the identical thing must involve the same cost. This implies that the time 3 payment in Exhibit 2a must equal the time 3 payment in Exhibit 2c. That is:

( [pic] $1 mil. = ( [pic][$1 mil./(1+[pic])] (4)

which implies that:

[pic]= [pic] (5)

Taking the square root of both sides of (5) and rearranging, we find that:

[pic] = [pic] ( 1 (6)

For example, if [pic] = 8% and [pic]= 10%, then by (6) we have:

[pic] = [pic]( 1 = 11.014% (7)

Equation (7) states that Corpus has, through the borrowing and lending transaction, locked in an 11.014% rate for its net $1 million borrowing from time 1 to time 3.

Strategy 4, approximate version to explain the intuition: At time 0, sell three-year bonds short. At time 1, cover (close out) the short position. At time 1, borrow for two years. If market interest rates rise over the coming year (from time 0 to time 1), the three-year bonds will fall in value, producing a profit on the short position. This profit will offset, in present value terms, the increase in the interest cost of borrowing at time 1 due to the rise in interest rates from time 0 to time 1. If interest rates fall over the coming year (from time 0 to time 1), the 3-year bonds will rise in value, producing a loss on the bond short position. This loss will offset, in present value terms, the lower interest cost on the borrowing at time 1. So, the net cost of borrowing (profit or loss on the 3-year bond short position + interest cost on the borrowing from time 1 to time 3) is the same whether interest rates rise or fall. The gain (loss) on the short position offsets the increase (decrease) in interest rates. Below we provide details on how this is done in the forward (or futures) markets.

Strategy 4 in detail (this section – pages 10 - 13 – is optional reading): At time 0, sell two-year bonds forward, with delivery at time 1; at time 1, conduct the appropriate spot lending or spot borrowing. This particular strategy is enormously popular (generally employed using futures) for both financial and non-financial businesses. The transactions appear complicated, but once understood, are not. The transactions are extremely easy to arrange and occur in a flash. Exhibits 2f and 2g summarize Strategy 4. An example appears on page 12.

A two-year bond here refers to a bond that has two years to maturity. Suppose that Spartan Corporation now (time 0) has zero-coupon Spartan “L-Bonds” that will mature in three years (so, one year from now, the bonds will be two-year bonds). Also assume that Spartan debt will, over the next 4 years, have the same default risk as will Corpus debt (this assumption is not necessary for Strategy 4, but significantly simplifies the discussion).

Now assume the following strategy. At time 0, Corpus sells forward some Spartan L-Bonds to “Buyer.” Under the forward contract, at time 1 (one year from now) Buyer will pay Corpus $1 million in return for the Spartan L-Bonds. Buyer is agreeing now to invest $1 million at time 1 with the return on this investment at time 3 (return = the time 3 maturity value, or coupon, of the Spartan L-Bonds). This transaction for Buyer is equivalent to the Bank’s position in the Strategy 1 situation (discussed earlier) in which Bank agrees at time 0 to advance $1 million to Corpus at time 1 with repayment at time 3. The time 0 forward contract should involve the same forward interest rate, [pic], as would a forward loan to Corpus. Therefore, the time 3 maturity value (coupon) on the Spartan bonds under the forward contract will be equal to [[pic] ( amount advanced at time 1] = [[pic] $1 million].

Exhibit 2d below describes the forward contract. There is no time 0 cash flow impact for Corpus or Buyer. At time 1, Corpus delivers, to Buyer, Spartan L-Bonds promising [pic] $1 million at time 3. Let [pic] signify the time 1 market value of those bonds. [pic] will depend on interest rates prevailing at time 1, and will therefore not be known until time 1.

Exhibit 2d. Transactions by Corpus and Buyer Under Forward Contract

| |Corpus’s Transactions |Buyer’s Transactions |

|Time 0 |Sells Spartan L-Bonds forward |Buys Spartan L-Bonds forward |

|Time 1 |Pays market price [pic] for L-Bonds, delivers the bonds to Buyer, |Pays $1 million to Corpus and receives delivery of |

| |and receives $1 million from Buyer. |L-Bonds (with time 1 market value [pic]) |

To see how [pic] depends on time 1 interest rates, define [pic] as the spot rate prevailing in the market at time 1 on a loan to Spartan or Corpus from time 1 to time 3 (funds advanced at time 1 and repaid at time 3). Rate [pic] is not known until time 1. Rate [pic] equals the promised rate that, at time 1, buyers in the market will demand on Spartan L-Bonds (which mature at time 3). So, the L-Bonds, which promise amount [pic] $1 million at time 3, will have a time 1 market value ([pic]) equal to the time 1 discounted value (using [pic] as the discount rate) of the [pic] $1 million time 3 L-Bond payoff. This amount is shown in equation (8a) below.

[pic] =[pic]= [pic] (8a)

Exhibit 2e describes the transactions Corpus will conduct at time 1 in order to implement hedging Strategy 4. Time 1 transaction [3] involves Corpus’s lending (buying bonds) or borrowing spot for two years (time 1 to time 3) at rate [pic] (whether there is lending or borrowing will depend on whether [pic] is positive or negative). Equation (8b) states the time 3 consequences of this time 1 lending or borrowing.

Exhibit 2e. Time 1 Net Cash Flow from Forward Contract and[pic] Transactions

| |Corpus’s Transactions |Net Cash Flow Impact on Corpus |

|[1] |Receives $1 million from Buyer |$1 million |

|[2] |Buys [pic] of L-Bonds in the market and delivers those bonds to Buyer (see |( [pic] |

| |equation (8a) for [pic]) | |

|[3] |[pic] ( The net amount from [1] and [2] | |

| |= $1 million ( [pic] | |

| |If [pic] > 0, Corpus lends [pic] in the market for two years at promised rate |([pic] |

| |[pic]; if [pic]< 0, Corpus borrows [pic]for two years at rate [pic]. | |

|TIME 1 NET CASH FLOW FROM [1], [2] AND [3] COMBINED = $0 |

At time 1, there is therefore no net cash flow impact from the combined strategies of the forward contract and [pic] investment (or borrowing) transaction (see Exhibit 2e above).

At time 2, there is no cash flow impact on Corpus from the forward transaction or from the time 1 transaction relating to [pic].

At time 3, there is a net cash flow impact from the combination of the forward contract and the [pic] investment (or borrowing) transaction; that time 3 cash flow impact is:

[pic]= [pic][pic]=[pic] [pic]

= [[pic] ( [pic]] $1 million (8b)

The right-hand side of equation (8b) is the positive return from lending [pic] at time 1 at rate [pic] if [pic] > 0, or the negative amount to pay off the borrowing of [pic] at time 1 at rate [pic] if [pic] < 0.

Recall that Corpus wants to borrow $1 million from time 1 to time 3 but to lock in the interest rate at time 0. The above transactions, combined with a $1 million spot loan negotiated at time 1 (at the time 1 spot rate), allows Corpus to do this. This is summarized in Exhibit 2f.

Exhibit 2f. Net Effect of Exhibit 2e Transactions and Time 1 Spot Loan

|Date |Time 1 Spot Loan |Exhibit 2e Transactions |Net Cash Flow |

|(1) |(2) |(3) |[=(2) + (3)] |

| | | |(4) |

|Time 0 |$0 |$0 |$0 |

|Time 1 |$1 million |$0 |$1 million |

|Time 2 |$0 |$0 |$0 |

|Time 3 |([pic] $1 mil. |[[pic] ( [pic]] $1 mil. |( [pic] $1 mil. |

Column (4) of Exhibit 2f shows the net effect of engaging in the hedging strategy in Exhibit 2e and waiting to time 1 to borrow the $1 million for two years on a spot basis. The net cash flow effect is that Corpus receives the $1 million at time 1 and, at time 3, repays the loan at an interest rate that is fixed at time 0 at the time 0 forward rate [pic], which is known at time 0.

Example: UltraTone Corporation, a rapidly growing electronics manufacturer, has just approved its five-year capital expenditure plan. It will want to borrow $10 million two years from now (time 2), and repay the loan four years from now (time 4). It will use Strategy 4 to lock in the interest rate that it will have to pay on the $10 million loan.

Let the debt rating of Zone, Inc. be the same as that of UltraTone. Assume that the current forward rate on Zone borrowing from time 2 to time 4 is [pic] = 7 percent. Let [pic] be the time 2 spot rate on Zone (or UltraTone) borrowing. Rate [pic] is not known until time 2; suppose that it turns out that [pic]= 8 percent. At time 0, UltraTone sells (to Buyer) a forward contract on Zone Bonds, with delivery at time 2. At time 2, UltraTone: [a] Receives $10 million from Buyer; [b] Buys Zone bonds in the market for [pic], and delivers the bonds to Buyer; and [c] Lends [pic] if [pic]> 0, and borrows [pic] if [pic] < 0, where:

[pic] =[pic]= [pic] = [pic] = $9,815,672 (8c)

[pic]([pic] = $10 mil.([pic]= $10 mil. ( [pic] (8d)

At time 2, UltraTone lends [pic] at interest rate [pic]= 8%. The time 4 promised return on this loan is (using [pic] in (8d) and simplifying):

[pic] =[pic][pic] = [[pic] ( [pic]] $10 million (8e)

The net effect of [a], [b] and [c] is a time 2 net cash flow of zero. However, in addition to [a], [b] and [c], UltraZone also at time 2 takes out a 2-year $10 million spot loan at rate [pic]= 8 percent. This loan produces a time 2 cash inflow of $10 million. At time 4, UltraTone will owe the following on the $10 million time 2 loan:

[pic] = [pic]$10 million (8f)

The time 2 loan of [pic]produces the time 4 cash inflow on the right-hand side of (8e). Repayment of the time 2 $10 million borrowing produces a time 4 cash outflow on the right-hand side of (8f). The time 4 net cash flow from all transactions is:

[pic]= [[pic]( [pic]] $10 million ( [pic] $10 million

= ( [pic] $10 million (a net cash outflow) (8g)

The outcome for UltraTone is a time 2 cash inflow of $10 million and a time 4 cash outflow of [pic] $10 million. That is, the net effect for UltraTone is forward borrowing at rate [pic] (7 percent), which is known at time 0. The level of future spot rate [pic] is irrelevant.

Under Strategy 4, the buyer of the bond forward contract is required, on the delivery date t, to take delivery of, and pay for, a bond, where the price paid for the bond is the forward price that was set when the forward contract was initially issued (price [pic]). The seller of the contract is required to deliver the bond on the delivery date t in exchange for amount [pic]. Let [pic] be the time t spot price of the bond in the market. If, [pic]> [pic], then the buyer of the forward contract will at time t pay more for the bond than it is worth in the market ([pic]), making the buyer worse off at time t by amount [[pic] ( [pic]]. The seller of the contract receives [pic] for the bond worth [pic], producing a time t gain on the contract of [[pic] ( [pic]]. On the other hand, if [pic]< [pic], the forward contract has made the buyer better off, and the seller worse off, at time t by amount [[pic] ( [pic]].

General Comments on Financial Hedging With Forward (or Futures) Contracts

Exhibit 2g below addresses the risks of borrowers and lenders, and how those risks can be hedged using a bond forward contract. The exhibit states that a borrower, or a lender, may want to hedge against an interest rate increase or interest rate decrease; the kind of hedge that is preferred will depend on the borrower’s, or the lender’s, particular situation.

Exhibit 2g. Financial Hedging with Forward Contracts

| |Future event to hedge |Effect of event |How to |Benefit of hedge* |Disadvantage of the hedge* |

| |against | |hedge | | |

|Borrower |Increase in interest |Higher future |Sell forward |Locks in future borrowing |Higher borrowing cost than |

| |rates |borrowing costs |contract |rate |without hedge if interest rates |

| | |(interest cost) | | |fall and bond market price |

| | | | | |exceeds forward price |

|Borrower |Decrease in interest |Increase in market |Buy forward |Produces gain on forward |Higher debt burden than without |

| |rates |value of existing |contract |contract to offset increase |hedge if interest rates rise and|

| | |debt burden | |in firm’s debt burden |bond market price is less than |

| | | | | |forward price |

|Lender |Decrease in interest |Lower interest |Buy forward |Locks in future lending rate|Lower in interest income from |

| |rates |income on future |contract | |new lending than without hedge |

| | |lending | | |if interest rates rise and |

| | | | | |forward price exceeds bond |

| | | | | |market price |

|Lender |Increase in interest |Decline in value of |Sell forward |Produces gain on forward |Lower bond portfolio value than |

| |rates |existing bond |contract |contract to offset decrease |without hedge if interest rates |

| | |portfolio | |in debt portfolio value |fall and bond price exceeds |

| | | | | |forward price |

* Benefit of the hedge, or disadvantage of the hedge, relative to not having the hedge.

It is a change in interest rates that drives the gains and losses on a bond forward contract, and whether a gain or loss accrues will depend on whether one is a buyer or seller of the contract. Because the bond forward price is set when the forward contract is created, that price reflects interest rate levels and expectations that prevail at that time. If interest rates rise (fall) after a bond forward contract is created, then bond prices in general will fall (rise), and the buyer of the bond forward contract will experience a loss (gain) on the contract. This loss for the buyer occurs because the buyer will be paying the forward price that was set before bond prices fell, and that forward price will exceed the bond spot price on the delivery date. On the other hand, the seller of the bond forward contract will gain as a result of a general interest rate increase, since the seller will be selling the bond at a forward price that exceeds the delivery date spot price of the bond.

Exhibit 2g indicates that borrowers can face two kinds of interest rate risk, the risk that interest rates will rise, and the risk that interest rates will fall. First consider a company that is concerned about a rise in future interest rates because it plans a future debt offering. To hedge against the risk of an increase in interest rates, the borrower can sell bonds forward. The seller of a bond future contract is obliged to sell a bond on the future delivery date at the forward price that is set today. If interest rates are higher on the delivery date than they are currently, the forward contract issued now will produce a gain for the seller of the contract; this gain will offset the future cost of borrowing at the higher interest rate.

A fall in interest rates also poses potential risks for a borrower since it will raise the market value of a borrower’s existing debts (a fall interest rates raises bond values and the market value of other interest bearing liabilities). To hedge the risk of a fall in interest rates, a borrower can buy bond forward contracts. The bond forward contracts increase in value as interest rate fall. This will at least in part offset the increase in the present value of the borrower’s outstanding debts.

Lenders also contend with the possibility of loss due to a rise, or a fall, in interest rates. A fall in interest rates reduces the income on future lending. A lending institution, or a company that invests its surplus cash in monetary assets, will be worse off if real interest rates earned on those assets fall. To hedge against that risk, the lender can buy bonds forward. As interest rates fall, bond prices rise and the buyer of bond forward will profit; this profit can offset the loss of income on future lending.

A rise in interest rates can also negatively impact a lender. The interest rate increase causes a fall in the market value of the lender’s existing portfolio of loans and other fixed interest rate assets. To hedge this risk, the lender can sell bonds forward (the value of that position rises as interest rates rise). The gain on the forward contracts can offset the loss on the lender’s portfolio.

Commodity Forward Contracts for Hedging

A seller of a commodity is generally concerned about a potential future fall in the price of the commodity, which will reduce revenue per unit sold on future sales. To hedge this risk, the seller can guarantee the future sales price of the commodity by selling in the forward markets. The forward contract obliges the seller to delivery the commodity in exchange for the currently set futures price. So, if the price of the commodity falls, the seller is better off having hedged with the forward contract. However, if the price of the commodity rises, the seller will be worse off having established the obligation to sell at the lower forward price. So, the forward contract eliminates the risk of a future decline in price of the commodity, but also eliminates the potential gain if the commodity price rises in the future. The buyer of a commodity has the opposite concern, the fear that the price of the commodity will rise. Engaging in a forward contract will eliminate the risk to the buyer of a commodity price increase, but it will also foreclose the chance of benefiting from a commodity price decrease.

Exhibit 3a. Commodity Hedging with Forward Contracts

| |Future event to hedge |Effect of event |How to hedge |Benefit of the |Disadvantage of the hedge* |

| |against | | |hedge* | |

|Seller of |Fall in price of |Decline in |Sell |Guarantees price received |Lower revenues than without |

|commodity |commodity |revenues |forward contract |for commodity |the hedge if market price of |

| | | | | |commodity above forward price |

|Buyer of commodity|Rise in price of |Increase in costs |Buy |Guarantees price |Higher costs than without the |

| |commodity | |forward contract |paid for commodity |hedge if market price falls |

| | | | | |below forward price |

* Benefit of the hedge, or disadvantage of the hedge, relative to not having the hedge.

Example: Now is 3/20/2004. Wildcat oil driller Tex Royale is a regular supplier of oil to Moon Refineries. Let [pic] signify the current (3/20/2004) forward price of oil to be delivered on 9/30/2004, and let [pic] be the 9/30/2004 spot price of a barrel of oil. Suppose that [pic] = $30. To hedge against a fall in the price of oil, Tex sells 100,000 barrels of oil forward at a forward price [pic] = $30 and delivery on 9/30/2004; and to hedge against a rise in the price of oil, Moon buys an identical forward contract for 100,000 barrels of oil forward to be delivered on 9/30/2004 for $30 a barrel. Under the forward contracts, on 9/30/2004, Moon will pay $3 million for 100,000 barrels of oil, and Tex will deliver 100,000 barrels of oil in exchange for $3 million.

Observe that, if 9/30/2004 spot price [pic] is below (above) the $30 forward price [pic], the forward contract makes Tex better (worse) off and Moon worse (better) off. So, for example, suppose that [pic] = $28. Tex has gained $200,000 from the forward contract because Tex will receive $30 per barrel under the forward contract rather than the $28 he would have received if he had not sold forward. In contrast, the forward contract has made Moon worse off by $200,000 because the forward contract requires Moon to pay $30 per barrel for the 100,000 barrels of oil, rather than the $28 it would have paid without the forward contract. On the other hand, if [pic]= $33, Tex is worse off by $300,000 and Moon is better off by $300,000 than would have been the case if neither had engaged in the forward contract. These figures are shown in Exhibit 3b below. The point here is that hedging using a forward contract fixes the future sale price, and thereby produces both a potential benefit and a potential loss (opportunity cost) for the buyer and for the seller of the contract.

Exhibit 3b. Buyer and Seller Gains and Losses on Forward Contract for 100,000 Barrels of Oil if Forward Price = $30 and Delivery Date is September 30, 2004

| |Spot price of oil on 9/30/2004 |

| |[pic]= $28 |[pic]= $30 |[pic]= $33 |

|Seller (Tex) |$200,000 |$0 |($300,000) |

|Buyer (Moon) |($200,000) |$0 |$300,000 |

Pricing Forward Contracts

Forward Price and Spot Price for a Financial Asset: Let [pic] and [pic] be, respectively, the prevailing time 0 spot price and forward price for asset X (assume competitive and efficient financial markets). The risk-free rate is signified by [pic], and PV[Inc] is the time 0 present of the dividends or interest payments on asset X from time 0 to time t (this income is received if the asset is owned from time 0 to time t, but is not received if one buys the forward contract). PV[Inc] is what it would cost to purchase a comparable income stream in the market.

[pic] = [pic] ( PV[Inc] (9)

It is shown in Appendix A.1 that equation (9) must hold or there are arbitrage opportunities. These arbitrage opportunities will be exploited until relation (9) holds.

Forward Price and Spot Price for a Commodity: Let [pic] and [pic] be, respectively, the prevailing time 0 spot and forward prices for commodity Y (assume competitive and efficient commodities markets). The risk-free rate is [pic]. PV[Stor. Costs] is the present value of the expected cost of storing commodity Y from time 0 to time t. PV[Con. Yield] is the time 0 present of the expected “convenience yield” provided by commodity Y to its owner over the time period from time 0 to time t. The convenience yield is received, and the storage costs are paid, by a time 0 spot buyer of the commodity but not by a time 0 forward buyer (since the forward buyer will not possess of the commodity until time t). [pic] and [pic] are related as follows.

[pic] = [pic] + PV[Stor. Costs] ( PV[Con. Yield] (10)

In Appendix A.2 we shown that arbitrage ensures (10).

FUTURES

As of June 2004, the worldwide interest rate futures market (outstanding contracts) was approximately $170 trillion (Bank of International Settlements data). Futures have become a basic tool for hedging interest rate risk. A futures contract is a standardized forward contract, and provides similar hedging possibilities. Forward contract were described earlier.

A futures contract stipulates that, at some particular future date (the contract “expiration date” or “delivery date”), a buyer will pay a specified price (“futures price”) and receive delivery of a given item (a commodity, currency, or financial asset) and a seller will receive the specified price and deliver the item. For example, on January 15, 2004, Mary and Bob agree that, on future date March 30, 2004, Mary will pay Bob $4 per bushel for 100 bushels of wheat (this is actually a forward contract, but is conceptually the same as a futures contract).

A futures contract is a standardized forward contract that is transacted through a futures exchange. The exchange sets up a contract that has two-subparts: the contract between the buyer and the exchange, and the contract between the seller and the exchange. At the contract’s expiration the buyer is required to buy the item at the original futures price, and the seller is required to deliver the commodity at that price (or there may be a cash settlement).

Delivery versus Cash Settlement: Suppose that, in June, Boris buys, and Sally sells, a September contract (that expires on September 30) on 100 ounces of gold at a futures price of $300 per ounce. Assume that both Boris and Sally maintain their futures positions until expiration. Futures contracts may require actual delivery of the underlying item, or simply involve cash settlement. Commodities usually involve actual delivery, whereas financial futures almost always involve cash settlement.

If the futures contract requires delivery, then on September 30 Boris will take deliver of 100 ounces of gold and pay $30,000 ($30,000 = 100 ounces ( $300 per ounce); and, on September 30, Sally will deliver 100 ounces of gold and receive $30,000.

If the futures contract requires cash settlement, then, on September 30, Boris and Sally will settle up by contributing or receiving cash. One party will gain (and receive cash) and the other will lose (and pay the same amount of cash). Suppose that, on September 30, the spot price of gold is $290 per ounce. Under the futures contract specifying $300 per ounce, Boris is obliged to pay, and Sally is entitled to receive, $10 per ounce more than the gold is worth. With cash settlement, Boris will pay the exchange $1,000 (100 ounces ( $10 per ounce) and Sally will receive from the exchange $1,000; and there will be no actual exchange of gold for cash.

The above description simplifies the actual process because futures markets involve a “marking to market” procedure that each day adjusts the contract buyer’s and the contract seller’s margin account balances to reflect daily profits and losses of each trader.

Marking to Market (optional): Buying or selling a futures contract requires establishing a margin account into which money or interest-bearing securities (e.g., U.S. Treasury bills) are deposited (the margin) to ensure performance. This account is increased or decreased each day as the trader’s position earns a profit or loss that day. An amount in the account above the required margin may be withdrawn by the trader. If the account falls below the required margin, the trader must deposit new cash or securities to comply with the margin requirement.

To illustrate, suppose that, on February 10, 200x, Buyer purchases, and Seller sells, a futures contract for 100,000 barrels of oil with delivery on May 15, 200x. The February 10 futures price of May 15 oil is $30 per barrel. These terms are stated in Exhibit 4. The $30 futures price is the price to be paid by Buyer, and to be received by Seller, on the settlement date, May 15, 200x.

Exhibit 4. Terms of Futures Contract

|Date of futures transaction: |February 10, 200x |

|Settlement date (when oil is delivered) |May 15, 200x |

|Futures price : |$30 per barrel |

|Number of barrels of oil: |100,000 |

|Settlement amount ($30 ( 100,000): |$3,000,000 |

Both Buyer and Seller have margin accounts at the New York Mercantile Exchange (NYMEX) that issued the futures contracts. Assume that each must initially (on February 10) deposit $200,000 in a margin account. As shown in Exhibit 5, on February 11 the futures price of May oil is $30.50, a $.50 increase from the previous day. That means that anyone buying a new May oil futures contract on February 11 would have to agree to pay $30.50 on May 15, 200x for each barrel of oil received on that date. Further, Buyer has earned a $50,000 gain and therefore Buyer’s margin account is increased by $50,000; the exchange replaces Buyer’s existing futures contract (with a $30 futures price) with one having a futures price of $30.50 and, like the old contract, having a May 15, 200X expiration date. Seller has suffered a $50,000 loss and Seller’s margin account is reduced by $50,000; and Seller’s existing futures contract (with a $30 futures price) is replaced by one having a futures price of $30.50 and a May 15, 200X expiration date. .

Exhibit 5. Gain and Loss on Futures Contract

| |Prevailing futures |Buyer’s long margin |Change in Buyer’s |Seller’s short margin |Change in Seller’s |

| |price of May oil |account |account from previous |account balance |account from previous |

| |(2) |balance |day |(5) |day |

|Date | |(3) |(4) | |(6) |

|(1) | | | | | |

|Feb. 10 |$30.00 |$200,000 |( |$200,000 |( |

|Feb. 11 |$30.50 | 250,000 | 50,000 | 150,000 | ( 50,000 |

|Feb. 12 |$30.25 | 225,000 | ( 25,000 | 175,000 | 25,000 |

|Feb. 13 |$30.25 | 225,000 |0 | 175,000 | 0 |

|Feb. 14 |$30.60 | 260,000 | 35,000 | 140,000 |( 35,000 |

| … | | | | | |

|May 14 |$30.90 | 290,000 | 5,000 | 110,000 |( 5,000 |

|May 15 |$31.00 | 300,000 | 10,000 | 100,000 |( 10,000 |

Consider the futures price when the May 15 futures contract expires, on May 15. The May 15 futures contract requires delivery of the oil and payment for the oil on May 15. But this is exactly what a spot contract negotiated on May 15 would require. So the prevailing May 15 oil futures price on May 15, $31.00, must equal the May 15 spot price for oil.

Financial Futures: A financial futures contract entitles the buyer to pay, at the contract expiration date, a specified amount for a financial asset having a value that is equal to, or tied to, the value of an actual financial asset (such as a U.S. Treasury bond) or financial quantity (such as a stock index or interest rate). Financial futures are done on a cash settlement basis.

Arbitrage ensures the relationship in (11) between the futures price and spot price of a financial asset. If you now (time 0) purchase a futures contract that expires in t periods (at time t), you receive the asset at time t, and you do not get the dividends or interest income on the asset from time 0 to time t.

[pic] = [pic] ( PV[Inc] (11)

To illustrate (11), if [pic] = $5,000, PV[Inc] = $500, t = 6, and [pic]= 1%, then [pic] = $4,776.84.

The proof of (11) is the same as that for forward markets equation (9). However, whereas equation (9) is correct for perfect forward markets for financial assets, equation (11) would only be a rough approximation for financial futures even if the market were perfect. This is because (9) and (11) assume that the buyer of the contract pays for the asset at time t, which is true for the forward contract but, because of the marking to market process, is not true for a futures contract. It can be shown that, due to marking to market, the interest rate and the spot price of the asset must have a zero correlation for (11) to hold. The degree to which Equation (11) holds depends on the type of financial asset (it holds better for stocks than for bonds).

Commodity Futures: A commodity futures contract entitles the buyer to pay, at the contract expiration date, a specified amount for a given quantity of a commodity (oil, gold, wheat, beef, etc.). Generally, commodities are delivery based rather than cash settlement based. The relationship between the spot and futures price of a commodity is as shown in equation (12).

[pic] = [pic] + PV[Stor. Costs] ( PV[Con. Yield] (12)

To illustrate (11), if [pic] = $10,000, PV[Stor. Costs] = $800, PV[Con. Yield] = $500, t = 4, and [pic]= .5%, then [pic] = $10,507.55.

Equation (12) is the same as forward markets equation (13), and the arguments underlying equation (13) also underlie equation (12). As with equation (11) for financial futures, equation (12) is only an approximation due to marking to market.

Equation (12) is useful to a buyer or seller of a commodity. At least roughly, a buyer (given that buyer’s particular Storage Costs and/or Convenience Yield) should buy in the spot market if the “=” in (12) is “>,” and should buy in the futures market if the “=” is “,” and should sell in the spot market if the “=” is instead “ 0, a rise in the value of B is associated with a rise in the expected value of A. In that case, in order to hedge against the risk of A, one must sell B. For example, if A is the common stock of Ascot Corporation, one could sell futures on the Ascot stock to hedge the Ascot stock risk.

If ( < 0, a rise in the value of B is associated with a decline in the expected value of A. In that case, in order to hedge against the risk of A, one must buy B. So, if A is the common stock of Ascot Corporation, one could sell shares of Ascot short.

Notice in (15a) that the dependent variable (the variable on the left-hand side of equation) is the expected change in the value of A, whereas the independent variable (the variable on the right-hand side of the equation) is the change in the value of B. The “expected” appears on the left-hand side because a change in B may not fully explain the change in A, i.e., there may be other variables that determine changes in A. To put it differently, A and B may not be perfectly correlated (+1 or ( 1). Equation (15a) says that if we know the change in B, we know the expected (mean) change in A. On the other hand, if any change in A is positively or negatively related strictly to a change in the value of B, then we could delete the word “expected” on the left-hand side of (15a) (retaining the “expected” would also be correct, but there would be zero variance to the change in A associated with any change in B).

The ( in (15a) is the expected change in the value of A due to a change in the value of B. The absolute value of ( equals the number of units of B that is needed to hedge the risk of a unit of A. To illustrate suppose that:

( = 2

( = ( .6.

Let’s see why owning .6 units of B makes the expected change in A always equal to (. Take a look at Exhibit 8. Let [pic]and [pic]be the expected change in the value of A, and the change in the value of B, respectively.

Exhibit 8. Change in Expected Value of Hedged Portfolio

|Expected change in |Change in |Expected change in value of hedged portfolio |

|the value of A |Value of B |(one unit A + .6 units of B) |

|= [pic]= ( + ([pic]= 2 ( .6[pic] |= [pic] |= [pic] + .6 [pic] |

|3.8 |( 3 |2 |

|3.2 |( 2 |2 |

|2.6 |( 1 |2 |

|2.0 |0 |2 |

|1.4 |1 |2 |

|.8 |2 |2 |

|.2 |3 |2 |

[pic]

You can create a similar table for a case in which ( > 0 and the hedge therefore involves selling B to hedge against the risk of owning A.

Example: Aura Corporation receives royalty income from its partnership interest in a silver mine (Asset “S”). It has observed the equation (15b) relationship between the expected change in the market value of one dollar invested in Asset S and the market value of one dollar invested in Stock Portfolio P (Asset “P”); the A and P are measured in units of $1. How much P must Aura purchase to neutralize the expected change in the value of S? For each dollar currently invested in Asset X, we have:

[pic] = ( + ( [pic]= $.02 ( .4 [pic] (15b)

Answer: For each dollar invested in S, Aura must invest $.40 in asset P because ( = ( .4 in (15b). For example, suppose that Asset S has a $1 million market value, and Aura purchases $400,000 of P as a hedge. Let’s see how much this portfolio change in value due to a [pic]change in the value of P.

Expected future change in value of $1 of current investment in S = $.02 ( .4([pic])

Now let’s consider the change in the value of the $1 million investment in Asset S, the $400,000 invested in P, and the combination of the two.

[pic] = 1 million ( [$.02 ( .4([pic])] = $20,000 ( 400,000 [pic]

[pic] = 400,000[pic]

[pic] = [pic]+ [pic]

= [$20,000 ( 400,000 [pic]] + 400,000[pic]

= $20,000.

So, for any change in the market value of P (i.e., any [pic]), the expected change in the market value of [Asset S + Portfolio P] will be $20,000. To illustrate, suppose that P increases by 6%. So each dollar invested in P rises by $.06, to $1.06 (that is, [pic]= $.06). Therefore:

[pic]= 1 million ( [$.02 ( .4([pic])] = 1 million ( [$.02 ( .4($.06))] = ( $4,000

Change in value of P = 400,000 ( $.06 = $24,000

Expected change in value of [Asset S +Portfolio P]= ( $4,000 + $24,000 = $20,000

Duration and Hedging Fixed Income Assets or Liabilities: From earlier handout Interest Rates and the Valuation of Debt, let PV[[pic] signify [pic] discounted using yield to maturity y:

PV[[pic] ( [pic] (16a)

Recall that PV[[pic] is not the present value (market value) of [pic]. The market value of [pic] is:

V[[pic]] = [pic] (16b)

where [pic] is the (current) time 0 spot rate for discounting the time t cash flow.

Duration is a measure of the average length of time from now to full repayment of the debt. For a bond with N payments over N time periods, Duration equals:

Duration = [pic]+ [pic]+ … + [pic] (17)

Where V is the market value of the bond (the discounted value of the interest and principal payments on the bond).

The percentage change in a bond’s price due to a one percent change in the yield to maturity (volatility) equals:

Volatility = [pic]= ([pic]= [pic] (18)

Rearranging (18) and noting that [pic]is positive (negative) when [pic]is negative (positive):

[pic]= [pic]= ([pic]( [pic] (19a)

= ( [Volatility ( [pic]] (19b)

Consider the impact of a change in market interest rates on two different bonds, Bond E and Bond F. Let [pic] and [pic] be the volatilities of Bond E and Bond F, respectively, and let [pic]and [pic] be the change in the yield to maturity of Bond E and Bond F that results from the change in market conditions. By (19b), the percent change in the value of the two bonds will be the same only if [[pic] ( [pic]] = [[pic]([pic]].

Example: Assume that Endive Corporation wants to hedge the interest rate risk of Asset A, a fixed 10-year royalty payment that it is receiving from another firm. For a change in the value of Asset A to produce a zero effect on the value of the firm’s equity, Endive will issue Debt D, the value of which will respond to a market interest rate change as would Asset A. Therefore, the resulting net effect on Equity = change in value of Asset A ( change in value of Debt D = 0. Assume the data in Exhibit 9 below.

Exhibit 9. Percentage Change in Asset A and Debt D Market Values

from a Change in Market Interest Rates

| | | |Percentage Change in Value |

| |Volatility |[pic] |( [Volatility ( [pic]] |

| |(1) |(2) |(3) |

|Asset A |4 |[pic] |( 4[pic] |

|Debt D |10 |[pic] = .5[pic] |( 5[pic] |

Column (2) of Exhibit 9 states that a change in the yield to maturity on Asset A, [pic], will be twice the change in the yield to maturity on Debt D, [pic]([[pic]/.5[pic]] = 2). This is because, when interest rates change, not all change by the same amount. The change in y depends on the assets duration, and pattern of payments over time.

Column (3) states that the percent change in the value of Asset A can be expected to be .8 times the percent change in the value of Debt D ([4[pic]/5[pic]] =.8). Therefore, we have the following relationship.

[pic] = .8 ( [pic] (20a)

Equation (20a) implies that for each dollar of Asset A value and each dollar of Debt D value:

[pic] = .8 ( [pic] (20b)

Thus, if a change in market interest rates causes the market value of a unit of Debt D to change from $1 to $.95, it will cause a fall in the value of a unit of Asset A to decline from $1 to $.96. That means that, for each dollar of Asset A market value, Endive must issue $.80 of debt to neutralize the risk of Asset A.

So, suppose that the current market value of Asset A is $10 million. Equation (20b) says that one must have .8 units (dollars) of D to hedge one unit (dollar) of A. So, Endive issues $8 million of Debt D to neutralize the risk of the $10 million of Asset A. Imagine a change in market interest rates that causes the value of Debt D to fall by $100,000 to $7.9 million, which is a change of $.0125 per dollar of Debt D. By (20b), each dollar of Asset A will fall by [.8 ( $.0125] = $.01. Since initial value of Asset A is $10 million, the fall in the value of Asset A = 10 million ( $.01 = $100,000, the same as the decline in Debt B’s value. So, the net effect on the firm’s equity due to the change in the values of Asset A and Debt D will be zero.

Example: Now is time 0, and times 1, 2, and 3 are, respectively, one year, two years, and three years from now. Carter, Inc. now has outstanding X-Bonds that were issued two years ago at par and an 8 percent yield, the then prevailing market rate. The X-Bonds will be retired over the next three years, the last payment being at time 3. The currently prevailing yield to maturity on the X-Bonds (y) is 10 percent. The X-Bonds have the promised payments shown in Exhibit 10 below; Carter owes $200, $500, and $700 at times 1, 2, and 3, respectively; these amounts (in $million) include principal and interest.

Exhibit 10: Promised Payments on X-Bonds (in $million)

|Date: |Time 1 |Time 2 |Time 3 |

|Promised payment: |$200 |$500 |$700 |

Answer the following.

(a) What is the current (time 0) market value of the X-Bonds?

(b) What is the duration of the X-Bonds?

(c) What would be the change in the market value of the X-Bonds (that is, the dollar value) if the yield to maturity on the debt were to rise by .5% (your solution is to be a dollar amount)?

Solution

(a) The current (time 0) market value of the X-Bonds equals the promised payments discounted using the yield to maturity.

[pic]= [pic] + [pic] + [pic] = $181.82 + $413.22 + $525.92 = $1,120.96

(b) Using equation (17), and the solution to (a), the duration of the X-Bonds is:

Duration = [pic]+ [pic]+[pic] = 2.307

(c) The change in the market value of the X-Bonds due to a change in y of .5% is (using (19a)):

[pic]= [pic][pic] [pic]

=[pic]( [pic]

= [pic]( $1,120.96

= ( $11.75 (rounded)

Options and Hedging: Options can be used for hedging. Each option allows the owner of the option to buy (a “call option”) or sell (a “put option”) one unit of a given asset. So, one call option on Intel stock would give the owner of the option to buy one share of Intel stock for a particular price (the “exercise price” or “strike price”). An option will expire at some given future date, called the “expiration date.” An “American option” allows the holder of the option to buy or sell the underlying asset at any time from the present to the expiration date. A “European option” permits the holder of the at any option to buy or sell the underlying asset only at the expiration date.

The delta (() for an option is the change in the market value of the option due to a small change in the market value of the underlying asset (that is, ( = partial derivative of the value of the option value with respect to the value of the underlying asset). Hedging with an option produces the following relationship between the change in the value of an underlying asset and the change in the value of the option:

[pic]= ( [pic] (21a)

We can divide both side of (21a) by ( and we have:

[pic]= [pic][pic] (21b)

Equation (21b) says that it will take (1/() options to hedge the risk of 1 unit of the underlying asset (e.g., one share). One could hedge X shares of Intel by owning [pic] Intel call options (using the ( applicable to the call option). Thus, (21a) implies that, in terms of dollar change in value:

1 option = ( shares

X options = ( X shares

(X/() options = X shares

For options on stock, a rise (fall) in the price of the stock will cause a rise (fall) in the value of call options and a fall (rise) in the value of put options. Therefore, to hedge a long position in ( shares, one could buy one put option or sell one call option (each would have its own (). Keep in mind that, as the value of the shares changes, ( changes, so the hedging position will have to continuously change.

Options can be used to hedge an asset or a liability. An increase (decrease) in market interest rates causes outstanding bond call options to decrease (increase) in value, and causes outstanding bond put options to increase (decrease) in value. So, to hedge the company’s outstanding debt against a fall in interest rates (which would raise the market value of the company’s debt), the firm could buy call options, or sell put options, on bonds; this would produce a rise in the company’s assets (the call options or put options) to offset the rise in the market value of the company’s debt. Similarly, for a company that plans to borrow in the future, to hedge against a rise in future interest rates, the company could buy puts, or sell calls, on bonds.

APPENDIX A: THE PRICING OF FORWARD CONTRACTS

A.1. Financial Forward Contracts: In equation [A.1] it is stated that the relationship between the forward price and the spot price of a commodity is:

[pic] = [pic] ( PV[Inc] [A.1]

Why must equation [A.1] hold? First suppose that:

[pic] < [pic] ( PV[Inc] [A.2a]

which implies that:

[pic] ( [pic] > PV[Inc] ( 0 [A.2b]

If relation [A.2b] held, an arbitrageur could conduct the time 0 and time t transactions shown in Exhibits A.1 and A.2 below.

Exhibit A.1. Time 0 Transactions

|Time 0 Transaction |Cash Flow |

|Buy a forward contract on asset X |$0 |

|Lend [pic] at the risk-free rate [pic]* |( [pic] |

|Sell asset X short for [pic] |[pic] |

| Net time 0 cash flow: |[pic] ([pic] > 0 |

* Lend for t periods, producing payoff [pic] at time t.

Exhibit A.2. Time t Transactions

|Time t Transaction |Cash Flow |

|Acquire asset X for [pic] (under the forward contract) | ([pic] |

|Receive loan proceeds [pic] |[pic] |

|Cover the time 0 short position using the acquired asset X |$0 |

| Net time t cash flow: |$0 |

As long as relation [A.2a] holds, the above transactions provide the arbitrageur with a positive time t cash flow equal to {[pic] ([pic]}, and a zero time 0 cash flow. Therefore, the shorting of asset X (which drives down [pic]) and purchase of asset X futures (which drives up [pic]) will continue until equation [A.1] holds. [Note: A current owner of financial asset X would also be ahead by selling X spot and buying X forward; for this, in Exhibit A.1 simply replace “Sell asset X short for [pic]” with “Sell asset X for [pic].” This spot sale and forward process by owners of asset X would also force prices to satisfy equation [A.1].

To illustrate [A.2a], assume that [pic] = $650, t = 3, [pic] = $1,200, PV[Inc] = $500, and [pic] = .5% (assume that each time period is one month).

[pic] = $640.35 < $1,200 ( $500 = [pic] ( PV[Inc] [A.3]

Exhibit A.3. Time 0 Transactions

|Time 0 Transaction |Cash Flow |

|Sell asset X short for [pic] |$1,200 |

|Buy a forward contract on asset X |$0 |

|Lend [pic] at the risk-free rate [pic]* |( $640.35 |

| Net time 0 cash flow: | $559.65 |

* Lend for t periods, producing payoff [pic] at time t.

Exhibit A.4. Time t Transactions

|Time t Transaction |Cash Flow |

|Receive loan proceeds [pic] |$650 |

|Acquire asset X for [pic] (under the forward contract) | ( $650 |

|Cover the time 0 short position using the acquired asset X |$0 |

| Net time t cash flow: |$0 |

Arbitrageurs will continue to conduct the above transactions until equation [A.1] holds.

Now suppose that, instead of disequilibrium in equation [A.2a], we have the following disequilibrium:

[pic] > [pic] ( PV[Inc] [A.4a]

which implies that:

[pic] ( PV[Inc] ( [pic] < 0 [A.4b]

Using arguments like those applicable to relations [A.2a] and [A.2b], it can be shown that relations [A.4a] and [A.4b] are not equilibrium relationships. Transactions will occur that produce equation [A.1]. These transactions will continue until there has been a sufficient decline in [pic] relative to [pic] (due to switching from forward positions to spot positions in asset X) to produce [A.1].

A.2. Commodity Forward Contracts: Equation (10) states that the relationship between the forward and spot price of a commodity is as follows.

[pic] = [pic] + PV[Stor. Costs] ( PV[Con. Yield] [A.5]

To show why [A.5] holds, for variety we will use a slightly different argument than the one we employed in examining (10a) and (12a). Suppose that:

[pic] < [pic] + PV[Stor. Costs] ( PV[Con. Yield] [A.6a]

which implies:

[pic] ( [pic] > PV[Con. Yield] ( PV[Stor. Costs] [A.6b]

The left-hand side of [A.6b] is the spot price of commodity Y minus the amount that one would have to invest at the risk-free rate in order to have [pic] at time t to enable execution of the purchase of commodity Y under a futures contract. This difference is the premium paid for immediate ownership of commodity Y rather than ownership at time t. The right-hand side of [A.6b] is the present value of the net benefit of immediate ownership (spot) relative to attaining ownership at time t (forward). If the premium paid for immediate ownership exceeds that net benefit of immediate ownership, investors will purchase the commodity forward, not spot. This will drive down [pic] relative to [pic] until [A.5] holds. Now assume that:

[pic] > [pic] + PV[Stor. Costs] ( PV[Con. Yield] [A.7a]

which implies:

[pic] ( [pic] < PV[Con. Yield] ( PV[Stor. Costs] [A.7b]

Relation [A.7b] states that the premium paid for immediate ownership is less than the net benefit of immediate ownership, which means than investors will buy the commodity spot rather than forward. This will cause [pic] to rise relative to [pic] until [A.5] is satisfied.

The quantity [convenience yield ( storage costs] is called “net convenience yield.” In the equations, PV[Con. Yield] ( PV[Stor. Costs] could be replaced with PV[net convenience yield].

If all buyers and sellers of commodity Y are homogenous with respect to PV[Stor. Costs] and PV[Con. Yield], equation [A.5] holds for all buyers and sellers. If buyers and sellers differ with respect to PV[Stor. Costs] or PV[Con. Yield], equation [A.5] holds for the “marginal” buyer and the “marginal seller” of the commodity. The marginal buyer and marginal seller are indifferent between a forward and spot position. For those who are not indifferent, [A.5] is an inequality. A buyer (given that buyer’s particular Storage Costs and Convenience Yield) prefers a spot purchase to a forward purchase if the “=” in [A.5] is instead, for that buyer, a “>” sign, and prefers a forward purchase if the “=” is instead a “ ................
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