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Berk Chapter 30: Risk Management

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  • 1. Chapter 30 Risk Management
  • 2. Chapter Outline
    • 30.1 Insurance
    • 30.2 Commodity Price Risk
    • 30.3 Exchange Rate Risk
    • 30.4 Interest Rate Risk
  • 3. Learning Objectives
    • Explain what is meant by the statement that, in a perfect market, insurance is actuarially fair.
    • Compute the value of an actuarially fair insurance premium.
    • Explain why insurance for large risks that are difficult to diversify has a negative beta; evaluate the impact of that beta on price.
    • Discuss five market imperfections that are sources of value for insurance.
    • List and define three costs of insurance.
  • 4. Learning Objectives (cont'd)
    • Describe three risk management strategies firms use to hedge their exposure to commodity price movements.
    • Discuss the use of currency forwards and options contracts to hedge exchange rate risk.
    • Discuss the use of the cash-and-carry strategy in currency hedging.
    • Describe situations in which a firm would prefer currency options to futures for hedging.
    • Use the Black-Scholes formula to compute the value of a currency option.
  • 5. Learning Objectives (cont'd)
    • Define interest rate risk and discuss tools to manage that risk.
    • Define and compute duration of a single asset and of a portfolio.
    • Use duration to measure the change in value attributable to a change in yields.
    • Explain the use of equity duration to manage interest rate risk.
    • Describe the use of swaps in managing interest rate risk; explain how the use of swaps separates the risk of interest changes from the risk of changes in the firm’s credit quality.
  • 6. 30.1 Insurance
    • Insurance is the most common method firms use to reduce risk.
    • Property Insurance
      • A type of insurance companies purchase to compensate them for losses to their assets due to fire, storm damage, vandalism, earthquakes, and other natural and environmental risks
  • 7. 30.1 Insurance (cont'd)
    • Business Liability Insurance
      • A type of insurance that covers the costs that result if some aspect of a business causes harm to a third party or someone else’s property
    • Business Interruption Insurance
      • A type of insurance that protects a firm against the loss of earnings if the business is interrupted due to fire, accident, or some other insured peril
  • 8. 30.1 Insurance (cont'd)
    • Key Personnel Insurance
      • A type of insurance that compensates a firm for the loss or unavoidable absence of crucial employees in the firm
  • 9. The Role of Insurance: An Example
    • Consider an oil refinery with a 0.02% chance of being destroyed by a fire in the next year.
      • If it is destroyed, the firm estimates that it will lose $150 million in rebuilding costs and lost business.
  • 10. The Role of Insurance: An Example (cont'd)
    • The risk from fire can be summarized with a probability distribution:
  • 11. The Role of Insurance: An Example (cont'd)
    • Given this probability distribution, the firm’s expected loss from fire each year is $30,000.
      • 99.98% × ($0) + 0.02% × ($150 million) = $30,000
      • While the expected loss is relatively small, the firm faces a large downside risk if a fire does occur.
        • The firm can manage the risk purchasing insurance to compensate its loss of $150 million.
  • 12. The Role of Insurance: An Example (cont'd)
    • Insurance Premium
      • The fee a firm pays to an insurance company for the purchase of an insurance policy
  • 13. Insurance Pricing in a Perfect Market
    • Actuarially Fair
      • When the NPV from selling insurance is zero because the price of insurance equals the present value of the expected payment
  • 14. Insurance Pricing in a Perfect Market (cont'd)
    • If r L is the appropriate cost of capital given the risk of the loss, the actuarially fair premium is calculated as follows:
      • Actuarially Fair Insurance Premium
        • r L depends on the risk being insured.
  • 15. Insurance Pricing in a Perfect Market (cont'd)
    • Consider again the oil refinery. The risk of fire is specific to this firm and, therefore, diversifiable.
      • By pooling together the risks from many policies, insurance companies can create very-low-risk portfolios whose annual claims are relatively predictable. In other words, the risk of fire has a beta of zero, so it will not command a risk premium. In this case, r L equals the risk-free interest rate.
  • 16. Insurance Pricing in a Perfect Market (cont'd)
    • Not all insurable risks have a beta of zero.
      • Some risks, such as hurricanes and earthquakes may be difficult to diversify completely.
      • For risks that cannot be fully diversified, the cost of capital r L will include a risk premium.
  • 17. Insurance Pricing in a Perfect Market (cont'd)
    • By its very nature, insurance for non-diversifiable hazards is generally a negative beta asset (it pays off in bad times).
      • Thus, the risk-adjusted rate r L for losses is less than the risk-free rate r f , leading to a higher insurance premium in the actuarially fair insurance premium equation.
        • While firms that purchase insurance earn a return r L < r f on their investment, because of the negative beta of the insurance payoff, it is still a zero-NPV transaction.
  • 18. Textbook Example 30.1
  • 19. Textbook Example 30.1 (cont'd)
  • 20. Alternative Example 30.1
    • Problem
      • As the owner of a concession booth in a major airport, you decide to purchase insurance that will pay $2 million in the event the airport terminal is destroyed by terrorists. Suppose the likelihood of such a loss is 0.05%, the risk-free interest rate is 3%, and the expected return of the market is 8%. If the risk has a beta of zero, what is the actuarially fair insurance premium? What is the premium if the beta of terrorism insurance is −3?
  • 21. Alternative Example 30.1 (cont’d)
    • Solution
      • The expected loss is 0.05% × $2 million = $1,000. If the risk has a beta of zero, we compute the insurance premium using the risk-free interest rate: ($1,000)/1.03 = $970.87. If the beta of the risk is not zero, we can use the CAPM to estimate the appropriate cost of capital.
  • 22. Alternative Example 30.1 (cont’d)
    • Solution (cont’d)
      • Given a beta for the loss, β L , of −3, and an expected market return, r mkt , of 8%:
      • r L = r f + β L (r mkt − r f ) = 3% − 3 (8% − 3%) = −12%
      • In this case, the actuarially fair premium is ($1,000)/(1 − 0.12) = $1,136.36. Although this premium exceeds the expected loss, it is a fair price given the negative beta of the risk.
  • 23. The Value of Insurance
    • In a perfect capital market, there is no benefit to the firm from any financial transaction, including insurance.
      • Insurance is a zero-NPV transaction that has no effect on value.
      • The value of insurance comes from reducing the cost of market imperfections.
  • 24. The Value of Insurance (cont'd)
    • Consider the potential benefits of insurance with respect to the following market imperfections:
    • Bankruptcy and Financial Distress Costs
      • By insuring risks that could lead to distress, the firm can reduce the likelihood that it will incur these costs.
  • 25. The Value of Insurance (cont'd)
    • Issuance Costs
      • When a firm experiences losses, it may need to raise cash from outside investors by issuing securities.
      • Insurance provides cash to the firm to offset losses, reducing the firm’s need for external capital thus reducing issuance costs.
  • 26. Textbook Example 30.2
  • 27. Textbook Example 30.2 (cont'd)
  • 28. Alternative Example 30.2
    • Problem
      • Suppose the risk of a railroad accident for a major railroad is 1.2% per year, with a beta of zero.
      • If the risk-free rate is 6%, what is the actuarially fair premium for a policy that pays $100 million in the event of a loss?
  • 29. Alternative Example 30.2
    • Problem (continued)
      • What is the NPV of purchasing insurance for an airline that would experience $25 million in financial distress costs and $15 million in issuance costs in the event of a loss if it were uninsured?
  • 30. Alternative Example 30.2
    • Solution
      • The expected loss is:
        • 1.2% × $100 million = $1,200,000
      • The actuarially fair premium is:
        • $1,200,000 ÷ 1.06 = $1,132,075
  • 31. Alternative Example 30.2
    • Solution (continued)
      • The total benefit of the insurance to the railroad is $100 million plus an additional $40 million in distress and issuance costs that it can avoid if it has insurance.
      • The NPV from purchasing the insurance is:
  • 32. The Value of Insurance (cont'd)
    • Tax Rate Fluctuations
      • When a firm is subject to graduated income tax rates, insurance can produce a tax savings if the firm is in a higher tax bracket when it pays the premium than the tax bracket it is in when it receives the insurance payment in the event of a loss.
  • 33. The Value of Insurance (cont'd)
    • Debt Capacity
      • Because insurance reduces the risk of financial distress, it can relax the tradeoff between leverage & financial distress costs and allow the firm to increase its use of debt financing.
  • 34. The Value of Insurance (cont'd)
    • Managerial Incentives
      • By eliminating the volatility that results from perils outside management’s control, insurance turns the firm’s earnings and share price into informative indicators of management’s performance.
  • 35. The Value of Insurance (cont'd)
    • Risk Assessment
      • Insurance companies specialize in assessing risk and will often be better informed about the extent of certain risks faced by the firm than the firm’s own managers.
  • 36. The Costs of Insurance
    • Market imperfections can raise the cost of insurance above the actuarially fair price.
  • 37. The Costs of Insurance (cont'd)
    • Insurance Market Imperfections
      • Three main frictions may arise between the firm and its insurer.
        • Transferring the risk to an insurance company entails administrative and overhead costs.
        • Adverse selection: A firm’s desire to buy insurance may signal that it has above-average risk.
  • 38. The Costs of Insurance (cont'd)
    • Insurance Market Imperfections
      • Three main frictions may arise between the firm and its insurer.
        • Agency costs
          • Moral Hazard: When purchasing insurance reduces a firm’s incentive to avoid risk.
            • For example, after purchasing fire insurance, a firm may decide to cut costs by reducing expenditures on fire prevention.
  • 39. The Costs of Insurance (cont'd)
    • Addressing Market Imperfections
      • Insurance companies try to mitigate adverse selection and moral hazard costs in a number of ways.
      • For example, they may
        • Screen applicants to assess their risk as accurately as possible
        • Investigate losses to look for evidence of fraud or deliberate intent
  • 40. The Costs of Insurance (cont'd)
    • Addressing Market Imperfections
      • Deductible
        • A provision of an insurance policy in which an initial amount of loss is not covered by the policy and must be paid by the insured
      • Policy Limits
        • The provisions of an insurance policy that limit the amount of loss that the policy covers regardless of the extent of the damage
  • 41. Textbook Example 30.3
  • 42. Textbook Example 30.3 (cont'd)
  • 43. The Insurance Decision
    • For insurance to be attractive, the benefit to the firm must exceed the additional premium charged by the insurer.
      • Insurance is most likely to be attractive to firms that are currently financially healthy, do not need external capital, and are paying high current tax rates.
        • They will benefit most from insuring risks that can lead to cash shortfalls or financial distress, and that insurers can accurately assess and monitor to prevent moral hazard.
  • 44. 30.2 Commodity Price Risk
    • Many risks that firms face arise naturally as part of their business operations.
      • For example, the risk from increases in the price of oil is one of the most important risks that faces an airline.
        • Firms can reduce, or hedge , their exposure to commodity price movements.
          • Like insurance, hedging involves contracts or transactions that provide the firm with cash flows that offset its losses from price changes.
  • 45. Hedging with Vertical Integration and Storage
    • Vertical Integration
      • Refers to the merger of a firm and its supplier or a firm and its customer.
        • Because an increase in the price of the commodity raises the firm’s costs and the supplier’s revenues, these firms can offset their risks by merging.
        • Vertical integration can add value if combining the firms results in important synergies.
        • Vertical integration is not a perfect hedge.
  • 46. Hedging with Vertical Integration and Storage (cont'd)
    • Long-term storage of inventory is another strategy for offsetting commodity price risk.
      • For example, an airline concerned about rising fuel costs could purchase a large quantity of fuel today and store the fuel until it is needed. By doing so, the firm locks in its cost for fuel at today’s price plus storage costs.
        • However, storage costs may be too high for this strategy to be attractive.
  • 47. Hedging with Vertical Integration and Storage (cont'd)
    • Long-term storage of inventory also requires a substantial cash outlay upfront.
      • If the firm does not have the required cash, it would need to raise external capital and would suffer issuance and adverse selection costs.
    • Maintaining large amounts of inventory would dramatically increase working capital requirements for the firm.
  • 48. Hedging with Long-Term Contracts
    • Consider Southwest Airlines.
      • In early 2000, when oil prices were close to $20 per barrel, the CFO developed a hedging strategy to protect the airline from a surge in oil prices. By the time oil prices soared above $30 per barrel later that year Southwest had signed contracts guaranteeing a price for its fuel equivalent to $23 per barrel.
  • 49. Hedging with Long-Term Contracts (cont'd)
    • However, had oil prices fallen below $23 per barrel in the fall of 2000, Southwest’s hedging policy would have obligated it to pay $23 per barrel for its oil.
      • Southwest accomplished it’s objective by locking in its cost of oil at $23 per barrel, regardless of what the price of oil did on the open market.
  • 50. Figure 30.1 Commodity Hedging Smoothes Earnings
  • 51. Textbook Example 30.4
  • 52. Textbook Example 30.4 (cont'd)
  • 53. Alternative Example 30.4
    • Problem
      • Consider a cereal manufacturer that will need 20 million bushels of corn next year.
      • The current market price of corn is $3 per bushel.
      • At $3 per bushel, the firm expects earnings before interest and taxes of $50 million next year.
  • 54. Alternative Example 30.4
    • Problem (continued)
      • What will the firm’s EBIT be if the price of corn rises to $3.50 per bushel?
      • What will EBIT be if the price of corn falls to $2.25 per bushel?
      • What will EBIT be in each scenario if the firm enters into a supply contract for corn for a fixed price of $3.25 per bushel?
  • 55. Alternative Example 30.4
    • Solution
      • At $3.50 per bushel:
        • EBIT = $50,000,000 − [($3.50 − $3.00) × 20,000,000] = $40,000,000
      • At $2.25 per bushel:
        • EBIT = $50,000,000 − [($2.25 − $3.00) × 20,000,000] = $65,000,000
      • At $3.25 per bushel:
        • EBIT = $50,000,000 − [($3.25 − $3.00) × 20,000,000] = $45,000,000
  • 56. Hedging with Long-Term Contracts (cont'd)
    • Long-term supply contracts have several potential disadvantages.
      • They expose each party to the risk that the other party may default and fail to live up to the terms of the contract.
        • Thus, while they insulate the firms from commodity price risk, they expose them to credit risk.
  • 57. Hedging with Long-Term Contracts (cont'd)
      • Long-term supply contracts cannot be entered into anonymously; the buyer and seller know each other’s identity.
        • This lack of anonymity may have strategic disadvantages.
      • The market value of the contract at any point in time may not be easy to determine, making it difficult to track gains and losses, and it may be difficult or even impossible to cancel the contract if necessary.
  • 58. Hedging with Futures Contracts
    • Futures Contract
      • An agreement to trade an asset on some future date, at a price that is locked in today
        • Futures contracts are traded anonymously on an exchange at a publicly observed market price and are generally very liquid.
        • Both the buyer and the seller can get out of the contract at any time by selling it to a third party at the current market price.
        • Futures contracts eliminate credit risk.
  • 59. Figure 30.2 Futures Prices for Light, Sweet Crude Oil, July 2009
  • 60. Hedging with Futures Contracts
    • Futures prices are not prices that are paid today.
      • Rather, they are prices agreed to today, to be paid in the future.
        • The futures prices are based on the supply and demand for each delivery date.
  • 61. Hedging with Futures Contracts (cont'd)
    • Eliminating Credit Risk
      • Futures exchanges use two mechanisms to prevent buyers or sellers from defaulting.
        • Traders are required to post collateral when buying or selling commodities using futures contracts.
          • This collateral serves as a guarantee that traders will meet their obligations.
      • Margin
        • Collateral that investors are required to post when buying or selling futures contracts
  • 62. Hedging with Futures Contracts (cont'd)
    • Eliminating Credit Risk
      • Marking to Market
        • Computing gain and losses each day based on the change in the market price of a futures contract
  • 63. Hedging with Futures Contracts (cont'd)
    • Marking to Market: An Example
      • Suppose a buyer who enters into the contract has committed to pay the futures price of $81 per barrel for oil.
        • If the next day the futures price is only $79 per barrel, the buyer has a loss of $2 per barrel on her position.
          • This loss is settled immediately by deducting $2 from the buyer’s margin account.
        • If the price rises to $80 per barrel on the following day, the gain of $1 is added to the buyer’s margin account.
  • 64. Hedging with Futures Contracts (cont'd)
    • Marking to Market: An Example
      • The buyer’s cumulative loss is the sum of these daily amounts and always equals the difference between the original contract price of $81 per barrel and the current contract price.
  • 65. Hedging with Futures Contracts (cont'd)
    • Marking to Market: An Example
      • If the price of oil is ultimately $59 per barrel, the buyer will have lost $22 per barrel in her margin account.
        • Thus her total cost is $59 + $22 = $81 per barrel, the price for oil she originally committed to.
          • Through this daily marking to market, buyers and sellers pay for any losses as they occur, rather than waiting until the final delivery date. In this way, the firm avoids the risk of default.
  • 66. Table 30.1 Example of Marking to Market and Daily Settlement for the July 2012 Light, Sweet Crude Oil Futures Contract ($/bbl)
  • 67. Deciding to Hedge Commodity Price Risk
    • The potential benefits of hedging commodity price risk include reduced financial distress and issuance costs, tax savings, increased debt capacity, and improved managerial incentives and risk assessment.
  • 68. Deciding to Hedge Commodity Price Risk
    • Speculate
      • When investors use futures to place a bet on the direction in which they believe the market is likely to move
        • A firm speculates when it enters into contracts that do not offset its actual risks.
        • Speculating increases the firm’s risk rather than reducing it.
  • 69. 30.3 Exchange Rate Risk
    • Floating Rate
      • An exchange rate that changes depending on supply and demand in the market
        • The supply and demand for each currency is driven by
          • Firms trading goods
          • Investors trading securities
          • The actions of central banks in each country
      • Most foreign exchange rates are floating rates.
  • 70. 30.3 Exchange Rate Risk (cont'd)
    • Fluctuating exchanges rates cause a problem known as the importer–exporter dilemma.
      • Consider a U.S. firm that imports parts from Italy.
        • If the supplier sets the price of its parts in euros, then the U.S. firm faces the risk that the dollar may fall, making euros, and therefore the parts, more expensive.
        • If the supplier sets its prices in dollars, then the supplier faces the risk that the dollar may fall and it will receive fewer euros for the parts it sells to the U.S. firm.
  • 71. Figure 30.3 Dollars per Euro ($/ € ), 1999-2009
  • 72. Textbook Example 30.5
  • 73. Textbook Example 30.5 (cont'd)
  • 74. Hedging with Forward Contracts
    • By entering into a currency forward contract, a firm can lock in an exchange rate in advance and reduce or eliminate its exposure to fluctuations in a currency’s value.
  • 75. Hedging with Forward Contracts (cont'd)
    • A currency forward contract specifies
      • An exchange rate
      • An amount of currency to exchange
      • A delivery date on which the exchange will take place
  • 76. Hedging with Forward Contracts
    • Forward Exchange Rate
      • The exchange rate set in a currency forward contract: it applies to an exchange that will occur in the future.
  • 77. Textbook Example 30.6
  • 78. Textbook Example 30.6 (cont'd)
  • 79. Figure 30.4 The Use of Currency Forwards to Eliminate Exchange Rate Risk
  • 80. Cash-and-Carry and the Pricing of Currency Forwards
    • Cash-and-Carry Strategy
      • A strategy used to lock in the future cost of an asset by buying the asset for cash today and “carrying” it until a future date
    • The cash-and-carry strategy also enables a firm to eliminate exchange rate risk.
  • 81. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • The Law of One Price and the Forward Exchange Rate
      • Currency forward contracts allow investors to exchange a foreign currency in the future for dollars in the future at the forward exchange rate.
  • 82. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
      • Currency Timeline
        • Indicates time horizontally by dates and currencies vertically
          • Note: An example of a currency timeline is on the following slide.
      • Spot Exchange Rate
        • The current foreign exchange rate
  • 83. Figure 30.5 Currency Timeline Showing Forward Contract and Cash-and-Carry Strategy
  • 84. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • An investor can convert euros to dollars today at the spot exchange rate, S $/ € .
    • By borrowing or lending at the dollar interest rate r $ , an investor can exchange dollars today for dollars in one year.
    • An investor can convert euros today for euros in one year at the euro interest rate r € .
  • 85. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • The cash-and-carry strategy consists of the following simultaneous trades
      • Borrow euros today using a one-year loan with the interest rate r €
      • Exchange the euros for dollars today at the spot exchange rate S $/ €
      • Invest the dollars today for one year at the interest rate r $
  • 86. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • In one year’s time, an investor will owe euros and receive dollars. That is, they have converted euros in one year to dollars in one year, just as with the forward contract.
      • Because the forward contract and the cash-and-carry strategy accomplish the same conversion, by the Law of One Price they must do so at the same rate.
  • 87. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • Combining the rates used in the cash-and-carry strategy leads to the following no-arbitrage formula for the forward exchange rate:
      • Covered Interest Parity
  • 88. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • Letting T equal the number of years, the no-arbitrage forward rate for an exchange that will occur T years in the future is
  • 89. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • Covered Interest Parity Equation
      • States that the difference between the forward and spot exchange rates is related to the interest rate differential between the currencies
  • 90. Textbook Example 30.7
  • 91. Textbook Example 30.7 (cont'd)
  • 92. Cash-and-Carry and the Pricing of Currency Forwards (cont'd)
    • Advantages of Forward Contracts
      • A forward contract is simpler, requiring one transaction rather than three.
      • Many firms are not able to borrow easily in different currencies and may pay a higher interest rate if their credit quality is poor.
  • 93. Textbook Example 30.8
  • 94. Textbook Example 30.8 (cont'd)
  • 95. Hedging with Options
    • Currency options are another method to manage exchange rate risk.
      • Assume that in December 2005, the one-year forward exchange rate was $1.20 per euro. A firm that will need euros in one year can buy a call option on the euro, giving it the right to buy euros at a maximum price.
  • 96. Hedging with Options (cont'd)
    • Suppose a one-year European call option on the euro with a strike price of $1.20 per euro trades for $0.05 per euro.
      • The table on the following slide shows the outcome from hedging with a call option.
  • 97. Table 30.2 Cost of Euros ($/€) When Hedging with a Currency Option with a Strike Price of $1.20/€ and an Initial Premium of $0.05/€
  • 98. Hedging with Options (cont'd)
    • If the spot exchange rate is less than the $1.20 per euro strike price of the option, then the firm will not exercise the option and will convert dollars to euros at the spot exchange rate.
  • 99. Hedging with Options (cont'd)
    • If the spot exchange rate is more than $1.20 per euro, the firm will exercise the option and convert dollars to euros at the rate of $1.20 per euro. Adding in the initial cost of the option gives the total dollar cost per euro paid by the firm.
  • 100. Hedging with Options (cont'd)
    • The following slide compares hedging with options to the alternative of hedging with a forward contract or not hedging at all.
  • 101. Figure 30.6 Comparison of Hedging the Exchange Rate Using a Forward Contract, an Option, or No Hedge
  • 102. Hedging with Options (cont'd)
    • If the firm does not hedge at all, its cost for euros is simply the spot exchange rate.
    • If the firm hedges with a forward contract, it locks in the cost of euros at the forward exchange rate and the firm’s cost is fixed.
    • If the firm hedges with options, it puts a cap on its potential cost, but will benefit if the euro depreciates in value.
  • 103. Hedging with Options (cont'd)
    • Options Versus Forward Contracts
      • A firm may use options instead of forward contract:
        • So the firm can benefit if the exchange rate moves in their favor and not be stuck paying an above-market rate
        • If the transaction they are hedging might not take place
  • 104. Textbook Example 30.9
  • 105. Textbook Example 30.9 (cont'd)
  • 106. Textbook Example 30.9 (cont'd)
  • 107. Hedging with Options (cont'd)
    • Currency Option Pricing
      • If the current spot exchange rate is S dollars per euro and the dollar and euro interest rates are r $ and r € , respectively, then the price of a European call option on the euro that expires in T years with a strike price of K dollars per euro is:
  • 108. Hedging with Options (cont'd)
    • Price of a Call Option on a Currency
      • Where
      • F T is the forward exchange rate from Eq. 30.3 and
  • 109. Textbook Example 30.10
  • 110. Textbook Example 30.10 (cont'd)
  • 111. 30.4 Interest Rate Risk
    • Interest Rate Risk Measurement: Duration
      • A security’s duration is computed as:
        • Where C t is the cash flow on date t , PV(C t ) is its present value (evaluated at the bond’s yield), and P= Σ t PV(C t ) is the total present value of the cash flows
          • Therefore, the duration weights each maturity t by the percentage contribution of its cash flow to the total present value, PV(C t ) ∕ P .
  • 112. Textbook Example 30.11
  • 113. Textbook Example 30.11 (cont'd)
  • 114. Table 30.3 Computing the Duration of a Coupon Bond
  • 115. 30.4 Interest Rate Risk (cont'd)
    • Interest Rate Risk Measurement: Duration
      • Duration and Interest Rate Sensitivity: If r, the APR used to discount a stream of cash flows, increases to r +  , where  is a small change, then the present value of the cash flows changes by approximately :
        • Where k is the number of compounding periods per year of the APR
  • 116. Textbook Example 30.12
  • 117. Textbook Example 30.12 (cont'd)
  • 118. Duration-Based Hedging
    • If the market value of a firm’s assets and liabilities are affected by changes in interest rates, the firm’s equity value will also be affected.
      • The firm’s sensitivity to changes in interest rates can be measured by computing the duration of its assets and liabilities.
  • 119. Duration-Based Hedging
    • Savings and Loans: An Example
      • Consider a typical S&L.
        • These institutions hold short-term deposits (checking and savings accounts, certificates of deposit, etc.). They also make long-term loans (car loans, home mortgages, etc.).
          • Most S&Ls face a problem because the duration of the loans they make is generally longer than the duration of their deposits.
  • 120. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • When the durations of a firm’s assets and liabilities are significantly different, the firm has a duration mismatch .
        • This mismatch puts the S&L at risk if interest rates change significantly.
  • 121. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • The following slide shows the market-value balance sheet for Acorn Savings and Loan:
  • 122. Table 30.4 Market-Value Balance Sheet for Acorn Savings and Loan
  • 123. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • The duration of a portfolio of investments is the value-weighted average of the durations of each investment in the portfolio.
        • A portfolio of securities with market values A and B and durations D A and D B , respectively, has the following duration:
  • 124. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • The duration of Acorn’s assets is:
      • The duration of Acorn’s liabilities is:
  • 125. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • The duration of Acorn’s equity is calculated as:
  • 126. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • Therefore, if interest rates rise by 1%, the value of Acorn’s equity will fall by about 40%.
        • This decline in the value of equity will occur as a result of the value of Acorn’s assets decreasing by approximately $16 million, while the value of its liabilities decrease by only $9.9 million. Acorn’s market value of equity therefore declines by $6.1 million or 40.67%.
          • 5.33% × $300 million = $16 million
          • 3.47% × $285 million = $9.9 million
            • ($16 million – $9.9million) ∕ $15 million = 40.67%
  • 127. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • To fully protect its equity from an overall increase or decrease in the level of interest rates, Acorn needs an equity duration of zero.
        • A portfolio with a zero duration is called a duration-neutral portfolio or an immunized portfolio , which means that for small interest rate fluctuations, the value of equity should remain unchanged.
        • Adjusting a portfolio to make its duration zero is referred to as immunizing the portfolio.
  • 128. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • To make its equity duration neutral, Acorn must reduce the duration of its assets or increase the duration of its liabilities.
        • The firm can lower the duration of its assets by selling some of its mortgages in exchange for cash.
  • 129. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • Acorn would like to reduce the duration of its equity from 40.7 to 0.
        • Because the duration of the mortgages will change from 8 to 0 if the S&L sells the mortgages for cash, Acorn must sell $76.3 million worth of mortgages.
          • (40.7 – 0) × 15 ∕ (8 – 0) = $76.3
  • 130. Duration-Based Hedging (cont'd)
    • Savings and Loans: An Example
      • If it Acorn does so, the duration of its assets will decline to:
      • Thus the equity duration will fall to:
  • 131. Table 30.5 Market-Value Balance Sheet for Acorn Savings and Loan After Immunization
  • 132. Duration-Based Hedging (cont'd)
    • A Cautionary Note
      • Duration matching has some important limitations.
        • The duration of a portfolio depends on the current interest rate.
          • As interest rates change, the duration of the portfolio changes.
            • Maintaining a duration-neutral portfolio requires constant adjusting as interest rates change.
  • 133. Duration-Based Hedging (cont'd)
    • A Cautionary Note
      • Duration matching has some important limitations.
        • A duration-neutral portfolio is only protected against parallel shifts in the yield curve.
          • If short-term interest rates were to rise while long-term rates remained stable, then short-term securities would fall in value relative to long-term securities, despite their shorter duration.
  • 134. Swap-Based Hedging
    • Interest Rate Swap
      • A contract in which two parties agree to exchange the coupons from two different types of loans
        • Interest rate swaps are an alternative way of modifying the firm’s interest rate risk exposure.
  • 135. Swap-Based Hedging (cont'd)
    • Interest Rate Swap
      • In a standard interest rate swap, one party agrees to pay coupons based on a fixed interest rate in exchange for receiving coupons based on the prevailing market interest rate during each coupon period.
        • An interest rate that adjusts to current market conditions is called a floating rate . Thus the parties exchange a fixed-rate coupon for a floating-rate coupon, which explains why this swap is also called a “fixed-for-floating interest rate swap.”
  • 136. Swap-Based Hedging (cont'd)
    • Interest Rate Swap
      • Consider a five-year, $100 million interest rate swap with a 7.8% fixed rate. Standard swaps have semiannual coupons, so that the fixed coupon amounts would be $3.9 million every six months.
        • ½ × 7.8% × $100 million = $3.9
  • 137. Swap-Based Hedging (cont'd)
    • Interest Rate Swap
      • The floating-rate coupons are often based on the six-month LIBOR.
        • Each coupon is calculated based on the six-month interest rate that prevailed in the market six months prior to the coupon payment date.
  • 138. Swap-Based Hedging (cont'd)
    • Interest Rate Swap
      • The following slide shows the cash flows of the swap under a hypothetical scenario for LIBOR rates over the life of the swap.
        • For example, at the first coupon date in six months, the fixed coupon is $3.9 million and the floating-rate coupon is $3.4 million (½ × 6.8% × $10 million = $3.4 million), for a net payment of $0.5 million from the fixed- to the floating-rate payer.
  • 139. Table 30.6 Cash Flows ($ millions) for a $100 million Fixed-for-Floating Interest Rate Swap
  • 140. Swap-Based Hedging (cont'd)
    • Interest Rate Swap
      • Each payment of the swap is equal to the difference between the fixed- and floating rate coupons.
        • Because the $100 million swap amount is used only to calculate the coupons but is never actually paid, it is referred to as the notional principal of the swap.
        • The fixed rate of the swap contract is set based on current market conditions so that the swap is a fair deal (i.e., has an NPV of zero) for both sides.
  • 141. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • The interest rate a firm pays on its loans can fluctuate for two reasons.
        • The risk-free interest rate in the market may change.
        • The firm’s credit quality can vary over time.
      • By combining swaps with loans, firms can choose which of these sources of interest rate risk they will tolerate and which they will eliminate.
  • 142. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • Consider Alloy Cutting Corporation (ACC).
        • It needs to borrow $10 million to fund this expansion. Currently, the six-month LIBOR is 4% and the ten-year interest rate for AA-rated firms is 6%. Given ACC’s low current credit rating, the bank will charge the firm a spread of 1% above these rates.
  • 143. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • ACC’s managers are considering whether they should borrow on a short-term basis and then refinance the loan every six months or whether they should borrow using a long-term, ten-year loan.
  • 144. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • ACC can use an interest rate swap to combine the best of both strategies.
        • First, ACC can borrow the $10 million it needs for expansion using a short-term loan that is rolled over every six months.
          • The interest rate on each loan will be r t +  t where r t is the new LIBOR and  t is the spread ACC must pay based on its credit rating at the time.
            • Given ACC’s belief that its credit quality will improve over time,  t should decline from its current 1% level.
  • 145. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • To eliminate the risk of an increase in r t in the future, ACC can enter into a ten-year interest rate swap in which it agrees to pay a fixed rate of 6% per year in exchange for receiving the floating rate.
  • 146. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • Combining the cash flows from the swap with ACC’s short-term borrowing, ACC’s net borrowing cost is computed as follows:
  • 147. Swap-Based Hedging (cont'd)
    • Combining Swaps with Standard Loans
      • ACC will have an initial net borrowing cost of 7% but this cost will decline in the future as its credit rating improves and the spread declines.
        • At the same time, this strategy protects ACC from an increase in interest rates.
  • 148. Table 30.7 Trade-Offs of Long-Term Versus Short-Term Borrowing for ACC
  • 149. Textbook Example 30.13
  • 150. Textbook Example 30.13 (cont'd)
  • 151. Swap-Based Hedging (cont'd)
    • Using a Swap to Change Duration
      • A swap contract will alter the duration of a portfolio according to the difference in the duration of the corresponding long-term and short-term bonds.
        • Swaps are a convenient way to alter the duration of a portfolio without buying or selling assets.
  • 152. Textbook Example 30.14
  • 153. Textbook Example 30.14 (cont'd)
  • 154. Chapter Quiz
    • How can insurance add value to a firm?
    • Identify the costs of insurance that arise due to market imperfections.
    • Discuss risk management strategies that firms use to hedge commodity price risk.
    • What are the potential risks associated with hedging using futures contracts?
    • How can firms hedge exchange rate risk?
    • Why may a firm prefer to hedge exchange rate risk with options rather than forward contracts?
    • How can firms hedge exchange rate risk?
    • Why may a firm prefer to hedge exchange rate risk with options rather than forward contracts?

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