CHAPTER 26: DERIVATIVES AND HEDGING RISK: CHAPTER 26: DERIVATIVES AND HEDGING RISK TOPICS: 26.1 Forward Contracts 26.2 Futures 26.3 Hedging 26.4 Interest Rate Futures Contracts 26.5 Duration Hedging 26.6 Swaps Hedging vs. speculation: The case of Orange County


Overview: Overview Risks to be managed, and the methods used to finance them. Commodity price risk (futures) Interest rate exposure (duration hedging/swaps) FX exposure (derivatives) Hedging Find two closely related assets Buy one and sell the other in proportions that minimize the risk of your net position If the assets are perfectly correlated, your net position is risk free


How risk is managed: How risk is managed Production costs: $1.50/bu Selling price in Sept.: Unknown What can the farmer do to reduce risk? 1. Do nothing 2. Buy Crop Insurance 3. Buy a put option 4. Enter a Forward/futures contract to sell Plant Harvest May Sept.


Volatility in oil prices: Volatility in oil prices


10-Year Treasury Yield 1953-2006 : 10-Year Treasury Yield 1953-2006


26.1 Forward Contracts: 26.1 Forward Contracts A forward contract is an agreement to buy / sell an asset at a particular future time for a certain price Forward contracts are customized and not usually traded on an exchange The long (short) position agrees to buy (sell) the asset on the specified date for the delivery price When the contract is entered into, the delivery price is chosen so that the value of the contract is zero to each party. You don’t pay any money upfront.


Examples of a forward: Examples of a forward Pizza forward contract. Order pizza by phone. Specify topping (type), size (large), delivery time and location and price - fixed when contract is established. Pay on delivery. Energy forward You buy 50,000 cubic feet (50 Mcf) of heating gas in summer from your heating company for $10 per thousand cubic feet (Mcf), deliverable from Jan. – March. Long in forward: You Short: Heating Co.


Problems with hedging with forwards: Problems with hedging with forwards Hedged with forwards is imperfect, since you do not know the quantity you will have to trade. There is credit risk with forward contracts. Bipartisan arrangement In the previous example, if heating gas price increase in the winter, your heating company will lose, and he might default.


26.2 Futures: 26.2 Futures Very similar to forwards in payoff profile, but addresses credit risk problem by “marking-to-market” every day. Highly standardized contracts (delivery location, contract size etc.), which permit exchange trading. (e.g. www.cbot.com) More institutional details: The exact delivery date is usually not specified in a futures contract; rather it is some time interval within the delivery month Actual delivery rarely occurs, instead parties close out positions by taking offsetting transactions prior to maturity. Cash settlement. There are commodities futures and financial futures (stocks, bonds and currencies).


Example: Corn Futures at CBOT: Example: Corn Futures at CBOT 4 Digit Price Quote: Fourth digit is 1/8 cent/bu


Profit and Loss: Profit and Loss Profit to a person who buys futures (i.e. goes long on futures) = Ultimate market price - Initial futures price = ST - Ft (ST: spot price at time of delivery T Ft: Futures price set at t payable at T) If an offsetting position is taken before the expiry, then profit/loss is Fnew - Ft ,where Fnew is the new futures price at the offsetting time. Profit to a person who sells futures (i.e. goes short on futures) = Initial futures price - Ultimate market price = Ft - ST


Marking to market/Margin: Marking to market/Margin Profit/loss is settled EVERY day on a margin account Minimize default risk Details Initial margin If the value of the margin account falls below the maintenance margin, the contract holder receives a margin call. You need to add $ to bring margin balance back to initial margin level (otherwise contract will be forced to close out.)


Slide13: What is the profit and loss to the investor, e.g. at days 2 and 3? (2) When does he receive a margin call? What to do when receiving a margin call? (3) What’s his ultimate gain/loss? Example: Marking-to-market Consider an investor who enters a futures contract to purchase 100 oz. of gold at the futures price of $275 per ounce. Suppose that the initial margin is set at $2,000 and the maintenance margin is set at $1,500. The contract is closed out after 6 days.


Futures vs. Options: Futures vs. Options Similarities Deferred delivery markets Limited number of contracts Standardized contracts Exchange is middleman Differences Options Longs have right, not obligation to buy/sell Frequent exercise (if appropriate) Futures Both longs and shorts have obligation to buy/sell Daily price limits Marked-to-market Delivery seldom occurs


26.3 Hedging with futures –Locking into future prices: 26.3 Hedging with futures –Locking into future prices A firm, which will be selling an asset in the future, can hedge by taking short futures positions (a short hedge). If the asset price falls, the firm _____ on the asset sale but _____ on the futures contract; if the asset price rises, the firm _____ on the sale but _____ on the futures contract Similarly a firm which purchases an asset can hedge by taking a long hedge Futures hedging does not necessarily improve the overall outcome objective of hedging is to reduce risk by making outcome less variable


Example: Short hedge: Example: Short hedge It is November 2003. The canola farmer is worried about the price of his crop (output). He sells canola futures; say 50 tonnes Feb 2004 at $300 per tonne. In February 2004, when the farmer harvests his crop, the market price of canola is $250 per tonne. The farmer's profits from futures = _____________ per tonne The farmer's proceeds from sale of canola = ___________ per tonne Total =_______ per tonne Suppose in February 2004, when the farmer harvests his crop, the market price of canola is $450 per tonne. The farmer's profits from futures = _____________ per tonne The farmer's proceeds from sale of canola = ___________ per tonne Total = ___________ per tonne


How do you determine forward/futures price? : How do you determine forward/futures price? Start with forward prices, assuming that the underlying asset does not provide a dividend Notations: T = maturity date of forward contract in years rf = annual risk free rate (suppose that you lend and borrow at riskfree rate) S0 = spot price of asset at time 0 F0 = forward price of asset at time 0 (not the value of the forward contract)


Slide18: If rf is a continuously compounded rate, If the underlying asset provides a known dividend (e.g., interest payment on a interest rate forward), let I0 = PV of dividend paid by the underlying asset from 0 to T,


Example: Example A forward contract on heating gas with an expiry of 6 months. Current price is $7.95 /Mcf. Annual riskfree rate is 5%. What’s the forward price?


26.4 Interest Rate Futures Contracts: 26.4 Interest Rate Futures Contracts Futures contract whose underlying security is a debt obligation. Let’s look at short-term interest rate futures Main uses are a) hedging against or speculating on interest rate movements b) locking into the forward term structure (lock into future interest rates).


Pricing of Bonds: A quick review: Pricing of Bonds: A quick review Consider a Government of Canada bond that pays a semiannual coupon of $C for the next T years: The yield to maturity is r Value of the bond under a flat term structure = PV of face value + PV of coupon payments Term structure: interest rates of different maturities.


Pricing of Bonds: Pricing of Bonds If the term structure of interest rates is not flat, then we need to discount the payments at different rates depending upon maturity = PV of face value + PV of coupon payments These r’s are called spot rate.


Pricing of Interest Rate Forward Contracts: Pricing of Interest Rate Forward Contracts An N-period forward contract on that Government Bond Can be valued as the present value of the forward price: Note that this is simply a variation of forward pricing formula: Let’s loosely call it the price (value) of the forward contract.


Example: Example Find the value of a 5-year forward contract on a 20-year Government of Canada bond. The coupon rate is 6 percent per annum and payments are made semiannually on a par value of $1,000. The quoted yield to maturity is 5%. (Find S0) Step 1: Find the forward price: Value of the bond 5-year from now. Step 2: Discount F0 back at time 0:


Use interest rate futures to lock into future interest rate: Use interest rate futures to lock into future interest rate Example: You own $10 million worth of 20 year 10% coupon bond (semiannual coupon payments). The term structure is flat at 5% (semi-annual). These bonds are therefore selling at $1,000. If the term structure shifts up uniformly to 5.5%, the new price per bond is: Since you have 10,000 of these bonds, you have lost You want to lock into the interest rates to prevent the loss. What should you do?


Example cont’d: Opposite position in futures : Example cont’d: Opposite position in futures Suppose government bond futures contract specifies 6-month delivery of $100,000 par value of 20 year 8% coupon bond. The current price for this futures contract is: ($788.96) After the term structure shift, it is: ($759.31/1.055) Each short futures contract gains Suppose you hedge by shorting N futures contracts: Gain on futures = Overall :  Approximately you lock into the 5% interest rate.


26.5 Duration Hedging: 26.5 Duration Hedging Interest rate risk — impact of changing market yields on price % price change in long-term pure discount bonds > % price change in short-term pure discount bonds


Interest rate risk cont’d: Interest rate risk cont’d % price change in lower coupon bonds > % price change in higher coupon bonds


Interest rate risk cont’d: Interest rate risk cont’d Returns on low coupon, long-term bonds are more sensitive to changes in interest rates than returns on high coupon, short-terms bonds. Rank bonds by their interest rate risk: How do we measure this sensitivity of bond prices to changes in interest rates?


(Macaulay) Duration: (Macaulay) Duration Duration measures how long, on average, a bondholder must wait to receive cash payments (a measure of the effective maturity of the bond given when its cash flows occur) CFt = cash flow at t


Example: Example Calculate the duration for a 3 year bond, P = 1,026.25, 8% annual coupon, r = 7% Duration =


Duration and Interest Rate Risk: Duration and Interest Rate Risk For a given change in yield, the larger a bond's duration the greater the impact on price (interest rate risk/sensitivity) Hence duration measures the sensitivity of bond prices to changes in interest rates It is the first-order approximation of price sensitivity to interest rate; For second-order, convexity (= same order of magnitude with dr2). Which bond has the higher duration (for each column, assume everything else being equal)?


Portfolio Duration: Portfolio Duration The duration of a portfolio P containing M bonds is: where wi is the percentage weight of bond i in P.


Examples: Examples D = 3.3, P=1,000, r = 10%, if r drops to 9%, what is the price change as measured by duration? Portfolio duration A bond mutual fund holds the following two zero-coupon bonds: (1) 5-year maturity and 5% yield with 40% of portfolio investment; (2) 10-year maturity and 6% yield with 60% portfolio investment. What’s the duration for the fund’s portfolio?


Immunization –Balance sheet hedging based on duration: Immunization –Balance sheet hedging based on duration Immunization is a hedging strategy based on duration, which is designed to protect against interest rate risk. Match the value changes in both sides of balance sheet: The drop in the value of assets can be (partially) offset by the drop in the value of liabilities. Immunization is accomplished by equating the interest rate exposure of assets and liabilities Asset Duration × assets = Liability Duration × liabilities


Example: Example You have just learned that your firm has a future liability of $1 million due at the end of two years. Suppose there are two different bonds available and r = 10%. Duration of liability = 2 years Bond 1: 7% annual coupon, T = 1 year, $1,000 par value; P1 = Duration (D1) = 1 year Bond 2: 8% annual coupon, T = 3 years, $1,000 par value P2 = duration (D2) = 2.78 years after some calculation


Example cont’d: Immunization strategies: Example cont’d: Immunization strategies If: Buy bond 1 and then another 1-year bond after a year - runs risk of lower rates available for second year - reinvestment risk Buy bond 2 and sell after 2 years - If rates rise before then, bond prices fall, so investment may not be enough to cover liability - price risk Invest in a combination of bonds 1 and 2 so that the exposure to interest rate risk will be the same between assets (your investment) and liability


Example cont’d: solution: Example cont’d: solution w1 % invested in 1 year bonds and (1 – w1) % in 3 year bonds. w1 *D1 + (1 – w1 )D2 = Total amount to be invested = Amount in 1-year bonds = Number of 1 year bonds = Amount in 3 year bonds = Number of 3 year bonds =


Example cont’d: Does the immunization strategy work?: Example cont’d: Does the immunization strategy work?


26.6 Swap Contracts: 26.6 Swap Contracts Private agreements to exchange future cash flows according to a predetermined formula Market size has increase from zero in 1980 to trillions of dollars today Many different kinds of swaps: we will focus on plain vanilla interest rate (fixed for floating) swaps Party A makes fixed rate payments to party B; in return B makes floating rate payments to A Payment size is based on notional principal Payments are netted Floating rate is usually 6 month LIBOR


A Swap: A Swap Motivation for A: e.g., A has to finance a fixed-rate liability


Global market size of Interest rate swap: Global market size of Interest rate swap The Bank for International Settlements reports that interest rate swaps are the largest component of the global OTC derivative market. The notional amount outstanding as of December 2006 in OTC interest rate swaps was $229.8 trillion. These contracts account for 55.4% of the entire $415 trillion OTC derivative market. By comparison, the NYSE trading amount is only $17 trillion for 2006.


Example: Swap: Example: Swap Notional principal is $35M. A pays 7.19% per year semi-annually to B, receives 6 month LIBOR + 30 bps (100 bps = 1%) from B Amount of fixed payment: If current 6 month LIBOR is 6.45%, floating payment is: Payments are netted so A pays B_______________ Bank serves as intermediary


Why use swaps?: Why use swaps? 1. Transform a liability: e.g. a fixed rate loan to floating rate loan Suppose B has a $35 M loan in which it pays 7.5%. Its net position after the swap is: Pays 7.5% to outside lenders Pays LIBOR + 30 BPS in swap to A Receives 7.19% in swap from A Effectively B pays ________ 2. Transform an asset Suppose B has a $35 M asset earning 6 month LIBOR – 20 bps. Its net position after the swap is: Receives 6 month LIBOR – 20 bps on asset Pays LIBOR + 30 bps on swap Receives 7.19% in swap from A Effectively B receives __________


Comparative advantage: Comparative advantage 3. Comparative advantage Early theories about why the swap market evolved relied on comparative advantage arguments potential gains arise from relative differences in fixed and floating rates B is less credit worthy than A, but has a comparative advantage in floating rates If one is to borrow fixed, the other floating, ideally A fixed, B floating so that the total cost of borrowing is Libor + 8.7% What if A wants floating and B wants fixed?


Solution: SWAP: Solution: SWAP Step 1: A borrow fixed: 8%; B borrow floating: Libor + 70 bps Step 2: SWAP A: Pays floating to B and receives fixed from B Terms? e.g., (1) Pays B Libor + 10 bps (2) Receives from B 8.05% Net for A: (1)+(2)+pays 8% to outside lenders =


Outcomes: Outcomes B: (1) Pays 8.05% to A (2) Receives LIBOR + 10 BPS (3) Pays 6 month LIBOR + 70 bps to lenders Net: Total gain (relative to B borrows floating and A borrows fixed directly): Gain to A: Gain to B:


Generalization: Generalization Total gain = |X-Y| X = difference in fixed maturity Y = difference in floating maturity The total gain is split among A, B, and (possibly) the swap bank. The split depends on bargaining power. There are infinite swap terms that allows you to achieve a preset split of gain.


Speculation vs. Hedging: The Case of Orange County: Speculation vs. Hedging: The Case of Orange County Speculation A bad motive (from a corporate finance perspective) for forwards, futures, options, etc. is to speculate. Loans can be fixed-rate, floating-rate, or inverse floaters (rate on the loan falls when interest rates rise). If you expect interest rates to rise, you preferences are (in order) Inverse floater, fixed-rate, floating-rate


Orange County: Orange County In December 1994, Orange County, California Fourth wealthiest county government in the US. Robert Citron lost $1.7B buying inverse floaters. He was betting interest rates would fall. Greenspan raised interest rates. Citron bought leveraged inverse floaters having coupons: Max(α * fixed rate – β*fed fund rate, 0) with β>1. Orange County filed for bankruptcy. Froze the funds of 185 Southern California school districts, towns and local agencies, casting doubt on pensions and payrolls for everyone from teachers to trash haulers. Largest US government bankruptcy in history.


Double trouble: Double trouble Robert Citron had been betting on inverse floaters for some time. The strategy worked well as Greenspan loosened monetary policy in the early 90’s. However, double trouble when interest rises: #1: The coupon payments paid by inverse floaters declines. #2: The smaller payments are now discounted at a higher rate.


Consequences: Consequences Robert Citron guilty of violating state investment laws Sentenced to one year of community service Who is him? Psychologists found he had math skills of seventh grader in the lowest 5% of the population in terms of ability to think and reason


Non-speculative Motives for risk management? : Non-speculative Motives for risk management? reduction of non-core risks allows the firm to focus on the underlying business rather than external surprises (currency shocks, etc.) This is probably the best justification for risk management. How to recognize speculation If a portfolio consistently earns higher returns than it “should” then either 1. The portfolio manager is better than average 2. The portfolio manager is taking more risk than average. Citron had been taking more risk.


Slide55: Assigned questions # 26.2 , 3, 5, 8, 9, 10, 11, 14, 17, 18, 20