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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. Overview. Risks to be managed, and the methods used to finance them. Commodity price risk ( futures )

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CHAPTER 26: DERIVATIVES AND HEDGING RISK

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  1. 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

  2. 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 to minimize the risk of your net position • If the assets are perfectly correlated, your net position is risk free

  3. 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.

  4. 26.1 Forward Contracts • A forward contract is an agreement to buy / sell an asset at a particular future time for a specified price called the delivery 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. • the forward price is the delivery price which makes the contract value zero (so the forward price is equal to the delivery price at the inception of the contract)

  5. Examples of a forward • Pizza forward contract. Order pizza by phone. Specify topping (type), size, 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.

  6. Payoffs From Forward Contracts • let ST denote the spot price of the asset at the delivery date T and let Ftbe the delivery price (price set at t payable at T) • The payoffs on the delivery date are: long position payoff: ST − Ft short position payoff: Ft − ST

  7. 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 sharply in the winter, your heating company will lose, and it might default.

  8. 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. • The futures price is analogous to the forward price: it is the delivery price for a futures contract • the futures price will converge to the spot price of the underlying asset when the contract matures • 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).

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

  10. Futures/forward price can change everyday • In options, X does not change • In futures, your profit/loss is based on fluctuation of futures price • Your daily profit/loss for long futures is: New futures price – the futures price that you agreed • If an offsetting position is taken before the expiry for a long futures, then profit/loss is Fnew- Ft ,where Fnew is the new futures price at the offsetting time. • Likewise, if an offsetting position is taken before the expiry for a short futures, then profit/loss is Ft – Fnew.

  11. Example • Consider an investor who enters a futures contract expiring one month from now to purchase 100 oz. of gold at the futures price of $275 per ounce. • If the spot price of gold is $290 on the expiry date, the profit/loss is _____. • If the investor closes out her position two week from now with a futures price of $280 on the same contract. The spot price is $290. Her profit/loss is ______.

  12. 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.)

  13. Example: Marking-to-market Consider an investor who enters a futures contract to purchase 100 oz. of gold at the futures price of $875 per ounce. Suppose that the initial margin is set at $6,000 and the maintenance margin is set at $4,500. The contract is closed out after 6 days. • 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?

  14. 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 Futures Both longs and shorts have obligation to buy/sell Daily price limits Marked-to-market Delivery seldom occurs Futures vs. Options

  15. Hedging with futures—Locking in price • There are two types of investors who use futures/forward • Speculators: try to profit from price movements • Hedgers: try to protect against price movement and to reduce risk by making outcome less variable • Short hedge (take a short position in futures) is used when you have asset to sell in the future • Conversely, long hedge (take a long position in futures) is used when you have asset to purchase in the future

  16. Short Hedge • Consider a firm which will be selling an asset at some future date T, and suppose there is a futures contract on that asset for delivery at T • The firm is exposed to the risk that the price of the asset might fall between now and T • If the firm takes a short position in a futures contract, its overall payoff is: futures payoff −(FT − F0) = −(ST − F0) payoff from selling asset at T ST total F0 i.e. the price of F0is locked in today • If the asset price falls, the firm loses on the asset sale but gains on the futures contract • If the asset price rises, the firm gains on the sale but loses on the futures contract

  17. 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

  18. Long Hedge • Suppose instead a firm wants to purchase an asset at some future date T • The firm is exposed to the risk that the price of the asset might rise between now and T • If the firm takes a long position in a futures contract, its overall payoff is: futures payoff FT − F0 = ST − F0 payment from purchasing asset at T -ST total F0 i.e. the price of F0is locked in today • if the asset price falls, the firm gains on the asset purchase but loses on the futures contract • And vice versa • Note that futures hedging does not necessarily improve the overall outcome: you can expect to lose on the futures contract roughly half of the time => the objective of hedging is to reduce risk by making the outcome less variable

  19. Interest Rate Futures Contracts • Futures contract whose underlying security is a debt obligation. • We’ll consider interest rate futures • Interest rate futures are used to lock into the forward term structure (lock into future interest rates).

  20. Term Structure of Interest Rates • The text coverage of this material is in Appendix 6A • Although in almost all cases in this course we consider a flat term structure ( interest rates of different maturities), it is important to keep in mind that this is a simplification • With a flat term structure, discount rates are the same for all maturities, but this is rarely (if ever) the case • For Oct. 31, 2007, the Bank of Canada reported government zero coupon government bond yields as follows: Maturity 1 yr 3 yr 5 yr 7 yr 10 yr 15 yr Yield 4.18 4.16 4.18 4.21 4.28 4.37 • This means, for example, that the price on Oct. 31 of a one year zero coupon government bond paying $1,000 at maturity was $1,000/1.0418 = $959.88, while the price of a ten year zero coupon government bond paying $1,000 at maturity was $1,000/1.042810 = $657.64 • The rates above, which can be used to determine prices at which bonds may be currently traded, are known as spot rates

  21. … 0 1 2 3 T Pricing of Government Bonds • Consider a Government of Canada bond that pays a semi-annual coupon of $C for the next T/2 years (Note that there is a total of T=2(T/2)payments): If the term structure is flat, i.e. r1 = r2 = · · · = rT = r , then the above formula simplifies to the familiar C ATr +F/(1+ r )T

  22. … 0 NN+1 N+2 N+3 N+T 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: • In the above, PV is the current value of the forward contract. • Pforward is the forward contract price (the price you’ll pay in the future). One implies the other.

  23. Example • Consider 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%. Assume that the term structure is flat. What is the value of the bond today? What is the forward price?

  24. Interest rate futures contracts and hedging • In practice, futures contracts on bonds are typically used rather than forward contracts • Futures contracts on bonds are referred to as interest rate futures contracts • The pricing relationships derived above for forward contracts will only be an approximation in this context • The exact delivery date is determined by the short party in a futures contract

  25. ,84-165 equals 84 16.5/32

  26. 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?

  27. 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 (value) for this futures contract is: After the term structure shift, it is: Each short futures contract gains Suppose you hedge by shorting K futures contracts: K = Size of exposure/size of futures contract Gain on futures = Overall : Approximately you lock into the 5% interest rate.

  28. Reasons in practice why interest hedging using futures may not work perfectly Different maturities (bonds in portfolio vs. futures contract) • Different coupon rates • Different risk (e.g. corporate bonds in portfolio, government bonds in futures contract)

  29. Interest rate risk • Interest rate risk — impact of changing market yields on price • Assume for simplicity a flat term structure. Consider these four bonds, each with $1,000 par value and coupons paid annually: Note: percentage price changes are calculated relative to the price when r = 10%, e.g. (877.93-875.66)/875.66 = +.2592%.

  30. Interest rate risk cont’d • Observations from the previous slide • Comparing A, B, and C: low coupon bond prices are more sensitive (i.e. higher percentage price change)to changes in r , given the same T • Comparing C and D: longer maturity bond prices are more sensitive to changes in r, given the same coupon • Rank bonds by their interest rate risk:

  31. Duration • How do we measure the sensitivity of bond prices to changes in interest rates? • This means that the percentage change in price for a given change in r is:

  32. Duration cont’d • Duration is defined as: • Or: • 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

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

  34. Duration and Interest Rate Risk A bit of algebra yields: • 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 a small change in interest rate, duration is quite an accurate estimate for percent price change • For a given change in yield, the larger a bond's duration the greater the impact on price (interest rate risk/sensitivity)

  35. Example • Which bond has the higher duration (treat each column separately, assume everything else being equal)?

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

  37. 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?

  38. Immunization –Balance sheet hedging based on duration • Immunization is a hedging strategy based on duration • 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

  39. 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

  40. 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 • Basic idea: if rates rise, the portfolio’s losses on the 3 year bonds will be offset by gains on reinvested 1 year bonds. • And vice versa. • How much should you invest in each bond?

  41. 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 =

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

  43. Immunization cont’d • As can be seen from the table above, the immunization strategy appears to perform fairly well • However, there are a number of assumptions needed for this to work. Some possible problems include: • the strategy assumes that there is no default risk or call risk for the bonds in the portfolio • The strategy assumes that the term structure is flat and any shifts in it are parallel • duration will change over time (even if r does not), so the manager may have to rebalance the portfolio (note that there is a tradeoff of accuracy from frequent rebalancing vs. transactions costs) • More complicated strategies exist to handle these types of problems, but immunization using duration is still a very widely used tool in practice

  44. Assigned questions # 26.1-5, 7-9, 12-14, 17

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