CHAPTER 26 DERIVATIVES AND HEDGING RISK TOPICS 26

CHAPTER 26: DERIVATIVES AND HEDGING RISK TOPICS: • 26. 1 • 26. 2 • 26. 3 • 26. 4 • 26. 5 Forward Contracts Futures Hedging Interest Rate Futures Contracts Duration Hedging 1

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 2

How risk is managed Plant May Harvest Sept. 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 3

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) 4

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. 5

Payoffs From Forward Contracts • let ST denote the spot price of the asset at the delivery date T and let Ft be 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 6

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. 7

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). 8

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

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. 10

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 ______. 11

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. ) 12

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. Day Futures Price $ Cash Flow $ Starting Cash added to Margin $ Ending Margin $ 1 875 0 0 6, 000 2 872 3 869 4 874 5 875 1, 100 7, 100 -1, 100 6, 000 6 884 900 6, 900 -6, 900 0 (1) 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? 13

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

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 15

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 − F 0) = −(ST − F 0) payoff from selling asset at T ST total F 0 i. e. the price of F 0 is 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 16

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 17

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 − F 0 = ST − F 0 payment from purchasing asset at T -ST total F 0 i. e. the price of F 0 is 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 18

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). 19

Term Structure of Interest Rates • The text coverage of this material is in Appendix 6 A • 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 20

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

Pricing of Interest Rate Forward Contracts • An N-period forward contract on that Government Bond … 0 N N+1 N+2 N+3 N+T 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. 22

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? 23

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 forward contracts will only be an approximation in this context – The exact delivery date is determined by the short party in a futures contract 24

, 84 -165 equals 84 16. 5/32 25

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% (semiannual). 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? 26

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. 27

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) 28

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%. 29

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: 30

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: 31

Duration cont’d • Duration is defined as: • Or: CFt = cash flow at t • 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) 32

Example Calculate the duration for a 3 year bond, P = 1, 026. 25, 8% annual coupon, r = 7% 1 2 3 80 80 1, 080 PV of cash Flow 74. 77 69. 88 881. 60 Relative value (Wt) 0. 0729 0. 0681 . 8591 Payment or cash flow Weighted maturity (t. Wt) Duration = 33

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) 34

Example • Which bond has the higher duration (treat each column separately, assume everything else being equal)? Bond Coupon Maturity Yield A 10% 10 years 10% B 5% 5 years 5% 35

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

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? 37

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 38

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; P 1 = Duration (D 1) = 1 year Bond 2: 8% annual coupon, T = 3 years, $1, 000 par value P 2 = duration (D 2) = 2. 78 years after some calculation 39

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? 40

Example cont’d: solution w 1 % invested in 1 year bonds and (1 – w 1) % in 3 year bonds. w 1 *D 1 + (1 – w 1 )D 2 = 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 = 41

Example cont’d: Does the immunization strategy work? r after 1 year 9% 10% 11% $438, 197 $442, 181 Value at t = 2 of 3 year bonds: Value from reinvesting coupons received at t = 1 $42, 997 $43, 388 Coupons received at t = 2; $39, 088 Selling price at t = 2; $479, 716 $475, 395 Total $999, 998 $1, 000, 052 Value at t = 2 from reinvesting 1 year bond proceeds: 42

Immunization cont’d • As can be seen from the table above, the immunization strategy appears to perform fairly well • However, there 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 43

Assigned questions # 26. 1 -5, 7 -9, 12 -14, 17 44