Hedging and Insuring Hedging inancial Risk Econ 422 Summer 2005 Both hedging and insuring are methods to manage or reduce inancial risk. Insuring involves the payment o a premium (a small certain loss) or the reduction or elimination o the possibility o a larger loss. Hedging involves a transaction that reduces the risk o inancial loss by giving up the possibility o a gain. Hedging oten involves the use o derivative securities. General Principles o Hedging Assume that you own risky asset A and want to reduce your risk exposure. ind an asset B whose price is (highly) correlated with that o A, i.e., there is a linear relationship between A s price and B s. Estimate the parameters o this relationship by running a regression o A s price on B s price: p A = α+ δp B + ε, δ = cov(p A,p B )/var(p B ) Delta measures the sensitivity o expected changes in A s price to expected changes in B s price: E[p A ] = α+ δe[p B ] General Principles o Hedging, Cont. I the prices o A and B are perectly correlated, with a hedge ratio o δ, then ε = 0 and you could construct a perect hedge by selling (short) δ units o B. hus, your portolio would have a long position o one unit o A and a short position o δ units o B. I the price o A rises by $1, the price o B (in this case) rises by ($1/δ). he value o your portolio changes by: $1 - δ ($1/ δ) = 0 You have eliminated the risk Delta is reerred to as the hedge ratio.
General Principles o Hedging, Cont. I the prices o A and B are not perectly correlated, then ε 0 and you could still construct a hedge by selling (short) δ units o B, but the hedge would not be perect. Your portolio would have a long position o one unit o A and a short position o δ units o B. I the price o A rises by $1, the price o B (in this case) rises by ($1/δ) on average, but not always. he value o your portolio changes on average by: $1 - δ ($1/ δ) = 0 but not always Hedging Using Returns In practice, hedging using regression is usually done with returns instead o prices, or certain statistical reasons: r A = α + δ r + B error he interpretation o δ remains the same: it is an estimate o the hedge ratio You have eliminated the risk on average, but not always. General Principles o Hedging (Example) XYZ Corp holds $12.5 million o IBM stock. It wants to reduce the risk associated with this asset, using a market index as a hedge instrument. We know based on the CAPM, that IBM s return is correlated with the market return and the relationship is r = α + βr + ε = + r + ˆ ε R = ˆ δ = ˆ β = 0.72 2 IBM M 0.78 0.72 M, 0.4 IBM o construct a hedge, sell 0.72 * $12.5 million = $9 million o the market index. Q: Is the hedge perect? x x IBM MK MK Risk o Hedge Portolio = 1 = share o wealth invested in IBM = 9 /12.5 = 0.72 = share o wealth sold short r = x r + x r = hedge portolio β β x p IBM IBM MK MK IBM = 0.72 = hedge ratio β = 1 + x IBM IBM MK β MK = Portolio beta = 1(0.72) 0.72(1) = 0 Hedge portolio has a zero beta: it is market neutral Market risk has been elminiated but speciic risk remains!
Hedging with Options Recall, rom the Black-Scholes option pricing ormula C = δ S bank loan δ = Nd ( ) = option hedge ratio 1 1 S = C+ bank loan δ S 1 = = # o options required to hedge stock C δ orward Contract An agreement to buy or sell an asset at a certain uture time or a certain price. ypically between two inancial institutions or between a inancial institution and a corporate client. Normally not traded on an exchange. One o the parties to a orward contract assumes a long position by agreeing to purchase the asset at the speciied price in the uture; whereas, the other party assumes a short position agrees to sell the asset on the same date or same price. orward Contract his contract creates or both buyer and seller both the right and the obligation to transact at the speciied terms. orward Contracts Usually held to maturity and settled at maturity One party looses, one party gains Speciied price = delivery price determined such that at time o contracting, value o contract is zero. Example: Contract today to buy a house or $200,000 with closing in two months. itle to the house and the money are exchanged in two months at terms agreed upon today in the contract Example: Bank A contracts to buy rom Bank B and Bank B contracts to sell to Bank A 10 million euros or $10.2 million, six months rom now. he loser may have an incentive to deault, so there is always a credit problem. hus, orward contracts are usually written between large institutions with good credit and an ongoing relationship Pre-paid variable orward contracts
Value o Contract Position Diagram or orward Contract Long Position Value o Contract Short Position utures utures are orward contracts managed by an exchange Oldest actively traded derivative instrument. Initially used with agricultural and commodities; e.g., pork bellies, cattle, sugar, wool, coee, rozen orange juice, copper, gold, aluminum, gold, tin. Value o asset Value o asset Underlying inancial assets include stock indices, currencies, reasuries. rade on the CBO (Chicago Board o rade) or CME (Chicago Mercantile Exchange). he buyer and seller don t meet or know each other. he exchange contracts separately with buyer and seller, then matches them. he exchange establishes margin accounts or each party to guarantee ulillment o contract he utures market is extremely liquid utures utures contract is an agreement between two parties to buy or sell an asset at a certain time (delivery) in the uture or a certain price. Contracts are standardized, speciying exactly what is to be delivered, where and when. Seller has some choice concerning the delivery date (usually within the inal month) he contract is marked to market, i.e., there is settlement ater every trading day. his arrangement reduces deault risk. Example: Marking to Market A 3 day utures contract (which is marked to market) and a 3 day orward contract (which is not) call or A to buy and B to sell 1000 ounces o silver three days rom now $6.00 per ounce. he price o silver moves to: $6.10 on day 1 $6.05 on day 2 $6.12 on day 3 Regarding the orward contract, A gains $0.12 *1000 = $120 and B looses $120 since silver is trading at a higher price on settlement day. he orward contract can either be settled by B actually selling to A the 1000 ounces o silver or $6,000 (when it is worth $6,120) or by making a payment o $120 to A.
Example: Marking to Market Regarding the utures contract, at the end o the trading on the irst day, B would pay and A would receive ($6.10 - $6.00) = $0.10*1000 = $100 as the settlement or the irst day s price rise o $0.10. Once this payment is made, the contract is rewritten with a price o $6.10. At the end o day 2, A would pay and B would receive ($6.10 - $6.05) = $0.05*1000 = $50 as settlement or the second day s price decline. he contract would be rewritten at $6.05. At the end o day 3, B would pay and A would receive a payment o ($6.12 - $6.05) = $0.07*1000 = $70. Note: he total net payment made by B is the same $120. I at the beginning o day 3, B had chosen to deliver the silver, B would sell 1000 ounces or $6050 (why?), but would have made net settlement payments o $50. A would have paid a net o $6,000 or the silver now worth $6,120. Marking to Market Marking to market largely eliminates the deault risk by the process o daily settlement. he utures exchange (CBO or CME) takes care o the marking to market accounting Although the net totals are the same as with the orward contract, the timing o the payments is dierent. he timing o payments can be very important. B looses $120 and A gains $120 o value Hedging with utures Consider a wheat armer who expects to harvest 100,000 bushels o wheat in one month. he price that will prevail in one month is uncertain, hence the armer is exposed to considerable inancial risk. he armer could reduce or eliminate this risk by entering into a utures contract to sell wheat in one month at a price, say $3.00/bushel, agreed to today. Consider a miller who will need to buy 100,000 bushels o wheat in one month. he miller is exposed to inancial risk because o the uncertainty about the price that will prevail in one month. he miller could eliminate this risk by entering into a utures contract to buy wheat in one month at the $3.00/bushel utures price. Hedging with utures Consider three possible prices or wheat one month rom now: $2.50, $3.00 and $3.50. he armer has hedged his 100,000 bushel crop with a utures contract at $3.00/bushel. he utures contract will be settled in cash rather than with delivery o wheat. he armer sells his crop to the local grain dealer at the spot or market price in one month. Show what the armer will receive or the crop, combining the revenue rom the actual sale with the settlement proceeds or the utures contract.
Hedging with utures Draw a position diagram or the armer showing the dependence o his/her wealth on the price o wheat i no hedging is used. Assume wheat is the armer s only asset, (100,000 bushels o it.) Show the armer s position ater taking a short position in a utures contract. (100,000 bu) Do the same thing or the miller. Are utures Risky? I you take a speculative position in a utures contract, either long or short, that position is risky. I you already have a long or short position in the physical commodity, then taking the opposite position in the utures contract is risk reducing. he risk o the utures contract depends on the context in which it is used. he Relationship Between Spot and utures Prices he spot price o a commodity at any date is the cash price o the commodity at that time. he orward or utures price is the price that would be established or a orward or utures contract. Certain no arbitrage conditions link spot and utures prices utures Prices or inancial utures Suppose that 1.5 years rom today you would like to own a inancial asset, e.g. 1000 shares o an S&P index und or a 10 yr. reasury Bond. One way to achieve this goal is to buy the asset now and hold it or 1.5 years. Another way to achieve this goal would be to buy a orward or utures contract or the asset. Using the irst approach approach you would pay the current spot price, P s, or the asset. You would also get any cash lows (e.g. dividends, coupon payments) generated by the asset during the 1.5 years that you hold it.
utures Prices or inancial utures Using the second approach you would pay the agreed upon utures price at the uture date = 1.5 years rom today. You would not get the intervening cash low rom the asset. In both cases you would own the asset at time =1.5. o prevent arbitrage, it must be the case that (1 + r ) = P PV(Cash low) S ( ) = (1 + r ) P PV(Cash low) S utures Pricing: Example Determine utures price o 10 yr -bond with delivery date in 6 months ( = 0.5) Suppose that the spot price or a 10 yr. reasury Bond is 100 he current six month interest rate is 0.06 (annual rate) he PV o the anticipated coupon interest payment over the next six months is 5 utures Pricing: Example Using the no-arbitrage condition, the price o the 6-month utures contract is = (1 + r ) ( P PV(Cash low)) S 0.5 =(1.06) (100 5) = 97.808 utures Pricing: Equity Index No arbitrage relationship: (1 + r ) = P PV(Cash low) S ( ) = (1 + r ) P PV(Cash low) Cash low or equity index is dividend payment S
utures and spot prices or commodities No-arbitrage condition or commodity utures. = P S +PV(Storage costs)- PV(Convenience yield) ( 1+ r ) Again compare the costs and beneits o buying a utures contract vs. buying the commodity and holding it. Convenience yield (non-negative) is the beneit associated with having possession o the physical commodity. It is likely to be inversely related to the size o existing inventories.