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Chapter-5 Determination of Forward and Future PricesKnown income: Deals with forward contract on financial assets which provide predictable cash income to the asset holders. (i.e. stocks paying dividend and coupon bearing bonds)F0 = (S0 -I) erT Where S0= Price of the asset underlying forward and future contractF0= Forward or future price todayI= PV of income from the asset during life of the forward contractT= time until delivery dateIf F0 > (S0 -I) erT: An arbitrageur can lock in profit by buying the asset and shorting a forward contract on the asset. If F0 < (S0 -I) erT: An arbitrageur can lock in a profit by shorting the asset and taking a long position in a forward contract.Ex. 10 month forward contract with S0 =$50, rf=8%, T= 10 months and dividend is paid $0.75 per share after 3 months, 6 months and 9 months. Find F0.I=PVI= 0.75 e-.08(3/12)+ 0.75 e-.08(6/12) + 0.75 e-.08(9/12) =$2.162F0= $(50-2.162) e-.08(10/12)= $51.14Future on Commodities: Deals with investment in goods such as gold and silver.Forward price of commodities (Investment asset with No storage cost and income):F0 = S0 erTForward price of commodities with storage cost: F0 = (S0 + U) erTU= PV of all storage cost net of income during life of a forward contract(Storage cost can be treated as negative income)If storage cost is proportional to the price of the commodity (i.e. negative yield):F0 = S0 e (r+u)TWhere u = storage cost per annumEx. one year future contract on investment asset with no income and $2 per unit storage cost with payment at the end of the year. Given S0=$450 per unit, rf= 7% per annum for all maturities.U= 2e-0.7x1 = $1.865F0 = $(450 + 1.865) e0.7x1= $484.63Consumption commodities:Consumption asset have no income but significant storage cost. Forward/ future price of the asset:F0 = (S0 + U) erTF0 > (S0 + U) erT: An arbitrageur can borrow S0+ U at rf to buy one unit of the commodity an use it to buy one unit of the commodity and pay the storage costs. He can short a future contract on one unit of the commodity. Profit: F0 – (S0 + U) erTF0 < (S0 + U) erT: An arbitrageur can sell the commodity, and invest the proceeds at rf . Take a long position in a future contract. Profit: (S0 + U) erT – F0Since, individuals and companies who own a consumption commodity usually use it and reluctant to sell in the spot market and buy a forward/ future contracts. Hence, for a consumption commodity, the following holds: F0 < (S0 + U) erT If storage costs are in the proportion of the spot price, the following holds:F0 < S0 e (r+u)TLHS-RHS imply convenience yield.Convenience yield(y):Benefits of holding a physical asset i.e. crude oil inventory rather than a future contract. It depends on market expectation concerning future availability of the good. Greater storage possibility –higher y. if user has a high inventory till chance of storage in near future then y=0. If low inventory and storage likely then high y.F0 eyT = (S0 + U) erTIf storage cost per unit is expressed as a proportion of the spot price thenF0 eyT = S0 e (r+u)TF0 = S0 e (r+u-y)TNote, for investment assets, y=0Cost of carry: It measures storage cost + r paid to finance the asset – income earned on the asset. For non-dividend paying stocks: cost of carry is r, since there is no storage cost and no income earned.For a stock index, it is r-q, since income earned at the rate of q on the asset.For currency, it is r-rf. For commodities, it is r-q+q which provide income at the rate q and requires storage costs at the rate u.For an investment asset, the future price isF0 = S0 ecTFor a consumption asset, it is F0 = S0 e(c-y)TChapter-3 Hedging Strategies Using FuturesFutures decrease the risk relating to fluctuations in prices of oil and stocks. A perfect hedge completely eliminates the risk but it is rare.Basic Principle:To take a position to neutralise risk as far as possibleIf price of a good decreases then short future position is taken to off-set the risk.Short Hedge: Short position in future contract. When a hedger already own asset & expects to sell it in future(i.e. a farmer owns grains). Can be used if asset not owned right now but will be in future.Exporter: May 15: S=$80 per barrelAug 15: Pfut = $79 per barrel & S= $75Gain = $(79-75) =$5per barrel Long Hedge: Appropriate, if a co. knows that it has to buy asset in future & wants to lock price now.Jan-15: 10000 pounds of copper required on May-15, SJan15=$3.40 per poundFMay15= $3.20 per pound. If long position and closes position on May-15SMay15= $3.25 per poundGain (appox.) = $(3.25-3.20)10000 = $5000Normally hedger would avoid the possibility to take delivery, and hence closes the position before delivery.For and against Hedging: Avoids unpleasant surprises & focus on core activities.India-Since need for hedging increased in commodities and forexTo manage exposures using market productsLess expensive by cos than by individuals (i.e. transaction cost)A shareholder may diversify risk more easily than a coA hedging co profitability may fluctuate If other cos don’t hedge then hedging co’s profit may fluctuate.Hedging may result in worse outcomeBasis Risk: Basis= S0 of asset to be hedged- Pfut of contract used If asset to be hedged same as that of the future contract then basis should be zero at the expiration, but may be positive or negative before the expiration.With time Ps and Pfut don’t change by the same amount.Strengthening of basis if basis increases and weakening of basis if basis decreasesAt time t1: S1, F1 and b1 At time t2 : S2, F2 and b2Then, b1 = S1 – F1 and b2 = S2 –F2If a hedger knows the asset to be sold at t2 and takes a short position at t1 Price realised for asset = S2Profit on future position = F1 – F2Effective price received for asset with hedging= S2 + F1 – F2 = F1+ b2Basis risk implies uncertainty associated with b2If a co knows that it will buy an asset at t2 and takes a long hedge at t1Changes in Basis:Cos using short position b/s it plans to sell an asset: An unexpected increase in basis improves co’s position b/s co gets higher price for asset after future gains/losses considered. An unexpected decrease in basis worsens co’s position.Cos using long position b/s it plans to buy an asset: An unexpected increase in basis worsens co’s position. An unexpected decrease in basis improves co’s position.Cross Hedging: Asset in hedger’s position is different from asset in future contract for hedging.S*2= future contract at t2S2 = asset being hedged at t2By hedging, co ensures price paid/received for the asset isS2+F1-F2 = F1 + (S*2 – F2) + (S2 – S*2)Two components of basis:(S*2 – F2) = Basis of asset hedged same as the asset in future contract(S2 – S*2) = Basis due to differences between two assetsChoice of Contract: Key factor affecting basis- choice of asset in future contract and choice of delivery monthIf asset being hedged is exactly matches asset in future contract- choice is easy. Otherwise, necessary to see available future contract closely connected with the price of asset being hedged.Influence by several factors- Expiration of hedge in delivery month. Later delivery month contract is chosen b/s future price erratic during delivery month. Also risk of having to take delivery of physical assets, which is costly and inconvenient. Choose delivery month as close as possible but later than expiration of hedge b/s basis increases as time difference between hedge expiration and delivery month increases.Ex. On jan-01, a US co expects 50 million yen at the end of July and contracts 12.5 million. Co shorts 04 Sept future Yen contract on 01 March at F= 0.9800. Contract closed at July end. Gain on future contract (Cent per Yen)= 0.9800 – 0.9250 = 0.0550 YenBasis = 0.9200 – 0.9250 = -0.0050 when contract closed out.Effective price obtained = Final S + Gains on future = 0.9200 + 0.0550 = 0.9750Also, Initial future price + final basis = 0.9800 + (-0.0050) = 0.9750Total amount received by the Co for 50 million Yen = 50 x 0.00975 = $ 487500Hedge Ratio: Size of position taken on future contracts to size of exposures. Important if there is cross hedging. A hedger should choose a value for the hedge ratio that has a minimum variance of the value of the hedge position.Minimum Variance Hedge Ratio: It is the slope of best fit line from linear regression of change in S against changes in F. It is the ratio of average change in S for a particular change in F.h* = ρ σs/σFh* = minimum hedge ratioσs = s.d. of changes in SσF = s.d. of changes in Fρ = correlation coefficientρ =1 & σs = σF then h* =1→ PF mirrors PSρ = 1, σs = 2σF then h = 0.5 → changes in PF twice as much as PSHedge effectiveness on- proportion of variance eliminated by the hedge and R2 from regression of changes in PS against change in PF equals ρIt is assumed that the parameters of ρ, σs and σF estimated from the historical data on change in PS and changes in PFOptimal number of Contracts for Hedging: N= h*QA/QFQA = size of position being hedgedQF = size of one future contractN = optimal no of future contract for hedgingh*QA = future contract on h*QA units of the assetEx. An airline expects to buy 2million gallons of jet fuel in one month and use heating oil futures for hedging. Given σs =0.0313, σF = 0.0263 and ρ= 0.928h* = 0.928x 0.0263/0.0313 = 078Optimal no of contracts = 0.78 x 2000000/42,000 ≈ 37Tailing the Hedge: Future contracts used for hedging as a series of one day hedges.Calculate correlation between % one day changes in the futures and spot prices i.e.ρ. and σs and σF are s.d. of % one day changes in spot and future prices.s.d. of one day price changes are Sσs & FσFone day hedge ratio = ρ Sσs/ FσFNo of contracts needed to hedge over the next day is N*= ρ SσsQA/ FσFQFUsing it sometimes referred to as tailing the hedgeN*= h VA/VFWhere VA = $ value of position being hedged (=SQA) VF = $ value of one future contract (=FQF) h = ρ σs/ σF(In theory, we should change future position every day to reflect latest values of VA and VF. In practice day to day changes in hedge are very small and are usually ignored.)Stock Index Future: Tracks changes in the value of artificial portfolio of stocks. Weight of a stock is the proportion of hypothetical portfolio invested in the stock so that % increase in the stock index is equal to the % increase in the value of the hypothetical portfolio. It tracks changes in the capital gains and losses from investing in the portfolio.(dividend not included).Weights to stocks are in proportion to their market prices, with adjustment being made for stock split. Weight calculated as the product of the values (market capitalisation) and no of shares outstanding then it reflects the changes in stock split, stock dividend and new equity issues.Hedging Equity Portfolio:VA = current value of portfolioVF = current value of one future contractIf portfolio mirrors index then optimal hedge ratio is 1 and the number of future stocks to be shortedN* =VA/VFIt assumes that the maturity of the future contract is close to the maturity of the hedge. In General: N*= β VA/VF Hence, h = βh ratio is the slope of the best-fit line when % one day changes in the portfolio are regressed against % one day changes in the future price of the index. β measures the slope of the best-fit line when the return from the portfolio is regressed against the return for the index.β =2 → excess return on portfolio twice as excess return on the index. Portfolio value is twice sensitive to the movements in the index. Necessary to use twice as many contracts to hedge the portfolioIt is assumed that the maturity of future contract is close to the maturity of the hedge.When portfolio doesn’t mirrors the index: CAPM is used. Ex. Portfolio worth $50,50,000 mirrors S & P index PFut =$1,010Each future contract is for delivery of $250 times the index.VA = $50,50,000VF = $1010 x 250 = $2,52,500So, 20 future contract to be shorted to hedge the portfolio.Reasons for using Equity Portfolio: Hedging- value for hedger’s position at the end of 3 months being 1% higher than at the beginning of 3 month period. (b/s rf =1% per 3 months). Hedging is justified if hedger feels that stocks in the portfolio are chosen well. Hedge using index future removes risk due to market movements and is only exposed to the performance of the portfolio relative to the market.Hedger planning to hold in the long-run requires short-run protection in uncertain market situations.Changing β of a portfolio: β=0→ hedger’s expected return is almost independent of the performance of the index. Sometimes future contracts are used to change β of a portfolio other than zero. If S & P index 1,000, S & P future price= 1,010, value of portfolio=$50,50,000, Beta of portfolio= 1.5VF = 250 x 1,010 = 2,52,500 and a complete hedge requires 1.5x 50,50,000/2,52,500 = 30 contracts needs to be shorted.To reduce beta from 1.5 to 0.75, 15 contracts needs to be shorted. To increase beta to 2.0, a long position in 10 contracts should be taken.In general, to change beta (portfolio) β to β* ,When β> β*, a short position in (β – β*) VA/VF contract are required.When β< β*, a long position in (β – β*) VA/VF contracts are required.Stack and Roll: Expiration date of a hedge is after the delivery date for all future contracts. Hedges can be rolled many times. Hedges can be rolled forward by closing out one future contract with another one having delivery date after that (taking same position). Let a Co uses short hedge to decrease risk associated with the price to be received for an asset at a later date T. The Co can use the following strategy:t1 :short future contract 1t2 :close out future contract 1 short future contract 2 t3 :close out future contract 2 short future contract 3 ??tn :close out future contract n-1 short future contract n T :close out future contract n Ex. Pg: 83 ................
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