Tài chính doanh nghiệp - Topic 12: Forward contracts and hedges, simulated correlated random variables

Financial ModelingTopic #12: Forward Contracts and Hedges, Simulated Correlated Random VariablesL. Gattis1Learning ObjectivesCompute no-arbitrage forward prices for equities, currencies, and commoditiesCompute the payoffs of forward contractsConstruct forward hedges using beta and correlationSimulate correlated random variables2Forward ContractA forward is a contract to buy or sell an asset in the future where the asset, price, quantity, delivery place and location, are specified in the contract.There is generally no up-front paymentAt maturity, the “seller” is obligated to deliver the asset to the “buyer” and is paid the forward priceForward contract terms are negotiated between counterparties in what is called the over-the-counter (OTC) market which is a network of institutionsFutures contracts are forward contracts that trade on exchanges such as the Chicago Mercantile Exchange. 3Forward PayoffsThe “buyer” takes delivery of a quantity (Q) of the asset and pays the forward price. The buyer’s payoff is the difference between the forward price (f0) and the spot price of the asset at delivery (St). The buyer is said to have a long position in the forward, since she profits if the spot price at maturity is above the forward price Long (Buyer) Forward Payoff = (St - f0)QThe “seller” is called the short party of the contract and profits if the spot price falls and the seller can receive a forward price which is above the spot price at maturity. Short (Seller) Forward Payoff = (f0 - St)Q41. Must Buy @ Forward Price2. Can Sell @ Spot Prices1. Can Buy @ Spot Price2. Must Sell @ Forward PriceForward Pricing and Cash and Carry Arbitrage: StockA cash and carry arbitrage is a risk-free strategy to buy an asset and simultaneously enter into a forward to sell it. E.g.,Buy Non-dividend paying stock in the cash market at S0 today Enter into forward contract to sell stock in t years at price F0,tFinance the purchase at interest rate r for t years.Risk-free profits = F0,t – S0*(1+r)t “No Arbitrage” Forward Price, F0,t = S0*(1+r)tr is the cost to “carry” (hold) the position until forward mat.The position is risk free and can be financed at the risk-free rateFor Dividend Payers, F0,t = S0*(1+r-d)t since dividends will reduce the total cost to carry the position.Cost of Carry for dividend payers = (r-d)5S&P 500 No Arb and Market Forward Prices6The forwards are selling at a discount to the spot price because the cost of carry (r-d) for S&P 500 equities is negative for all maturities = S0*(1+r-d)t Cash and Carry Arbitrage: CommodityThe cost to carry a commodity such as oil, gold, hogs, and cocoa include storage costs (s), possible lease income (l) and other convenience benefits (c) No Arbitrage Forward Price, F0,t = S0*(1+r+s-l-c)tExample, Live Cow Cost of CarryFinancing costs = “r”Storage and feed cost = pay “s” (as a percentage of the spot price)Lease Cow to Dairy = recive ”l”Benefits of physical carry (milk) = receive other benefit “c”Example, Gold Cost of CarryFinancing costs = rStorage costs = s = 0 (Can hold in Certificate Form)Lease payments from Short Sellers = l (What are short sales?)Convenience Yield = c = 0 7Commodity Forward ExampleThe spot price of gold is $1,265.80/ounce gold Lease rates is .41% annualized costs and are assumed to be zero (gold can be held in certificate form)The 1-year, 10-month CME futures price is $1287.10 and Carry Arbitrage: CurrencyWhen “carrying” a currency, the entire amount of the FX can be invested into a foreign sovereign bondE.g., suppose the spot price of the euro = S0,$/€Buy 1 Euro and Finance, Cost = 1*S0,$/€(1+r$)Invest euro in €1 Bond and earn €1(1+r€)Sell €1 Euro Forward and receive F$/€ at maturityNo Arb Condition: F$/€= S0,$/€(1+r$)/(1+r€)No Arbitrage Forward Price for any maturity, tDirect Quote $/FX = F0,t,$/FX = S0,$/FX*((1+r$)/(1+rFX))tIndirect Quote FX/$ = F0,t,FX/$ = S0,FX/$*((1+rFX)/(1+r$))t9Currency Forward ExampleThe Australian Dollar is selling for $.8959 in spot markets (2/14/14) and the 1 year forward is $.8474( The 1-year U.S. Treasury yield is .375%, The 2-year Aussie yield is 2.82% Arb Forward Pricing SummaryThe Forward Price must be equal to the Cost of Carrying the Position, otherwise arbitrage is possibleAny asset that can be cheaply stored (without spoilage) can be priced using a no-arb argument“Carry market” assets include Stocks, Bonds, Currencies, Gold, Silver, Oil, etc.Some assets such as electricity are too costly to store and their forward prices simply reflect expected supply and demand.11Hedging with ForwardsIf you own an asset and want to guarantee a price to sell it in the future, you could short (sell) a forward and lock into the forward priceShort (Seller) Forward Hedger: Ranchers (beef), Miners (gold), Exporters (fx), Energy Companies (oil), investors (stocks, bonds)If you plan to buy an asset and want to guarantee a price to buy it in the future, you could long (buy) a forward and lock into the forward priceLong (Buyer) Forward Hedgers: Airlines (Jet fuel), Restaurants (beef), Jewelers (gold), Importers (fx), investors (stock), Industrials (Fuel), Hershey (Cocoa) 12Hedging Heating Oil CostsSay it is January and “New York Harbor Heating Oil No. 2” is selling for about $1.80 per gallon for May delivery on the NYMEXIf a factory owner requires 100,000 gallons in May, they could enter into a contract to buy 100,000 gallons at $1.80 for May delivery and be hedged (no exposure to price changes)13Hedging Jet Fuel Costs with Heating OilSuppose Southwest Airlines would like to hedge the cost of 100,000 gallons of Jet Fuel in May.Unfortunately, the Jet Fuel forward market does not provide sufficient liquidity (market depth) to take such a large position at a reasonable price. However, there is a liquid market in Heating Oil and Regular Gasoline.Southwest could enter into a forward to buy Oil or Gas to “cross hedge” the cost of Jet Fuel.A “cross hedge” is a hedge using an asset that is correlated to underlying asset (but not the underlying asset needed)The hedge is effective as long as Jet fuel is highly correlated to Gas or Oil14Hedging Jet Fuel Costs5 years of monthly jet fuel, Gas, and Heating Oil Prices. Monthly price returns and correlation matrixCompute monthly returns and correlation matrixJet Fuel and Heating Oil have highest correlation (.97)Compute Jet Fuel / Heating Oil beta, =Slope(Jet Returns, Oil)Cross Hedging Strategy: Enter into contract to buy 93,750 gallons of heating oil. Optimal Hedge Qty (H*) when Beta (between asset and hedge) = 1H* = Underlying Asset Spot Price x Quantity / hedge asset spot priceCurrent Jet Fuel Spot Price = $1.50 (Underlying Asset)Current Oil Spot Price = $1.60 (Highest Correlated Hedge Asset)H* = $1.50 * 100,000 / $1.60 = 93,750 gallons (Oil position to hedge jet fuel)Result: if jet fuel and heating oil are both up 20% (beta = 1)Pay 20% more for jet fuelCurrent Jet Fuel Spot Price = $1.5 Jet Fuel St = $1.5*1.2=$1.8Incremental Jet Fuel Cost=(1.5-1.8)*100,000=-$30,000Offset by gain on Oil Forward ContractCurrent Oil Spot Price = $1.60Oil Forward = $1.60 (Assume zero Cost of Carry)Oil St = $1.60*1.2=$1.92Long Forward Payoff =(St-F)Q= (1.92-1.60)*93,750=+$30,000Buy Oil @ the forward price and Sell in Spot (earning $30,000), then take forward payoff to offset additional cost of Jet Fuel (-$30,000)15Hedging Jet Fuel CostsOptimal (Risk minimizing) cross hedgeSelect hedge asset with highest correlationHedge is effective if asset and hedge move together (highly correlated)If beta ≠ 1, adjust optimal hedge quantity by multiplying H* by betaOptimal Hedge QuantityH* = Spot of Asset x Quantity / Spot of hedge * Beta (1.0367)H* = 1.50 * 100,000 / 1.60 * 1.0367 = 93,750 * 1.0367 = 97,190Result: if jet fuel is up 20.73% (1.0367*20%) and heating oil is up 20%Pay 20.73% more for jet fuelCurrent Spot Price = $1.5 St = $1.5*1.2073=$1.81095Δcost=(1.5-1.81095)*100,000=-$31,095Receive gain on Oil Forward ContractCurrent Spot Price = $1.60Forward = $1.60 (Assume zero Cost of Carry)St = $1.60*1.20=$1.92Payoff =(St-F)Q= (1.92-1.60)*97,190=+$31,100Since Jet Fuel is more volatile than Oil, must increase the number of contracts by the beta16Optimal HedgeOptimal Hedge (H*)H* is the number of shares, commodities, or Index Portfolios to hedge underlying positionThe expected risk is reduced by the expected correlation between the asset and hedgeThe actual effectiveness of the hedge depends on the actual correlation and beta (basis)17Cross HedgesHedging an asset with a different assetExamplesAirlines hedging jet fuel costs with crude oil forwardCorporation hedging Danish Krone revenues with Euro forwardHershey hedging Ghana cocoa costs with Ivory Coast cocoa forwardHedging single stock with an index such as the S&P 500Steps to establish risk minimizing hedge is:Choose hedge instrument with highest correlation to your asset (lowest basis risk: risk that basis change)Compute beta to determine optimal number of contracts18rprS&Pβ =1.4ρ=1.0rprS&Pβ =1.4ρ < 1.0Perfect Corr.Imperfect Corr.Hedging Sale of IBMSuppose you own 1,000 shares of IBM that are currently selling for $199.63 in spot markets. You want to lock into a price to sell in one yearYou could short a forward for 1,000 shares and the risk is eliminatedHowever, if no IBM forward contract is available, you can cross hedgeCross Hedge: Hedging a position with imperfectly correlated hedge instrumentYou could short the S&P500 or NASDAQ as long as they are correlated to IBMHedge is effective as long as asset and hedge instrument are correlatedE.g.; IBM Stock and S&P 500 both go down 10% – Stock losses (St,IBM-S0) offset by short forward gains (F0-St,SP500)Enter into forward contract at time = 0 19Select Forward InstrumentHedge using the higher correlated S&P50020Select Forward InstrumentHedge using the more correlated S&P50021Simulating Correlated Random VariablesWhen simulating more than one variable using a Monte Carlo routine or @Risk, you must account for the correlation among variables.Not specifying a correlation is the same as specifying a zero correlation (independent variables)@Risk allows you to specify a correlation matrix and results in the creation of correlated random variables.Technical note on Cholesky decomposition for generating correlated random variables VBA Code: Random ExampleAdd three simulated random walk asset return formulasAdd output (portfolio return=sumproduct of weights at means)Run Simulation (10,000 iterations) and Copy Mean and Standard Dev You can insert “=RISKMEAN(Output cell)” and “=RISKSTdDEV(Output cell)”Add correlation MatrixRe-Run23To add a correlation matrix in @RiskCreate a correlation matrix in excel@Risk Model windowSelect inputs you want to correlateRight click, correlate, Create newBe sure to re-arrange correlations to match your worksheet correlations (@Risk uses alphabetical order), select column and move to correct order as shown in worksheetRight click on matrix, copy coefficients from excel (select range)Select correlation matrix from worksheetSelect “location” for @Risk to output the matrix in the worksheet – this is the one that @Risk usesCorrelated Random Example24Learning ObjectivesCompute no-arbitrage forward prices for Equities: Spot*(1+r-d)^tCommodities: Spot*(1+r+s-l-c)^tCurrencies ($/FX): Spot*((1+r_$)/(1+r_FX))^tCompute the payoffs of forward contractsLong Forward: (St-F)*QShort Forward: (F-St)*QConstruct forward hedges using beta and correlationSelect hedge asset with higher correlation, using beta to adjust quantitySimulate correlated random variables25