Day 3: Pricing/valuation of forwards and futures Selected discussion from Chapter 9 (pp. 287 – 312) FIN 441 Spring 2011
Cost of carry (carry abitrage) model - Introduction • What does “futures price” or “forward price” mean? • What does “value” mean when discussing futures or forward contracts? • Unlike stocks, bonds, options, etc., price and value are entirely different concepts for futures/forward contracts.
Cost of carry model: Initial value of futures or forward • Price of futures or forward is merely the agreed-upon price at which the future delivery will be made. • Value refers to how much is paid by buyer to enter into contract. • At inception of futures or forward contract, the value is always “zero!”
Cost of carry model: Notation • Vt(0,T) = Value of forward contract created at time 0, as of time t, with expiration at time T. • Vt(T) = Value of corresponding futures contract. • F(0,T) = price at time t of forward contract with expiration at time T. • ft(T) = price at time t of corresponding futures contract.
Cost of carry model: Value of forward contract • F(T,T) = ST • Forward price of forward created at time of expiration. • Forward price = spot price of asset (S). • Trivial case, but HAS to be true (or arbitrage). • VT(0,T) = ST – F(0,T) • Value of forward contract at expiration. • Example: entered into “long” forward to buy asset for $500 in 1 month. One month later, spot price of asset is $550. How much (opportunity) profit did you earn on the forward contract? • Vt(0,T) = St – F(0,T)(1+r)-(T-t) • Value of forward contract prior to expiration. • Use the same example, but suppose it’s 10 days into contract and spot price is at $540 and annual interest rate is 10% (assume 365 days per year). • Value = $42.60 (SHOW IT!!!!)
Cost of carry model: Forward price (relative to spot) • Based on last equation and fact that forward contract has zero value at creation, • V0(0,T) = S0 – F(0,T)(1+r)-T = 0 • Solving last equation for F(0,T), • F(0,T) = S0(1+r)T • Forward price is the spot price compounded to the contract’s expiration. • If not true, then an arbitrage opportunity exists. • Example • What must the spot price of gold be if 2-year forward contracts for gold are priced at $850? Assume 2-year T-bills yield 1.5% per year? (see kitco.com)
Forward currency price • Illustration of interest rate parity. • Fundamental relation between spot and forward prices and interest rates in 2 countries. • F(0,T) = S0(1 +ρ)-T(1+r)T • ρ = foreign currency interest rate • r = domestic currency interest rate • S0 = spot rate (in foreign currency/unit of domestic currency) • NOTE: can “reverse” the notation! • Suppose you believe that you can earn higher interest rate in another county. • Example: What should be the forward price (in US$) of NZ$100,000 to be delivered in 1 year if 1-year US T-bills yield 3.25% per year, the NZ 1-year rate is 7.5%, and the spot rate is NZ$1.3333/US$1?
An example of a forward currency strategy • Buy NZ$133,333 for US$100,000. • Enters into forward contract to buy US$ with NZ$ in 1 year. • After 1 year, treasurer holds NZ$143,333 (i.e., 133,333 x 1.075). • Convert back to US$. What forward rate at time of original investment would guarantee that treasurer could convert NZ dollars into $103,250 (i.e., 100,000 x 1.0325)? • F(0,T) = NZ$1.3882/US$1 (or US$0.7204/NZ$1) = 1.33333(1.075)/(1.0325). • Moral: even though I can earn higher interest rate in another currency, the forward rate is more expensive (US$0.72 vs. spot of US$0.75). • See example on page 299 for similar example.
Cost of carry model: Value of futures contract • fT(T) = ST • Futures price at expiration of contract. • Same result as with forward price. • vt(T) = ft(T) – ft-1(T) before contract is marked-to-market. • Suppose I bought November crude oil at $99.00 at market open on Oct 1, but at 1 PM, November crude is trading at $98.50. • The contract is worth negative 50 cents per barrel to me. • vt(T) = 0 as soon as the contract is marked-to-market. • Suppose settlement price for November crude is $99.30. • My account is credited with the $0.30 per barrel gain, so the futures contract itself has zero value once this happens.
Cost of carry model: Futures price relative to spot • ft(T) = S0(1+r)T • This fact is true immediately after each daily settlement (i.e., after marking to market). • Thus, futures price = forward price.
Cost of carry for underlying that generates cash flows • Standard example: single-stock futures or stock index futures. • Cash flow on underlying = dividends. • Assume dividend (DT) on stock paid at expiration of futures contract. • f0(T) = S0(1+r)T – DT • Futures price is equal to the compounded spot price of stock minus the amount of the dividend. • If dividends are paid at multiple times between time 0 and time T, DT represents the future value of all expected dividend payments.
Cost of carry with cash flows on underlying (cont’d) • f0(T) = (S0 – D0)(1+r)T • D0 is present value of expected dividends. • f0(T) = S0e(r – δ)T • If dividends are paid continuously (like on large index), use continuous compounding with δ as the continuous dividend yield. • Stock futures pricing examples. • Value of forward contract on underlying with cash flows (see page 297).
Additional issues in futures/forward pricing • Storage costs • Denoted by “s” (page 300). • Futures price equals compounded spot price plus storage costs. • Risk premium • Under certainty or risk neutrality, price of an asset (today) equals expected price of asset at future date minus storage costs and incurred interest costs. • If investors are risk averse AND there is uncertainty about future asset prices, today’s price reflects a risk premium, so the spot price also includes a discount to reflect risk premium. • Final definition of “cost of carry” (page 302): • Denoted by θ. • Combination of storage costs and “net interest.” • Cost of carry applies to most assets traded on futures exchanges.
Future/forward pricing equilibrium • Futures price equals spot price plus cost of carry. • f0(T) = S0 + θ • Explain what happens if this equality does not hold. • Pricing Implications: • Positive cost of carry (true for most commodities) • Contango (futures price > spot price) • When can cost of carry be negative? • Can futures price < spot price? • Convenience yield = Premium earned by those who hold inventories of a commodity that is in short supply. • Backwardation (inverted market) • More common for assets with negligible storage costs (financials). • Occasional occurrence in commodity markets (see first point).
Futures prices and risk premia • We’ve considered the idea of risk premiums for investing in “spot” assets. • Are there risk premiums for investing in futures contracts? • No risk premium hypothesis • f0(T) = E(ST) • Futures price is an unbiased expectation of future spot price. • Risk premium hypothesis • f0(T) < E(ST) = f0(T) + E(φ) = E(fT(T)) • Futures prices are (downward) biased expectations of future spot prices. • Suggested by economists arguing that spot and futures markets are dominated by those who are “naturally long” in the underlying (i.e., farmers who own wheat, corn, etc.). • Risk premium could be negative if hedgers are predominantly buyers of futures contracts.
Next 3 classes • Hedging with futures (Chapter 11) • Price risk • Short vs. long hedge • Basis and basis risk • Hedge ratio • Liquidating a hedge • Hedging a “spot” transaction vs. hedging an “ongoing” transaction. • Introduction to class project